Resea ch A icle
On an Ex ended Time-Va ying Be e on–Hol Equa ion Subjec o
Ha es ing Moni o ing and Popula ion Excess Penal y
Manuel De la Sen ,
1
San iago Alonso-Quesada ,
1
Asie Ibeas ,
2
and Ai o J. Ga ido
3
1
Ins i u e o Resea ch and De elopmen o P ocesses, Depa men o Elec ici y and Elec onics, Facul y o Science and Technology,
Uni e si y o he Basque Coun y (UPV/EHU), Leioa 48940, Bizkaia, Spain
2
Depa men o Telecommunica ions and Sys ems Enginee ing, Uni e si a Au `
onoma de Ba celona, UAB 08193,
Ba celona, Spain
3
Depa men o Au oma ic Con ol and Sys ems, Ins i u e o Resea ch and De elopmen o P ocesses,
Facul y o Enginee ing o Bilbao, Uni e si y o he Basque Coun y (UPV/EHU), Po. Ra ael Mo eno, 3, Bilbao 48013, Spain
Co espondence should be add essed o Manuel De la Sen; [email p o ec ed]
Recei ed 22 Decembe 2022; Re ised 15 Ma ch 2023; Accep ed 4 Ap il 2023; Published 28 Ap il 2023
Academic Edi o : Ewa Pawluszewicz
Copy igh ©2023 Manuel De la Sen e al. Tis is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion
License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
Tis pape conside s a mo e gene al e en ually ime- a ying Be e on–Hol equa ion o species e olu ion which can include
a ha es ing ac ion and a penal y o o e popula ion numbe s. Te ha es ing ac ion may be posi i e ( ypically consis ing o
hun ing o shing) o nega i e which e e s o epopula ion wi hin he en i onmen . One conside s also a penal y o quad a ic
ype on he o e popula ion and he in oduc ion o a e m ela ed o Allee e ec o ake accoun o small le els o popula ion. Te
in insic g ow h a e is assumed ei he o exceed uni y o o be unde uni y. In he second case, he ex inc ion poin is a locally
s able a ac o while he o he posi i e equilib ium poin is uns able con a ily o he commonly s udied case o in insic g ow h
a e exceeding uni y whe e he abo e oles a e in e ed. Tis consequence implies ha he ex inc ion poin is also globally
asymp o ically s able o any gi en ni e ini ial condi ion. In he case when he e en ual o e popula ion is penalized wi h
a su cien ly la ge coe cien which exceeds a p esc ibed h eshold, o quan i y such an excess, only a globally asymp o ically
s able ex inc ion a ac o is p esen and no o he posi i e equilib ium poin s exis . In he case o a posi i e mode a e quad a ic
e alua ion e m o such an o e popula ion, one o wo posi i e equilib ium poin s coexis wi h he ex inc ion one. Te smalle
one is uns able con a ily o he ex inc ion equilib ium which is locally asymp o ically s able. I i exis s a second la ges posi i e
equilib ium poin , being dis inc o he abo e-gi en one, hen i can be uns able o locally s able depending on he pa ame-
e iza ion. Also, some me hods o moni o ing he popula ion e olu ion h ough con ol laws on he ha es ing ac ion a e
discussed.
1. In oduc ion
Be e on–Hol equa ion is an use ul disc e e equa ion o
modelling he e olu ion o species which ep oduce by eggs
such as bi ds, shes and insec s [1]. I can be conside ed he
coun e pa o he logis ic equa ion in he Ve huls ´s
con inuous- ime model. Te basic Be e on–Hol model is
pa ame e ized by wo posi i e sequences, namely, he ca -
ying capaci y o he en i onmen which depends on e-
sou ces a ailabili y, empe a u e, humidi y, e c., and he
in insic g ow h a e which is associa ed wi h he species
ep oduc ion capabili y, he su i o ship chance e c. Te
in insic g ow h a e has ypically o exceed uni y o a oid
ex inc ion. In he mos gene al case, hose pa ame e s can be
changed o sequences o desc ibe po en ial di e en be-
ha io s o he popula ion e olu ion in di e en pe iods, o
ins ance, seasonali y. Te e a e o he wo ypical pa ame e s
o be e en ually conside ed which gene alize he model such
as he independen consump ion which desc ibes e-
c ui men a ia ions due o un o eseen dis u bances and
e en ual epopula ion o in e change o popula ion wi h
neighbou en i onmen s and he ha es ing p ocess
Hindawi
Disc e e Dynamics in Na u e and Socie y
Volume 2023, A icle ID 5052799, 21 pages
h ps://doi.o g/10.1155/2023/5052799
associa ed wi h shing o hun ing and which depends on
egula ion based on he a ailable spawning s ock and
o eseen ec ui men . Te basic ime-in a ian Be -
e on–Hol equa ion has wo equilib ium poin s which a e
he ex inc ion poin , which is locally uns able, and he
ca ying capaci y le el which is globally asymp o ically
s able. Te so-called Cushing–Henson conjec u e es ab-
lished ha , i he equa ion is modelled by pe iodic pa-
ame e izing sequences o ca ying capaci ies and in insic
g ow h a es, hen he a e aged pe iodic sequence o pop-
ula ion lies below he a e age o he co esponding a e age
o he ca ying capaci ies. Te conjec u e has been igo ously
p o ed o be ue by S e ic [2]. Some ex ensions o he basic
model conce ning he Cushing–Henson conjec u es o he
Be e on–Hol q-di e ence equa ion ha e been discussed in
[3]. A con ol heo y poin o iew on he Be e on–Hol
equa ion has been adop ed and discussed in [4–6] while
gi ing a design p ocedu e o he en i onmen ca ying
capaci y o moni o ing he sui able sequence o alues o
ollow o he popula ion e olu ion. Te applicabili y o he
p oposed me hod is claimed o semiopen en i onmen s
such like ce ain she ies. I is discussed in [7] how, in
p ac ice, he in insic g ow h a e can be dependen on he
en i onmen ca ying capaci y. Also, i is discussed in [8] an
impulsi e ex ended compe i ion Be e on–Hol model be-
ween species om he s abili y poin o iew. Te use ulness
o he Be e on–Hol and o he ma hema ical models in
ma i ime biology is desc ibed in [9]. In [10–13], he ha -
es ing ac ion is in es iga ed in an ex ended Be e on–Hol
model. No mally, ha es ing e e s o shing o hun ing
which is subjec ed o au ho i ies egula ion bu i can also be
o al o pa ially illegal while associa ed o u i e un-
con olled ac ions. See also some e e ences he ein and
[6, 8]. O he ela ed s udies conside al e na i e gene al-
iza ions conce ned wi h pe iodic beha io s, associa ed, o
ins ance, o seasonali y [14, 15], global dynamics analysis o
some ex ended equa