TESE DE DOUTORAMENTO
CONTRIBUTIONS TO
MATHEMATICAL ANALYSIS OF
NON-LINEAR MODELS WITH
APPLICATIONS IN POPULATION
DYNAMICS
C is ina Lois P ados
ESCOLA DE DOUTORAMENTO INTERNACIONAL DA
UNIVERSIDADE DE SANTIAGO DE COMPOSTELA
PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS
SANTIAGO DE COMPOSTELA
2021
DECLARACIÓN DA AUTORA DA TESE
CONTRIBUTIONS TO MATHEMATICAL ANALYSIS OF
NON-LINEAR MODELS WITH APPLICATIONS IN
POPULATION DYNAMICS
Dna. C is ina Lois P ados
P esen o a miña ese, seguindo o p ocedemen o axei ado ao Regulamen o, e
decla o que:
1. A ese aba ca os esul ados da elabo ación do meu aballo.
2. De se o caso, na ese aise e e encia ás colabo acións que i o es e a-
ballo.
3. Con i mo que a ese non inco e en ningún ipo de plaxio dou os/as
au o es/as nin de aballos p esen ados po min pa a a ob ención dou os
í ulos.
4. A ese é a e sión de ini i a p esen ada pa a a súa de ensa e coincide a
e sión imp esa coa p esen ada en o ma o elec ónico.
E comp omé ome a p esen a o Comp omiso Documen al de Supe isión no
caso de que o o ixinal non es ea na Escola.
En San iago de Compos ela, a 21 de xuño de 2021.
Asdo. C is ina Lois P ados
AUTORIZACIÓN DOS DIRECTORES/TITORA DA TESE
CONTRIBUTIONS TO MATHEMATICAL ANALYSIS OF
NON-LINEAR MODELS WITH APPLICATIONS IN
POPULATION DYNAMICS
D. Edua do Liz Ma zán,
Dna. Rosana Rod íguez López
INFORMAN:
Que a p esen e ese se co esponde co aballo ealizado po Dna. C is ina Lois
P ados, baixo a nosa di ección/ i o ización, e au o izamos a súa p esen ación,
conside ando que eúne os equisi os esixidos no Regulamen o de Es udos de
Dou o amen o da USC, e que como di ec o es/ i o a des a non inco en nas
causas de abs ención es ablecidas na Lei 40/2015.
De aco do co indicado no Regulamen o de Es udos de Dou o amen o da USC,
decla amos amén que a p esen e ese de dou o amen o é idónea pa a se de-
endida en base á modalidade Monog á ica con ep odución de publicacións,
nas que a pa icipación da dou o anda oi decisi a pa a a súa elabo ación e as
publicacións axús anse ao Plan de In es igación.
En Vigo/San iago de Compos ela, a 21 de xuño de 2021.
Asdo. Edua do Liz Ma zán Asdo. Rosana Rod íguez López
Aos meus pais,
que, ensinándome e apoiándome,
consegui on que chega a a a aquí.
ACKNOWLEDGEMENTS/AGRADECEMENTOS
pe sonal/pe soal
Es a ese de dou o amen o non é só o esul ado do es o zo ealizado du an e
es es ca o úl imos anos, eu amén a conside o o e lexo do camiño pe co ido
da man dos que me o on acompañando dende os meus inicios e, po supos o,
daqueles que se o on unindo, e son eles aos que an dedicadas as seguin es
liñas.
Xa ai moi o empo que as miñas p imas saben do meu gus o polas Ma e-
má icas, pois pa ece que cando e a unha nena eci aba a áboa de mul iplica
en oz al a e amén lles pedía que me puxe an exe cicios de cálculo (segu o
que is o non é o peo que iñan que aguan a ...). Sen emba go, a miña p imei a
ocación oi se p o eso a de Educación Física. Tal ez o mo i o e a que nesa
ma e ia non iñamos debe es. A e dade é que non me gus aba moi o ae a e-
as pa a a casa, segu o que os meus pais o poden co obo a , pois eñen pasado
ho as e ho as sen ados comigo pa a que as ixe a. A eles eño que ag adece -
lles oda a dedicación que me p es a on, coa que consegui on que mello a a as
miñas calidades como es udan e.
Foi en Bacha ela o cando me deca ei de que a miña paixón e an as Ma e-
má icas. No p imei o cu so, a p o eso a a aba de ensina nos a esencia des a
ciencia exac a, conseguindo así espe a o meu gus o po ela. No segundo ano
mudou o p o eso , pe o non o meu in e ese, que se pode dici que se iu e-
o zado. A Juan An onio e a Jose Jo ge (Jo a) eño que da lles as g azas po
o ien a me e apoia me nos meus úl imos anos de ins i u o, pe o amén po
acompaña me na miña e apa uni e si a ia.
Ago a én un dos pia es undamen ais que sus en an a miña e apa na uni e -
sidade, sen o que es ou segu a de que moi os dos éxi os académicos consegui-
dos non se ían posibles. Son os compañei os que es i e on ao meu ca ón nos
p imei os anos que pasamos na nosa segunda casa, a Facul ade de Ma emá i-
cas. Es ou alando dos in eg an es do “G upo Abie o” ou “G upo Ha eliano”
e as apazas do g upo “Aplicadas”. Mención especial pa a Pila , Ángela e Uzal.
Es e pia aumen ou de amaño no ecuado da ca ei a, cando a p omoción
2012/2016 se con e eu nunha piña que a opou a manei a de des u a do ano
máis du o des es es udos. Que bos eco dos! Non me esquezo de Rosana e
Fe nando Cos al, os p o eso es que ixe on que me in e esa a pola Análise Ma-
emá ica e as Ecuacións Di e enciais. Tamén que o des aca nes e momen o o
apoio dos meus pais e a miña mad iña, que semp e con ia on na miña elección
e ixe on odo o que es aba nas súas mans pa a que es a e apa o a posible.
T as o G ao en Ma emá icas, es udei o Más e en Ma emá icas e o que eu
despois supoño que o podedes imaxina . Teño que ag adece a odas esas pe -
ix
de eloped by C is ina Lois P ados du ing he Mas e deg ee s udies, since hey
a e he s a ing poin o new achie emen s in ixed poin heo y, see he ini ial
pa ag aphs o he co esponding sec ions o u he de ails. In his chap e ,
we ha e also added some ex a in o ma ion o he In oduc ion and o Sec ions
2.3and 2.4. In Chap e 3, we ha e included addi ional backg ound in he In-
oduc o y sec ion and we ha e p o ided mo e insigh on he p oblems ound
conce ning he applicabili y when wo king wi h some well-known ixed poin
esul s and he model in o conside a ion. The Discussion sec ion is new in bo h
Chap e s 2and 3.
The s uc u e o Chap e s 2and 3is simila , since bo h o hem s a wi h
an In oduc ion and conclude wi h a Discussion; bu di e s in he cen al sec-
ions. In he in oduc o y sec ions, we show he ma hema ical in e es o ou
esea ch and, in Chap e 3, we also include he ma hema ical o mula ion o
he non-au onomous Lo ka-Vol e a ype sys em ha we will in es iga e. In
he Discussion sec ion, we highligh some ele an aspec s o ou s udy and we
compa e i wi h o he ela ed esea ch wo ks. The in e media e sec ions/sub-
sec ions a e de o ed o show he con ibu ions o ou esea ch wo k. In Chap-
e 2, we gi e o imp o e K asnosel’skii ype comp ession-expansion esul s
o se con ac ions and conical domains de e mined by balls o s a con ex
se s. In Chap e 3, we ocus on he exis ence o posi i e pe iodic solu ions o
Lo ka-Vol e a sys ems wi h gene al p ey g ow h and unc ional esponse o
p eda o s. We use an ope a o app oach based on he homo opy e sion o
K asnosel’skii expansion ixed poin heo em.
Chap e 1: Backg ound de ini ions and esul s I
We s a by p esen ing he ma hema ical amewo k whe e we will de elop o
apply ixed poin heo y echniques. We i s conside an ini ial alue p oblem
o a simple i s o de di e en ial equa ion, and we illus a e he p ocedu e
needed o apply he classical K asnosel’skii comp ession-expansion ixed poin
esul , showing he hypo hesis ha should be sa is ied by he associa ed non-
linea ope a o . Then, we p o ide o he examples o ini ial o bounda y alue
p oblems ha jus i y he necessi y o gene alize o eplace some o he classical
hypo heses owed o K asnosel’skii.
We con inue by s a ing he no ions and esul s in which we ha e based ou
esea ch wo k on ixed poin heo y. In Sec ion 1.1, we ecall some concep s
and esul s ela ed o compac maps and se con ac ions, ha is, he egula -
i y condi ions equi ed o he mappings in ol ed. In Sec ion 1.2, we include
he K asnosel’skii comp ession-expansion ixed poin heo em and some o i s
gene aliza ions. They a e di ided in wo subsec ions, one o he gene aliza-
ions in e ms o he mapping hypo heses, and he o he o hose eplacing he
comp ession and expansion condi ions.
x i
Chap e 2: K asnosel’skii ype comp ession-expansion ixed poin heo em o se con-
ac ions and s a con ex se s
In he amewo k o ixed poin heo y, many gene aliza ions o he classical e-
sul due o K asnosel’skii a e known, so we begin wi h an in oduc o y sec ion,
whe e we ecall some ex ensions o his esul in di e en di ec ions, and we
also compa e wo common app oaches used in hei p oo s: di ec a gumen s
o opological deg ee.
One o he ex ensions consis s in elaxing he condi ions imposed on he
mapping, wo king wi h se con ac ions ins ead o con inuous and compac
mappings. These esul s wo k o conical domains de e mined by balls o ,
equi alen ly, by he no m. The e o e, hey a e no use ul o dis inguish ixed
poin s wi h he same no m. To o e come his necessi y, we gene alize hese
esul s o s a con ex se s ha can be de e mined by unc ionals mo e gene al
han a no m.
The i s s ep o he gene aliza ion is gi en in Sec ion 2.2, whe e we de e -
mine he condi ions ha we will equi e o he s a con ex se s. Then, in Sec ion
2.3, we p o e he main esul s. We i s deal wi h he comp essi e case, whe e
we adap o his mo e gene al amewo k he p oo de eloped by Po e . Then,
he expansi e case is educed o he comp essi e one by means o a change o
a iable. We imp o e he exis ing esul o balls and p o ide a new heo em
o s a con ex se s. Finally, in Sec ion 2.4, we look o unc ionals ha de ine
an admissible s a con ex se and, consequen ly, a e app op ia e eplace he
no m.
To illus a e he heo y, we gi e an applica ion o he ini ial alue p oblem
o a sys em o implici i s o de di e en ial equa ions.
We conclude he chap e wi h a discussion o ou indings in he con ex
o ixed poin heo y, wi h special a en ion o K asnosel’skii ype ixed poin
esul s. We begin explaining he ele ance o he ob ained gene aliza ions o
comp ession-expansion ype esul s, we men ion he complexi y ha may ap-
pea when dealing wi h se con ac ions and s a con ex se s and, we commen
he u ili y o de ining he localiza ion domains by means o unc ionals.
Chap e 3: Applica ions o ixed poin heo y o pe iodic p eda o -p ey di e en ial equa-
ions
In he li e a u e, he e exis many se e al pape s de o ed o he s udy o he
classical Lo ka-Vol e a equa ions and i s gene aliza ions. Fo ins ance, Ts e ko
(1996) conside s he classical sys em wi h pe iodic coe icien s and p o es he
exis ence o pe iodic solu ions ia he opological me hod o ixed poin index.
A di e en app oach was de eloped in (Teixei a Al es and Hilke , 2017), whe e
hey include logis ic p ey g ow h and hun ing coope a ion be ween p eda o s
in he au onomous model and, by means o a quali a i e s udy, hey obse e
he p esence o oscilla ions as a consequence o coope a ion. Inspi ed by he
x ii
gene al o mula ion p oposed by Teixei a Al es and Hilke (2017) and he pe-
iodici y o he model s udied in (Ts e ko , 1996), we conside he ollowing
non-au onomous Lo ka-Vol e a popula ion model
x0=a( )xg(x) − ϕ( ,x,y)xy;
y0= −b( )y+c( )ϕ( ,x,y)xy;
o which we equi e some condi ions o he unc ions a,b,c,gand ϕ, includ-
ing pe iodici y in he ime a iable. Fo pa icula exp essions o he p ey
g ow h gand he unc ional esponse ϕ, we eco e he models conside by
Teixei a Al es and Hilke (2017) and Ts e ko (1996). As a con inua ion o hei
esea ch wo k, we a e also in e es ed in he exis ence o pe iodic solu ions.
To ha pu pose, we use an ope a o app oach as in (Ts e ko , 1996), bu
ins ead o applying opological me hods, we use he homo opy e sion o K as-
nosel’skii ixed poin heo em. To ou knowledge, o hese pa icula p eda o -
p ey models, he applica ion o ixed poin esul s, whose hypo heses a e gi en
di ec ly in e ms o he mapping, has no been conside ed. O he mo e popu-
la e sions o K asnosel’skii ixed poin heo em ha e been applied o simila
models, bu hey do no wo k o ou pa icula o mula ion. Thus, we con-
ibu e o ill his gap in he li e a u e.
In addi ion, we s udy some in e es ing p ope ies o he localized solu ions
ha can be seen as a small con ibu ion o he quali a i e s udy o his ype o
sys ems. We i s s a e su icien condi ions o ensu e ha he pe iodic solu ion
does no educe o a s eady s a e. Then, unde uniqueness condi ions, we p o e
ha he noncons an solu ions a e posi i e, he e o e he p ey and p eda o
popula ions do no end in ex inc ion.
To conclude, we summa ize ou main con ibu ions, we compa e ou ixed
poin heo y app oach wi h o he ela ed esea ch wo k and we commen some
aspec s o he p elimina y quali a i e s udy.
esea ch line ii
The esea ch wi hin his pa o he hesis co esponds o ha in he join wo ks
wi h Edua do Liz Ma zán and F ank M. Hilke : (Liz and Lois-P ados, 2020b),
(Lois-P ados and Hilke , submi ed) (Chap e 5) and (Liz and Lois-P ados,
2020a) (Chap e 6). The con en s and esul s which had been p e iously de el-
oped by o he au ho s a e mainly comp ised in Chap e 4and small po ions
o Chap e s 5(Subsec ions 5.2.1and 5.2.2) and 6(Sec ion 6.1). The p elimina y
con en s in Chap e 4we e compiled om he li e a u e and mos o hem do
no appea in he h ee men ioned esea ch a icles. By con as , he majo i y
o Chap e s 5and 6uses he con en s o hese h ee esea ch wo ks; in Chap e
5, we ha e included some ex a in o ma ion in he ini ial pa ag aphs o he In-
oduc ion sec ion, and we ha e eo ganized he dis inc pa s o he in ol ed
a icles in o de o ge a simila s uc u e o bo h o hem; in Chap e 6, he e is
x iii
a new subsec ion, whe e we desc ibe in de ail he di e en smoo h bi u ca ions
o ixed poin s.
We no ice ha Chap e s 5and 6 ollow a simila s uc u e: we s a wi h an
In oduc ion whe e we show se e al ma hema ical and biological mo i a ions
o he esea ch s udy and, in his sec ion o Chap e 6, we also include he ma h-
ema ical o mula ion o he blood cell p oduc ion model p oposed by Laso a,
while in Chap e 5we jus p o ide he ecological desc ip ion o some ishe ies
managemen policies and shi hei ma hema ical o maliza ion o a sepa a e
sec ion. The subsequen sec ions/subsec ions a e de o ed o de e mine he
asymp o ic dynamics o he models conside ed. Fo ha pu pose, we comple-
men an analy ical app oach by means o he quali a i e heo y o dynamical
sys ems wi h some nume ical ools such as 1-pa ame e bi u ca ion diag ams.
By using his in o ma ion, we plo 2-pa ame e bi u ca ion diag ams, which
gi e a global pic u e o he long- e m dynamics wi h espec o he a ia ion o
wo pa ame e s. We conclude he chap e s wi h a discussion sec ion whe e we
compa e ou s udy wi h o he ela ed esea ch wo ks. I is wo h men ioning
ha , in Chap e 5, we conside gene al unc ions sa is ying some ypical con-
di ions o disc e e- ime one-dimensional popula ion maps, and we also wo k
wi h some pa icula cases in o de o ob ain mo e in o ma ion om analy ical
esul s and nume ical simula ions. Howe e , in Chap e 6, we jus deal wi h
a case s udy, which is lexible enough o ul ill di e en ypical condi ions o
popula ion maps, depending on he choice o he pa ame e s. Fo ins ance, he
associa ed map can be mono one, unimodal o e en bimodal.
Chap e 4: Backg ound de ini ions and esul s II
We begin by in oducing, in a ma hema ical and ecological con ex , he ypes
o one-dimensional di e ence equa ions o be conside ed. In he ma hema ical
se ing, we dis inguish be ween smoo h and piecewise-smoo h con inuous o
discon inuous equa ions. In he ecology con ex , pa icula ly ha o unman-
aged single-species popula ions wi h disc e e- ime ep oduc i e seasons, we
ecall di e en s ock- ec ui men ela ionships and we s a e some ypical ma h-
ema ical condi ions ul illed by he maps desc ibing hese ela ions.
Then, we p esen he no ions and esul s we use o ca y ou he quali a-
i e s udy o asymp o ic dynamics. In Sec ion 4.2, we ecall some basic con-
cep s ha a e use ul when de e mining long- e m dynamics o smoo h and
piecewise-smoo h maps, hen we s a e some auxilia y esul s which ensu e
global s abili y o an equilib ium. We use some p o o ypes o s ock- ec ui men
models o illus a e hei applicabili y. In Sec ions 4.3and 4.4, we desc ibe in de-
ail a numbe o bi u ca ions and ea u es o 1-pa ame e bi u ca ion diag ams
which appea h oughou he hesis. The bi u ca ions a e local o global and
he a ie y o complexi y o local bi u ca ions inc eases when he egula i y o
he piecewise-smoo h map is dec eased. The ea u es one can ind in he anal-
xix
ysis o bi u ca ion diag ams sugges unexpec ed dynamic beha io s which can
ha e impo an consequences o he managemen o ecological sys ems.
Chap e 5: Combina ions o cons an quo a and h eshold based ha es ing s a egies
We s udy wo disc e e- ime models o single-species popula ions subjec o
di e en ha es ing ules. Bo h s a egies combine cons an ca ches o ob ain
p edic able yield and a h eshold o minimum biomass le el o p o ec he
popula ion. In Sec ion 5.1, we in oduce hese con ol ules in he con ex o
ishe ies managemen .
F om his poin o iew, he simples ule allows o ha es a maximum an-
nual quo a Hi he popula ion size a e ep oduc ion is abo e he h eshold
T; and, i i is below he h eshold, no ha es ing is applied. We e e o his
s a egy as h eshold cons an ca ch ha es ing (TCC). I we equi e he addi-
ional condi ion ha a minimum biomass le el Tmus emain a e ha es ing,
hen he s a egy becomes mo e p o ec i e and di icul o apply, we e e o i
as p ecau iona y h eshold cons an ca ch (PTCC).
Disc e e- ime ma hema ical models o hese s a egies lead in a na u al way
o piecewise-smoo h maps, whose dynamics a e challenging because mul iple
non-smoo h bi u ca ions may appea . As he map associa ed o he TCC ule
is discon inuous and ha co esponding o PTCC is con inuous, he quali a i e
s udy o TCC dynamics is mo e complex. Thus, we i s s udy he PTCC ule
in Sec ion 5.3; hen he TCC ule in Sec ion 5.4. In bo h cases, we combine
analy ical and nume ical esul s o p o ide a comp ehensi e o e iew o he
dynamics, which depend on he wo ele an ha es ing pa ame e s Hand T.
In Sec ion 5.3, we p o ide a ho ough analy ical desc ip ion o dynamics
and bi u ca ions o gene al compensa o y popula ion models. In he o e com-
pensa o y case, whe e we ound mo e complica ed dynamics, we explain he
dynamical beha io in some egions on he 2-pa ame e plane (H,T), bu we
choose he Ricke model as a case s udy o gi e a global pic u e o he dynamics.
In Sec ion 5.4, he discon inui y o he map associa ed o TCC induces complex
dynamical beha io o he s ic ly inc easing compensa o y case. Thus, we e-
s ic ou s udy o his ype o s ock- ec ui men ela ions, o which we use he
Be e on-Hol model as a pa icula example. Fo he TCC ule, we de o e an
addi ional subsec ion o he s udy o wo equen ly conside ed managemen
objec i es: a e age yield and ha es equency.
We conclude he chap e wi h a discussion o ou indings in he con ex o
piecewise-smoo h di e ence equa ions and ha es ing con ol ules.
xx
Chap e 6: Laso a disc e e model o blood cell p oduc ion
In an a emp o explain some expe imen al e idences o chao ic beha io in
blood cell popula ions, And zej Laso a p oposed in (Laso a, 1977) he ollowing
disc e e- ime one-dimensional model:
xn+1= (1−σ)xn+ (cxn)γe−xn,n∈N∪{0}.
This model includes a des uc ion a e pa ame e σ∈(0,1)and a gamma-
Ricke unc ion as a ep esen a ion o he quan i y o cells p oduced in he bone
ma ow. The gamma-Ricke map inco po a es wo o he pa ame e s γ,c > 0.
Wi h he aim o explaining he in luence o he des uc ion a e on he dy-
namics, Laso a ixed he pa ame e alues as c=0.47 and γ=8, and chose
some alues o σ. These alues allowed him o desc ibe he dynamics in some
ele an clinical cases: no mal condi ions (σ=0.1: depending on he ini ial con-
di ion, solu ions ei he go o ex inc ion o con e ge o a posi i e equilib ium),
non-se e e disease (σ=0.4: he posi i e equilib ia a e uns able and he e is a
2-pe iodic a ac o ), and se e e disease (σ=0.8, whe e Laso a obse ed he
p esence o a 3-pe iodic o bi and, he e o e, chao ic beha io ).
In his chap e , we s udy he model in de ail, by means o an analy ical ap-
p oach combined wi h some nume ical simula ions. In pa icula , we e isi he
esul s which appea in he o iginal pape , bu we also disco e new in e es ing
phenomena. In he analy ical pa , we ind su icien condi ions o ex inc ion
and s abili y (including a sha p global s abili y condi ion o γ⩽1) depending
on he in ol ed pa ame e s. Then, we show some 1-pa ame e bi u ca ion di-
ag ams using ei he γo σas bi u ca ion pa ame e s, while keeping c=0.47
as in (Laso a, 1977). These diag ams allow us o disco e he ich dynamics
o he model, which exhibi s ea u es such as s abili y swi ches (bubbles), ex-
inc ion windows, hyd a e ec s and sudden collapses, among o he s. We also
p esen a 2-pa ame e bi u ca ion diag am as an illus a ion ha summa izes
he long- e m dynamics.
Finally, we compa e ou esul s wi h hose in he p e ious wo k ca ied ou
by Laso a, and we addi ionally in e p e hem in he con ex o popula ion dy-
namics. The conside ed equa ion is also sui able o model he dynamics o
popula ions wi h disc e e ep oduc i e seasons, adul su i o ship, o e com-
pensa o y densi y dependence and Allee e ec s. In his con ex , ou esul s
show he ich dynamics o his ype o models and poin ou he sub le in-
e play be ween adul su i o ship a es and s eng h o densi y dependence
(including Allee e ec s).
xxi
RESUMO ESTENDIDO
Os con idos des a ese de dou o amen o son o esul ado do aballo ealiza-
do pola au o a C is ina Lois P ados du an e os seus es udos co esponden es
ao P og ama de Dou o amen o en Ma emá icas da Uni e sidade de San iago
de Compos ela, en colabo ación cos seus di ec o es Edua do Liz Ma zán (Uni-
e sidade de Vigo) e Rosana Rod íguez López (Uni e sidade de San iago de
Compos ela); así como cos p o eso es esponsables das dúas es adías de in es-
igación ealizadas, F ank M. Hilke (Osnab ück Uni e si y, Alemaña) e Radu
P ecup (Babe¸s-Bolyai Uni e si y, Romanía).
Es e documen o comp ende dúas liñas de in es igación di e en es que se
desen ol en no ámbi o da análise ma emá ica dos modelos non lineais. As
maio es di e enzas obsé anse no ipo de ecuacións que se conside an e na me-
odoloxía u ilizada. Na Liña de In es igación I (Resea ch Line I), median e a
xene alización dalgúns esul ados da eo ía de pun o ixo, melló ase a localiza-
ción das solucións a p oblemas de alo inicial ou p oblemas de on ei a pa a
di e en es ipos de ecuacións di e enciais, en pa icula , pa a as o dina ias; ade-
mais con ibúese á aplicación da eo ía de pun o ixo a modelos de poboación
pa a es a clase de ecuacións. Na Liña de In es igación II (Resea ch Line II), o
noso obxec i o p incipal é desc ibi o compo amen o asin ó ico e as bi u ca-
cións de algúns sis emas dinámicos disc e os unidimensionais. Séguese unha
me odoloxía máis aplicada, aballando con modelos de poboación que xo den
na xes ión de ecu sos pesquei os ou no modelado da p odución de glóbulos
e mellos.
Dado que os con idos das liñas de in es igación mencionadas son bas an e
di e en es, decidimos di idi es e manusc i o en dúas pa es au ocon idas, que
inclúen a súa p opia lis a de e e encias. Sen emba go, seguen unha es u u a
simila : comezan con algúns esul ados p elimina es que se en pa a sen a as
bases ma emá icas de cada pa e da ese dou o al e ema an cunhas conclusións
e algunhas ideas pa a con inua as liñas de in es igación des e manusc i o. Os
capí ulos in e medios ecollen os con idos cen ais da in es igación.
No que segue, damos unha b e e desc ición dos con idos dos capí ulos co-
esponden es a cada liña de in es igación.
liña de in es igación i
A in es igación le ada a cabo nes a liña co espóndese cos aballos ealizados
conxun amen e con Rosana Rod íguez López e Radu P ecup: (Lois-P ados, P e-
cup e Rod íguez-López, 2020; Lois-P ados e Rod íguez-López, 2020) (Capí ulo
2) e (Lois-P ados e P ecup, 2020) (Capí ulo 3). A maio ía dos con idos des a
pa e da ese o on ecompilados dos es a igos mencionados e edis ibuídos
xxiii
nos di e en es capí ulos. As nocións e os esul ados p elimina es p e iamen e
es ablecidos po ou os au o es es án maio i a iamen e ecollidos no Capí ulo
1. Ademais, no Capí ulo 2(Sección 2.2e Subsección 2.4), apa ecen algúns con-
idos que C is ina Lois P ados desen ol eu du an e os seus es udos de mes-
ado, dado que son a pun o de pa ida de no as achegas no ámbi o da eo ía
de pun o ixo, pa a máis de alles éxanse os pa ág a os iniciais das seccións
co esponden es. Po ou a banda, na In odución e nas Seccións 2.3e2.4des e
capí ulo, incluíuse in o mación adicional á que apa ece nos a igos. No Capí u-
lo 3, amén se amplía a desc ición do con ex o no que se desen ol e o aballo
ealizado (Sección 3.1) e o écense máis de alles sob e os p oblemas que apa e-
cen ao aplica e sións coñecidas do esul ado de pun o ixo de K asnosel’skii
ao modelo que es amos a conside a . A Discusión é no a en ambos Capí ulos
2e3.
A es u u a dos Capí ulos 2e3é simila , xa que ambos comezan cunha
In odución e ema an cunha Discusión, pe o di e éncianse nas seccións cen-
ais. Nas seccións in odu o ias, abo damos o in e ese da nosa in es igación
dende un pun o de is a ma emá ico, no Capí ulo 3 amén se inclúe a o mu-
lación ma emá ica do modelo non-au ónomo de Lo ka-Vol e a co que imos
aballa . Nas seccións de Discusión, des acamos os aspec os máis ele an es
do es udo ealizado e compa ámolo con ou as in es igacións elacionadas. As
seccións/subseccións in e medias amosan a in es igación que se le ou a ca-
bo. No Capí ulo 2, p obamos ou mello amos esul ados de pun o ixo de ipo
comp esi o-expansi o de K asnosel’skii pa a aplicacións con ac i as e domi-
nios cónicos de e minados po bolas ou conxun os es elados. No Capí ulo 3,
cen ámonos na exis encia de solucións pe iódicas pa a modelos de ipo Lo ka-
Vol e a con e mo xené ico de p edación. Facemos uso do ope ado non lineal
asociado, que se a opa nas condicións do caso expansi o da e sión homo ópi-
ca do esul ado clásico de K asnosel’skii.
Capí ulo 1: De inicións e esul ados p elimina es I
Comezamos p esen ando o con ex o ma emá ico no que se desen ol e án e
aplica án as écnicas de pun o ixo. En p imei a ins ancia, conside amos un
p oblema de alo inicial pa a unha ecuación di e encial o dina ia sinxela e
ilus amos o p ocedemen o a segui pa a aplica o esul ado clásico de K as-
nosel’skii de ipo comp esi o-expansi o, mos ando as hipó eses que debe e-
i ica o ope ado non lineal asociado. Despois, engadimos ou os exemplos
de p oblemas de alo inicial ou de on ei a, que xus i ican a necesidade de
xene aliza ou cambia algunhas das hipó eses clásicas equi idas po K asno-
sel’skii.
A con inuación lemb amos algunhas nocións e esul ados nos que baseamos
a nosa in es igación no eido da eo ía de pun o ixo. Na Sección 1.1, es ablece-
mos concep os e esul ados elacionados con aplicacións compac as e con ac-
i as, que son as condicións de egula idade que lles esiximos aos ope ado es.
xxi
Na Sección 1.2, incluímos o esul ado de K asnosel’skii de ipo comp esi o-
expansi o e algunhas das súas xene alizacións. Di idímolas en dúas subsec-
cións, unha pa a as xene alizacións en e mos das hipó eses sob e as aplica-
cións e ou a pa a aquelas que modi ican as condicións de comp esión e expan-
sión.
Capí ulo 2: O eo ema de pun o ixo de K asnosel’skii de ipo comp esi o-expansi o
pa a aplicacións con ac i as e conxun os es elados
No ámbi o da eo ía de pun o ixo, exis en moi as xene alizacións do esul ado
clásico de K asnosel’skii, polo que comezamos cunha sección in odu o ia, on-
de lemb amos algunhas das ex ensións exis en es en a ias di eccións, e amén
compa amos dúas écnicas comunmen e u ilizadas nas súas demos acións: a -
gumen os di ec os e eo ía do g ado opolóxico.
Unha das ex ensións consis e en elaxa as condicións que se lle impoñen
á aplicación, aballando con aplicacións con ac i as en luga de ope ado es
con inuos e compac os. Es es esul ados son álidos pa a dominios cónicos de-
e minados po bólas, ou de xei o equi alen e, po no mas. En consecuencia,
es es esul ados non se en pa a localiza pun os ixos coa mesma no ma. Pa-
a palia es a necesidade, ob i emos unha xene alización des es esul ados a
conxun os es elados que poden i dados po uncionais máis xe ais que a
no ma.
O p imei o paso pa a di a xene alización dáse na Sección 2.2, onde se de e -
minan as condicións que lles imos esixi aos conxun os es elados. A con inua-
ción, na Sección 2.3, aise a demos ación dos esul ados p incipais. T abállase
en p imei o luga co caso comp esi o, no que se adap a a demos ación de Po -
e a es e ámbi o máis xe al. Despois, na p oba do caso expansi o aise unha
edución ao caso comp esi o median e un cambio de a iable. Nes e caso, me-
lló ase o esul ado xa coñecido pa a bólas e conséguese un esul ado no o pa a
conxun os es elados. Finalmen e, na Sección 2.4, búscanse uncionais que de i-
nan un des es conxun os es elados, e que en consecuencia, poidan se usados
pa a subs i uí a no ma.
Pa a ilus a es a eo ía, aplicamos os esul ados ob idos a un p oblema de
alo inicial pa a un sis ema de ecuacións di e enciais implíci as de p imei a
o de.
Rema amos es e capí ulo cunha discusión sob e as nosas achegas á eo ía de
pun o ixo, con especial a ención aos esul ados do ipo dos de K asnosel’skii.
Comezamos explicando a ele ancia que pode e es a xene alización dos esul-
ados de ipo comp esi o-expansi o, explicamos amén cal é a complexidade
que se p esen a ao aballa con aplicacións e conxun os máis xe ais e, po úl-
imo, poñemos en alo a posibilidade de de ini os dominios de localización
median e uncionais.
xx
on he dynamics o disc e e ime popula ion models (see Liz, 2010b). He e,
we y o ob ain simila esul s o in e en ion ules ha conside a h eshold
alue o minimum biomass le el unde which no ha es ing is allowed. Mo e
p ecisely, we wo k in he ollowing goals:
G3 Dynamical s udy o one-dimensional piecewise-smoo h popula ion models applied
o h eshold ha es ing: We s a by s udying he dynamics o di e en sus-
ainable in e en ion s a egies applied o s icly inc easing o unimodal
maps wi h a mos one posi i e ixed poin , such as he classical Be e on-
Hol and Ricke models. I is in e es ing o s udy he in luence o he
unde lying ha es ing pa ame e s on he dynamics, o ha pu pose we
look o egions o global s abili y, bis abili y, chaos, o pa ame e alues
whe e he e is a change on he popula ion asymp o ic dynamics (bi u -
ca ions). Once we inish wi h he quali a i e s udy, we ind i signi ican
o compa e he esul s wi h hose ob ained o some ela ed adi ional
s a egies.
G4 Mo e lexible popula ion models: A simila bu mo e ambi ious objec i e is o
desc ibe he dynamics o he same in e en ion s a egies applied o mo e
gene al maps, no necessa ily mono one no unimodal and wi h 0,1o
2posi i e ixed poin s (see, o example, Liz, 2018a). As he model lex-
ibili y usually inc eases he di icul y o he dynamical analysis, we can
s a s udying he dynamics o hese models wi hou ha es ing, which
implies a educ ion in he numbe o pa ame e s.
xxxii
METHODOLOGY
This chap e o he hesis is de o ed o desc ibe he me hods o app oaches
used in he de elopmen o he goals G1-G4. In gene al, he esea ch begins
wi h a de ailed e iew o some ela ed classical and ecen bibliog aphical e e -
ences in o de o acqui e some necessa y knowledge abou ele an esul s and
echniques in he ela ed ield. A e ha , we s a o wo k on he unde lying
p oblem by ollowing hese app oaches and, some imes, we also need o look
o o o de elop o he s a emen s and me hods.
In pa icula , in ela ion wi h he objec i e G1, we p o ide a gene aliza ion
o he comp ession-expansion K asnosel’skii ixed poin heo em by using clas-
sical a gumen s o his heo y, ollowing he app oach used by K asnosel’skii
and o he au ho s. Fo he es o he goals, unless we use dis inc me hods,
somehow we ollow a simila app oach in which we o mula e he model and
hen we look o he p ope esul s and echniques ha can be used o i s anal-
ysis. In he ollowing, we p o ide mo e de ails abou he me hodology ha we
use o a ain each o he objec i es. We use speci ic esul s and echniques o
goals G1 and G2. In he case o goals G3 and G4, some me hods a e applied o
bo h o hem, so we explain he simila i ies and di e ences in M3.
M1 Fo he gene aliza ion o he comp ession-expansion K asnosel’skii ixed
poin heo em o se con ac ions and s a -con ex se s, we i s obse e
ha , in he comp essi e case, he app oach ca ied ou by Po e (1974)
o se con ac ions can be adap ed o mo e gene al domains de e mined
by s a -con ex se s ins ead o balls. Thus, we s a by de e mining he
p ope ies we ha e o impose on he s a -con ex se s in o de o ep oduce
he p oo gi en by Po e in his mo e gene al amewo k and, hen, we
adap his main esul s. A e ha , we obse e ha he app oach due o
Các and Ga ica (1979), in he expansi e case, canno be p ope ly adap ed
o ou amewo k. So, we look o o he me hodologies and we inally
use he idea in (P ecup, 2006), ha is, a change o a iable o educe he
expansi e case o he comp essi e one. I is wo h men ioning ha we do
no use opological deg ee echniques.
M2 In he applica ion o ixed poin heo y o pe iodic p eda o -p ey models,
we s a by ans o ming he o dina y di e en ial equa ion in o i s equi -
alen in eg al o mula ion, so we ob ain an ope a o Twhose ixed poin s
a e he solu ions o he p eda o -p ey model. Then, we look o a special
se whe e we in end o localize a pe iodic solu ion, and o a p ope ixed
poin heo em ha could be applied o his ype o p oblems. Finally,
we also use some basic a gumen s in he analysis o o dina y di e en ial
equa ions o p o ide su icien condi ions ha ensu e he non-cons ancy
and posi i eness o he localized pe iodic solu ion.
xxxiii
M3 In o de o ealize a de ailed s udy o he long- e m dynamics o he one-
dimensional di e ence equa ions ha we conside in Chap e s 5and 6,
we i s de e mine he numbe o posi i e ixed poin s o he associa ed
map depending on he pa ame e alues, and we also s udy i s local and
global s abili y. By using his in o ma ion, one can classi y some o he bi-
u ca ions in which ixed poin s a e in ol ed. Then, we con inue wi h he
analy ical s udy o o he in e es ing phenomena, such as, he bi u ca ions
o 2-cycles, he bounda y-collision bi u ca ions, he egions o bis abili y
o he sudden collapses. Once we conclude wi h he analy ical s udy, we
use 2-pa ame e BD o p o ide a global pic u e o he long e m dynam-
ics, and 1-pa ame e BD o illus a e he mos ele an dynamical ea u es.
This is he gene al p ocedu e ha we ollow o a ain goals G3 and G4,
howe e , we use di e en esul s and a gumen s o s udy he dynamics.
On he one hand, he in e en ion s a egies ha we ha e men ioned in
objec i e G3 p oduce a piecewise-smoo h map whose long- e m beha -
io p esen s some ea u es ha canno occu o smoo h models. On he
o he hand, he map in goal G4 is smoo h bu mo e lexible, which in
some sense inc eases he di icul y o he dynamical s udy.
xxxi
CONTENTS
acknowledgemen s ix
abs ac / esumo xiii
p e ace x
esumo es endido xxiii
aims and objec i es xxxi
me hodology xxxiii
i esea ch line i 1
1 backg ound de ini ions and esul s i 3
1.1Condi ions o in eg al mappings . . . . . . . . . . . . . . . . . . . 5
1.2K asnosel’skii ype ixed poin esul s . . . . . . . . . . . . . . . . 8
1.2.1Gene aliza ions o se con ac ions . . . . . . . . . . . . . . 10
1.2.2No m ype, ec o and homo opy e sions . . . . . . . . . 10
2 k asnosel’skii ype esul s in s a con ex se s 13
2.1In oduc ion ............................... 14
2.2Gene al amewo k........................... 17
2.3Main esul s ............................... 20
2.3.1Comp essi ecase........................ 22
2.3.2Expansi ecase ......................... 31
2.4Admissible se s de ined by unc ionals . . . . . . . . . . . . . . . . 35
2.5Applica ion o a i s o de implici di e en ial sys em . . . . . . 41
2.6Discussion ................................ 48
3 applica ions o p eda o -p ey di e en ial equa ions 51
3.1In oduc ion and model desc ip ion . . . . . . . . . . . . . . . . . 52
3.2In eg al e sion o he sys em and ela ed no a ions . . . . . . . . 54
3.3Exis ence, localiza ion and o he p ope ies o solu ions . . . . . . 56
3.3.1Exis ence and localiza ion . . . . . . . . . . . . . . . . . . . 56
3.3.2P ope ies o solu ions . . . . . . . . . . . . . . . . . . . . . 66
3.3.3Time independen p eda o s unc ional esponse ϕ.... 71
3.4Discussion ................................ 75
conclusions and u u e p ospec s i 77
e e ences i 81
ii esea ch line ii 85
4 backg ound de ini ions and esul s ii 87
4.1Condi ions o single-species popula ion models . . . . . . . . . . 88
4.2S abili y concep s and esul s . . . . . . . . . . . . . . . . . . . . . 91
4.3Bi u ca ion ypes ............................ 96
xxx
4.41-pa ame e bi u ca ion diag ams ea u es . . . . . . . . . . . . . 100
5 combina ions o cc and h ha es ing s a egies 103
5.1In oduc ion ............................... 104
5.2Modelsdesc ip ion ........................... 110
5.2.1Cons an ca ch ule....................... 110
5.2.2Th eshold ha es ing ule . . . . . . . . . . . . . . . . . . . 111
5.2.3P ecau iona y h eshold cons an ca ch ule . . . . . . . . 112
5.2.4Th eshold cons an ca ch ule . . . . . . . . . . . . . . . . . 113
5.3P ecau iona y h eshold cons an ca ch (PTCC) . . . . . . . . . . . 115
5.3.1Fixed poin s: loca ion, s abili y and local bi u ca ions . . . 115
5.3.2Complex dynamics: essen ial a ac ion and chao ic be-
ha io ............................... 127
5.3.3Impac o ha es pa ame e s on popula ion dynamics . . 129
5.4Th eshold cons an ca ch (TCC) . . . . . . . . . . . . . . . . . . . . 139
5.4.1Fixed poin s: loca ion, s abili y and ela ed bi u ca ions . 139
5.4.2Complex dynamics: abso bing in e als . . . . . . . . . . . 144
5.4.32-cyclesSBsandBCBs ..................... 146
5.4.4Impac o ha es pa ame e s on popula ion dynamics . . 148
5.4.5A e age yield and ha es equency . . . . . . . . . . . . . 156
5.5Discussion ................................ 158
6 laso a disc e e model o blood cell p oduc ion 165
6.1In oduc ion and model desc ip ion . . . . . . . . . . . . . . . . . 166
6.2Fixed poin s: loca ion, s abili y and bi u ca ions . . . . . . . . . . 167
6.2.1P elimina y esul s....................... 167
6.2.2Exis ence and s abili y . . . . . . . . . . . . . . . . . . . . . 169
6.2.3Bi u ca ions o ixed poin s . . . . . . . . . . . . . . . . . . 172
6.3Impac o pa ame e s a ia ion on long- e m dynamics . . . . . . 173
6.4Discussion ................................ 177
conclusions and u u e p ospec s ii 181
e e ences ii 186
glossa y 193
ans e o copy igh /publish ag eemen 197
xxx i
Pa I
RESEARCH LINE I
1
BACKGROUND DEFINITIONS AND RESULTS I
Fo he sake o comple eness, in his chap e we p o ide some no ions and
esul s on ixed poin heo y.
We begin in oducing he ype o p oblems we will deal wi h. In gene al,
gi en an ini ial o bounda y alue p oblem o di e en ial equa ions, we a e
in e es ed in i s in eg al cha ac e iza ion o , in ob aining a mapping Twhose
ixed poin s a e in co espondence wi h solu ions o he ini ial o bounda y
alue p oblem. Once we ge he associa ed mapping, ixed poin esul s con-
s i u e a use ul ool o p o e he exis ence and o localize he ixed poin s o
T. Th oughou his pa o he hesis, his p ocedu e o ind solu ions o di e -
en ial equa ions is e e ed o as (classical) ope a o app oach. We illus a e he
p ocess wi h a simple ini ial alue p oblem:
x0( ) = ( ,x( )), ∈[0,1];
x(0) = 0;(1)
whe e : [0,1]×R−→ Ris bounded and con inuous. Fo some pa icula
exp essions o he map , he e exis echniques ha allow o ind he solu ions
o (1). In gene al, by using he con inui y o , we can asse ha he e exis s a
leas a solu ion, bu i may no be easy o ind p ocedu es o sol e he p oblem
explici ly. Thus, i is in e es ing o ge he equi alen p oblem o which we can
apply some exis ence and localiza ion ixed poin esul s. Fo ha pu pose, le
us conside he non-linea ope a o T:C([0,1],R)−→ C([0,1],R)gi en by
T(x)( ) = Z
0
(s,x(s))ds,x∈C([0,1],R), ∈[0,1]. (2)
Each solu ion o he ini ial alue p oblem (1) in C([0,1],R)is in co espon-
dence wi h a ixed poin o he ope a o Tgi en by (2), ha is, an elemen
x∈C1([0,1],R)such ha x( ) = T(x)( ) o all ∈[0,1]. The simple exp ession
o he map T, oge he wi h he egula i y o , allows us o use, o ins ance, he
classical comp ession-expansion Theo em o K asnosel’skii (see K asnosel’skii,
1964, Chap e 4) o localize ixed poin s o T. Howe e , he abo e-men ioned
ixed poin esul canno be applied di ec ly o he map Tin (2) wi hou he
es ablishmen o addi ional s uc u es and he imposi ion o ce ain es ic ions
on he mapping , as we explain below.
We i s should equip he se C([0,1],R)wi h a no m in o de o ob ain a Ba-
nach space. Fo ins ance, he pai (C([0,1],R),kxk∞)is a Banach space, whe e
kxk∞:= max{|x( )|, ∈[0,1]}and |·|deno es he absolu e alue in R.
Secondly, he esul does no wo k in he whole space C([0,1],R), so we ha e
o conside a pa icula subse o he Banach space which is called a cone, o
3
4 backg ound de ini ions and esul s i
example, C([0,1],R+). Thus, i we equi e ha ([0,1]×R+)⊂R+, hen i is
gua an eed ha Tmaps C([0,1],R+)in o i sel .
Thi dly, he es ic ion o T o he cone C([0,1],R+)should be a compac
mapping. This is indeed he case since is bounded and con inuous.
Finally, he comp ession o expansion condi ions should be ul illed. They
depend on he mapping Tand he elemen s o he cone wi h kxk∞= and
kxk∞=R, whe e < R a e wo posi i e eal numbe s.
Once we ha e checked ha all he hypo heses a e ul illed, he esul p o ides
he exis ence o a ixed poin x∈C([0,1],R+)wi h ⩽kxk∞⩽R, and, he e o e,
a localiza ion o his ixed poin is gi en.
The men ioned classical esul s due o K asnosel’skii can be applied o a
huge amoun o p oblems. Howe e , i we elax o eplace some o he equi ed
hypo heses, we can imp o e he localiza ion o ixed poin s o e en apply he
esul s o a wide ange o ini ial o bounda y alue p oblems.
The gene aliza ions o K asnosel’skii ixed poin heo em can be s a ed by
elaxing di e en hypo heses: he p ope ies o he domain o T; he compac -
ness o he mapping and he comp ession-expansion condi ions. In Sec ions
1.1and 1.2, we s a e some gene aliza ions o wo o hese condi ions, bu o
hose ega ding he domain o Twe e e he eade o Chap e 2.
In Sec ion 1.1, we s a desc ibing he concep o compac mapping and ecall-
ing he A zelà-Ascoli cha ac e iza ion o ela i ely compac se s in he Banach
space (C(I,Rn),k·k∞), whe e (I,d)is a compac me ic space, ha will be use-
ul in applica ions. Then, we e iew some concep s ha gene alize he p e ious
ones: he measu e o noncompac ness and α-Lipschi z maps. We also p o ide
some esul s ha s a e use ul p ope ies o bo h concep s.
Sec ion 1.2is de o ed o ecall some K asnosel’skii ype ixed poin heo ems
ha we use in his pa o he monog aph. We i s e iew he concep o
cone and he classical K asnosel’skii comp ession-expansion esul . Then, we
o ganize he gene aliza ions in wo di e en subsec ions. In Subsec ion 1.2.1,
we ecall K asnosel’skii ype esul s o se con ac ions. These gene aliza ions
allow o localize solu ions o implici i s o de ini ial alue p oblems o he
o m:
x0( ) = ( ,x( )) + g( ,x0( )), ∈[0,1];
x(0) = 0;(3)
whe e ,g: [0,1]×R−→ Ra e con inuous and gsa is ies some Lipschi z ype
condi ion. I is wo h men ioning ha he non-linea ope a o Tassocia ed o
his p oblem is no necessa ily compac , bu i can be a se con ac ion, see
Chap e 2 o u he de ails. In Subsec ion 1.2.2, we s a e h ee ixed poin e-
sul s p ese ing he compac ness hypo hesis o he mapping T, bu conside ing
di e en comp ession-expansion condi ions. In Chap e 3, we s udy hei appli-
cabili y o a class o Lo ka-Vol e a ype equa ions. The i s esul is a gene al-
iza ion due o Güo and Lakshmikan ham (1998), wi h comp ession-expansion
condi ions o no m ype, which localizes he solu ions in mo e gene al domains
1.1 condi ions o in eg al mappings 5
de e mined by a cone and wo open se s. This gene aliza ion wo ks o some
pa icula Lo ka-Vol e a models wi h ime-delays, bu i canno be applied
o simple o mula ions o hese p eda o -p ey sys ems. The second esul is
he ec o e sion gi en in (P ecup, 2007), which was applied o localize posi-
i e pe iodic solu ions o a di e en ial sys em wi h linea and non-linea e ms.
Tha sys em is qui e simila o Lo ka-Vol e a ype equa ions; howe e , i seems
ha he ec o e sion p esen s some di icul ies in i s applicabili y o simple
o mula ions o hese popula ion models, as i happens wi h he o iginal esul
owed by K asnosel’skii. Besides, he homo opy e sion o he o iginal Theo em
o K asnosel’skii, s a ed a he end o Subsec ion 1.2.2, wo ks o hese ype o
p eda o -p ey equa ions.
1.1 condi ions o in eg al mappings
In his sec ion, we s a e some basic no ions and esul s ela ed wi h he egula -
i y o he mapping. These concep s will appea in he hypo hesis o ou ixed
poin esul s.
We i s ecall he concep s o ( ela i ely) compac se and compac mapping.
We also p o ide he A zèla-Ascoli heo em ha gi es a cha ac e iza ion o el-
a i ely compac se s in (C(I,Rn),k·k∞)whe e (I,d)is a compac me ic space.
This esul will help o p o e he compac ness o mappings in applica ions, in
pa icula , we will use Co olla y 1.1.4. Fo u he de ails, see (P ecup, 2002,
Chap e 1).
De ini ion 1.1.1.Le (X,d)be a me ic space. We say ha Xis a compac se
i , o each ε > 0,Xadmi s a ini e co e ing by open balls o adius ε. Mo e
p ecisely, o each ε > 0, he e exis Nε∈Nand xi∈X,i=1,...,Nε, such ha
X⊂
Nε
[
i=1
B(xi,ε),
whe e B(xi,ε) = {y∈X:d(xi,y)< ε}.
A subse Do Xis ela i ely compac i i s closu e Dis a compac se (as a me ic
subspace o X).
De ini ion 1.1.2.Le X,Ybe me ic spaces and D⊂X. The map T:D⊂X−→ Y
is compac i , o all A⊂Dbounded, T(A)is a compac se .
Theo em 1.1.3(A zelà-Ascoli).Le (I,d)be a compac me ic space and conside he
Banach space (C(I,Rn),k·k∞). A se D⊂C(I,Rn)is ela i ely compac i and only
i he ollowing condi ions a e sa is ied:
1.Dis bounded, ha is, he e exis s M > 0 such ha kxk∞< M o all x∈D;
2.Dis uni o mly equicon inuous, ha is, o all ε > 0, he e exis s δ > 0 such ha
o e e y x∈D,
|x( ) − x(s)|< ε, o all ,s∈I,wi h d( ,s)< δ.
2
KRASNOSEL’SKII TYPE COMPRESSION-EXPANSION
FIXED POINT THEOREM FOR SET CONTRACTIONS AND
STAR CONVEX SETS
The con en s o his chap e a e comp ised in he esea ch a icles (Lois-P ados
and Rod íguez-López, 2020)1and (Lois-P ados, P ecup, and Rod íguez-López,
2020)2. In bo h manusc ip s, we deal wi h gene aliza ions o K asnosel’skii ype
comp ession-expansion esul s o se con ac ions and domains de e mined by
a cone and wo s a con ex se s; and we show he applicabili y o he esul s o
ini ial o bounda y alue p oblems. We o ganize he con en s o he chap e as
ollows.
We begin wi h he In oduc ion sec ion, whe e we gi e a con ex ualiza ion o
ou s udy in he amewo k o ixed poin heo y. In pa icula , we e iew some
gene aliza ions o he classical Theo em o K asnosel’skii and we explain how
we con ibu e o hem. We also jus i y he use o a di ec app oach by means o
classical ixed poin a gumen s a he han opological deg ee echniques.
The heo e ical esul s we ha e de eloped a e con ained in Sec ions 2.2-2.4.
In Sec ion 2.2we desc ibe he ype o s a con ex se s we use o localize he
1C is ina Lois-P ados (Ins i u o de Ma emá icas, Uni e sidade de San iago de Compos ela,
Spain) & Rosana Rod íguez-López (Ins i u o de Ma emá icas, Uni e sidade de San iago de
Compos ela, Spain), “A gene aliza ion o K asnosel’skii comp ession ixed poin heo em
by using s a con ex se s”, P oceedings o he Royal Socie y o Edinbu gh - A (ISSN: 14737124,
03082105),150, pp. 277-303,2020. The inal au hen ica ed e sion is a ailable online a :
h ps://doi.o g/10.1017/p m.2018.119.
JCR 2019 (ca ego y; impac ac o ; ela i e posi ion; qua ile): Ma hema ics; 1.009; Q2;111/325.
SJR 2019 (ca ego y; impac ac o ; qua ile; H index): Ma hema ics (miscellaneous); 1.08; Q1;
52.
PhD s uden con ibu ions: The esea ch idea was concei ed by my supe iso R. Rod íguez-
López, hen C. Lois-P ados de eloped he majo i y o con en s o he Mas e deg ee inal dis-
se a ion. A he beginning o he PhD s udies, she elabo a ed Sec ion 3.1wi hin he published
manusc ip . My supe iso R. Rod íguez-López p o ided help, suppo and ideas h ough all
he elabo a ion and publica ion p ocess.
2C is ina Lois-P ados & Radu P ecup (Depa men o Ma hema ics, Babe¸s Bolyai Uni e si y,
Romania) & Rosana Rod íguez-López, “K asnosel’skii ype comp ession-expansion ixed poin
heo em o se -con ac ions and s a con ex se s”, Jou nal o Fixed Poin Theo y and Applica ions
(ISSN: 16617738,16617746),22,2020. The inal au hen ica ed e sion is a ailable online a :
h ps://doi.o g/10.1007/s11784-020-00799-0.
JCR 2019 (ca ego y; impac ac o ; ela i e posi ion; qua ile): Ma hema ics; 1.741;29/325; Q1.
PhD s uden con ibu ions: In his manusc ip , we imp o ed he wo main esul s ob ained in
he Mas e deg ee inal disse a ion o C. Lois-P ados (supe ised by R. Rod íguez-López). C.
Lois-P ados de eloped almos all he con en s o he submi ed e sion o he a icle, ollowing
some sugges ions om he supe iso (imp o emen o he comp ession esul ) and p o esso
R. P ecup (ideas o he expansion esul and he applica ion sec ion). Asked by a e iewe ,
p o esso R. P ecup en iched he applica ion esul s, now wo king o sys ems a he han o
a singula equa ion.
13
14 k asnosel’skii ype esul s in s a con ex se s
ixed poin s and we also de i e some use ul p ope ies. Then, we a e in a
posi ion o p o e he main ixed poin esul s in Sec ion 2.3. Finally, in Sec ion
2.4, we show ha we can use unc ionals o desc ibe he unde lying s a con ex
se s, and hese unc ionals a e mo e gene al han a no m.
In Sec ion 2.5, we show he applicabili y o he comp ession ype esul o
an ini ial alue p oblem o a sys em o i s o de di e en ial equa ions. The
associa ed in eg al ype mapping is noncompac bu a se con ac ion; and he
ou e bounda y o he localiza ion domain is de ined by se e al unc ionals
mo e gene al han a no m.
We conclude wi h he Discussion sec ion, whe e we summa ize ou main
achie emen s and compa e ou esea ch wo k wi h o he ela ed s udies in he
amewo k o ixed poin heo y.
2.1 in oduc ion
In his sec ion, we in oduce he esea ch wo k de eloped wi hin his chap e
in he amewo k o ixed poin heo y. We p esen he con en s di ided in
blocks.
Applicabili y o K asnosel’skii ype ixed poin esul s
K asnosel’skii ype comp ession-expansion ixed poin heo ems a e a powe ul
ool o p o e he exis ence o posi i e solu ions o se e al classes o p oblems
and also o ob ain mul iple solu ions. E be and Wang (1994), To es (2003), and
Zima (2004) ha e applied a gene aliza ion o Theo em 1.2.5w i en in e ms
o he no m o di e en second o de bounda y alue p oblems. Fo ins ance,
To es (2003) conside s a second o de equa ion wi h pe iodic bounda y condi-
ions and a Ca a heodo y non-linea e m. He also shows se e al applica ions,
such as one o equa ions wi h jumping nonlinea i ies. O’Regan and P ecup
(2005) ha e applied o Hamme s ein in eg al equa ions a gene aliza ion o The-
o em 1.2.5wi h comp ession-expansion condi ions gi en in e ms o wo no ms.
O he au ho s ha e applied his ype o esul s o i s o de di e en ial sys ems.
In (Wang, 2011), i is p o ed he exis ence and mul iplici y o ω-pe iodic solu-
ions o a non-au onomous singula sys em; while Bolojan and P ecup (2014)
ha e s udied an implici sys em wi h nonlocal condi ions.
Gene aliza ions o he K asnosel’skii comp ession-expansion ixed poin heo em
We al eady men ioned, in he in oduc o y pa ag aphs o Chap e 1, ha he
classical K asnosel’skii comp ession-expansion ixed poin heo em can be gen-
e alized by elaxing di e en hypo hesis. In Subsec ion 1.2.1, we ecall some ex-
ensions o he Theo em o K asnosel’skii wo king o mo e gene al mappings,
mo e p ecisely, o se con ac ions. In Subsec ion 1.2.2, we conside he no m
ype, ec o and homo opy e sions in which he comp ession-expansion condi-
2.1 in oduc ion 15
ions di e om he o iginal ones. Mo eo e , one o hese ex ensions conside s
mo e gene al domains. This is he esul due o Güo and Lakshmikan ham
(1998) which localizes he ixed poin in he mo e gene al egion C∩(Ω2 Ω1),
whe e Cis a cone in a Banach space (X,k·k)and Ω1,Ω2a e bounded open se s.
Ano he gene aliza ion in his di ec ion can be ound in (P ecup, 2006), whe e
he ixed poin is loca ed in he egion {x∈C: < kxk1,kxk2< R}, which
is de e mined by he cone and wo no ms. These wo gene aliza ions wo k
o con inuous and compac mappings, bu change he comp ession-expansion
condi ions and conside mo e gene al domains. This is also he case o Theo-
em 4.1in (Ande son, A e y, and Hende son, 2010), which conside s con inu-
ous conca e/con ex unc ionals ha a e in ol ed in he Legge -Williams ype
condi ions and he localiza ion domains. In he wo k by Kwong (2008), we ind
se e al ex ensions o Theo em 1.2.5.
In his chap e , we deal wi h a gene aliza ion ha p ese es he comp ession-
expansion ype condi ions o he K asnosel’skii o iginal esul , bu wo ks o
se con ac ions and mo e gene al domains, which a e de e mined by a cone
and wo s a con ex se s. The mo i a ion o wo king wi h s a con ex se s
comes om he necessi y o dis inguish be ween wo solu ions in case ha hey
ha e he same no m, see Figu e 1. I is wo h men ioning ha he esul s owed
by P ecup (2006) could also o e come his p oblem. Howe e , ou esul s wo k
o mo e gene al mappings and we claim ha he s a con ex se s can p o ide a
be e localiza ion han he se s de e mined by wo no ms. Mo eo e , in Sec ion
2.4, he s a emen s o Co olla y 2.4.4is gi en in e ms o wo no ms and i is
simila o Theo em 3in (P ecup, 2006), wi h he di e ence ha ou esul wo ks
o mo e gene al mappings, bu he o he does no equi e he comple eness o
he space.
Di ec app oach s deg ee heo y
K asnosel’skii p o ed his heo em di ec ly, using only classical a gumen s o
ixed poin heo y, pa icula ly Schaude ’s ixed poin heo em (K asnosel’skii,
1960,1964). Howe e , i is well-known ha his esul can also be deduced
as a consequence o he opological deg ee heo y (see G anas and Dugundji,
2013). Mo eo e , many o he gene aliza ions o he classical esul s due o
K asnosel’skii ha e been p o ed using opological deg ee heo y, such as hose
in (Ande son, A e y, and Hende son, 2010; Güo and Lakshmikan ham, 1998).
Ne e heless, om a heo e ical pe spec i e, a di ec app oach wi hou using
deg ee a gumen s could be use ul when ying o ex end he esul s om com-
pac mappings o mo e gene al ones. Such a possibili y is shown in (O’Regan
and P ecup, 2001, Chap e 10), whe e some comp ession-expansion esul s a e
es ablished o a amily o mappings o which he opological deg ee has no
been de eloped. Also, o applica ions, i seems mo e con enien o use he
Theo em o K asnosel’skii a he han he deg ee heo y, because he ixed
16 k asnosel’skii ype esul s in s a con ex se s
(A)
SR
S
•
•
(B)
C∩F2
C∩F1
•
•
Figu e 1: Illus a ion o he egions whe e he K asnosel’skii comp ession-expansion
ixed poin heo em (A)and ou main esul s (B)loca e he ixed poin s. The
cone C={(x,y)∈R2:x,y⩾0}is ep esen ed in g ay colo ; he black
do s a e he ixed poin s and he black a c ep esen s he poin s wi h he
same no m as he ixed poin s. The magen a and blue cu es a e he se s
whe e he comp ession o expansion condi ions a e sa is ied. The egions in
be ween hese cu es co espond o hose whe e he esul s localize he ixed
poin s. In (A)bo h ixed poin s a e inside his egion, while in (B) he egion
isola es one o hem.
poin esul o e s di ec ly he comp ession-expansion condi ions ha ha e o
be ul illed.
The di ec app oach owed by K asnosel’skii was also ollowed by Po e
(1974) and Các and Ga ica (1979), who espec i ely ex ended he comp ession
and expansion esul om con inuous and compac mappings o se con ac-
ions (see Theo ems 1.2.7,1.2.8). No ice ha bo h K asnosel’skii and Po e
ob ained a solu ion localized in conical annula se s de e mined by he no m,
bu he esul ob ained by Các and Ga ica p o ides a less e ined localiza ion,
since hey can only asse ha he e exis s a posi i e ixed poin in he cone.
Ou main esul s can be seen as gene aliza ions o he esul s due o Po e
(1974) and Các and Ga ica (1979) o mo e gene al domains; and ou p oo s also
ollow a di ec app oach wi hou using opological deg ee echniques.
O e iew o he esea ch de elopmen
Ou s udy is ocused on he gene aliza ion o he comp ession-expansion e-
sul s o se con ac ions o conical domains de e mined by wo s a con ex
se s. In his ega d, we i s equi e some addi ional condi ions o hese non
necessa ily con ex se s. An essen ial p ope y is he ollowing one: each ay
a eling om ze o and passing h ough any o he poin in he s a con ex
se mee s a unique elemen in i s bounda y. We no ice ha , o he heo e i-
cal a gumen s, we use an equi alen condi ion gi en in e ms o a con inuous
unc ional. Once we ha e de e mined he amewo k whe e we will de elop
2.2 gene al amewo k 17
ou esea ch, o p o e he comp ession esul we adap Po e ’s ideas o s a
con ex se s. Ne e heless, o he expansi e case we do no ollow he p oo
gi en by Các and Ga ica, bu we use he idea o educing he expansi e case
o he comp essi e one, as shown in (P ecup, 2006) o compac mappings. In
his way, we imp o e he localiza ion o ixed poin s p o ided by he exis ing
heo em o balls and s a e a new esul o s a con ex se s. The possibili y
o use s a con ex se s ins ead o balls allows o de e mine he conical domain
by unc ionals mo e gene al han a no m. The use o such unc ionals seems
o be e y use ul o applica ions, so we de i e some su icien condi ions ha
hey shall sa is y. Finally, i is impo an o show he applicabili y o he new
esul s. Fo ha pu pose, we conside an ini ial alue p oblem o a sys em o
i s o de implici di e en ial equa ions, we apply he comp ession esul o
he associa ed noncompac in eg al mapping, and we use unc ionals o de ine
a s a con ex se .
2.2 gene al amewo k
We ha e al eady men ioned ha ou esul s imp o e he localiza ion o ixed
poin s in Theo em 1.2.5by using s a con ex se s. Thus, we begin his sec ion by
ecalling his concep and by s a ing some addi ional condi ions we will equi e
o he se s conside ed. We p o ide wo cha ac e iza ions o he unde lying s a
con ex se s, one easie o isualiza ion and he o he mo e use ul o he heo-
e ical aspec s. Then we p o e he equi alence be ween hese cha ac e iza ions
and we deduce some help ul p ope ies o hese se s.
We no ice ha mos o he sec ion con en s we e de eloped o he Mas e
deg ee disse a ion o C is ina Lois P ados. Howe e , i was du ing he PhD
pe iod when he equi alence be ween he cha ac e iza ions in Condi ion 1was
comple ed wi h P oposi ion 2.2.2.
De ini ion 2.2.1.Le (X,k·k)be a Banach space, E⊂Xand x0∈E. We say ha
Eis an x0-s a con ex se i
λx0+ (1−λ)x∈E, o all λ∈[0,1]and x∈E.
In case ha x0=0,Eis simply called a s a con ex se .
Condi ion 1.In he ollowing, we conside a Banach space (X,k·k),Ca cone
in Xand E⊂Xa s a con ex se (which i ially sa is y E∩C6=∅). In addi ion,
he s a con ex se shall ul ill:
1.Eis bounded and closed;
2. I Fis he bounda y o Ein X, hen 0 /∈F;
and one o he ollowing equi alen hypo heses:
3. o e e y x∈E {0}, he e exis s a unique βx> 0 wi h βxx∈F;
18 k asnosel’skii ype esul s in s a con ex se s
4. he e exis s a con inuous mapping ∂:E {0}−→ F,x7−→ ∂(x), such ha
∂(x) = ∂(λx),∀x∈E,∀λ∈(0,1];
∂(x) = x,∀x∈F.
The ollowing esul p o es ha hypo hesis 3in Condi ion 1implies s a e-
men 4, bu in he gene al con ex o an x0-s a con ex se .
P oposi ion 2.2.2.Le (X,k·k)be a Banach space, x0∈Xand E⊂Xbe a bounded
closed x0-s a con ex se such ha i s bounda y Fdoes no con ain x0,and
o all x∈E {x0}, he e is a unique λx> 0 wi h λxx+ (1−λx)x0∈F. (6)
Then he e exis s a con inuous mapping ∂:E {x0}→Fsuch ha
∂(x) = ∂(λx + (1−λ)x0), o all x∈E {x0},λ∈(0,1];
∂(x) = x, o all x∈F.(7)
P oo . As x0∈E F, he e exis s γ > 0 such ha B:= B(x0,γ)⊂E F. Le Sbe
he bounda y o B, i.e., S:= {x∈X:kx−x0k=γ}. We de ine he mapping ∂
as he composi ion η◦η0, whe e η0is he adial p ojec ion
η0:E {x0}→S,η0(x)=γ
kx−x0k(x−x0),
and
η:S→F,η(x)=λxx+(1−λx)x0.
F om (6), he mapping ηis well-de ined. Also, i is easy o see ha condi ion
(7) is sa is ied. Clea ly, η0is con inuous, so i emains o p o e he con inui y
o η. To his aim, i su ices o p o e he con inui y o he unc ion
λ:S−→ R+,λ(x)=λx.
Fo ha pu pose, le {yn}n∈Nbe any sequence in Scon e ging o some y∈S.
Since Eis bounded and λ(S)⊂[0,+∞), hen he e exis s m∈R+such ha
λ(S)⊂[0,m]. The e o e, he sequence {λ(yn)}n∈Nis included in he compac
in e al [0,m], so any o i s limi poin s is ini e. Le lbe any limi poin o
{λ(yn)}n∈N. F om
η(yn) = λ(yn)yn+ (1−λ(yn))x0∈F,
we ind ha
ly +(1−l)x0∈F.
This, in iew o (6), gi es l=λ(y). Hence λ(yn)→λ(y)as n→+∞. The e o e,
λis con inuous as wished.
2.2 gene al amewo k 19
The nex esul p o es he emaining implica ion.
P oposi ion 2.2.3.Le Cbe a cone a Banach space (X,k·k)and Ebe a s a con ex
se sa is ying Condi ion 1wi h hypo hesis 4. Then, Ealso sa is ies hypo hesis 3, ha
is, o all x∈E {0}, he e exis s a unique eal numbe βx> 0 such ha βxx∈F. In
pa icula , i x∈C∩(E {0}), hen βxx∈C∩F.
P oo . The exis ence o such a numbe is clea since Cis a cone and Eis closed,
bounded, wi h non-emp y in e io , and a s a con ex se . Suppose ha he e
exis β1
x,β2
x∈R+such ha β1
x6=β2
xand β1
xx,β2
xx∈F. Wi hou loss o
gene ali y, we assume ha β2
x> β1
x, hen 0 < β1
x
β2
x< 1 and, he e o e,
∂(β1
xx) = ∂β1
x
β2
x
β2
xx=∂(β2
xx) = β2
xx∈F. (8)
Mo eo e , since β1
xx∈F, we ha e ∂(β1
xx) = β1
xx∈F. By using (8), we ob ain
β1
xx=β2
xx. Taking he no m, we ge β1
xkxk=β2
xkxk, and, since x6=0, we
conclude ha β1
x=β2
x, i.e., he elemen βxin he s a emen is unique.
Besides, i x∈Cand βx> 0, hen i is clea ha βxx∈Cby using he
de ini ion o cone.
Apa om he con inuous map ∂gi en by hypo hesis 4in Condi ion 1, he
map de ined by he scala numbe e e ed o in he equi alen hypo hesis 3
also plays an impo an ole in he heo e ical de elopmen . Thus, i is in e es -
ing o deduce some o i s p ope ies.
P oposi ion 2.2.4.Le Cbe a cone and Ebe a s a con ex se sa is ying Condi ion 1.
Then he mapping
β:C∩(E {0})−→ [1,+∞)
x7−→ β(x) := βx;
whe e βxis he unique posi i e eal numbe such ha βxx∈F∩C, is con inuous and
β(x)→+∞as x→0.
P oo . Fi s o all, we p o e ha he image o βis a subse o [1,+∞). Le
x∈C∩(E {0}), hen i is sa is ied ha βxx=∂(x)∈C∩F. Thus, by using ha
Eis a s a con ex se sa is ying Condi ion 1, we ge
βx=k∂(x)k
kxk⩾1.
The con inui y o β ollows easily since i can be exp essed as a composi ion
o con inuous unc ions:
β:C∩(E {0})−→ [1,+∞)
x7−→ β(x) = d(0,∂(x))
d(0,x)=(d0◦∂)(x)
d0(x);
20 k asnosel’skii ype esul s in s a con ex se s
whe e ∂is con inuous by hypo hesis and d0(x) := d(0,x) = kxk,x∈X, is
con inuous because o he dis ance p ope ies.
Finally, i emains o p o e ha β(x) ends o in ini y as xgoes o 0. Thus,
o all M∈R+, we look o δ∈R+such ha
β(x)> M, o all x∈C∩(E {0})wi h ||x|| < δ.
I M∈(0,1), hen β(x)> M is i ially sa is ied o all x∈C∩(E {0}), since
β∈[1,+∞).
I M⩾1, we ake 0 < δ := d(0,F)/M < +∞, wi h d(0,F) := in {d(0,y) : y∈F}.
Le x∈C∩(E {0})wi h ||x|| < δ, hen we p o e ha β(x)> M:
β(x) = d(0,∂(x))
d(0,x)⩾d(0,F)
d(0,x)=d(0,F)
||x|| >d(0,F)
δ=M.
We conclude his sec ion by showing ano he use ul p ope y o bo h map-
pings ∂and β, he ac ha hey can be con inuously ex ended o C {0}.
Rema k 2.Le Cbe a cone and Ebe a s a con ex se sa is ying Condi ion 1.
As Eis closed, a s a con ex se and 0∈˚
E, hen he es ic ion o he mapping
∂ o C∩(E {0})can be con inuously ex ended o C {0}as ollows
∂C:C {0}−→ F
x7−→ ∂C(x) :=
∂(x),x∈E {0};
∂d(0,F)
||x|| x,x∈C (C∩˚
E).
Figu e 2illus a es he beha io o ∂Cin a pa icula case.
By using ∂C, we can also con inuously ex end β o C {0}as:
βC(x) = d(0,∂C(x))
d(0,x), o all x∈C {0}.
2.3 main esul s
In his sec ion, we ex end he K asnosel’skii comp ession-expansion ixed poin
heo em o se con ac ions and s a con ex se s. Fo ha pu pose, he i s
s ep will be o e o mula e he comp ession-expansion condi ions in (4)-(5) o
s a con ex se s sa is ying Condi ion 1. A e ha , we a e in a posi ion o
p o e he mo e gene al esul s. Since we use di e en app oaches o p o e he
comp essi e and expansi e cases, we de o e one subsec ion o each one.
Le us conside Ca cone and Ei,i=1,2, s a con ex se s sa is ying Condi ion
1. We es ablish he ollowing no a ions o i=1,2:
•˚
Eiand Fia e he in e io and he bounda y o Ei, espec i ely.
2.3 main esul s 21
C∩F
•(x1,y1)
•
∂(x1,y1)
•
(x2,y2)
•∂C(x2,y2)
Figu e 2: Illus a ion o mappings ∂and ∂C o he cone C={(x,y)∈R2:x,y⩾0}.
The blue cu e ep esen s he in e sec ion o he cone wi h he bounda y o
he s a con ex se . The black segmen lines a e ays a eling om (0,0) o
he poin s (xi,yi)∈C {0}, o i=1,2.
•∂i:Ei {0}−→ Fiis he con inuous mapping gi en in Condi ion 1and ∂C
i
he con inuous ex ension o C {0}p o ided in Rema k 2.
•βi:C∩(Ei {0})−→ [1,+∞)is he con inuous map de ined in P oposi ion
2.2.4and βC
iis he co esponding con inuous ex ension o C {0}p o ided
in Rema k 2.
Nex , we assume ha E1⊂E2,F1∩F2=∅, and conside a mapping
T:C∩E2˚
E1→C.
We say ha Tis a comp ession o he se C∩E2 ˚
E1(see Figu e 3(A)) i :
(C1)x−T(x)/∈C, o all x∈C∩F1;
(C2)T(x) − (1+ε)x /∈C, o all ε > 0 and x∈C∩F2.
We say ha Tis an expansion o he se C∩E2 ˚
E1(see Figu e 3(B)) i :
(E1)T(x) − (1+ε)x /∈C, o all ε > 0 and x∈C∩F1;
(E2)x−T(x)/∈C, o all x∈C∩F2.
Be o e going h ough he de ails o he p oo o he gene alized esul s, we
p o ide some a gumen s o jus i y ha , unless he e exis s a bounded homeo-
mophism h ha ans o ms a s a con ex se Ei(i=1,2) sa is ying Condi ion 1
in a bounded closed ball Bi, we canno adap Theo ems 1.2.7and 1.2.8 o hese
s a con ex se s by simply using he classical idea o composing he mapping
Twi h he men ioned homeomo phic ans o ma ion. The main easons a e:
•Tis a k-se con ac ion: while i we i s apply a con inuous and compac
mapping (0-se con ac ion) and hen a con inuous and bounded map,
he composi ion inhe i s he compac ness p ope y, i k∈(0,1), hen he
co esponding condi ion is no gene ally p ese ed.
28 k asnosel’skii ype esul s in s a con ex se s
• I x∈C∩(E2 ˚
E1), as i is a bounded se and Tis a k-se con ac ion, he e
exis s M2∈R+such ha
T(x)
=kT(x)k⩽M2, o all x∈C∩(E2 ˚
E1).
The e o e, we conclude ha he e exis s R1∈R+wi h he abo e p ope y.
In addi ion, he e exis s R2∈R+such ha R2=sup{d(0,x) : x∈C∩E2},
since C∩E2is a bounded se .
Thus, we conside R=max{R1,R2}∈R+and B≡BR={x∈C:d(0,x)⩽R}.
We ha e chosen R∈R+such ha T(BR)⊆BR. Besides, he se BRis bounded,
closed and con ex, because i is he in e sec ion o he closed and con ex se C
wi h he bounded, closed and con ex se B(0,R) = {x∈X:d(0,x)⩽R}.
S ep 2:We p o e ha T|BRis a k-se con ac ion.
We i s show ha Tis con inuous. I is clea ly con inuous on C {0}and he
con inui y a x=0 ollows om simila a gumen s o hose in Lemma 2.3.1.
Then, we shall p o e ha he e exis s k<1such ha
α(T(A)) ⩽kα(A), o all A⊂BR.
In o de o p o e i , we de ine some help ul auxilia y mappings:
T1:BR∩E1−→ BRis gi en by
T1(x) :=
δh,x=0;
1
β1(x)T(β1(x)x) + δh,x∈BR∩Eδ
1 {0};
1
β1(x)T(β1(x)x) + d(x,F1)h,x∈BR∩E1 ˚
Eδ
1.
T2:BR BR∩˚
E1−→ BRis gi en by
T2(x) :=
T(x),x∈BR∩E2 ˚
E1;
T∂C
2(x),x∈BR BR∩˚
E2.
We i s deal wi h T1, which can be exp essed as he sum o wo mappings
T1
1and T2
1. The mapping
T1
1:BR∩E1−→ BR
x7−→ T1
1(x) :=
δh,x∈BR∩Eδ
1;
d(x,F1)h,x∈BR∩E1 ˚
Eδ
1
is a 0-se con ac ion. Indeed, le A⊂BR∩E1, hen Ais bounded and
T1
1(A) = T1
1A∩BR∩Eδ
1∪T1
1A∩BR∩E1 ˚
Eδ
1.
By using ha T1
1A∩BR∩E1 ˚
Eδ
1⊂co({0}∪{δh}), and he p ope ies 1,
2,3,6,7o P oposi ion 1.1.6, we can conclude
α(T1
1(A)) ⩽max{α({δh}),α(co({0}∪{δh}))}=0.
2.3 main esul s 29
Fu he mo e, he mapping
T2
1:BR∩E1−→ BR
x7−→ T2
1(x) :=
0,x=0;
1
β1(x)T(β1(x)x),x6=0
is a k2
1-se con ac ion wi h k2
1< 1, because i is he es ic ion o BR∩E1o ˜
Tin
Lemma 2.3.1wi h E=E1and F=F1.
Finally, by using s a emen 2o P oposi ion 1.1.9, we can asse ha T1is a
k1=0+k2
1-se con ac ion wi h k1< 1.
Now, we show ha T2is a k2-se con ac ion wi h k2=k, because i can
be w i en as he composi ion T◦S, whe e S:BR BR∩˚
E1−→ BR∩E2 ˚
E1,
which is gi en by
S(x) :=
x,x∈BR∩E2 ˚
E1;
∂C
2(x) = β2d(0,F2)
||x|| xd(0,F2)
||x|| x,x∈BR BR∩˚
E2;
is a 1-se con ac ion. The e o e, s a emen 1o P oposi ion 1.1.9is sa is ied
and, as a consequence, T2is a k-se con ac ion. We now p o e ha Sis a 1-se
con ac ion. Fo his, le us conside λ:BR BR∩˚
E1−→ R+gi en by
λ(x) :=
1,x∈BR∩E2 ˚
E1;
β2d(0,F2)
||x|| xd(0,F2)
||x|| ,x∈BR BR∩˚
E2;
which is a con inuous unc ion and sa is ies
supλ(x) : x∈BR∩BR ˚
E1⩽1.
Hence, by using s a emen 3in P oposi ion 1.1.9, we can conclude ha Sis a
1-se con ac ion, since he iden i y map also ul ills his p ope y.
Finally, applying Co olla y 1.1.10 o T1and T2, we ge ha T|BRis a k-se
con ac ion wi h k=max{k1,k}< 1.
The e o e, he hypo heses o Theo em 1.2.6a e sa is ied and T|BRhas a leas
one ixed poin x∈BR.
The ixed poin is in he conical domain C∩(E2 ˚
E1)
We inish he p oo by showing ha x∈C∩(E2 ˚
E1). To ha pu pose, we
assume ha he ixed poin belongs o one o he o he ou se s in ol ed in he
de ini ion o T:
• Suppose ha x=0.
Since T(0) = 0, hen δkhk=0and his is no possible because δ,khk> 0.
30 k asnosel’skii ype esul s in s a con ex se s
• Assume ha x∈BR∩(Eδ
1 {0}).
Consequen ly, T(x) = 1
β1(x)T(β1(x)x) + δh =x, so
khk⩽1
δkxk+1
δβ1(x)kT(β1(x)x)k
and i is a con adic ion wi h he selec ion o hin he de ini ion o he
mapping T:C−→ C.
• Le x∈BR∩E1 Eδ
1.
Since xis a ixed poin , hen T(x) = 1
β1(x)T(β1(x)x) + d(x,F1)h=x, so
x−1
β1(x)T(β1(x)x) = d(x,F1)h∈C,
due o d(x,F1)⩾0and h∈C. Mo eo e , β1(x)∈[1,+∞), hus
β1(x)x−T(β1(x)x)∈C, whe e β1(x)x∈C∩F1,
which con adic s he hypo hesis (C1) o Tbeing a comp ession o he
cone C.
• Suppose ha x∈BR (BR∩E2).
Le us de ine yx=d(0,F2)
||x|| x, hen
T(x) = T∂C
2(x)=T(∂2(yx))
=T(β2(yx)yx)=Td(0,∂2(yx))
kyxk
d(0,F2)
kxkx.
As kyxk=d(0,F2), hen T(x) = Td(0,∂2(yx))
kxkx. Take ε=kxk
d(0,∂2(yx)) −1,
we ha e ha ε>0since kxk
d(0,∂2(yx)) > 1. Mo eo e , we can exp ess xas
(1+ε)d(0,∂2(yx))
kxkxand
T(x) = Td(0,∂2(yx))
kxkx= (1+ε)d(0,∂2(yx))
kxkx.
By using ha d(0,∂2(yx))
kxkx∈C∩F2and
Td(0,∂2(yx))
kxkx− (1+ε)d(0,∂2(yx))
kxkx=0∈C,
we a i e o a con adic ion wi h he hypo hesis (C2)o Tbeing a com-
p ession o he cone C.
Thus, we ha e shown ha he ixed poin o Tbelongs o C∩E2 ˚
E1. Since
Tand Tcoincide on his se , hen we conclude ha Thas a ixed poin in he
men ioned se .
2.3 main esul s 31
2.3.2Expansi e case
We begin his sec ion by imp o ing he expansion ype esul in Theo em 1.2.8.
The p oo de eloped by Các and Ga ica (1979) ollowed he ideas used by K as-
nosel’skii. Ne e heless, by using he idea in (P ecup, 2006) o educe he ex-
pansi e case o he comp essi e one, we can p o ide a be e localiza ion o he
ixed poin in a conical annula se . Finally, he same idea is used o ex end he
esul o s a con ex se s.
Theo em 2.3.2.Le (X,k·k)be a Banach space, Cbe a cone in X, ,R∈R,0 < < R,
and T:C ,R−→ Cbe a k-se con ac ion sa is ying he expansion condi ion (5). Then,
Thas a ixed poin in C ,R.
P oo . We conside an auxilia y mapping ˜
T:C ,R−→ Cgi en by
˜
T(x) := 1
θ(x)T(θ(x)x),
whe e θ(x) = ( +R)/kxk−1, o e e y x∈C ,R. Then, we show ha i sa is ies
he hypo heses o Theo em 1.2.7due o Po e , ha is, we p o e sepa a ely ha
he mapping ˜
Tis well-de ined, i sa is ies he comp ession condi ions gi en in
(4) and i is a se con ac ion.
˜
Tis well-de ined
We need o show ha θ(x)x∈C ,R o e e y x∈C ,R. Since θ(x)> 0
and x∈C, we can asse ha θ(x)x∈C. The e o e, i emains o p o e ha
⩽kθ(x)xk⩽R. I x∈C ,R, hen ⩽kxk⩽R, whence ⩽ +R−kxk⩽R, ha
is, ⩽θ(x)kxk⩽R. Due o his, we can asse ha θ(x)x∈C ,Rand, inally, ˜
T
is well-de ined.
˜
Tsa is ies he comp ession condi ion (4)
I kxk= , hen kθ(x)xk=Rand, i kxk=R, hen kθ(x)xk= . As a
consequence, Tsa is ying (5) sa is ies ha ˜
T e i ies (4).
˜
Tis a se con ac ion
I is clea ha ˜
Tis con inuous since T,θa e con inuous and θ > 0. The e o e,
i emains o p o e ha , o e e y A⊂C ,R(Ais bounded), we ha e
α(˜
T(A)) ⩽˜
kα (A), (12)
o some cons an 0⩽˜
k < 1, independen o A. Fo ha pu pose, we p oceed
simila ly he p oo o Lemma 2.3.1o (Po e , 1974, Lemma 3.1). We begin by
dis inguishing wo cases.
I α(A) = 0, hen Ais compac . As ˜
Tis con inuous, hen ˜
T(A)is compac
and, he e o e,
α(˜
T(A)) ⩽α(˜
T(A)) = 0=˜
k α(A), o any ˜
k⩾0.
32 k asnosel’skii ype esul s in s a con ex se s
I α(A)6=0, le us assume ha k6=0, bu , i i is no he case, we conside
0 < ˆ
k < 1 as close o kas we wish. We p oceed as ollows:
S ep 1:We co e Aby a ini e numbe o subse s such ha he es ic ions o ˜
T
o each o hem a e α-Lipschi z wi h a sui able cons an .
As 0< <R, hen δ ,R=R
−
R> 0 and, o each n∈N, we can conside
εn
,R:= δ ,R/n.
Le n∈N,n⩾1, be a bi a ily ixed, o each in ege numbe m⩾0, we
de ine he ollowing se s:
An
m:= x∈A:θ(x)∈h
R+m εn
,R,
R+ (m+1)εn
,Ri.
Since A⊂C ,R, we can asse ha
R⩽θ(x)⩽R
. Mo eo e , no icing ha
R+0 εn
,R=
Rand
R+ [(n−1) + 1]εn
,R=R
, we ge
A⊂
n−1
[
m=0
An
m.
F om his, by using some p ope ies o he measu e o noncompac ness in
P oposi ion 1.1.6, i ollows ha
α˜
T(A)⩽α n−1
[
m=0
˜
T(An
m)!=max
m∈{0,...,n−1}α˜
T(An
m).
S ep 2:Fo each m∈{0,...,n−1}, we show ha ˜
T|An
mis α-Lipschi z wi h
cons an k
R+ (m+1)εn
,R/
R+m εn
,R.
We s udy some p ope ies o he ollowing auxilia y mappings:
1
θ|An
m
:An
m−→ C,x7−→ 1
θ(x);
Sn
m:An
m−→ C,x7−→ Sn
m(x) := θ(x)x.
On he one hand, o each x∈An
m, i is sa is ied ha 1/θ(x)⩽1/(
R+m εn
,R),
hen
sup1
θ(x):x∈An
m⩽1
R+m εn
,R
.
On he o he hand, o e e y B⊂An
m, we ha e
Sn
m(B) = {θ(x)x:x∈B}
⊂hλ
R+m εn
,R+ (1−λ)
R+ (m+1)εn
,Rix:λ∈[0,1],x∈B
=coh
R+m εn
,RiB∪h
R+ (m+1)εn
,RiB.
2.3 main esul s 33
Now, by using he p ope ies o he measu e o noncompac ness, we can asse
α(Sn
m(B))⩽αh
R+m εn
,RiB∪h
R+ (m+1)εn
,RiB
=
R+ (m+1)εn
,Rα(B).
Consequen ly, Sn
mis α-Lipschi z wi h cons an /R + (m+1)εn
,R.
As Tis a k-se con ac ion and ˜
T|An
m= (T◦Sn
m)/θ|An
m, we inally ge ha ˜
T|An
m
is α-Lipschi z wi h cons an k
R+ (m+1)εn
,R/
R+m εn
,R.
S ep 3:˜
Tis a se con ac ion.
Indeed, o each m∈{1,...,n−1}, i ollows ha
R+ (m+1)εn
,R
R+m εn
,R
<
R+εn
,R
R
.
Thus, aking in o accoun he di e en s a emen s which ha e been p o ed, we
ha e
α˜
T(A)⩽max
m∈{0,...,n−1}α˜
T(An
m)
⩽max
m∈{0,...,n−1}
R+ (m+1)εn
,R
R+m εn
,R
α(An
m)
⩽
R+εn
,R
R
k α(A).
Since εn
,R→0as n→∞, one has ha , o nla ge enough, he numbe
˜
k:=
R+εn
,R
R
k
is as close o kas we wish. The e o e, we can gua an ee ha ˜
k∈(0,1).
Exis ence o a ixed poin o Tin C ,R
To inish he p oo , we apply Theo em 1.2.7 o he mapping ˜
T. Hence, ˜
Thas
a ixed poin ˜x∈C ,R, ha is,
˜x=˜
T(˜x) = 1
θ(˜x)T(θ(˜x)˜x),
o , equi alen ly,
θ(˜x)˜x=T(θ(˜x)˜x).
This shows ha he poin ˆx:= θ(˜x)˜xis a ixed poin o Tin C ,R.
The nex esul ex ends Theo em 2.3.2 o s a con ex se s. We no ice ha he
p oo is qui e simila , he main di e ence appea s in he de ini ion o he map
θ, which ans o ms he expansion condi ions in comp essi e ones.
34 k asnosel’skii ype esul s in s a con ex se s
Theo em 2.3.3.Le (X,k·k)be a Banach space, Cbe a cone in X, and E1,E2be s a
con ex se s ul illing Condi ion 1. I T:C∩E2 ˚
E1−→ Cis a k-se con ac ion and
an expansion o he se C∩E2 ˚
E1, hen Thas a ixed poin in C∩E2 ˚
E1.
P oo . We conside he auxilia y mapping ˜
T:C∩E2 ˚
E1−→ Cgi en by
˜
T(x) := 1
θ(x)T(θ(x)x),
whe e θ(x) = βC
1(x) + β2(x) − 1, o x∈C∩E2 ˚
E1. Then, we show ha i
sa is ies he hypo heses o Theo em 2.3.1, ha is, we p o e sepa a ely ha ˜
Tis
well-de ined, i is a comp ession o he se C∩E2 ˚
E1and i is a se con ac ion.
˜
Tis well-de ined
We need o show ha θ(x)x∈C∩E2 ˚
E1, o e e y x∈C∩E2 ˚
E1. Since
θ(x)> 0 and x∈C, we clea ly ha e θ(x)x∈C. To p o e ha θ(x)x∈E2 ˚
E1, we
no e he equi alence be ween λx ∈E2 ˚
E1and he inequali y βC
1(x)⩽λ⩽β2(x).
Now, le us conside x∈E2 ˚
E1, hen βC
1(x)⩽1⩽β2(x), and we can asse ha
βC
1(x)⩽βC
1(x) + β2(x) − 1⩽β2(x), ha is, βC
1(x)⩽θ(x)⩽β2(x), which shows
ha θ(x)x∈E2 ˚
E1. The e o e, we conclude ha ˜
Tis well-de ined.
˜
Tis a comp ession o he se C∩E2 ˚
E1
Fo ha pu pose, we p o e ha , i x∈F1, hen θ(x)x∈F2, and, i x∈F2, hen
θ(x)x∈F1. Consequen ly, i T ul ills (E1) hen ˜
T ul ills (C2); and i T ul ills
(E2) hen ˜
T ul ills (C1). This way, ˜
Tis a comp ession o he se C∩E2 ˚
E1.
Indeed, i x∈F1, hen βC
1(x) = 1and so θ(x) = β2(x). Hence, acco ding
o P oposi ion 2.2.4,θ(x)x∈F2. Simila ly, i x∈F2, hen β2(x) = 1and so
θ(x) = βC
1(x). Thus, by using he de ini ion o βC
1,θ(x)x∈F1.
˜
Tis a se con ac ion
On he one hand, as βC
1,β2and Ta e con inuous and θ(x)6=0, o all
x∈C∩E2 ˚
E1, one has ha ˜
Tis con inuous.
On he o he hand, o e e y A⊂C∩E2 ˚
E1, we ha e o p o e ha
α˜
T(A)⩽˜
kα(A), (13)
o some cons an 0⩽˜
k < 1, independen o A.
The p oo o (13) is iden ical o ha o o mula (12) in he p oo o Theo em
2.3.2, once we ha e shown he exis ence o wo posi i e numbe s and Rwi h
< R such ha
R⩽θ(x)⩽R
, o all x∈C∩E2 ˚
E1. (14)
Indeed, as 0∈E1 F1, he e exis s > 0 such ha B(0, )⊂˚
E1. Since E1⊂E2
and E2is bounded, hen he e exis s R > such ha E2⊂B(0,R). The e o e, we
2.4 admissible se s de ined by unc ionals 35
ha e ha C∩E2 ˚
E1⊂C∩(B(0,R) B(0, )). Hence, o any x∈C∩E2 ˚
E1,
we ob ain
x,θ(x)x∈C∩(B(0,R) B(0, )),
implying
⩽kxk,θ(x)kxk⩽R,
which immedia ely yield (14).
Exis ence o a ixed poin o Tin C∩E2 ˚
E1
To inish he p oo , we apply Theo em 2.3.1 o he mapping ˜
T. Thus, ˜
Thas a
ixed poin ˜x∈C∩E2 ˚
E1. Then
˜x=˜
T(˜x) = 1
θ(˜x)T(θ(˜x)˜x),
o , equi alen ly,
θ(˜x)˜x=T(θ(˜x)˜x).
This shows ha he poin ˆx:= θ(˜x)˜xis a ixed poin o Tin C∩E2 ˚
E1.
2.4 admissible se s de ined by unc ionals
The ixed poin heo ems due o K asnosel’skii, Po e , o Các and Ga ica p o-
ide he exis ence o ixed poin s o a mapping Tin ce ain subse s o a Banach
space (X,|| ·||). An in e es ing cha ac e is ic o he se s in ol ed in hese esul s,
ei he in he comp ession-expansion condi ions equi ed o he mapping To
in he se whe e he ixed poin s a e loca ed, is ha hey a e de e mined by a
no m. These se s can be exp essed in e ms o :
B =C∩{x∈X:||x|| ⩽ },S =C∩{x∈X:||x|| = },
whe e is a posi i e eal numbe .
In Sec ion 2.3, we gene alized he men ioned esul s o s a con ex se s,
which in gene al canno be exp essed by using a no m. Thus, o applica ions, i
is in e es ing o de e mine some condi ions o e a unc ional ϕ:X−→ [0,+∞)
such ha , o a eal numbe > 0, he se s
E := {x∈X:ϕ(x)⩽ },F := {x∈X:ϕ(x) = },
sa is y Condi ion 1. Thus, in his subsec ion, we i s de e mine some sui able
condi ions o be imposed he unc ional ϕ. Then, we de i e some co olla ies o
Theo ems 2.3.1and 2.3.3.
We no ice ha , i ϕ=|| ·||, hen C∩E ≡B and C∩F ≡S . Howe e , his
app oach allows o conside unc ionals wi h weake p ope ies in compa ison
36 k asnosel’skii ype esul s in s a con ex se s
wi h he no m. In o de o see ha we wo k wi h weake hypo heses, once
we know how o choose a sui able unc ional ϕ, we p o ide an example o a
unc ional sa is ying he new es ic ions ha is no a no m. Mo eo e , we will
show ha we canno elax he con inui y hypo hesis, a leas by conside ing an
uppe /lowe semicon inuous unc ional ϕ.
P oposi ion 2.4.1.Le (X,k·k)be a Banach space, x∈X, ∈R, >0, and
ϕ:X−→ [0,+∞)be a unc ional sa is ying:
(F1)i ϕ(x)⩽ , hen ϕ(λx)⩽ o all λ∈[0,1];
(F2)ϕis con inuous;
(F3) o all x∈E ,ϕ(x) = 0i and only i x=0;
(F4)ϕ(λx) = λϕ(x) o all λ∈(0,+∞)and x∈E ;
(F5) he e exis s m∈R,m > 0 such ha
mkxk⩽ϕ(x) o all x∈Xwi h kxk> ϕ(x),
o limin
kxk→+∞
ϕ(x)> .
Unde hese assump ions, E and F sa is y Condi ion 1.
P oo . We p oceed s ep by s ep, i.e., we p o e each o he p ope ies equi ed
o E and F by using he app op ia e hypo hesis o e ϕ. Indeed:
•(F1)is equi alen o E being a s a con ex se .
•(F2)implies ha E is closed and ϕ−1([0, )) is open in X. Indeed, we can
w i e E =ϕ−1([0, ]). As [0, ]is a closed subse o ([0,+∞),|·|), whe e
|·|is he absolu e alue o eal numbe s, hen E is closed since i is he
p eimage o a closed se by a con inuous unc ion. Besides, we can asse
ha ϕ−1([0, )) is open in Xsince [0, )is an open subse o [0,+∞).
•(F3)implies ha 0 /∈F , because ϕ(x) = > 0 o all x∈F .
• By hypo hesis (F4), we p o e wo condi ions o e he se s E and F .
Fi s , we show ha F is he bounda y o E . We ha e jus p o ed ha E
is closed and E F =ϕ−1([0, )) is open, so he bounda y o E is a subse
o F . Le us conside y∈F a bi a ily ixed, we wan o p o e ha y
is a bounda y poin o E . As y∈F ⊂E , we mus p o e ha , o all
ε > 0,B(y,ε)∩(X E )6=∅. We ake z=1+ε
2kyky∈B(y,ε), hen i is
sa is ied ha ϕ(z) = 1+ε
2kykϕ(y), and, as 1+ε
2kyk> 1, we can
conclude ha ϕ(z)> ϕ(y) = . The e o e, z∈X E and yis a poin in
2.4 admissible se s de ined by unc ionals 37
he bounda y o E .
Secondly, we p o e ha he mapping
∂:E {0}−→ F ,x7−→ ∂(x) :=
ϕ(x)x
sa is ies he desi ed condi ions. By using hypo heses (F3)and (F4), we can
asse ha ∂is well-de ined. Le x∈E {0},ϕ(∂(x)) = ϕ
ϕ(x)x= ,
so ∂(x)∈F . Then, (F2)implies ha ∂is con inuous. Now, i x∈F , hen
ϕ(x) = and, he e o e, ∂(x) = x. By using again he hypo hesis (F4), we
can asse ha ∂(λx) = ∂(x) o all x∈E and λ∈(0,1].
• Finally, (F5)implies ha E is a bounded se .
Example 2.4.2.Le (C([0,1],R),k·k∞)be he Banach space in Example 1.2.3 o
n=1. We conside he unc ional ϕ:C([0,1],R)−→ [0,∞)gi en by
ϕ(x) := amin
∈[0,1]|x( )|+bkxk∞, o all x∈C([0,1],R), (15)
whe e a,ba e posi i e eal numbe s. I is easy o see ha ϕsa is ies he hy-
po heses (F1)-(F5)in P oposi ion 2.4.1. Howe e , his unc ional does no ul ill
he iangula inequali y, hen i is no a no m. Indeed, le us conside he unc-
ions x( ) = o all ∈[0,2];y( ) = 1− i ∈[0,1],y( ) = −1i ∈[1,2],
hen we ha e
ϕ(x+y) = a+3b > ϕ(x) + ϕ(y) = 3b.
Nex , we make he ollowing ques ion: is i possible o ex end his s udy o
uppe o lowe semicon inuous unc ionals? In he ollowing, we p o e ha
he answe is nega i e, since his assump ion is no enough.
Assume ha ϕis uppe semicon inuous. We show ha E F is open, bu E
is no necessa ily closed. Indeed:
• Le ∈R, > 0, hen ϕ−1([0, )) is open. We ake y∈ϕ−1([0, )) a bi a -
ily ixed and show ha i is an in e io poin , i.e., he e exis s δ∈R,δ > 0
such ha
B(y,δ)⊂ϕ−1([0, )). (16)
As ϕ(y)< , we can ake ε > 0 such ha ϕ(y) + ε < . Besides, by
using ha ϕis uppe semicon inuous, he e exis s δy
εsuch ha , o all
x∈B(y,δy
ε),0⩽ϕ(x)< ϕ(y) + ε < . Thus, B(y,δy
ε)⊂ϕ−1([0, )), and i
p o es ha ϕ−1([0, )) is an open se .
44 k asnosel’skii ype esul s in s a con ex se s
Nex , we show ha (H4,c1)gua an ees ha condi ion (C1)is ul illed. Indeed,
i we assume he con a y, hen he e exis s y∈Cwi h
yj
∞⩽ j, o all
j∈{1,...,n}, and kykk∞= k o some k∈{1,...,n}, such ha
y( )⩾T(y)( ), o all ∈[0,1].
Le 0∈[0,1]be such ha yk( 0) = kykk∞= k. F om he p e ious inequali y,
by using he de ini ion o T, we ob ain
k=yk( 0)⩾Tk(y)( 0) = k 0,Z 0
0
y(s)ds+gk( 0,y( 0)) ⩾ k+gk,
which con adic s (H4,c1). Hence, (C1)holds.
We conclude by p o ing ha (C2)is also sa is ied as a consequence o hy-
po hesis (H4,c2). I we assume he con a y, hen he e exis ε>0and y∈C
wi h ϕj(yj)⩽Rj o all j∈{1,...,n}, and ϕk(yk) = Rk o some k∈{1,...,n},
such ha
T(y)( )⩾(1+ε)y( ), o all ∈[0,1].
Le 0∈[0,1]be such ha yk( 0) = kykk∞. Then, using he las inequali y, he
exp ession o T, and (23), we ob ain
k+gk⩾ k 0,Z 0
0
y(s)ds+gk( 0,y( 0))> yk( 0)⩾Rk
ak+bk
,
which con adic s (H4,c2). Hence, (C2)holds.
The e o e, since all he assump ions o Theo em 2.3.1a e ul illed, we ha e
he ollowing exis ence and localiza ion esul .
Theo em 2.5.1.Unde condi ions (H1)-(H3)and (H4,c1)-(H4,c2), p oblem (20)has
a non-nega i e and inc easing solu ion x∈C1([0,1],Rn)such ha
k⩽
x0
k
∞, o a leas one k∈{1,...,n},and
ϕi(x0
i)⩽Ri, o all i∈{1,...,n}.(24)
Pa icula case n=1
In pa icula , i we assume ha n=1, and he ollowing mono onici y condi-
ion on and g:
(H5)Fo each ∈[0,1], he unc ions ( ,·)and g( ,·)a e inc easing in R+;
hen condi ions (H4,c1)and (H4,c2), wi h 1,R1,a1,b1simply deno ed by ,R,
a,b, u n in o
(H5,c1) ( ,0) + g( , )> , o all ∈[0,1];
(H5,c2) ,R
b+g ,R
b⩽R
a+b, o all ∈[0,1].
Le us p esen wo examples. The i s one, which is in ac an explici ly
sol able equa ion, is gi en o es he condi ions (H1)-(H3),(H5,c1)-(H5,c2).
2.5 applica ion o a i s o de implici di e en ial sys em 45
Example 2.5.2.Le us conside he equa ion
x0( ) = λx( ) + αx0( ) + β, ∈[0,1].
In his case
( ,s) = λs and g( ,s) = αs +β, o all s∈R, ∈[0,1],
whe e we assume ha λ,α⩾0,β>0and λ+α < b
a+b. I
< β
1−αand R1
a+b−λ+α
b⩾β, (25)
hen he assump ions o Theo em 2.5.1a e ul illed.
One we ha e de e mined some su icien condi ions on he pa ame e s, we
deal wi h some pa icula alues and show ha he pa icula solu ion o p ob-
lem (20) sa is ies condi ion (24) in Theo em 2.5.1. Fi s , we ha e ha condi ion
(25) is sa is ied o a=b=1, =1,R=6,β=1and λ=α=1/6. In his case,
he exac solu ion o he p oblem wi h x(0) = 0is x( ) = 6e
5−1, ∈[0,1],
whose de i a i e is x0( ) = 6
5e
5, ∈[0,1]. Then
kx0k∞=6
5e1
5and ϕ(x0) = 6
51+e1
5,
and i is easy o see ha condi ion (24) holds.
The second example deals wi h an equa ion ha canno be explici ly sol ed.
Howe e , o some pa icula pa ame e alues, his equa ion educes o he
sol able implici p oblem gi en in Example 2.5.2. Thus, in his pa icula case,
we can see ha he su icien condi ions equi ed o he pa ame e s co espond
o hose in Example 2.5.2.
Example 2.5.3.Conside he equa ion
x0( ) = λx( ) + αx0( ) + β+γsin(x0( )), ∈[0,1]. (26)
In his case,
(s) = λs and g(s) = αs +β+γsins(s∈R),
whe e we assume ha α,β,γand λa e non-nega i e.
Now, we explain how o ul ill he condi ions (H1)-(H3),(H5,c1)-(H5,c2)in
Theo em 2.5.1.
Clea ly, condi ion (H1)holds.
Nex , i α < 1 −γ, hen |g0(s)|=|α+γcoss|⩽α+γ < 1. The e o e, (H2)is
sa is ied o k=α+γ < 1.
To gua an ee condi ion (H3), we need g(R+)⊂R+, which akes place i
β⩾γ.
46 k asnosel’skii ype esul s in s a con ex se s
Fu he mo e, condi ion (H5)is ul illed i gis inc easing in R+, and his
happens i α⩾γ. This condi ion, oge he wi h α < 1 −γ, gi es γ⩽α < 1 −γ.
Then, ob iously, γhas o sa is y 0⩽γ < 1
2.
Finally, we ha e o check condi ions (H5,c1)and (H5,c2). Fo he i s , we
need > 0 such ha g( )> , ha is, α +β+γsin > . This clea ly happens
i α +β−γ> , o , equi alen ly, < β−γ
1−α, which equi es β > γ since has o
be posi i e. Condi ion (H5,c2) eads as
λR
b+αR
b+β+γsin R
b⩽R
a+b. (27)
We show ha he e exis s Rla ge enough ha sa is ies his inequali y. Indeed,
i we di ide by R
b, we ob ain
λ+α+βb
R+γsinR
b
R
b
⩽b
a+b.
The limi o he le hand side, when R ends o ∞, being λ+αgua an ees he
exis ence o Rp o ided ha λ+α < b
a+bo , equi alen ly, λ < b
a+b−α. In iew
o λ⩾0, i equi es ha α < b
a+b.
The e o e, he condi ions o Theo em 2.5.1a e ul illed i he non-nega i e
pa ame e s α,β,γand λsa is y:
γ⩽α < 1 −γ,λ+α < b
a+b, and γ < min1
2,β.
Unde hese condi ions, o e e y
< β−γ
1−α,
he e exis s a solu ion x∈C1([0,1],R)o equa ion (26) wi h x(0) = 0 ha is
non-nega i e, inc easing and wi h kx0k∞⩾ .
I , in addi ion, a numbe Ris chosen such ha inequali y (27) holds, hen he
solu ion xalso sa is ies
amin
∈[0,1]x0( ) + bmax
∈[0,1]x0( )⩽R.
We inally show ha a simila app oach does no wo k o expansion ype
condi ions, ha is, Theo em 2.3.3does no apply o he ini ial alue p oblem
(20), a leas i we use a gumen s like hose de eloped o he comp essi e case.
Indeed, i we ake
E1={y∈C([0,1],R) : ϕ(y)⩽ },E2={y∈C([0,1],R) : kyk∞⩽R},
2.5 applica ion o a i s o de implici di e en ial sys em 47
whe e ϕis he unc ional de ined in Example 2.4.2and ,Ra e posi i e numbe s
wi h < bR; and we p oceed simila ly o he comp ession case, we a i e o
he ollowing su icien expansion ype condi ions:
(H4,E1)max
∈[0,1],y∈[0,
b] ( ,y) + max
∈[0,1],y∈[
a+b,
b]g( ,y)⩽
a+b;
(H4,E2)min
∈[0,1],y∈[0,R] ( ,y) + min
∈[0,1]g( ,R)> R.
These condi ions ensu e ha E1and E2a e s a con ex se s sa is ying Con-
di ion 1,0∈E1⊂˚
E2and (E1)-(E2)hold; bu , un o una ely, hey a e no
compa ible wi h hypo hesis (H2), and hus Theo em 2.3.3can no be applied.
We p o e his incompa ibili y in he au onomous case, ha is, when and gdo
no depend on , and condi ions (H4,E1)-(H4,E2) ead as
max
y∈[0,
b] (y) + max
y∈[
a+b,
b]g(y)⩽
a+b;
min
y∈[0,R] (y) + g(R)> R.
Sub ac ing he wo inequali ies yields
g(R)−max
y∈[
a+b,
b]g(y)> R −
a+b+max
y∈[0,
b] (y) − min
y∈[0,R] (y). (28)
F om < bR, we ha e [0, /b]⊂[0,R], whence
min
y∈[0,R] (y)⩽min
y∈[0,
b] (y)⩽max
y∈[0,
b] (y).
Hence, he igh -hand side in (28) is g ea e han o equal o R− / (a+b), so
g(R)−max
y∈[
a+b,
b]g(y)> R −
a+b. (29)
On he o he hand, i ˆy∈[ / (a+b), /b]is such ha g(ˆy) = max
y∈[ /(a+b), /b]g(y),
hen, by using (H2), we ha e
g(R)−max
y∈[
a+b,
b]g(y) = g(R)−g(ˆy)⩽k(R−ˆy)
⩽kR−
a+b< R −
a+b.
This oge he wi h condi ion (29) clea ly yields a con adic ion.
Thus, we did no succeed in applying he expansion ype esul o (20). How-
e e , we claim ha i may wo k o o he ypes o ini ial o bounda y alue
p oblems, since he expansion condi ions a e compa ible wi h many o he p ob-
lems in ol ing compac ope a o s, as illus a ed ex ensi ely in he li e a u e.
48 k asnosel’skii ype esul s in s a con ex se s
2.6 discussion
In his sec ion, we summa ize and discuss he esea ch wo k we ha e de eloped
h oughou he chap e . We p esen he con en s classi ied in blocks o a be e
dis ibu ion.
In e es o he new ex ensions o K asnosel’skii ixed poin heo em
In Sec ion 2.1, we conside some o he se e al gene aliza ions o K asnosel’skii
comp ession-expansion ixed poin heo em. Then, in Sec ions 2.2and 2.3, we
ex end he esul s o se con ac ions o s a con ex se s. In he ollowing, we
jus i y he use ulness o he ob ained esul s.
The ini ial mo i a ion o wo k in his pa icula gene aliza ion was he pos-
sibili y o localize di e en solu ions wi h he same no m. We no ice ha he e
exis o he esul s wi h his po en ial, like hose owed by Güo and Lakshmikan-
ham (1998) and P ecup (2006). Ou esul s gene alize he ones in (P ecup,
2006) (see Sec ion 2.4 o de ails), bu do no wo k o he gene al domains de-
e mined by wo open se s in (Güo and Lakshmikan ham, 1998). Howe e , he
key poin o ou esul s is ha hey conside se con ac ions ins ead o con in-
uous and compac maps, and i allows us o deal wi h mo e gene al p oblems,
such as he sys em o implici i s o de di e en ial equa ions in Sec ion 2.5.
The esul s in (P ecup, 2006) ex end he applicabili y o K asnosel’skii ype
ixed poin heo ems o bounda y alue p oblems o pa ial di e en ial equa-
ions, such as semi-linea ellip ic p oblems. The key ing edien o hese esul s
is he possibili y o wo k in mo e gene al domains de e mined by wo no ms.
Since ou ixed poin heo ems can conside un ionals mo e gene al han a
no m, we claim ha hey can be also use ul o deal wi h his ype o p oblems.
I is wo h men ioning ha , apa om he gene aliza ion o he esul s o
conical domains de e mined by wo s a con ex se s, we also imp o e he ex-
pansion ype ixed poin heo em o balls owed by Các and Ga ica (1979). In
he wo k (Po e , 1974), i is said ha Theo em 2.3.2 ollows om simila a gu-
men s o hose de eloped o he comp essi e case, bu he p oo is no gi en.
Besides, Các and Ga ica (1979) ollow K asnosel’ski ideas in he p oo o he
expansi e case in Theo em 1.2.5, bu hey needed o impose mo e es ic i e
expansion condi ions and he balls did no play a ole in he localiza ion o
ixed poin s. Thus, we ealized ha a change o a iable which ans o ms he
expansi e case in he comp essi e one helps o elax he expansion hypo heses
and o localize he ixed poin s in a conical shell de e mined by wo balls.
Complexi y o he hypo heses in ou main esul s
A he beginning o Sec ion 2.3, we show ha he gene aliza ion o he exis ing
esul s o balls o s a con ex se s canno be simply adap ed by using he clas-
sical idea o composing wi h a homeomo phic ans o ma ion. We p o ided
2.6 discussion 49
di e en easons ela ed wi h he se con ac ion p ope y, he domain and
ange o he mapping and he comp ession-expansion condi ions.
Besides, in Subsec ion 2.3.1, whe e we adap Po e ’s esul s o s a con ex
se s, we obse e ha he di icul y when wo king wi h k-se con ac ions comes
om he ac ha he geome ic ans o ma ions can change uncon ollably he
cons an k. Mo eo e , he use o s a con ex se s in oduces much mo e com-
plica ed geome ic ans o ma ions connec ed o hei e o-ac i i y p ope y,
which ha e o be pu in o acco dance wi h he cons an k, as Lemma 2.3.1
shows.
Rele ance o de ining he s a con ex se s by using unc ionals
We al eady men ioned ha E be and Wang (1994), To es (2003), and Zima
(2004) apply gene aliza ions o Theo em 1.2.5in e ms o he no m o di e -
en second o de bounda y alue p oblems. These ixed poin esul s enable
o localize solu ions in he gene al domain C∩(Ω2 Ω1), whe e Cis a cone
and Ω1,Ω2a e open se s. Howe e , o applica ions, hey choose he simples
possibili y and de e mine he open se s by using he no m. In his way, hey
a e missing he good localiza ion quali ies o his ype o esul s. In ou case,
he comp ession-expansion esul s wo k o s a con ex se s mo e gene al han
balls, so o ake ad an age o his gene al amewo k in applica ions, we de e -
mine condi ions o e a unc ional in o de o de ine admissible s a con ex se s
while being mo e gene al han a no m.
To conclude, we ecall ha he e exis o he esea ch s udies dealing wi h
gene aliza ions o K asnosel’skii ixed poin heo em in e ms o unc ionals.
We men ioned in he In oduc ion he wo k by Ande son, A e y, and Hende -
son (2010), whe e he condi ions equi ed do no mee hypo heses (F1)-(F5),
since hey conside con ex/conca e unc ionals.
3
APPLICATIONS OF FIXED POINT THEORY TO PERIODIC
PREDATOR-PREY DIFFERENTIAL EQUATIONS
This chap e includes he con en s o he esea ch a icle (Lois-P ados and P e-
cup, 2020)1, in which we con ibu e o he applica ion o ixed poin esul s o
Lo ka-Vol e a ype models. The chap e is o ganized as ollows:
We s a wi h an in oduc o y sec ion, whe e we i s e isi di e en o -
mula ions o p eda o -p ey sys ems, we p o ide he o mula ion o he model
in o conside a ion and we also desc ibe ou ini ial mo i a ion o de elop he
p esen esea ch. Nex , we explain he in e es o ou s udy in he amewo k
o ixed poin heo y, whe e we con ibu e o he applica ion o K asnosel’skii
ype ixed poin heo ems. Finally, we b ie ly e iew he esea ch de eloped
h oughou his chap e .
Then, in Sec ion 3.2, we p epa e he unde lying model o he applica ion
o ixed poin echniques, so we ob ain i s in eg al e sion and we s a e some
use ul no a ions.
The cen al pa o he esea ch is comp ised in Sec ion 3.3, which is di ided
in o h ee subsec ions. In he i s one, we s a e and p o e he main esul
abou he exis ence and localiza ion o pe iodic solu ions as a consequence o
he homo opy e sion o K asnosel’skii ixed poin heo em; and we discuss
he possibili y o localiza ion o some pa icula exp essions o he model. In
Subsec ion 3.3.2, we s udy wo in e es ing p ope ies o he localized solu ions.
In he las subsec ion, he exis ence esul s a e imp o ed o he pa icula case
in which he p eda o unc ional esponse does no depend on ime.
Finally, we include a Discussion sec ion, whe e we summa ize ou main con-
ibu ions o he s udy o Lo ka-Vol e a ype sys ems, by means o ixed poin
heo y echniques, and we compa e ou app oach wi h o he simila esea ch
wo ks. We also de o e some lines o alk abou he quali a i e p ope ies o he
localized solu ions.
1C is ina Lois-P ados (Ins i u o de Ma emá icas, Uni e sidade de San iago de Compos ela,
Spain) & Radu P ecup (Depa men o Ma hema ics, Babe¸s Bolyai Uni e si y, Romania), “Pos-
i i e pe iodic solu ions o Lo ka–Vol e a sys ems wi h a gene al a ack a e”, Nonlinea Anal-
ysis: Real Wo ld Applica ions (ISSN: 14681218),52,2020. The inal au hen ica ed e sion is
a ailable online a : h ps://doi.o g/10.1016/j.non wa.2019.103024.
JCR 2019 (ca ego y; impac ac o ; ela i e posi ion; qua ile): Applied Ma hema ics; 2.072;
37/261; Q1.
PhD s uden con ibu ions: The au ho s ha e equally con ibu ed o he de elopmen o he a -
icle con en s, hey we e con inuously collabo a ing du ing he esea ch s ay o C. Lois-P ados
a Babe¸s Bolyai Uni e si y. The ini ial idea o s udying he in ol ed Lo ka-Vol e a ype model
was concei ed by C. Lois-P ados.
51
52 applica ions o p eda o -p ey di e en ial equa ions
3.1 in oduc ion and model desc ip ion
In his sec ion, we include he o mula ion o he non-au onomous p eda o -
p ey popula ion model s udied in his chap e , as well as some mo i a ions
and in e es s o ou esea ch wo k. We o ganize he con en s by di iding hem
in blocks wi h hei co esponding desc ip i e headline.
Re iew o di e en Lo ka-Vol e a ype sys ems and esea ch mo i a ion
Lo ka-Vol e a ype sys ems a e commonly used o desc ibe in e ac ions be-
ween wo species, p ey and p eda o . In he au onomous case, hese models
ha e a Kolmogo o s uc u e, being o he o m
x0=x F(x,y);
y0=y G(x,y);
and mos o hem sa is y he ollowing condi ions:
Fy(x,y)< 0,Gx(x,y)> 0 and Gy(x,y)⩽0
(see, e.g., B aue and Cas illo Chá ez, 2001, Sec ion 5.4). Fo non-au onomous
Kolmogo o ype sys ems, we e e he eade o he pape by Zanolin (1992).
The o iginal model p oposed by Lo ka (1925) and Vol e a (1926) is gi en by
x0=ax −λxy;
y0= −by +cλxy.(30)
Fo a his o ical no e on his classical model see (Bacaë , 2011). As sugges ed
by Vol e a himsel , a mo e ealis ic p ey g ow h is he logis ic one, which was
conside ed by se e al au ho s. Fo ins ance, Rosenzweig and MacA hu (1963)
p oposed he ollowing model:
x0=ax 1−x
K−φ(x)y;
y0= −by +cφ(x)y.
Some gene aliza ions o he Rosenzweig-MacA hu model a e gi en in (Van
de Ho and Fay, 2016), whe e, in pa icula , i is conside ed he logis ic g ow h
o bo h p ey and p eda o popula ions (see also Bu oni, G oppi, and So esina,
2011).
In his pape , we look o pe iodic solu ions o he ollowing pe iodic Lo ka-
Vol e a ype sys ems wi h a gene al p ey g ow h gand a gene al unc ional
esponse o p eda o s ϕ:
x0=a( )xg(x) − ϕ( ,x,y)xy;
y0= −b( )y+c( )ϕ( ,x,y)xy;(31)
3.1 in oduc ion and model desc ip ion 53
whe e a,b,c∈C(R,R+)a e ω-pe iodic wi h he same pe iod ω > 0,a,b6≡ 0,
mins∈[0,ω]c(s)> 0;ϕ∈C(R×R+×R+,R+)is such ha ϕ(·,x,y)is ω-pe iodic
o e e y (x,y)∈R+×R+; and g∈C(R+,R)is dec easing, wi h g(0)⩽1. We
no ice ha x,y ep esen he p ey and p eda o popula ions, espec i ely.
In pa icula , we use
g(x)≡1(linea g ow h o he p ey), o
g(x)≡1−x
K(logis ic g ow h o he p ey)
and one o he ollowing exp essions o he unc ional esponse ϕ,
ϕI( ,x,y)≡λ( ) + α( )y;
ϕII( ,x,y)≡λ( ) + α( )y
1+β( )(λ( ) + α( )y)x.
He e, we assume ha α,β,λ∈C(R,R+)a e ω-pe iodic unc ions and λ,β6≡ 0.
The pa icula exp essions o bo h ϕIand ϕII in ol ing cons an coe icien s
(λ,β > 0 and α⩾0) a e used in he li e a u e o simula e he e ec s o hun -
ing coope a ion be ween p eda o s (see Be ec, 2010; Teixei a Al es and Hilke ,
2017).
The mo i a ion o s udy he exis ence o ω-pe iodic solu ions o sys em (31)
comes om he esea ch ca ied ou by Teixei a Al es and Hilke (2017). They
conside cons an coe icien s and he unc ional esponse ϕ(x,y) = λ+α y,
whe e λis he a ack a e and α ep esen s he coope a ion e m. This model
does no sus ain p eda o -p ey oscilla ions in he absence o hun ing coope a-
ion (α=0), so hey can asse ha he obse ed oscilla ions a e clea ly gene -
a ed by he coope a i e beha io .
Fixed poin heo y and p eda o -p ey models
The exis ence o ω-pe iodic solu ions o Lo ka-Vol e a ype models has been
s udied by means o di e en ixed poin heo y app oaches. Fo ins ance,
Ts e ko (1996) and Zanolin (1992) use opological a gumen s, such as index
heo y o Mawhin’s coincidence deg ee. Howe e , L , Lu, and Yan (2010) and
Tang and Zou (2006) conside mo e complex p eda o -p ey models including
pe iodic ime-delays, bu hey apply di ec ly he no m ype gene aliza ion o
K asnosel’skii ixed poin heo em gi en in Theo em 1.2.9.
To ou knowledge, he e a e no esea ch wo ks a ailable in which K as-
nosel’skii ype comp ession o expansion esul s a e applied o simple models
such as he one in Ts e ko (1996). One possible eason is ha he mos com-
mon e sions o K asnosel’skii ixed poin heo em do no apply o his class
o Lo ka-Vol e a equa ions.
In his chap e , we con ibu e o ill his gap in he applica ion o K as-
nosel’skii ype esul s o Lo ka-Vol e a popula ion models. Fo he non-
au onomous p eda o -p ey sys em (31), which is a gene aliza ion o he model
60 applica ions o p eda o -p ey di e en ial equa ions
whe e we ha e used ha λ,x0,y0> 0.
On he one hand, o each ∈[0,ω], using condi ion (G2)o e ϕ, one has
kxk∞⩾x( )> N1(x,y)( )
=Z +ω
H1( ,s)[a(s)x(s)(1−g(x(s))) + ϕ(s,x(s),y(s))x(s)y(s)]ds
⩾Z +ω
H1( ,s)ϕ(s,x(s),y(s)) x(s)y(s)ds
⩾m1q1q2kxk∞kyk∞Zω
0
ψ(s,x(s) + y(s))ds
⩾m1q1q2kxk∞kyk∞Zω
0
ψ(s,k(x,y)kω)ds,
which, a e di iding by kxk∞, yields
1 > m1q1q2kyk∞Zω
0
ψ(s,R)ds. (42)
On he o he hand, in a simila way, om y( )> N2(x,y)( ), o all ∈[0,ω],
we deduce ha
1 > m2c q1q2kxk∞Zω
0
ψ(s,R)ds. (43)
Now, by adding inequali ies (42), (43) and using kxk∞+kyk∞=k(x,y)kω=R,
we ob ain
2 > m3RZω
0
ψ(s,R)ds,
which con adic s ou assump ion (35). Thus, condi ion (41) is ul illed.
Applica ion o Theo em 1.2.11
We ha e p o ed ha all he condi ions o Theo em 1.2.11 a e sa is ied, he e-
o e he non-linea ope a o Nhas a ixed poin (x,y)∈Cwi h ⩽k(x,y)k⩽R.
This ixed poin (x,y)is a con inuous ω-pe iodic solu ion o he Lo ka-Vol e a
ype sys em (31).
3.3.1.1Applicabili y p oblems o Theo ems 1.2.5,1.2.9and 1.2.10
In his subsec ion, we show he app op ia eness o he homo opy e sion o he
Theo em o K asnosel’skii o Lo ka-Vol e a ype sys ems, as compa ed o he
o he mo e popula e sions.
We i s show ha he classical K asnosel’skii comp ession-expansion ixed
poin esul conside ed in Theo em 1.2.5is no applicable o he Banach space
Cω(R,R2),k·kω, he cone Cin Example 1.2.4and he in eg al mapping N
associa ed o he Lo ka-Vol e a ype sys em (31).
3.3 exis ence,localiza ion and o he p ope ies o solu ions 61
One o he comp ession o expansion hypo heses o he Theo em o K as-
nosel’skii equi es ha , o some eal numbe τ > 0,
(x,y) − N(x,y)/∈C, o all (x,y)∈Cwi h k(x,y)kω=τ. (44)
Howe e , i we choose pai s o he o m (0,y)∈C, wi h k(0,y)kω=kyk∞=τ,
hen, since N(0,y) = (0,0), one has
(0,y) − N(0,y) = (0,y)∈C.
Consequen ly, condi ion (44) does no hold.
Nex , we use simila a gumen s o show ha he no m ype e sion p o ided
in Theo em 1.2.9can nei he be applied o sys ems o he o m (31). We ollow
he app oach in (L , Lu, and Yan, 2010; Tang and Zou, 2006), whe e hey
also use he Banach space Cω(R,R2),k·kωand he cone Cin Example 1.2.4.
Mo eo e , o some posi i e eal numbe s 1< R1, 2< R2, hey conside he
open and bounded se s
Ω1:= (x,y)∈Cω(R,R2),k·kω:kxk∞< 1,kyk∞< 2;
Ω2:= (x,y)∈Cω(R,R2),k·kω:kxk∞< R1,kyk∞< R2.
One o he hypo heses in condi ions (NC),(NE) equi es ha , o some i∈{1,2},
kN(x,y)kω⩾k(x,y)kω, o all (x,y)∈C∩∂Ωi, (45)
whe e o τj= jo τj=Rj( o bo h j∈{1,2}) he elemen s o ∂Ωisa is y
kxk∞⩽τ1,kyk∞=τ2o kxk∞=τ1,kyk∞⩽τ2.
Thus, i we choose pai s o he o m (0,y)∈C∩∂Ωi, wi h kyk∞=τ2, since
N(0,y) = (0,0), one has
0=kN(0,y)kω<k(0,y)kω:= kyk∞=τ2.
Consequen ly, condi ion (45) does no hold.
Finally, we conside he classical Lo ka-Vol e a sys em wi h con inuous and
ω-pe iodic coe icien s and we show ha he ec o e sion in Theo em 1.2.10
s ill canno be applied, a leas by ollowing he p oo o Theo em 3.3.1. I
is wo h men ioning ha he esul could wo k o his ype o p oblems i
we used a dis inc easoning, bu he di icul y in insic o he compu a ions
obscu es he isualiza ion o his possible applica ion.
Le us conside he eal numbe s 0 < 1< R1,0 < 2< R2, he ollowing
cones in he Banach space (C(R,R),k·k∞)
C1:= {x∈C(R,R) : x( ) = x( +ω),x( )⩾q1kxk∞, o all ∈[0,ω]};
C2:= {y∈C(R,R) : y( ) = y( +ω),y( )⩾q2kyk∞, o all ∈[0,ω]};
62 applica ions o p eda o -p ey di e en ial equa ions
and he mapping N= (N1,N2) : (C1) 1,R1×(C2) 2,R2−→ C1×C2gi en by
N1(x,y)( ) := Z +ω
H1( ,s)λ(s)x(s)y(s)ds;
N2(x,y)( ) := Z +ω
H2( ,s)c(s)λ(s)x(s)y(s)ds.
The non-applicabili y o he esul is deduced once we show ha bo h condi-
ions (V1)and (V2)canno be sa is ied o one o he index alues i∈{1,2}. Le
us see, o ins ance, wha happens o i=1and condi ion (V1). I we p oceed
like in he p oo o Theo em 3.3.1, his condi ion is sa is ied p o ided ha he
ollowing inequali ies hold
M1λR2ω < 1 < m1λ 2q1q2ω,
bu his is no possible since m1⩽M1,λ⩽λ, 2< R2and q1q2∈(0,1).
3.3.1.2Viabili y o condi ions (33)-(35)
In his subsec ion, we s udy he exis ence o eal numbe s 0 < < R sa is y-
ing he equi ed hypo heses in Theo em 3.3.1. Fo gene al exp essions o he
p ey g ow h gand he p eda o s unc ional esponse ϕ, we p o e he exis ence
o small enough >0 ul illing condi ions (33) and (34). Howe e , we s udy
condi ion (35) in he pa icula cases o linea and logis ic g ow h, whe e we
also speci y how o choose sui able alues o and R. Unde he simples ex-
p essions o gand ϕ, we also discuss whe he i is possible o no o localize
mul iple solu ions and compa e ou esul s wi h hose in (Ts e ko , 1996).
Rema k 4.Le us conside he Lo ka-Vol e a model (31) wi h a unc ional
esponse ϕsa is ying condi ions (G1)-(G2). I
g(0)> 1 −1
M1a, (46)
hen he e exis s a numbe > 0 such ha condi ions (33) and (34) hold.
Indeed, by using he con inui y o ga 0, i (46) is sa is ied, hen he e exis s
0> 0 such ha g( )⩾1−1/ M1a, o , equi alen ly, condi ion (33) holds o
e e y ∈(0, 0).
F om (46), we also ha e g(0)> 1 −2/ M1a, o , equi alen ly,
(1−g(0))M1a < 2,
which, oge he wi h he inc easing cha ac e o η equi ed in condi ion (G1),
gua an ees (34) o any small enough > 0.
No ice ha condi ion (46) is i ially sa is ied when g(0) = 1, which is he
case o bo h linea and logis ic g ow h o he p ey popula ion.
3.3 exis ence,localiza ion and o he p ope ies o solu ions 63
We conside now he pa icula exp ession o gwhich co esponds o he lin-
ea o logis ic g ow h o he p ey popula ion, and we show how he condi ions
o e ,R > 0 in Theo em 3.3.1look like. Mo eo e , we s udy he exis ence o
such numbe s and R, when ϕ≡ϕIo ϕ≡ϕII.
Fo ha pu pose, le us s a by p o ing ha ϕIand ϕII sa is y all he condi-
ions p e iously equi ed o a gene al ϕ. I is clea ha bo h unc ions belong
o C(R×R+×R+,R+)and a e ω-pe iodic in he i s a iable. Conce ning
condi ions (G1)and (G2), o unc ion ϕI, we can ake
η=ϕIand ψ≡λ,
while o unc ion ϕII, we can se
η=ϕIand ψ( ,z) = λ( )
1+β( )(λ( ) + α( )z)z.
Addi ionally, we ix he ollowing no a ions
λ:= min
s∈[0,ω]λ(s),λ:= max
s∈[0,ω]λ(s),α:= max
s∈[0,ω]α(s)and β:= max
s∈[0,ω]β(s).
Linea g ow h
In his case, he localiza ion esul p o ided in Theo em 3.3.1 eads as ollows.
Co olla y 3.3.2.Assume ha g≡1and condi ions (G1),(G2)o e ϕa e sa is ied.
I he e exis ,R∈R,0 < < R, such ha
Zω
0
η(s, , )ds ⩽2
M3
, (47)
and (35)hold, hen he sys em
x0=a( )x−ϕ( ,x,y)xy;
y0= −b( )y+c( )ϕ( ,x,y)xy
(48)
has an ω-pe iodic solu ion (x,y)∈C, being C he cone in Example 1.2.4, such ha
⩽k(x,y)kω=kxk∞+kyk∞⩽R.
Nex , o each pa icula exp ession o ϕ, we gi e su icien condi ions o
(47) and (35) o hold.
Case I: When ϕ=ϕI, condi ions (47), (35) ead as
Zω
0
λ(s) + α(s) ds ⩽2
M3
,R⩾2
m3Rω
0λ(s)ds (49)
and a e espec i ely sa is ied p o ided ha
λ+α ⩽2
M3ω,R⩾2
m3ω λ,
64 applica ions o p eda o -p ey di e en ial equa ions
o , equi alen ly,
⩽−λ+qλ2+4 α 2/(M3ω)
2α ,R⩾2
m3ω λ. (50)
The e o e, unde condi ion (50), which is sa is ied o small enough and su -
icien ly la ge R, Co olla y 3.3.2applies. We no ice ha he exis ence o small
enough > 0 was al eady p o ed in Rema k 4. Howe e , he exp essions in
(50) ell us how o choose sui able alues o and R.
The simplici y o he condi ions o e ,Rgi en in (50) allows us o easily
discuss he possibili y o localize mul iple solu ions, eaching a nega i e con-
clusion, as we will jus i y.
Rema k 5.Since ω-pe iodic unc ions a e also nω-pe iodic, o e e y na u al
numbe n⩾2, i makes sense o ask o wha ex en he numbe s and R
depend on he pe iod. Fo each na u al numbe n⩾1, we deno e by nand Rn
he numbe s and Rsa is ying he condi ions (50), when ωis eplaced by nω.
Making some compu a ions, i can be shown ha
n+1< n< Rn< Rn+1, o all n∈N,n⩾1;
lim
n→∞
n=0, lim
n→∞
Rn=∞.
The e o e, by aking a mul iple o he pe iod ω, i may happen o localize he
same ω-pe iodic solu ion o e e y n⩾1, so ha he bes localiza ion egion
is ha o n=1. I is wo h men ioning ha one should ake in o accoun ha
he e ms M3and m3in ol ed in he inequali ies depend on he pe iod.
Rema k 6.In he pa icula case o ϕ=ϕIand α=0, sys em (48) educes o
he model
x0=a( )x−λ( )xy;
y0= −b( )y+c( )λ( )xy;(51)
which was s udied in (Ts e ko , 1996) by means o index heo y. E en in his
pa icula case, ou esul based on Theo em 1.2.11 gi es a be e localiza ion
o ω-pe iodic solu ions, namely in he annula conical se
C ,R:= {(x,y)∈C: ⩽kxk∞+kyk∞⩽R},
whe e
=2
M3Rω
0λ(s)ds,R=2
m3Rω
0λ(s)ds.
Case II: When ϕ=ϕII, condi ions (47), (35) become
Zω
0
λ(s) + α(s) ds ⩽2
M3
,RZω
0
λ(s)
1+β(s)(λ(s) + α(s)R)Rds ⩾2
m3
(52)
3.3 exis ence,localiza ion and o he p ope ies o solu ions 65
and a e espec i ely sa is ied p o ided ha
(λ+α )⩽2
M3ω,Rλ
1+β(λ+α R)R⩾2
m3ω. (53)
The i s inequali y in (53) is sa is ied o ⩽−λ+qλ2+4 α 2/(M3ω)/(2α)
as i happens in Case I. Nex , we s udy he exis ence o Ras equi ed by he
second inequali y in (53). To his aim, we conside he auxilia y unc ion
(z) := Az
1+Bz +Cz2,z∈R+ {0},
whe e A:= m3ω λ,B:= β λ and C:= β α. Le us show ha he e exis s R>0
ul illing he las inequali y in (53) i , and only i ,
A⩾2(2√C+B). (54)
One can easily p o e ha limz→0 (z) = limz→+∞ (z) = 0and (z)> 0, o all
z>0. Addi ionally, has a unique c i ical poin a 1/√C>0and he p e ious
p ope ies ensu e ha a ains a maximum a zmax := 1/√C. The e o e, i
(zmax)⩾2,
o , equi alen ly, A⩾2(2√C+B), hen i is possible o choose Rclose enough
o equal o zmax, such ha he equi ed inequali y holds. Mo eo e , unde
assump ion (54), we can p ecise he in e al whe e we can choose R. I is
[z1,z2], whe e z1,z2a e he solu ions o he equa ion (z) = 2, namely
z1=A−2B −p(A−2B)2−42C
4C ,z2=A−2B +p(A−2B)2−42C
4C . (55)
3.3.1.3Logis ic g ow h
In his pa icula case, we can ew i e Theo em 3.3.1as ollows.
Co olla y 3.3.3.Assume ha g(x)≡1−x/K and condi ions (G1),(G2)o e ϕa e
sa is ied. I he e exis ,R∈R,0 < < R, such ha
⩽K
a M1
, (56)
a M1
K +M3 Zω
0
η(s, , )ds ⩽2, (57)
and (35)hold, hen he sys em
x0=a( )x1−x
K−ϕ( ,x,y)xy;
y0= −b( )y+c( )ϕ( ,x,y)xy
(58)
has an ω-pe iodic solu ion (x,y)∈C, being C he cone in Example 1.2.4, such ha
⩽k(x,y)kω=kxk∞+kyk∞⩽R.
66 applica ions o p eda o -p ey di e en ial equa ions
I is clea how condi ion (56) can be sa is ied. In addi ion, i is no necessa y
o s udy again he exis ence o R > 0 ul illing condi ion (35), since i does no
depend on he exp ession o g, and he e o e we can ollow he a gumen s o
Subsec ion 3.3.1.2. Thus, le us s a e some su icien condi ions on > 0, such
ha (57) holds when ϕ≡ϕIo ϕ≡ϕII. We ha e al eady men ioned ha we
can conside η=ϕI o bo h exp essions o ϕ. Then, o bo h cases, condi ion
(57) eads as
M1a
K +M3 Zω
0
(λ(s) + α(s) )ds ⩽2,
being i ially sa is ied p o ided ha
M1a
K +M3ω λ+α ⩽2. (59)
We know, om Rema k 4, ha he e exis s a small enough > 0 ul illing
condi ion (57). Howe e , condi ion (59) ells us how o ob ain sui able alues
o > 0.
3.3.2P ope ies o solu ions
We de o e his subsec ion o s udy some in e es ing p ope ies o he ω-pe iodic
solu ions o he Lo ka-Vol e a sys em (31). Fi s , in Subsec ion 3.3.2.1, we de-
e mine he s eady s a es o cons an solu ions o he model. Then, we use
his in o ma ion and he localiza ion esul s in Subsec ion 3.3.1 o ind su i-
cien condi ions ha ensu e he noncons ancy o he solu ions. Finally, unde
uniqueness hypo heses, we show ha , i he localized solu ion is noncons an ,
hen he p ey and p eda o popula ions do no ge ex inc .
3.3.2.1S eady s a es
The s eady s a es o sys em (31) a e poin s (x0,y0)∈R+×R+such ha
x0(a( )g(x0) − ϕ( ,x0,y0)y0)=0;
y0(c( )ϕ( ,x0,y0)x0−b( )) = 0;(60)
o all ∈R. I is clea ha (0,0)is a solu ion o (60). To de e mine o he
possible solu ions, we dis inguish h ee cases:
Case 1:x0=0,y0> 0. Unde hese condi ions, he i s equa ion in (60) is
ob iously sa is ied, while om he second one we ha e y0b( ) = 0, o all
∈[0,ω], which is no possible o y0> 0 and b6≡ 0. The e o e, he e a e no
s eady s a es o he o m (0,y0), wi h y0> 0.
Case 2:x0> 0,y0=0. Now he second equa ion in (60) i ially holds, while
he i s one gi es a( )g(x0) = 0, o all ∈[0,ω]. As a6≡ 0, one mus ha e
3.3 exis ence,localiza ion and o he p ope ies o solu ions 67
g(x0) = 0. The e o e, a poin o he o m (x0,0),x0> 0, is a s eady s a e i and
only i g(x0) = 0.
Case 3:x0> 0,y0> 0. Unde hese condi ions, sys em (60) is equi alen o
ϕ( ,x0,y0) = a( )g(x0)
y0
=b( )
c( )x0
, o all ∈[0,ω].
We collec hese esul s abou he exis ence o s eady s a es o sys em (31) in
he ollowing p oposi ion.
P oposi ion 3.3.4.A poin (x0,y0)∈R+×R+is a s eady s a e o sys em (31)i ,
and only i , one o he ollowing condi ions holds:
1.x0=0and y0=0;
2.x0> 0,g(x0) = 0and y0=0;
3.x0> 0,y0> 0 and
ϕ( ,x0,y0) = a( )g(x0)
y0
=b( )
c( )x0
, o all ∈[0,ω]. (61)
Linea g ow h
Acco ding o his p oposi ion, in case o conside ing he linea g ow h o he
p ey popula ion, s a emen 2is no possible, and s eady s a es o he o m
(x0,y0)wi h x0,y0> 0 exis i and only i
ϕ( ,x0,y0) = a( )
y0
=b( )
c( )x0
, o all ∈[0,ω]. (62)
Logis ic g ow h
I one conside s he logis ic g ow h o he p ey popula ion, hen om s a emen
2in P oposi ion 3.3.4we ob ain he s eady s a e (K,0), and s eady s a es o he
o m (x0,y0)wi h x0,y0> 0 exis i and only i
ϕ( ,x0,y0) = a( )1−x0
K
y0
=b( )
c( )x0
, o all ∈[0,ω]. (63)
Gene al p ey g ow h
Coming back o he gene al sys em (31), le us no e ha , i he e is no any
cons an k > 0 such ha
a( ) = kb( )
c( ), o all ∈[0,ω], (64)
hen he sys em has no s eady s a es (x0,y0)wi h x0,y0> 0. The e o e, unde
condi ion (64), he o bi s o all ω-pe iodic solu ions (x,y), wi h x( ),y( )> 0
o all ∈[0,ω], do no educe o poin s.
68 applica ions o p eda o -p ey di e en ial equa ions
3.3.2.2Noncons an solu ions
He e, we deduce su icien condi ions o ensu e ha he pe iodic solu ion ob-
ained in Theo em 3.3.1does no educe o a s eady s a e, le ing i he possibil-
i y o be a limi cycle. We s a e a gene al esul which is use ul o he pa icula
case o linea p ey g ow h, bu canno be applied o p ey popula ions wi h lo-
gis ic g ow h. The e o e, we s udy he la e case sepa a ely, and we p o ide
an example o illus a e he heo e ical a gumen s.
Theo em 3.3.5.Assume ha condi ion (61)does no hold and
g(x)6=0, o all x > 0. (65)
Then any ω-pe iodic solu ion (x,y)o sys em (31)wi h k(x,y)kω> 0 does no educe
o a s eady s a e.
P oo . In iew o he inequali y k(x,y)kω> 0, he esul ollows once we ha e
p o ed ha he unique s eady s a e is (0,0).
F om P oposi ion 3.3.4, we know ha he e a e no s eady s a es o he o m
(0,y0)wi h y0> 0; mo eo e , i g(x)6=0, o all x > 0, hen he e a e no s eady
s a es o he ype (x0,0)wi h x0> 0; and, i condi ion (61) does no hold, hen
he e a e no posi i e (wi h bo h posi i e componen s) s eady s a es. The e o e,
he unique s eady s a e o he sys em is (0,0), as wished.
Linea g ow h
In case o linea g ow h o he p ey popula ion, condi ion (65) in Theo em 3.3.5
i ially holds. Consequen ly, gi en wo eal numbe s 0 < < R sa is ying
he equi ed hypo heses in Co olla y 3.3.2, i condi ion (62) is no ul illed,
hen, applying Theo em 3.3.5, we can asse ha he e exis s a noncons an
ω-pe iodic solu ion (x,y)∈Cwi h ⩽k(x,y)kω⩽R.
In pa icula , when ϕ=ϕI, he hypo heses in Co olla y 3.3.2 ead as (49),
which can always be sa is ied o small enough and su icien ly la ge R. The e-
o e, i we ensu e ha condi ion (62) does no hold, hen he e is an ω-pe iodic
solu ion which is noncons an . Fo ins ance, i a,b,c,λa e cons an s, bu he
coope a ion coe icien αis noncons an , hen condi ion (62) is no ul illed.
Logis ic g ow h
Howe e , o he logis ic g ow h o he p ey popula ion, Theo em 3.3.5does
no apply since g(K) = 0. Ne e heless, i condi ion (63) does no hold, hen
he s eady s a es o sys em (58) a e (0,0)and (K,0). Recall ha we can always
conside small enough > 0 such ha condi ions (56)-(57) in Co olla y 3.3.3
a e ul illed. The e o e, i he exis s a numbe R > 0 such ha K > R > > 0
and condi ion (35) holds, hen he o bi o he ω-pe iodic solu ion gi en by
Co olla y 3.3.3does no educe o a poin .
3.3 exis ence,localiza ion and o he p ope ies o solu ions 69
Fo he pa icula cases o ϕ=ϕIand ϕ=ϕII, i is possible o choose a eal
numbe 0 < R < K p o ided ha
K > 2
m3Rω
0λ(s)ds o z1< K,
espec i ely, whe e z1is gi en in (55).
In pa icula , when ϕ=ϕI, i a,b,c,λa e cons an s and αis noncons an ,
hen condi ion (63) does no hold. The e o e, o e e y
K > 2
m3ωλ,
he ω-pe iodic solu ion o sys em (58), gi en by Co olla y 3.3.3, is noncons an .
We conclude his subsec ion wi h an example whe e all he coe icien s, ex-
cep c, a e noncons an .
Example 3.3.6.Le he coe icien s o sys em (58), wi h ϕ=ϕI, be
a( ) := sin2π ,b( ) := cos2π ,c( ) := c∈(0,1);
λ( ) := θa( ) + (1−θ)b( ) (θ∈(0,1)),α( ) := (1−b( ))b( ).
We s a wi h a b ie in e p e a ion o his pa icula sys em, which could be
sui able o model he in e play be ween p ey and p eda o popula ions in
which we ake in o accoun he ollowing ac o s: ecological seasonal e ec s;
he p ey g ow h a e aa ains i s maximum alue when he p eda o mo al-
i y a e b eaches i s minimum and ice e sa; he a ack a e o p eda o s λ
is a con ex combina ion o he p ey g ow h and he p eda o mo ali y a es;
and he hun ing coope a ion coe icien α anishes when he mo ali y a e o
p eda o s a ains i s maximum o minimum.
Fo his pa icula sys em, condi ion (63) does no hold. Indeed, we can show
ha he e a e no cons an s k > 0 sa is ying condi ion (64). Fo =0and any
k > 0, we ob ain
a(0) = sin20=06=k
c=kcos20
c=kb(0)
c.
So, condi ion (64) does no hold o =0and, consequen ly, condi ion (63) is
no ul illed. The e o e, i
K > 2
m3R1
0λ(s)ds
=4e(√e−1)
c,
hen he 1-pe iodic solu ion gi en by Co olla y 3.3.3is noncons an .
76 applica ions o p eda o -p ey di e en ial equa ions
Con ibu ions o he di ec applica ion o ixed poin heo ems o Lo ka-Vol e a models
In Sec ion 3.1, we obse e ha he exis ence o pe iodic solu ions o simple gen-
e aliza ions o Lo ka-Vol e a ype sys ems has been s udied by means o ixed
poin heo y based on opological a gumen s. In ou iew, o applica ions, he
use o opological echniques may complexi y he de ails o he esea ch de el-
opmen in compa ison wi h he use o some classical es ic ions in ixed poin
heo y, i.e., hose gi en di ec ly in e ms o he unde lying mapping. Thus,
we s udy he applicabili y o di e en e sions o K asnosel’skii ixed poin
heo em, whose hypo heses a e di ec ly exp essed in e ms o he non-linea
ope a o , o he pe iodic p eda o -p ey sys em (31).
In Subsec ion 3.3.1, we use he homo opy e sion o p o e he exis ence o
pe iodic solu ions o his class o models. We ollow a simila app oach o
ha in (L , Lu, and Yan, 2010; Tang and Zou, 2006), whe e hey show he
applicabili y o he no m- ype e sion o mo e complex p eda o -p ey models
including ime-delays. Howe e , we p o e ha bo h classical and no m- ype
e sions do no apply o sys em (31). Mo eo e , o he pa icula case o he
classical Lo ka-Vol e a sys em wi h pe iodic coe icien s, we also show some
p oblems wi h he applicabili y o he ec o e sion, which had been used o
deal wi h simila pe iodic sys ems (see P ecup, 2007).
On he o he hand, we imp o e he localiza ion esul s in (Ts e ko , 1996),
whe e sys em (51) was s udied by means o index heo y. Besides, ou app oach
holds o a mo e gene al o mula ion o Lo ka-Vol e a ype sys ems.
Fi s s eps on he quali a i e s udy o sys em (31)
Ou main esul allows o localize con inuous and pe iodic solu ions o he
gene al sys em (31), which desc ibes a pe iodic in e play be ween p ey and
p eda o popula ions. Howe e , hese pe iodic solu ions could be in ac s eady
s a es, i.e, solu ions ha do no a y wi h ime, as we showed o he au-
onomous case (see Subsec ion 3.3.3). Mo eo e , he localiza ion domain does
exclude he possibili y o ex inc ion o some o he species. The e o e, i is
in e es ing o de e mine su icien condi ions ha ensu e he noncons ancy o
he pe iodic solu ion and he posi i eness o bo h popula ions. We de elop his
s udy in Subsec ion 3.3.2, whe e we can obse e ha he su icien condi ions
depend on he exp essions o he p ey g ow h gand he p eda o s unc ional
esponse ϕ. Thus, we analyzed hei iabili y o he linea and logis ic p ey
g ow h and he exp essions o ϕI,ϕII, which conside hun ing coope a ion
be ween p eda o s.
This is a p elimina y s udy o he solu ions p ope ies, which le s he possi-
bili y o localize a limi cycle. A u he s ep on he quali a i e s udy o his
class o models would be o de e mine he s abili y p ope ies o he s eady
s a es and he localized pe iodic solu ion.
CONCLUSIONS AND FUTURE PROSPECTS I
In his esea ch line, we ha e con ibu ed o bo h he gene aliza ion and appli-
ca ion o K asnosel’skii ype comp ession-expansion ixed poin esul s. In ou
iew, one o he mos ele an aspec s o ou esea ch a e he app oaches we
used, since one o hem allows o con inue wi h he ex ension o he localiza-
ion egion p o ided by ou ixed poin esul s and he o he ills a gap in he
applica ion o K asnosel’skii ype ixed poin heo ems o Lo ka-Vol e a pop-
ula ion models. We de o e he ollowing lines o desc ibe he esea ch skills
we ha e gained wo king on he objec i es o his pa o he hesis. Then, we
gi e de ails abou some ideas o con inue wi h he in es iga ion, which could
po en ially inc ease he in e es o ou wo k.
A. Conclusions
In Chap e s 2and 3wi hin Resea ch Line I, we include he esea ch we ha e
de eloped in he amewo k o ixed poin heo y. The mos ele an esul s
we ha e ob ained a e summa ized and compa ed wi h o he p esen in he
ela ed li e a u e a he end o each chap e . In his sec ion, we complemen
his discussion by e iewing he lea ning p ocess in which I go in ol ed when
I was wo king in goals G1 and G2.
I begin wi h Chap e 2, whe e we achie e objec i e G1. As we al eady
men ioned, an impo an pa o he esea ch ca ied ou o gene alize K as-
nosel’skii comp ession-expansion ixed poin heo em o se con ac ions and
s a con ex se s has been de eloped du ing my Mas e deg ee s udies. Fo
ins ance, i was du ing ha pe iod when my supe iso Rosana Rod íguez
López p oposed me he idea o adap he esul s o balls owed by Các and
Ga ica (1979) and Po e (1974) o mo e gene al se s, which con ain he ays
connec ing he poin s on i s bounda y o he o igin. Thus, be o e he beginning
o my PhD s udies, I had de eloped his mo e gene al se ing and p o ed he
comp ession ype esul . In his way, I had acqui ed he capaci y o analyze
he p oo o a esul and o adap i o a mo e gene al amewo k. We had
also p o ed he expansion ype esul as a gene aliza ion o he wo k in (Các
and Ga ica, 1979), bu he localiza ion domain was no so p ecise. Mo eo e ,
he bounda y alue p oblem we had conside ed o illus a e he applicabili y
o ou esul s sa is ied he hypo heses o some classical esul s, so we had no
jus i ied p ope ly he necessi y o ou gene aliza ions. I was du ing he PhD
s udies when we p o ed bo h comp ession and expansion esul s unde he
gene al hypo heses conside ed in his documen . Fo he expansi e case, p o-
esso Radu P ecup sugges ed us o use a change o a iable o educe i o he
comp essi e one. Fo bo h p oo s, I needed o become amilia wi h he pa -
77
icula beha io o se con ac ions. Besides, ega ding he applica ions o he
ob ained esul s, I wo ked o he i s ime wi h a noncompac mapping and
pe cei ed he ele ance o de ining he s a con ex se s by means o unc ionals.
The con en s o Chap e 3con ibu e o each he goal G2 and we e com-
ple ely de eloped du ing he PhD s udies. In his case, I e iewed some exis -
ing li e a u e and decided o s udy p eda o -p ey models including coope a-
ion be ween p eda o s by means o ixed poin heo y. Ou main aim was o
apply di ec ly some K asnosel’skii ype esul s, since, in ou iew, his me hod-
ology is clea e and easie o ep oduce han opological app oaches ha ha e
been used o localize solu ions o simila models. The main di icul ies we e
o o e come he p oblems in he applica ion o some popula e sions o K as-
nosel’skii ixed poin heo em and also o ind he app op ia e esul o ou
p oblem o in e es . Mo eo e , I also lea ned how o discuss he possibili y o
localize mul iple solu ions. Finally, i is also in e es ing o commen ha , o
ob ain some basic quali a i e p ope ies o pe iodic solu ions, we combined he
localiza ion esul s wi h some knowledge acqui ed du ing my deg ee s udies
abou o dina y di e en ial equa ions.
B. Fu u e p ospec s
In his sec ion, we p o ide some ideas ha we would like o de elop as a
con inua ion o he esea ch ca ied ou in his pa o he hesis. I is wo h
men ioning ha we ha e al eady s a ed he esea ch on he u u e p ospec I,
which is pa o a join wo k wi h my supe iso Rosana Rod íguez López.
i.gene aliza ion o k asnosel’skii comp ession-expansion ixed
poin heo em o se con ac ions o mo e gene al domains
Along his pa o he hesis, we p o ided di e en easons o conside new
gene aliza ions o K asnosel’skii comp ession-expansion ixed poin heo em,
such as he highe applicabili y and he be e localiza ion hey can p o ide.
Mo eo e , in he In oduc o y sec ion o Chap e 2, we also commen ed he
ad an ages o using classical ixed poin a gumen s a he han opological
echniques, bo h in applica ions and o ob ain heo e ical ex ensions. In he
la e case, we also conside ha he classical app oach exhibi s he key poin s
o he p oo s in a clea e way, allowing o easily iden i y he gene aliza ion
po en ial o he esul s.
We ecall ha we ha e ob ained a gene aliza ion o he classical comp ession-
expansion Theo em o K asnosel’skii o se con ac ions which localizes he
solu ions in conical domains de e mined by s a con ex se s. A deepe analysis
o he p oo s e eals ha hei key poin s a e he p ope ies o se con ac ions
and he shape o he localiza ion domain. Mo eo e , we also obse e ha we
a e no aking ad an age o all he possibili ies o e ed by he se con ac ions,
78
and we claim ha he obs acle o ob ain be e localiza ion esul s comes om
he peculia i ies o he cone.
The e o e, o con inue wi h he ex ension o hese esul s, we decided o p e-
se e he ideas in he p oo o bo h comp ession and expansion cases, as well
as he egula i y o he mapping, bu o cons uc a mo e gene al amewo k o
he localiza ion domains. We no ice ha , o applica ions, we mus ake ca e o
he ex ension o he cone, since i s cha ac e is ics a e ele an o de e mine he
p ope ies o he localized solu ion.
Apa om he heo e ical in e es o his esea ch idea, i would also be
in e es ing o jus i y i s po en ial in applica ions.
ii.quali a i e s udy o he pe iodic lo ka- ol e a sys em wi h
gene al p ey g ow h and p eda o s unc ional esponse
In Sec ion 3.1, we men ioned ha he mo i a ion o s udy he p eda o -p ey
sys em (31) came om he esea ch de eloped in (Teixei a Al es and Hilke ,
2017). Howe e , we do no ollow hei me hodology and, inspi ed by he wo k
in (Ts e ko , 1996), we decided o use an ope a o app oach o localize pe iodic
solu ions.
Once we ha e p o ed he exis ence o pe iodic solu ions, we also s a ed o
s udy some o hei p ope ies. We i s de e mined su icien condi ions o
ensu e ha he localized solu ions do no educe o s eady s a es. Then, unde
uniqueness hypo heses, we also p o ed ha he noncons an pe iodic solu ions
a e posi i e, so none o he species ends in ex inc ion. In his way, we sligh ly
con ibu ed o he quali a i e s udy o he nonau onomous sys em (31). How-
e e , he e a e s ill many in e es ing aspec s o he dynamical beha io o he
model which dese e o be s udied. Fo ins ance, in Sec ion 3.4, we p oposed
o s udy he s abili y o s eady s a es and he localized pe iodic solu ion, which
is in e es ing o unde s and he long- e m beha io o he popula ions. To ha
pu pose, as a beginne in his ield, he i s s ep would be a ca e ul e ision
o he exis ing li e a u e on he opic. Mo eo e , i can also be use ul o con-
side he app oach in (Fa ia and Oli ei a, 2019), whe e hey p o e he global
a ac ion o a pe iodic o bi o a nonau onomous model o hema opoiesis.
In he esea ch de eloped in (Teixei a Al es and Hilke , 2017) o an au-
onomous model, a e he s abili y s udy, hey con inue wi h an analysis o
bi u ca ions, ha is, hey obse e how he s abili y beha io o he sys em
changes as a pa ame e is a ied. I would also be in e es ing o de elop
a simila s udy o he mo e gene al nonau onomous Lo ka-Vol e a sys ems
(31). The e a e plen y o esea ch wo ks de o ed o he s udy o bi u ca ions
in au onomous sys ems; howe e , he analogous esea ch o nonau onomous
di e en ial equa ions seems o be unde cons uc ion and i does no exis a
whole heo e ical amewo k in which we can base ou esea ch. In spi e o
ha , we can s a looking a he wo ks in (Langa, Robinson, and Suá ez, 2002;
79
Rasmussen, 2007), whe e wo di e en gene aliza ions o some bi u ca ion no-
ions a e de eloped o he nonau onomous case.
80
REFERENCES I
Ande son, D. R., R. I. A e y, and J. Hende son (2010). “Func ional expansion-
comp ession ixed poin heo em o Legge -Williams ype.” In: Elec onic
Jou nal o Di e en ial Equa ions 2010:63, pp. 1–9.
Bacaë , N. (2011). “Lo ka, Vol e a and he p eda o -p ey sys em (1920-1926).”
In: A sho his o y o ma hema ical popula ion dynamics. Sp inge , pp. 71–76.
Be ec, L. (2010). “Impac s o o aging acili a ion among p eda o s on p eda o -
p ey dynamics.” In: Bulle in o Ma hema ical Biology 72:1, pp. 94–121.
Bolojan, O. and R. P ecup (2014). “Implici i s o de di e en ial sys ems wi h
nonlocal condi ions.” In: Elec onic Jou nal o Quali a i e Theo y o Di e en ial
Equa ions 69, pp. 1–13.
B aue , F. and C. Cas illo Chá ez (2001). Ma hema ical models in popula ion biology
and epidemiology. Tex s in Applied Ma hema ics. New Yo k: Sp inge .
Bu oni, G., M. G oppi, and C. So esina (2011). “E ec s o p ey o e -unde -
c owding in p eda o -p ey sys ems wi h p ey-dependen ophic unc ions.”
In: Nonlinea Analysis: Real Wo ld Applica ions 12:5, pp. 2871–2887.
Các, N. P. and J. A. Ga ica (1979). “Fixed poin heo ems o mappings in o -
de ed Banach spaces.” In: Jou nal o Ma hema ical Analysis and Applica ions
71:2, pp. 547–557.
Da bo, G. (1955). “Pun i uni i ans o mazioni a condominio non-compac o.”
In: Rendicon i del Semina io Ma ema ico della Uni e si à di Pado a 24, pp. 84–
92.
Deimling, K. (1985). Nonlinea unc ional analysis. Be lin: Sp inge -Ve lag.
E be, L. H. and H. Wang (1994). “On he exis ence o posi i e solu ions o
o dina y di e en ial equa ions.” In: P oceedings o he Ame ican Ma hema ical
Socie y 120:3, pp. 743–748.
Fa ia, T. and J. J. Oli ei a (2019). “A no e on global a ac i i y o he pe iodic
solu ion o a model o hema opoiesis.” In: Applied Ma hema ics Le e s 94,
doi: 10.1016/j.aml.2019.02.009.
G anas, A. and J. Dugundji (2013). Fixed Poin Theo y. Sp inge Monog aphs in
Ma hema ics. New Yo k: Sp inge -Ve lag.
Güo, D. and V. Lakshmikan ham (1998). Nonlinea p oblems in abs ac cones.
Vol. 5. No es and Repo s in Ma hema ics in Science and Enginee ing. Cali-
o nia: Academic P ess.
K asnosel’skii, M. A. (1960). “Fixed poin s o cone-comp essing and cone-ex-
panding ope a o s.” In: So ie Ma hema ics Doklady 1, pp. 1285–1288.
— (1964). Posi i e solu ions o ope a o equa ions. Ne he lands: P. Noo dho L d.
Kwong, M. K. (2008). “On K asnoselskii’s cone ixed poin heo em.” In: Fixed
Poin Theo y and Applica ions 2008, doi: 10.1155/2008/164537.
81
Langa, J. A., J. C. Robinson, and A. Suá ez (2002). “S abili y, ins abili y, and bi-
u ca ion phenomena in non-au onomous di e en ial equa ions.” In: Non-
linea i y 15:3, pp. 1–7.
Lois-P ados, C. and R. P ecup (2020). “Posi i e pe iodic solu ions o Lo ka-
Vol e a sys ems wi h a gene al a ack a e.” In: Nonlinea Analysis: Real
Wo ld Applica ions 52, doi: 10.1016/j.non wa.2019.103024.
Lois-P ados, C., R. P ecup, and R. Rod íguez-López (2020). “K asnosel’skii
ype comp ession-expansion ixed poin heo em o se con ac ions and
s a con ex se s.” In: Jou nal o Fixed Poin Theo y and Applica ions 22:3, doi:
10.1007/s11784–020–00799–0.
Lois-P ados, C. and R. Rod íguez-López (2020). “A gene aliza ion o K as-
nosel’skii comp ession ixed poin heo em by using s a con ex se s.” In:
P oceedings o he Royal Socie y o Edinbu gh , Sec ion A: Ma hema ics 150:1,
pp. 277–303.
Lo ka, A. J. (1925). Elemen s o Physical Biology. Bal imo e: Williams and Wilkins.
L , X., S. Lu, and P. Yan (2010). “Exis ence and global a ac i i y o posi i e
pe iodic solu ions o Lo ka-Vol e a p eda o -p ey sys ems wi h de ia ing
a gumen s.” In: Nonlinea Analysis: Real Wo ld Applica ions 11:1, pp. 574–583.
O’Regan, D. and R. P ecup (2001). Theo em o Le ay-Schaude ype and i s applica-
ions. Singapo e: Go don and B each.
— (2005). “Comp ession-expansion ixed poin heo em in wo no ms and ap-
plica ions.” In: Jou nal o Ma hema ical Analysis and Applica ions 309:2, pp.
383–391.
Po e , A. J. B. (1974). “A ixed poin heo em o posi i e k-se con ac ions.”
In: P oceedings o he Edinbu gh Ma hema ical Socie y 19:1, pp. 93–102.
P ecup, R. (2002). Me hods in nonlinea in eg al equa ions. Ne he lands: Sp inge .
— (2006). “Posi i e solu ions o semi-linea ellip ic p oblems ia K asnosel’skii
ype heo ems in cones and Ha nack’s inequali ies.” In: AIP Con e ence P o-
ceedings 835, pp. 125–132.
— (2007). “A ec o e sion o K asnosel’skii ixed poin heo em in cones and
posi i e pe iodic solu ions o nonlinea sys ems.” In: Jou nal o Fixed Poin
Theo y and Applica ions 2, pp. 141–151.
Rasmussen, M. (2007). “Nonau onomous bi u ca ion pa e ns o one-dimen-
sional di e en ial equa ions.” In: Jou nal o Di e en ial Equa ions 234:1, pp.
267–288.
Rosenzweig, M. L. and R. H. MacA hu (1963). “G aphical ep esen a ion and
s abili y condi ions o p eda o -p ey in e ac ions.” In: The Ame ican Na u al-
is 97:895, pp. 209–223.
Tang, X. and X. Zou (2006). “On posi i e pe iodic solu ions o Lo ka-Vol e a
compe i ion sys ems wi h de ia ing a gumen s.” In: P oceedings o he Ame -
ican Ma hema ical Socie y 134:10, pp. 2967–2974.
Teixei a Al es, M. and F. M. Hilke (2017). “Hun ing coope a ion and Allee
e ec s in p eda o s.” In: Jou nal o Theo e ical Biology 419, pp. 13–22.
82
To es, P. J. (2003). “Exis ence o one-signed pe iodic solu ions o second-o de
di e en ial equa ions ia a K asnosel’skii ixed poin heo em.” In: Jou nal
o Di e en ial Equa ions 190:2, pp. 643–662.
Ts e ko , D. (1996). “A pe iodic Lo ka-Vol e a sys em.” In: Se dica Ma hema ical
Jou nal 22:2, pp. 109–116.
Van de Ho , Q. and T. H. Fay (2016). “A p eda o -p ey model wi h p eda o
popula ion sa u a ion.” In: Ma hema ics and S a is ics 4:4, pp. 101–107.
Vol e a, V. (1926). “Va iazioni e lu azioni del nume o d’indi idui in specie
animali con i en i.” In: Memo ia della Reale Accademia Nazionale dei Lincei 2,
pp. 31–113.
Wang, H. (2011). “Posi i e pe iodic solu ions o singula sys ems o i s o de
o dina y di e en ial equa ions.” In: Applied Ma hema ics and Compu a ion
218:5, pp. 1605–1610.
Zanolin, F. (1992). “Pe manence and posi i e pe iodic solu ions o Kolmogo o
compe ing species sys ems.” In: Resul s in Ma hema ics 21, pp. 224–250.
Zima, M. (2004). “Fixed poin heo em o Legge -Williams ype and i s appli-
ca ions.” In: Jou nal o Ma hema ical Analysis and Applica ions 299:1, pp. 254–
260.
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Pa II
RESEARCH LINE II
92 backg ound de ini ions and esul s ii
De ini ion 4.2.2.A closed in a ian se Ris called epelling i he e exis s a
neighbo hood Uo Rsuch ha e e y poin in U Ris mapped ou side Uin a
ini e numbe o i e a ions.
An a ac ing se A⊂Jis a closed in a ian se o which he e exis s a
neighbo hood Uo Asuch ha h(U)⊂Uand ∩∞
i=0hi(U) = A. An a ac o Ais
an a ac ing se wi h a dense o bi .
The basin o a ac ion o an a ac o Ais he se o all poin s con e ging o
A, ha is, he poin s x∈Jsuch ha lim
n→∞
hn(x)∈A.
Acco ding o (A u in e al., 2019, Sec ion 1.4), discon inuous one-dimensional
maps can ha e ou ypes o a ac o s: k-cycles and k-band chao ic a ac o s,
k⩾1; and wo o he ypes o a ac o s associa ed wi h quasi-pe iodic o bi s.
In gene al, we use he no a ion oscilla o y a ac o o hose a ac o s which a e
no ixed poin s o he map h.
Fo a ixed poin x∗∈J, de ined by h(x∗) = x∗, we use some addi ional
no ions.
De ini ion 4.2.3.A epelling ixed poin is also called uns able.
The ixed poin x∗is s able i o e e y neighbo hood Vo x∗, he e exis s
ano he neighbo hood Uo x∗such ha hn(x)∈V o all x∈U,n⩾1. We say
x∗is locally asymp o ically s able (LAS) i i is s able and an a ac o . I x∗is an
a ac o wi h U=J, we say ha x∗is a global a ac o . I a global a ac o is
s able, hen we e e o i as globally asymp o ically s able (GAS).
We use he e m s abilizing when an uns able ixed poin becomes LAS unde
a ia ion o a pa ame e ; and des abilizing when a s able equilib ium loses i s
s abili y as a pa ame e is changed.
I x∗is no an a ac o , bu limn→∞hn(x) = x∗ o all x∈[x∗,x∗+ε)o
x∈(x∗−ε,x∗]and some ε > 0, hen we say x∗is semi-s able.
I x∗is no an a ac o bu limn→∞hn(x) = x∗ o Lebesgue almos e e y xin
a neighbo hood Uo x∗, we say ha x∗is an essen ial a ac o . I U=J, hen we
call x∗an essen ial global a ac o . In he con ex o popula ion dynamics, when
he ixed poin is x∗=0, hen we e e o his phenomena as essen ial ex inc ion.
Th oughou his monog aph, he i s s ep on he s udy o asymp o ic dy-
namics is o ind pa ame e es ic ions which ensu e he exis ence o posi i e
ixed poin s x∗o he one-dimensional map h, which is usually an easy ask.
Then, we s udy he s abili y o he equilib ium poin s which a e loca ed a he
smoo h b anches o he map hby means o he alue h0(x∗). I is also s aigh -
o wa d, specially in he hype bolic case, ha is, when |h0(x∗)|6=1. Fo ixed
poin s which a e locally asymp o ically s able, i is na u al o s udy hei global
s abili y. Al hough he s a emen “LAS implies GAS” is no gene ally ue, i
holds when he map hsa is ies some addi ional condi ions. In he wo k by
Coppel (1955), we ind a simple condi ion which ensu es he global s abili y o
an equilib ium o con inuous maps de ined on bounded in e als. I he in e -
al is bounded o unbounded, we ha e o equi e some addi ional condi ions
(F anco, Pe án, and Segu a, 2020):
4.2 s abili y concep s and esul s 93
Lemma 4.2.1.I he map h ela ed o he di e ence equa ion (76)is con inuous in J,
hen a ixed poin x∗is GAS i and only i h2(x)6=x o all x∈J {x∗}and x∗is s able
o he dynamical sys em associa ed o hn, o some n∈N.
In gene al, i is di icul o ensu e analy ically he absence o 2-cycles, bu
based on nume ical simula ions we can use he esul o conjec u e GAS. In he
li e a u e we can ind se e al pape s de o ed o ind condi ions on he map h
ha a e easie o check, an example is he ollowing simple esul which wo ks
o con inuous maps and is a gene aliza ion o (B a e man and Liz, 2012) s a ed
in (Liz and Lois-P ados, 2020a):
Lemma 4.2.2.I he map h ela ed o he one-dimensional disc e e- ime di e ence
equa ion (76)is con inuous on J= (a,b)(−∞⩽a < b ⩽∞) and has a unique ixed
poin x∗such ha
1.x < h(x)< x∗ o all x∈(a,x∗);
2.a < h(x)< x o all x∈(x∗,b);
hen x∗is GAS on J.
This esul ensu es he global asymp o ic s abili y o he unique posi i e ixed
poin o compensa o y models, in pa icula , we ob ain he ollowing esul .
Co olla y 4.2.4.Le us conside he Be e on-Hol model gi en by (80)and he Ricke
model gi en by (81)wi h 0 < ⩽1, hen he posi i e ixed poin K(K⩽xc) is GAS.
O he esul s in his line make use o Lemma 4.2.1in hei p oo , his is he
case o he ollowing esul by Cull and Cha ee (2000) (see also (Cull, 2007)).
Lemma 4.2.3.Le φ: [0,∞)−→ [0,∞)be a dec easing unc ion which is posi i e on
(0,l)and so ha φ2(x) = x o all x∈[0,∞). I he map h ela ed o he di e ence
equa ion (76)is con inuous on J= [0,∞), has a posi i e ixed poin x∗and
1.φ(x)> h(x)on (0,x∗);
2.φ(x)< h(x)on (x∗,l);
3.h(x)> x on (0,x∗);
4.h(x)< x on (x∗,∞);
5.h(x)> 0 on (0,∞);
hen x∗is GAS on (0,∞).
This esul applies o compensa o y and o e compensa o y models, in pa -
icula o hose wi h 0(x)⩾−1 o all x∈[0,∞)and a mos one poin in
[0,∞)such ha 0(x)=−1. We illus a e i wi h he Ricke model gi en by
(81).
94 backg ound de ini ions and esul s ii
Co olla y 4.2.5.Le us conside he Ricke model gi en by (81). I 0 < ⩽2, hen
0(x)⩾−1 o all x > 0. Mo eo e , 0(x) = −1i and only i =2and x=K=1. In
pa icula , i 0< ⩽2, hen he posi i e equilib ium K=1is GAS in (0,∞). While,
i > 2, hen K=1is uns able.
P oo . Le us conside > 0 and x > 0, since
0(x) = (1− x)e (1−x)and 00(x) = − (2− x)e (1−x),
i ollows ha 00(x)< 0 o all x∈(0,2/ ), 00(2/ ) = 0and 00(x)> 0 o all
x > 2/ . Hence, 0a ains i s global minimum a 2/ wi h 0(2/ ) = −e −2.
The e o e, 0(x)<−1i and only i > 2 and 0(x)=−1i and only i =2,
x=1.
Now, we assume 0 < ⩽2. The global a ac ion o he posi i e ixed poin
K=1 ollows om Lemma 4.2.3wi h φ(x) = 2K −x=2−xand h(x) = (x) o
all x∈(0,∞);l=2and x∗=K=1. Le us p o e ha he equi ed condi ions
a e ul illed (see Figu e 9).
Fi s we ha e φ2(x) = 2− (2−x) = xand i is well-known ha he Ricke
map sa is ies condi ions 3-5.
I emains o p o e ha φand ul ill condi ions 1,2. On he one hand, since
φ(0) = 2 > (0) = 0,φ(1) = (1) = 1,φ0(x) = −1 o all x > 0 and 0(x)<−1
o all x>0,x6=1; hen (x)< φ(x) o all x∈(0,K) = (0,1). On he o he
hand, since φ(1) = (1) = 1,φ0(x) = −1and 0(x)⩾−1 o all x > 0 ( 0(x) = −1
i and only i =2,x=1), hen (x)> φ(x) o all x > K =1.
Finally, i > 2, hen we ha e 0(K) = 0(1) = (1− )<−1, so he equilib ium
is uns able.
01234
0.0
0.5
1.0
1.5
Figu e 9: Diag am showing he Ricke map (x) = xe (1−x), ⩽2(compensa o y,
=0.7∈(0,1]in black; o e compensa o y, =1.5∈(1,2]in blue) en eloped
by φ(x) = 2−x( ed line). We also plo y=x(g ay, dashed).
No ice ha we ha e ob ained sha p global s abili y o he Ricke map, ha
is, we p o ed ha he posi i e equilib ium is GAS o all he alues o he
pa ame e > 0 o which he ixed poin is LAS.
The nex esul p o ides a simple c i e ion o compa e global s abili y o wo
con inuous maps (see (El-Mo shedy and Jiménez López, 2008, Theo em B)).
Lemma 4.2.4.Assume ha he con inuous map h ela ed o he di e ence equa ion
(76)has a GAS equilib ium x∗in J= (0,∞), and le H: (0,∞)−→ (0,∞)be a
con inuous map sa is ying
4.2 s abili y concep s and esul s 95
1.x < H(x)⩽max{h(x),x∗}, o all x∈(0,x∗);
2.x > H(x)⩾min{h(x),x∗}, o all x∈(x∗,∞);
hen x∗is a GAS ixed poin o xn+1=H(xn).
Fo smoo h maps h∈C3, a ypical condi ion equi ed in he global s abili y
esul s is he nega i e sign o he Schwa zian de i a i e. This e m was i s
o mula ed in 1869 by He mann A. Schwa z in his wo k on con o mal map-
pings, bu i was no un il he wo k by Singe (1978) ha i was in oduced in
he s udy o one-dimensional dynamical sys ems. The ollowing esul ollows
om Theo em 2.7and Addendum in (Singe , 1978).
Lemma 4.2.5.Le he map h: [0,∞)−→ [0,∞) ela ed o he one-dimensional
di e ence equa ion (76)be S-unimodal, hen he posi i e ixed poin x∗=Kis GAS in
(0,∞)i and only i h0(K)⩾−1.
By using ha he Ricke map is S-unimodal and he condi ions p o ed in
Co olla y 4.2.5 o i s de i a i e when 0< ⩽2, we can also apply Lemma
4.2.5 o p o e he sha p global s abili y o he Ricke model.
A gene aliza ion o Lemma 4.2.5is gi en in (El-Mo shedy and Jiménez López,
2008), whe e he condi ion on he nega i e Schwa zian de i a i e is es ic ed
o a sui able subin e al o J.
Lemma 4.2.6.Le he map h ela ed o he one-dimensional disc e e- ime di e ence
equa ion (76)be con inuous in a closed in e al Jand ha e a unique ixed poin x∗∈J,
such ha h(x)> x o all x∈J,x < x∗and h(x)< x o all x∈J,x > x∗. Assume
ha he e exis c,d∈J,c < x∗< d such ha :
1.h|(c,d)has a mos one c i ical poin ;
2.h(x)⩽h(c) o all x∈J,x⩽c;
3.h(x)⩾h(d) o all x∈J,x⩾d.
I his dec easing a x∗,−1⩽h0(x∗)< 0 and (Sh)(x)< 0 o all x∈(c,d)excep o
a mos one c i ical poin o h; hen x∗is GAS in J.
Fo he emainde o his sec ion, assume ha 0 < σ < 1 and le he map
h ela ed o he one-dimensional disc e e- ime di e ence equa ion (76) be a C3
unc ion on J= [0,∞)gi en by h(x) = (1−σ)x+ (x). Fo gene al S-unimodal
maps , he co esponding map hdoes no necessa ily inhe i all he p ope ies
o , his is he case o he condi ion on he Schwa zian de i a i e (see (Liz and
F anco, 2010) o de ails). Thus, he p e ious esul s canno be di ec ly applied
o he sys em (78). This gap on he li e a u e was illed by Liz and F anco
(2010, Theo em 1) and he nex esul om (Liz and Lois-P ados, 2020a) is a
gene aliza ion.
Lemma 4.2.7.Assume ha 0 < σ < 1 and : [0,∞)−→ [0,∞)sa is y he ollowing
condi ions:
96 backg ound de ini ions and esul s ii
1. σ(x)=(1/σ) (x)has a unique posi i e ixed poin x∗> 0, σ(0) = 0and
lim
x→0+ 0
σ(x)> 1 (i can be ∞);
2. has a unique c i ical poin xc; mo eo e , 0(x)> 0 o all x∈(0,xc)and
0(x)< 0 o all x > xc;
3. 00(x)< 0 o all x∈(0,xc);
4.(S )(x)< 0 o all x > xc.
Then he unique posi i e equilib ium x∗o equa ion (78)is GAS in (0,∞)i and only
i
h0(x∗) = (1−σ) + 0(x∗)⩾−1. (84)
Mo eo e , i condi ion (84)does no hold, hen x∗is uns able.
P oo . Equa ion (78) can be w i en in he o m o equa ion (4) in (Liz and
F anco, 2010), ha is:
xn+1=αxn+ (1−α) σ(xn),
wi h α=1−σand σ(x) = (1/σ) (x).
I is clea ha condi ions 2-4hold o σbecause
0
σ(x) = (1/σ) 0(x), 00
σ(x) = (1/σ) 00
σ(x), and (S σ)(x) = (S )(x).
The e a e wo ele an di e ences wi h espec o Theo em 1in (Liz and
F anco, 2010). On he one hand, condi ion 4 he e equi ed (S )(x)< 0 o all
x6=xc. Howe e , a simple inspec ion o he p oo shows ha he less es ic i e
condi ion (S )(x)< 0 o all x > xcis enough o ge he esul . On he o he
hand, he es ic ion xc< x∗is equi ed in (Liz and F anco, 2010, Theo em 1).
In case xc⩾x∗, he map h(x) = (1−σ)x+ (x)de ining he igh -hand side
o (78) sa is ies he hypo hesis o Lemma 4.2.2. Fi s , he ixed poin s o ha e
exac ly hose o σ, he e o e hhas a unique posi i e ixed poin . Second, since
h(x) = xi and only i x∈{0,x∗}, lim
x→0+h0(x) = lim
x→0+(1−σ) + σ 0
σ(x)> 1 and
h0(x)> 0 o all x∈(0,x∗), hen x<h(x)< x∗ o all x∈(0,x∗). Thi d, as
hsa is ies h(x∗) = x∗,h0(x) = (1−σ) + (x)> 0 and h00(x) = 00(x)< 0 o
all x∈(x∗,xc)and h0(x) = (1−σ) + (x)<(1−σ)< 1 o all x > xc, hen
0 < h(x)< x o all x > x∗. Thus, we can conclude ha x∗is GAS in (0,∞).
4.3 bi u ca ion ypes
We dis inguish be ween wo ca ego ies o bi u ca ions: local and global. We
use he e m local o hose occu ing in a neighbo hood o a ixed poin and
he e m global o all he es . In Chap e s 5and 6, once we ha e de e mined
he s abili y o ixed poin s, we use ha in o ma ion o s udy i s local and
hen global bi u ca ions.
4.3 bi u ca ion ypes 97
In he ca ego y o local bi u ca ions, he e exis s a well de eloped classi ica-
ion heo y when, a he bi u ca ion poin , he equilib ium lies in a smoo h o
con inuous b anch o he map h. We begin e iewing hese ypes o bi u ca-
ions and hen we con inue wi h he ones which ake place when a ixed poin
collides wi h a discon inui y poin .
We use he e m smoo h bi u ca ions (SBs) o hose bi u ca ions ha a e yp-
ical o smoo h dynamical sys ems, which a e associa ed wi h he mul iplie
h0(x∗)o a ixed poin x∗passing h ough he alues ±1. We ecall ha , when
|h0(x∗)|6=1we say ha x∗is hype bolic. We can ind he ollowing classi ica ion
wi h illus a ions in (Wiggins, 1990, Chap e 3):
• A saddle-node o old SB occu s when wo coexis ing LAS and uns able
ixed poin s become a unique equilib ium wi h mul iplie 1and hen dis-
appea .
• A ansc i ical SB occu s when wo coexis ing LAS and uns able cu es o
ixed poin s become a unique equilib ium wi h mul iplie 1, and hen he
cu es o ixed poin s in e change hei s abili y.
• A pi ch o k SB occu s when h ee b anches o ixed poin s collide and hen
only one emains. In he supe c i ical case, wo LAS ixed poin s collide
wi h an uns able ixed poin and he emaining ixed poin is LAS. In he
subc i ical case, wo uns able ixed poin s collide wi h a LAS ixed poin ,
and he emaining ixed poin is uns able. A he bi u ca ion poin , he e
is a unique equilib ium wi h mul iplie 1.
• A lip o pe iod-doubling SB occu s when a ixed poin x∗becomes hype -
bolic wi h mul iplie −1and hen he ixed poin changes i s s abili y
and a 2-cycle {p1,p2}appea s, whe e p1< x∗< p2. In he supe c i ical
case, he LAS ixed poin becomes uns able and he 2-cycle is LAS. In he
subc i ical case, he uns able ixed poin becomes LAS and he 2-cycle is
uns able. We use he e m pe iod-hal ing SB when he bi u ca ion akes
place in he opposi e di ec ion.
Fo smoo h maps h∈C ( ⩾2 o old and ansc i ical; ⩾3 o pi ch o k
and lip), he book by Wiggins (1990) s a es a lis o su icien condi ions o he
map hunde which each o hese ypes o SBs akes place. Simila bi u ca ions
ela ed o he mul iplie o a ixed poin passing h ough ±1, bu occu ing
unde some degene acy condi ions, a e e e ed as degene a e SBs (see (A u in
e al., 2019, Sec ion 2.2) o de ails).
Be o e desc ibing local bi u ca ions o ixed poin s o piecewise-smoo h (con-
inuous o discon inuous) maps, we need o adop some special no a ions in-
oduced in (Be na do e al., 2008). Fo ha pu pose, we conside hland h
98 backg ound de ini ions and esul s ii
wo smoo h maps on J, a poin d1∈In (J)such ha hl0(d1)6=h 0(d1), and we
assume ha he map hassocia ed o he di e ence equa ion (76) is de ined by
h(x) =
hl(x),x < d1;
hl(x)o h (x),x=d1;
h (x),x > d1;
(85)
which is no di e en iable a d1. The poin d1whe e his no di e en iable is
called a b eak poin . I a b eak poin is a ixed poin o h, hen we say i is a
bounda y ixed poin . We ecall ha he map hin (85) is de ined by wo smoo h
maps hland h , so he e may a e ixed poin s o hland h ha a e no ixed
poin s o h, we e e o hem as i ual ixed poin s; while he ixed poin s o h
a e called admissible ixed poin s.
The local bi u ca ions a a bounda y ixed poin whe e in oduced by Nusse
and Yo ke (1992) and occu when, unde in ini esimal pa ame e a ia ion, a
ixed poin collides wi h a b eak poin and he collision leads o a quali a i e
change on he dynamics, wi h he ixed poin ansi ioning om being admis-
sible o being i ual o ice e sa. We e e o hese ypes o bi u ca ions as
bo de -collision bi u ca ions (BCBs).
A classi ica ion o BCBs in piecewise-smoo h con inuous disc e e- ime di e -
ence equa ions is gi en in (Be na do e al., 2008, Chap e 3). In Sec ion 3.4,
he e is a classi ica ion o one-dimensional sys ems, which is gi en in e ms o
he de i a i es o hl,h a he bounda y ixed poin . We pay special a en ion
o he con en s in Subsec ion 3.1.2, whe e hey desc ibe and illus a e he ou
basic dynamical scena ios which ake place a a BCB:
• A old BCB occu s when wo coexis ing admissible ixed poin s collide a
a b eak poin and become wo i ual ixed poin s.
• A pe sis ence BCB occu s when an admissible and a i ual ixed poin
collide a a b eak poin and in e change hei oles. No o he pe iodic
poin s a e c ea ed o des oyed a he bi u ca ion poin .
• A lip o pe iod-doubling BCB occu s when an admissible ixed poin x∗
collides wi h a b eak poin and a 2-cycle {p1,p2}, wi h p1< x∗< p2,
appea s.
• A pe iod-mul iplying BCB occu s when an admissible ixed poin collides
wi h a b eak poin and an m-cycle appea s, wi h m > 2.
We obse e ha new dynamic ansi ions a he bi u ca ion poin s occu due
o he lack o di e en iabili y o he map h, such as he pe sis ence o pe iod-
mul iplying BCBs. Mo eo e , he e exis mo e complex bi u ca ions, whe e
a e he bo de collision o an admissible ixed poin , he asymp o ic dynamics
become chao ic. I is well-known ha such a ansi ion in gene al does no
occu in smoo h sys ems.
4.3 bi u ca ion ypes 99
The p e iously men ioned bi u ca ions can also be obse ed in he long- e m
dynamics o a piecewise-smoo h discon inuous dynamical sys em, bu he dis-
con inui y cha ac e inc eases again he a ie y o ixed poin local bi u ca ions.
We de o e he nex lines o desc ibe some addi ional BCBs aking place a dis-
con inui y poin s o sys ems ha we will conside in Chap e 5(see (A u in
e al., 2019, Chap e s 2and 3) o de ails):
• An exis ence BCB occu s when a i ual ixed poin o hbecomes admis-
sible a e a collision wi h a b eak poin . No o he o bi s a e c ea ed o
des oyed a he bi u ca ion poin .
• A pe iod-adding BCB occu s when a LAS admissible ixed poin collides
wi h a b eak poin ; a he collision poin , he e is a homoclinic o bi (we
ecall he de ini ion a he end o his subsec ion); a e he collision he
ixed poin becomes i ual, and an a ac ing m-cycle becomes admis-
sible, gi ing ise o a pe iod-adding scena io. The pe iod-adding scena io
e e s o he o de o pe iodici y egions in he pa ame e space whe e be-
ween wo cycles o pe iods nand l he e is an (n+l)-cycle (see G anados,
Alsedà, and K upa, 2017, o de ails).
In he ca ego y o global bi u ca ions, we conside wo di e en g oups:
•Bounda y-collision bi u ca ions, which a e caused by he collision o an a -
ac o wi h an uns able ixed poin o m-cycle (m∈N,m⩾2). These
bi u ca ions whe e in oduced as c ises by G ebogi, O , and Yo ke (1982)
o he case o a chao ic a ac o . They use he e m bounda y c isis when
he uns able o bi is on he bounda y o he chao ic a ac o and he col-
lision causes e mina ion o he a ac o ; and in e io c isis when he colli-
sion occu s wi hin he basin o a ac ion. They men ion ha he in e io
c isis o en esul s in a sudden expansion o he basin.
•Basin bounda y me amo phoses, which a e hose ela ed wi h ans o ma-
ions o he basins o a ac ion. We will ind ansi ions om a simply-
connec ed o a mul iply-connec ed basin. The ansi ions can be mo e
complex, as hose in (G ebogi, O , and Yo ke, 1983) om a egula basin
o a ac al one.
We ha e al eady men ioned ha a homoclinic o bi appea s a he pe iod-
adding BCBs, and we will also obse e he p esence o his ype o o bi s in
bounda y-collision bi u ca ions and basin bounda y me amo phoses. We ecall
ha a homoclinic o bi is o med by a homoclinic poin , i s p eimages and i s
( ini e) o wa d o bi . A poin xis homoclinic o an n-cycle p={p1,...,pn}i
( n)m(x) = pk o some m∈Nand k∈{1,...,n}, and xbelongs o he uns able
mani old o pk. Fo u he de ails, see (De aney, 1989, Sec ion 1.16) o (Liz,
2010a, Appendix C).
Finally, i is wo h men ioning ha , in gene al, we s udy bi u ca ions o a -
ac o s, bu bi u ca ions o o he in a ian se s also dese e o be in es iga ed
since hey can in luence he asymp o ic dynamics as well.
100 backg ound de ini ions and esul s ii
4.41-pa ame e bi u ca ion diag ams ea u es
The las s ep in he analysis o asymp o ic dynamics is o illus a e he in luence
o pa ame e s. Fo ha pu pose, we i s e lec he in o ma ion we ha e on
s abili y o ixed poin s and bi u ca ions in a 2-pa ame e BD. Then we plo
se e al 1-pa ame e BDs o ge mo e insigh and comple e he global pic u e o
he dynamics.
In his sec ion we e iew some ele an ea u es o phenomena ha ha e
been obse ed in 1-pa ame e BDs o a wide ange o disc e e one-dimensional
dynamical sys ems.
Long- ansien s and hys e esis
Le us i s discuss some consequences o ocusing on asymp o ic dynamics
in ecology. In (Has ings e al., 2018), he e is a e iew summa y on he long
ansien phenomena, de ined as a dynamical egime ha pe sis s o mo e
han a ew and as many o en gene a ions, bu which is no he s able long-
e m dynamics ha would e en ually occu . In he pa icula amewo k o
ma hema ical analysis o he dynamical sys ems gi en by (76), he s udy o
non-asymp o ic dynamics has no much in e es , since long- ansien s a e jus
i e a ions o an ini ial condi ion by he map h. Howe e , he p esence o long-
ansien s can obscu e he decisions on he managemen o ecological sys ems,
so i becomes in e es ing o ca ego ize di e en ways in which ansien s can
a ise.
Wi h his possibili y in mind, we s a de ining he long- e m dynamics phe-
nomenon o hys e esis, which can also ha e unexpec ed consequences o he
managemen in biological sys ems. In (Blackwood, Has ings, and Mumby,
2012), o a sys em wi h mul iple s able s a es, he au ho s say ha his ea u e
occu s when he p e ious his o y o he sys em in luences he con e gence o
an ini ial condi ion o one o he s able s a es o he o he s. They also desc ibe
he phenomenon in o he wo ds, saying ha he c i ical pa ame e condi ions
unde which some poin s con e ging o one s able s a e swi ch and con e ge
o ano he one a e di e en om he condi ions ha will allow he con e gence
o such poin s o he o iginal s a e. See Figu e 22 (A).
Bubbles, bis abili y and hyd a e ec
We now deal wi h h ee ea u es ha ha e been obse ed in bi u ca ion dia-
g ams o bo h semelpa ous and i e opa ous popula ion models.
We begin wi h a phenomenon ela ed o he pe iod-doubling sequence o
bi u ca ions as a uni e sal ou e o chaos, which can be b oken and e e sed
as shown in (Bie and Boun is, 1984) o simple non-linea disc e e dynamical
sys ems in ol ing he a ia ion o wo o mo e pa ame e s. As a consequence,
he bi u ca ion diag ams o m closed loop-like s uc u es simila o bubbles
4.41-pa ame e bi u ca ion diag ams ea u es 101
and he e ec is usually e e ed o as bubbling. De ini ion 3in (Liz and Ruiz-
He e a, 2012) gi es a o mal de ini ion o he concep o bubble. The complex-
i y o he bubble s uc u e depends on he poin a which he pe iod-doubling
sequence is e e sed. The simples bubble occu s when an equilib ium loses
i s asymp o ic s abili y h ough a pe iod-doubling bi u ca ion and, a he nex
bi u ca ion poin , a pe iod-hal ing bi u ca ion occu s, so ha he local asymp-
o ic s abili y o he ixed poin is egained, we e e o i as p ima y bubble. I he
pe iod-doubling sequence o bi u ca ions is e e sed a e a egion o chao ic
dynamics, we use he e m chao ic bubble.
The bis abili y ea u e e e s o he coexis ence o wo a ac o s, and i occu s
due o old bi u ca ions (ei he smoo h o bo de -collisions). Bis abili y has
impo an consequences in popula ion dynamics because, in his case, he long-
e m beha io o he solu ions s ongly depends on he ini ial condi ion.
Ano he o mal de ini ion s a ed in (Liz and Ruiz-He e a, 2012) is he hyd a
e ec . This e m was used by Ab ams (2009) and he e e ences he ein o he
phenomenon o a popula ion inc ease in esponse o an inc ease in i s mo ali y
a e. This ea u e has been i s ecognized by Ricke (1954) o he well-know
Ricke model which assumes ha mo ali y p ecedes ep oduc ion. Tha is
one o he h ee mechanisms unde lying hyd a e ec which a e de eloped in
(Ab ams, 2009) o non-o e lapping popula ions models. Liz and Ruiz-He e a
(2012) s udied he hyd a e ec o i e opa ous popula ion models, in which a
pe cen age o he adul popula ion su i es he ep oduc i e season.
Ex inc ion windows and sudden collapses
The ollowing ea u es a e ypical o models wi h Allee e ec , whe e popula ions
canno su i e in he long- e m i i s abundance is below a c i ical size, hey
a e called ex inc ion windows and sudden collapses.
The e m ex inc ion window e e s o he su i al-ex inc ion dynamics de-
sc ibed by Sinha and Pa hasa a hy (1996) as an unusual s uc u e wi h al-
e na ing egions o su i als and ex inc ion unde a ia ion o pa ame e s, so
ha he popula ion can pe sis unde e y low and ai ly high alues o he
pa ame e , hough i is no able o su i e a in e media e pa ame e alues.
The sudden collapse phenomenon appea s also in (Sinha and Pa hasa a hy,
1996), bu a desc ip ion o he ea u e is gi en in (Sch eibe , 2001) whe e i
is said ha he e exis s a c i ical pa ame e alue abo e which popula ions
a e d i en o ex inc ion o all ini ial densi ies and below which pe sis ence is
possible. By pe sis ence we mean ha popula ion emains a densi ies bounded
away om ze o.
Pe iodic-windows, s a -like in e sec ions and e ec i ely chao ic beha iou
We inally desc ibe some ypical ea u es o bi u ca ion diag ams o piecewise-
smoo h maps wi h la b anches (see (Sinha, 1994) o u he de ails).
108 combina ions o cc and h ha es ing s a egies
ha es ing s a egies. The conside a ion o cons an quo as ins ead o ixed
mo ali y a es can educe he isk o o e capaci y when he s ock dec eases,
since PH s imula es in es men when s ock size is la ge. The esul s in (Hje ne
and Hansson, 2001) show ha long- e m yield o he p ecau iona y combi-
na ion o TH and CC ( e e ed o as quasi cons an ca ch, QCC o sho ) is
10% less han ha o PH a MSY. They conside ha i is a small di e ence
and emphasize ha he much lowe in e annual a iabili y o QCC gi es he
oppo uni y o be e ishe y capaci y u iliza ion. We no ice ha hey i s con-
side ed he TCC con ol ule, and hen QCC as a p ecau iona y modi ica ion o
educe ishe y closu es. To ou knowledge, he TCC s a egy has been consid-
e ed so a only by Hje ne and Hansson (2001) and in a sligh ly di e en o m
by AlSha awi and Rhouma (2009). We also no ice ha Pun (2010) ci es he a i-
cle (Bu e wo h, 1987) o alk abou h eshold managemen s a egies in which
ca ch becomes cons an when he s ock size is g ea e han a a ge le el. We
inally p o ide a di e en eason o conside his ype o con ol ules. S eine ,
C iddle, and Adkinson (2011) looked o a solu ion o he e enue decline o
B is ol Bay sockeye salmon managed wi h a ixed escapemen con ol ule. In
con as o o he ishe ies, he landings had emained high, bu he p ices ell as
a consequence o compe i ion esul ing om he inc eased p oduc ion o ou
and salmon a med species in Chile. They p o ide se e al easons o conside
implemen ing he PTCC s a egy a he han TH o PTH: he ac ha PTCC
induces lowe ha es s allows o imp o e he quali y o he ish deli e ed, hus
p oducing a high ex essel p ice pe pound; he lowe a iabili y in ha es p o-
ides mo e e iciency o he managemen ope a ions because he ha es le els
will be known a he beginning o he season.
H
(A)
escapemen
ca ch
?
No ha es
T T +H
H
(B)
escapemen
ca ch
?
No ha es
T
H
No ha es
?
Ex inc ion
?
escapem.
T H
ca ch
(C)
Figu e 11: Di e en ha es ing s a egies ha combine cons an ca ches wi h h eshold
e e ence poin s. We ep esen he ca ch ( ed solid line) as a unc ion o
he popula ion biomass. The blue line ep esen s he iden i y map, Tis he
h eshold and H he maximum allowed quo a. (A): P ecau iona y h eshold
cons an ca ch ha es ing. The o he panels show h eshold cons an ca ch
ha es ing wi h (B): H < T and (C): H > T.
5.1 in oduc ion 109
Con ex ualiza ion in he amewo k o piecewise-smoo h one-dimensional di e ence
equa ions
As a as we know, we o mula e and s udy he PTCC and TCC ules in he
amewo k o one-dimensional disc e e- ime dynamical sys ems by means o
an analy ical app oach o he i s ime. As hese ha es ing s a egies a e
based on h eshold popula ion sizes, he associa ed map will be composed o
di e en b anches (co esponding o high o low/no ha es ing) de ined on
in e als which a e sepa a ed a he biomass e e ence poin s (also called b eak
poin s in he ma hema ical li e a u e). As he maps a e no di e en iable a
he h eshold poin s, hey gi e ise o piecewise-smoo h dynamical sys ems
(A u in e al., 2019; Be na do e al., 2008). They can exhibi so-called non-
smoo h bi u ca ions ha di e subs an ially om hose ha occu in smoo h
dynamical sys ems, e.g., bo de -collision bi u ca ions (Nusse and Yo ke, 1992).
Non-smoo h bi u ca ions a e ela ed o in a ian se s colliding wi h a b eak
poin , which is gi en by he ha es ing h eshold. In ecen yea s, a lo o
p og ess has been made in unde s anding he dynamics o piecewise-smoo h
maps, e.g., (Bane jee e al., 2000; Radi and Ga dini, 2018; Sushko, Ga dini,
and Ma suyama, 2014). Howe e , e en hough hey eme ge qui e na u ally
in he con ex o h eshold-based ha es ing, hei ma hema ical analysis in
he con ex o ishe ies models is jus a he beginning (Bischi, Laman ia, and
T amon ana, 2014; F anco and Hilke , 2013,2014; Hilke and Liz, 2019,2020;
Liz and Lois-P ados, 2020b; Lois-P ados and Hilke , submi ed; Segu a, Hilke ,
and F anco, 2016,2020).
O e iew o he esea ch de elopmen
Ou s udy is ocused on s abili y and bi u ca ions; in his ega d, we conside
he wo ele an ha es pa ame e s Hand Tand ob ain 1-pa ame e and 2-
pa ame e bi u ca ion diag ams ha help o unde s and how a con inuous a i-
a ion o any o hem in luences he dynamics, and he in e play be ween bo h
pa ame e s. We iden i y egions whe e global a ac ion, pe iodic a ac o s,
mul i-s abili y and complex beha io a e likely o occu , paying special a en-
ion o non-smoo h bi u ca ions. We i s s udy he long- e m beha io o he
con inuous map associa ed wi h PTCC, which will se e as a baseline agains
which we can compa e he e ec s induced by he discon inui y in TCC. Fo
he PTCC ule, we comple ely de e mine he asymp o ic dynamics and bi u -
ca ions o gene al compensa o y popula ion models; we also unde s and he
long- e m beha io in some egions o he pa ame e plane (H,T) o gene al
o e compensa o y maps, o which we conside he Ricke map as a case s udy
o gi e a global pic u e o he dynamics. Fo he TCC ule, we jus deal wi h
s ic ly inc easing s ock- ec ui men maps, while in his case all ini ial condi-
ions would con e ge o an equilib ium o he PTCC ule, he discon inui y
poin o TCC gi es ise o highly complex dynamics, including mul iple a ac-
110 combina ions o cc and h ha es ing s a egies
o s, di e en pe iodic cycles, homoclinic o bi s and e en chao ic oscilla ions.
The bi u ca ions in which hese dynamical pa e ns eme ge and disappea in-
ol e bo de - and bounda y-collision bi u ca ions as well as basin bounda y
me amo phoses. Hence, he discon inui y in he ha es con ol ule p oduces
ich dynamics ha , o ou knowledge, ha e no been obse ed in con inuous
ha es ing models applied o popula ion maps ha a e s ic ly inc easing in
he absence o ha es ing. The esea ch on PTCC o ally ocuses on dynami-
cal sys ems heo e ical aspec s, we no ice ha o some pa icula pa ame e
alues he long- e m beha io is in luenced by he dynamics o cons an -ca ch
o h eshold ha es ing. Ne e heless, some cha ac e is ic dynamics o CC o
TH a e no comple ely p ese ed, e.g., he ex inc ion a ac o o CC o he
con e gence o almos all solu ions o a pe iodic o bi con aining To TH. Fo
he TCC ule, we ho oughly combine he quali a i e s udy o popula ion dy-
namics wi h managemen -o ien ed in e p e a ions; addi ionally we s udy he
in luence o ha es ing pa ame e s in long- e m a e age yield and ha es e-
quency. We pay special a en ion o he in luence o he h eshold poin ( ha is,
he e ec ha TCC has in compa ison o CC) in popula ion, a e age yield and
ha es equency beha io .
5.2 models desc ip ion
In his sec ion, o he sake o comple eness, we i s s a e he ma hema ical
o mula ion and ecall some esul s on asymp o ic dynamics o he classical
con ol ules ha can be conside ed as pa icula cases o TCC o PTCC, ha is,
cons an quo a (subsec ion 5.2.1) and h eshold ha es ing (subsec ion 5.2.2). In
subsec ions 5.2.3and 5.2.4, we p o ide he ma hema ical exp ession o PTCC
and TCC, espec i ely. We compa e he TCC con ol ule wi h he p ecau ion-
a y e sion gi en by PTCC and he simila s a egy conside ed by A u in e al.
(2019).
5.2.1Cons an ca ch ule
Fo la e e e ence, we s a e some well-known esul s o he CC ha es ing
ule. Fo a gene al map : [0,∞)−→ [0,∞)and he cons an quo a H > 0, he
model eads
xn+1=FCC(xn) = max{0, (xn) − H}. (86)
Le us conside he map g(x) = (x) − H,x∈[0,∞), hen we can ew i e FCC in
he o m: FCC(x) = max{0,g(x)}.
The ollowing esul es ablishes a c i ical alue H20 o he ha es ing quo a.
Ha es ing abo e his le el (H > H20) will d i e he popula ion o ex inc ion.
Below his le el, se e al long- e m dynamics can occu . The simples dynamics
ake place o a map sa is ying condi ions (A1)-(A2)(de ined in Sec ion 4.1)
wi h xc=∞, o which he popula ion will su i e p o ided he ini ial con-
5.2 models desc ip ion 111
di ion is la ge enough. By con as , he mos complex dynamics occu when
xc< K and he e is bis abili y be ween he LAS ixed poin 0and o he a ac o .
The p oposi ion uses ha he e exis s a unique ˜x>0such ha 0(˜x) = 1. This
ollows om assump ions (A1)-(A2)and he Mean Value Theo em. Mo eo e ,
˜x∈(0,K).
P oposi ion 5.2.1.Assume ha H>0and sa is ies (A1)-(A2). Deno e by ˜x he
unique solu ion o 0(x) = 1.
1. I H < H20 := (˜x) − ˜x, hen ghas wo posi i e equilib ia x∗
−and x∗
+wi h
0<x∗
−<˜x < x∗
+< K. While x∗
−is always uns able and 0is locally asymp o i-
cally s able, he ixed poin x∗
+can be ei he s able o uns able:
a) I xc=∞, hen x∗
+is LAS wi h basin o a ac ion (x∗
−,∞)and [0,x∗
−)is
he basin o a ac ion o 0.
b) I K < xc<∞, hen x∗
+is LAS wi h basin o a ac ion (x∗
−,g−1(x∗
−)) and
[0,x∗
−)∪(g−1(x∗
−),∞)is he basin o a ac ion o 0.
c) I xc< K, hen x∗
+can be LAS wi h basin o a ac ion (x∗
−,g−1(x∗
−)) o
uns able. I x∗
+is uns able, he dynamics depend on he posi ion o g2(xc)
wi h espec o x∗
−:
I g2(xc)> x∗
−, hen I= [g2(xc),g(xc)] is abso bing and he ini ial condi-
ions in (x∗
−,g−1(x∗
−)) en e Iin ini e ime; besides, [0,x∗
−)∪(g−1(x∗
−),∞)
is he basin o a ac ion o 0.
I g2(xc)< x∗
−, hen 0is an essen ial global a ac o .
2. I H=H20, hen ˜xis he unique posi i e ixed poin o g. The equilib ium ˜xis
semi-s able and 0is locally asymp o ically s able. I xc=∞,[˜x,∞)and [0, ˜x)
a e hei espec i e basins o a ac ion; while i xc<∞ hey a e [˜x,g−1(˜x)] and
[0, ˜x)∪(g−1(˜x),∞), espec i ely.
3. I H > H20, hen ghas no posi i e ixed poin s and 0is GAS.
We e e he eade o (Sch eibe , 2001) o a igo ous p oo and analysis o
he esul s in P oposi ion 5.2.1.
5.2.2Th eshold ha es ing ule
We conside now he TH ule, which can be seen as he an ipode o he CC
ha es ing s a egy. Fo a gene al map : [0,∞)−→ [0,∞)and a h eshold
le el T > 0, he dynamics o TH a e go e ned by he di e ence equa ion:
xn+1=FTH(xn) = min{ (xn),T}=
(xn), (xn)⩽T;
T, (xn)> T.(87)
In he ecen wo k by Hilke and Liz (2019), we ind a igo ous heo e ical
s udy on he in luence o he h eshold Ton he dynamics o (87). Unde some
112 combina ions o cc and h ha es ing s a egies
gene al condi ions o he map , hey show ha h eshold ha es ing can ne e
ha e a des abilizing e ec on he managed popula ion. The ollowing esul
summa izes he indings in (Hilke and Liz, 2019, Sec ion 2.1).
P oposi ion 5.2.2.Assume ha T > 0 and sa is ies (A1).
1. I T⩽K, hen Tis he unique posi i e equilib ium o (87)and i is GAS.
2. I T > K, he dynamics o he managed sys em (87)depend on he dynamics o
he unmanaged sys em xn+1= (xn):
a) I Kis GAS o he unmanaged sys em, hen Kis he unique posi i e equi-
lib ium o (87)and i is GAS.
b) I Kis uns able bu he unmanaged sys em has a ini e numbe o pe iodic
o bi s, hen dec easing h eshold induces a sequence o pe iod-hal ing bi u -
ca ions un il he equillib ium becomes s able.
c) I Kis uns able and he unmanaged sys em is chao ic, o S-unimodal maps
he e is a unique pe iodic o bi ha is an essen ial global a ac o o he
managed sys em (87). Dec easing T om (xc) o K, he e is Li-Yo ke chaos
(see De ini ion 3.1, Aulbach and Kieninge , 2001) as long as he pe iod o
he a ac ing cycle is no a powe o 2. Once he dynamics become simple
due o smalle h eshold alues, we ha e he si ua ion conside ed in he
p e ious case.
5.2.3P ecau iona y h eshold cons an ca ch ule
Applying he PTCC ha es ing ule o a semelpa ous popula ion model gi en
by (77), we ob ain he di e ence equa ion
xn+1=FPTCC(xn) =
(xn), (xn)⩽T;
T,T < (xn)⩽T+H;
(xn) − H, (xn)> T +H.
(88)
The map g(x) = (x) − Hallows o w i e FPTCC in he o m
FPTCC(x) =
(x), (x)⩽T;
T,g(x)⩽T < (x);
g(x),g(x)> T.
The piecewise smoo h con inuous map FPTCC can also be w i en in a line as
FPTCC(x) = min{ (x), (x) − min{H, (x) − T}}=min { (x),max{g(x),T}};
and depends on he wo ha es ing pa ame e s Hand T. Th oughou he anal-
ysis o he PTCC ha es ing s a egy (88), we will assume ha H > 0 and T > 0.
I is wo h no icing ha we ge he unmanaged map o he CC and TH ules
as pa icula o limi cases:
5.2 models desc ip ion 113
• I H=0o T⩾sup{ (x),x⩾0}, hen FPTCC ≡ .
• In he limi case T=0, (88) becomes he usual cons an ca ch policy,
de ined by (86).
• I T < sup{ (x),x⩾0}⩽H+T, hen he PTCC ha es ing s a egy be-
comes he pu e h eshold ha es ing ule (87).
The ypical shape o FPTCC can be seen in Figu e 12. Roughly speaking, he
g aphs o and ga e joined by la segmen s de ined by T, which ypically
esul s in i e in e als o smoo hness o FPTCC i is unimodal, and h ee i
is s ic ly inc easing.
(A)
4
(A)
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
(B)
T
?
6
H
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
FIG. 2: Illus a ion o he g aph o he piecewise smoo h map F(blue solid line). We also plo he line y=x( ed, dashed)
and he g aph o (black, dashed). (A): Unimodal Ricke map (x)=xe
2.6(1x), wi h H=0.5andT=0.7. (B): Mono one
Be e on-Hol map (x)=2x/(1 + x), wi h H=0.2 and T=0.7.
TABLE I: Main no a ions
Symbol/concep Meaning
H(maximum) ha es ing quo a
T h eshold ha es ing pa ame e
PTCH p ecau iona y h eshold cons an -ca ch ha es ing
TH (pu e) h eshold ha es ing
p oduc ion map go e ning (1)
gg(x)= (x)H
Fmap de ining he PTCH ule (2)
Ricke map (x)=xe
(1x), >0
xcc i ical poin o ( 0(xc)=0)
˜xpoin such ha 0(˜x)=1
x(smalles ) poin such ha 0(x)=1
Kposi i e ixed poin o
p, q posi i e ixed poin s o g(0 <pq<K)
b eak poin poin a which Fis no di↵e en iable
bounda y ixed poin b eak poin which is a ixed poin o F
admissible ixed poin ixed poin o F
i ual ixed poin ixed poin o one map de ining Fbu no o F
BCB bo de -collision bi u ca ion
SB smoo h bi u ca ion
III. FIXED POINTS: LOCATION, STABILITY, AND BIFURCATIONS
In his sec ion, we s udy he ixed poin s o Fdepending on he pa ame e alues Tand H. In he i s subsec ion
we ocus on he numbe o ixed poin s and hei loca ion, in he second one we s udy hei s abili y p ope ies, and
in he hi d subsec ion we desc ibe he local bi u ca ions o ixed poin s, ha is, we de e mine he c i ical alues o
he pa ame e s o which ixed poin s a e c ea ed o des oyed, o s abili y swi ches occu .
One impo an consequence o condi ions (A1) and (A2) is ha he e is a unique ˜x>0 such ha 0(˜x) = 1.
Mo eo e , ˜x2(0,min{K, xc}). This p ope y is a di ec consequence o he Mean Value Theo em and he conca i y
o on (0,x
c). The poin ˜xplays an impo an ole in he s udy o ixed poin s.
(B)
4
(A)
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
(B)
T
?
6
H
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
FIG. 2: Illus a ion o he g aph o he piecewise smoo h map F(blue solid line). We also plo he line y=x( ed, dashed)
and he g aph o (black, dashed). (A): Unimodal Ricke map (x)=xe
2.6(1x), wi h H=0.5andT=0.7. (B): Mono one
Be e on-Hol map (x)=2x/(1 + x), wi h H=0.2 and T=0.7.
TABLE I: Main no a ions
Symbol/concep Meaning
H(maximum) ha es ing quo a
T h eshold ha es ing pa ame e
PTCH p ecau iona y h eshold cons an -ca ch ha es ing
TH (pu e) h eshold ha es ing
p oduc ion map go e ning (1)
gg(x)= (x)H
Fmap de ining he PTCH ule (2)
Ricke map (x)=xe
(1x), >0
xcc i ical poin o ( 0(xc)=0)
˜xpoin such ha 0(˜x)=1
x(smalles ) poin such ha 0(x)=1
Kposi i e ixed poin o
p, q posi i e ixed poin s o g(0 <pq<K)
b eak poin poin a which Fis no di↵e en iable
bounda y ixed poin b eak poin which is a ixed poin o F
admissible ixed poin ixed poin o F
i ual ixed poin ixed poin o one map de ining Fbu no o F
BCB bo de -collision bi u ca ion
SB smoo h bi u ca ion
III. FIXED POINTS: LOCATION, STABILITY, AND BIFURCATIONS
In his sec ion, we s udy he ixed poin s o Fdepending on he pa ame e alues Tand H. In he i s subsec ion
we ocus on he numbe o ixed poin s and hei loca ion, in he second one we s udy hei s abili y p ope ies, and
in he hi d subsec ion we desc ibe he local bi u ca ions o ixed poin s, ha is, we de e mine he c i ical alues o
he pa ame e s o which ixed poin s a e c ea ed o des oyed, o s abili y swi ches occu .
One impo an consequence o condi ions (A1) and (A2) is ha he e is a unique ˜x>0 such ha 0(˜x) = 1.
Mo eo e , ˜x2(0,min{K, xc}). This p ope y is a di ec consequence o he Mean Value Theo em and he conca i y
o on (0,x
c). The poin ˜xplays an impo an ole in he s udy o ixed poin s.
Figu e 12: Illus a ion o he g aph o he piecewise smoo h map FPT CC (blue solid
line). We also plo he line y=x( ed, dashed) and he g aph o (black,
dashed). (A): Unimodal Ricke map (x) = x e2.6(1−x), wi h H=0.5and
T=0.7.(B): S ic ly inc easing Be e on-Hol map (x) = 2x/(1+x), wi h
H=0.2and T=0.7.
5.2.4Th eshold cons an ca ch ule
When ha es ing a popula ion ha is g owing acco ding o (77) wi h he TCC
ule, we ob ain
xn+1=FTCC(xn) :=
(xn), (xn)< T;
max{0, (xn) − H}, (xn)⩾T.(89)
Le us conside he map g(x) = (x) − H, hen we can ew i e he map FTCC in
he o m
FTCC(x) =
(x), (x)< T;
0,T⩽ (x)< H;
g(x), (x)⩾max{T,H}.
(90)
114 combina ions o cc and h ha es ing s a egies
The piecewise-smoo h map FTCC depends on he wo ha es ing pa ame e s
Tand H. Fo he s udy o TCC ha es ing ule (89), we assume H > 0 and
T > 0. As pa icula o limi cases, we ob ain he unmanaged map and he
CC ule, asking he same condi ions o he ha es ing pa ame e s as in PTCC.
By con as , we canno ge he TH ha es ing s a egy as a pa icula case o
TCC.
The ypical shape o FTCC applied o s ic ly inc easing maps is shown in
Figu e 13. The main di e ences wi h espec o FPTCC a e he exis ence o a
poin o discon inui y a xT= −1(T)and he absence o a la segmen de ined
by T.
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
(A) (B)
?
6
H
T
?
6
H
T
Figu e 13: Illus a ion o he g aph o he piecewise-smoo h discon inuous map FTCC
(blue solid line). We also plo he line y=x( ed, dashed) and he g aph o
he Be e on-Hol map (x) = 3x/(1+x).(A) : T=1,H=0.4.(B) : T=0.3,
H=0.4.
Rela ed ha es ing s a egy
We poin ou ha AlSha awi and Rhouma (2009, Sec ion 4) p oposed a ha es -
ing s a egy e y simila o TCC. I eads
xn+1=
(xn),xn< x h;
(xn) − h,xn⩾x h;(91)
whe e hey conside ed speci ically he Be e on-Hol map o ,x h ∈(0,K)is
a h eshold popula ion size and h∈(0, (x h)) is he cons an quo a. The di -
e ence wi h (89) is ha TCC compa es he h eshold wi h he popula ion size
(xn)a e ep oduc ion, whe eas (91) compa es he h eshold wi h he popula-
ion size xnbe o e ep oduc ion. Howe e , o he s ic ly inc easing popula ion
maps conside ed in his pape , he e is a co espondence be ween (89) and (91).
To see his, we no e ha is bijec i e and ha we jus ha e o es ablish he
ollowing ela ion: T:= (x h),H:= ho , equi alen ly, x h := −1(T),h:= H
o T∈(0,K),H∈(0,T). The co espondence allows us o use some esul s
ob ained by AlSha awi and Rhouma (2009) on he exis ence o (s able o un-
s able) 2-cycles and o desc ibe he bi u ca ions o F2
TCC. Fo non-mono one
popula ion maps like he Ricke map, he e can be wo b eak poin s in he ha -
es ule (89), and he e is no co espondence be ween (89) and (91). Tha is,
5.3 p ecau iona y h eshold cons an ca ch (p cc)115
he wo ha es s a egies could di e quali a i ely in hei dynamics. As he
ha es ule in (91) e e s o a measu emen o popula ion size ha is u he
in he pas , his in oduces a ime lag ha , o o e compensa o y popula ion
maps, could lead o delayed densi y-dependen e ec s which a e known o
change dynamics quan i a i ely and quali a i ely (F anco and Hilke , 2014).
5.3 p ecau iona y h eshold cons an ca ch (p cc)
We o ganize he sec ion as ollows: Subsec ion 5.3.1is de o ed o he s udy o
ixed poin s. We s a gi ing he loca ion o posi i e equilib ia depending on
he ele an pa ame e s. Then we p o ide s abili y esul s o he posi i e ixed
poin s: while o compensa o y models all solu ions con e ge o an equilib-
ium, in he o e compensa o y case he e a e s abili y swi ches and he global
pic u e is mo e complica ed; we gi e some gene al esul s o global s abili y
and s udy in mo e de ail he Ricke map, which is a p o o ype o disc e e
popula ion models, especially in he con ex o ishe ies (Quinn and De iso,
1999; Ricke , 1954). Finally, we desc ibe all posible bi u ca ions o ixed poin s
(smoo h and bo de -collision bi u ca ions). In Subsec ion 5.3.2, we ocus on a
pa icula egion o he pa ame e plane o which chaos and essen ial a ac-
ion can occu . We ecall ha he la e means ha an equilib ium is no glob-
ally a ac ing, bu solu ions con e ge o i wi h p obabili y one. The ob ained
esul s allow us o de e mine some bounda y-collision bi u ca ions. In Subsec-
ion 5.3.3, we add ess wo case s udies: a simple compensa o y model, whe e
only bi u ca ions o ixed poin s appea , and an o e compensa o y model ha
exhibi s iche dynamics; in bo h cases nume ical bi u ca ion diag ams help
o unde s and he in luence o he pa ame e s. In he o e compensa o y case
we pay special a en ion o bi u ca ions o 2-cycles, bis abili y egions and he
in luence o la b anches on he dynamics.
5.3.1Fixed poin s: loca ion, s abili y and local bi u ca ions
In his subec ion, we s udy he ixed poin s o FPTCC depending on he pa am-
e e alues Hand T. We i s ocus on he numbe o posi i e ixed poin s and
hei loca ion; secondly we s udy hei s abili y p ope ies; and hi dly we de-
sc ibe he local bi u ca ions o ixed poin s, ha is, we de e mine he c i ical
alues o he pa ame e s o which ixed poin s a e c ea ed o des oyed, o
s abili y swi ches can occu . We no ice ha exis ence and localiza ion esul s
hold o bo h compensa o y and o e compensa o y models, while he s abili y
ones depend on he ype o s ock- ec ui men ela ionship.
We ecall ha a consequence o condi ions (A1)-(A2), he Mean Value The-
o em and he conca i y o in (0,xc)is he exis ence o a unique poin ˜xin
(0,min{K,xc})such ha 0(˜x) = 1. The poin ˜xplays an impo an ole in he
s udy o ixed poin s.
116 combina ions o cc and h ha es ing s a egies
5.3.1.1Exis ence and localiza ion o posi i e ixed poin s
The ollowing esul p o ides he numbe o posi i e ixed poin s o he map
FPTCC de ined in (88). We assume ha 0 < T < sup{ (x) : x > 0}and H > 0.
P oposi ion 5.3.1.Assume ha (A1)-(A2)hold. Deno e by ˜x he unique solu ion
o 0(x) = 1and by x∗
−,x∗
+(0 < x∗
−⩽˜x⩽x∗
+< K) he posi i e ixed poin s o
g(x) = (x) − H, when hey exis . The ollowing asse ions hold:
1. I T⩾K, hen Kis he unique posi i e equilib ium o (88).
2. I ˜x⩽T < K, hen FPTCC has a unique posi i e ixed poin , which is Ti
H⩾ (T) − T; o x∗
+∈(T,K)i H < (T) − T.
3. I 0 < T < ˜x, hen:
a) FPTCC has a unique posi i e ixed poin x∗
+∈(˜x,K)i H < (T) − T.
b) FPTCC has h ee posi i e ixed poin s (Tand he wo ixed poin s o g) i
(T) − T < H < (˜x) − ˜x.
c) Tis he unique posi i e ixed poin o FPTCC i H > (˜x) − ˜x.
d) FPTCC has wo posi i e ixed poin s i ei he H= (T) − T(Tand x∗
+) o
H= (˜x) − ˜x(Tand ˜x).
P oo . In iew o (88), Kis a ixed poin o FPTCC i and only i K= (K)⩽T.
Mo eo e , i K⩽T, hen Kis he only posi i e ixed poin o FPTCC because
FPTCC(x)⩾min{ (x),T}> x i x < K, and FPTCC(x)⩽ (x)< x i x > K.
I T < K, hen he e a e wo possibili ies o he posi i e ixed poin s o FPTCC:
he h eshold Tand he posi i e equilib ia o g.
I is ob ious om he de ini ion o FPTCC ha Tis a ixed poin i and only i
(T)⩽T+H, ha is, H⩾ (T) − T. The condi ion T < (T)holds since T < K.
Since g0(x) = 0(x)and g00(x) = 00(x),gcan ha e a mos wo posi i e ixed
poin s x∗
−⩽x∗
+. Then, by he Mean Value Theo em, x∗
−⩽˜x⩽x∗
+, whe e ˜xis he
only poin o which 0(˜x) = 1. I also ollows ha T < x∗
−⩽x∗
+< K, because
x∗
−=FPTCC(x∗
−) = g(x∗
−) = (x∗
−) − H > T, and g(x) = (x) − H⩽x−H < x o
all x⩾K.
Now, s a emen s 2and 3 ollow easily. We include he p oo o 2and omi he
de ails o he o he asse ion since i is analogous, so assume ha ˜x⩽T < K.
I H⩾ (T) − T, hen he h eshold Tis he unique equilib ium o FPTCC
because g(T) = (T) − H⩽Tand, o T⩾xc, i ollows om g0(x)< 1 o all
x > T; while o T < xc, i ollows om g0(x)> 0,x < xcand g0(x)< 0,x > xc.
See Figu e 14 as an illus a ion o he p oo .
I H< (T) − T, hen FPTCC has a posi i e ixed poin x∗
+∈(T,K)because
g(T) = (T) − H > T and g(K) = (K) − H=K−H < K. This ixed poin
is unique because, i he e we e wo ixed poin s x∗
−< x∗
+, hen T < x∗
−and
g(x)< x o all x < x∗
−would imply ha g(T)< T, a con adic ion.
5.3 p ecau iona y h eshold cons an ca ch (p cc)117
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(A) (B)
T T
Figu e 14: Illus a ion o he piecewise-smoo h con inuous map FPTCC (blue solid
line). We also plo he line y=x( ed, dashed) and he Ricke map
(x) = x/(1+x)(black, dashed). (A) : =2.6,T=0.7∈(xc,K)≈
(0.3846,1),H=1 > (T) − T≈0.827.(B) : =1.5,T=0.4∈(˜x,xc)≈
(0.3969,0.6667),H=0.6> (T) − T≈0.5838.
Figu e 15 illus a es he numbe o ixed poin s in he pa ame e plane (H,T).
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Ha es ing quo a, H
Th eshold, T
1 ixed poin K
1 ixed poin T
1 ixed poin
x∗
+∈(T,K)
3equilib ia
˜x
Figu e 15: Numbe o posi i e equilib ia o FPTCC wi h (x) = xe2.6(1−x). The e a e
wo posi i e ixed poin s o he pa ame e alues in he bounda ies colo ed
in ed: H= (T) − T,0 < T < ˜x; and H= (˜x) − ˜x,0 < T < ˜x. The e is only
one posi i e ixed poin o pa ame e s a he bounda ies colo ed in blue:
H= (T) − T, ˜x < T < 1; and T=1.
5.3.1.2S abili y o posi i e ixed poin s
This sec ion is de o ed o s udy he s abili y p ope ies o he posi i e ixed
poin s o FPTCC. I is wo h men ioning ha , o a map sa is ying condi ions
(A1)-(A2),0is a ixed poin o FPTCC, and i is always uns able.
We i s conside he case T⩾K, o which we ob ain a global s abili y esul
i is a gene al map sa is ying condi ion (A1). The p oo ollows he one o he
analogous esul o p opo ional h eshold ha es ing (Hilke and Liz, 2019,
P oposi ion A.3).
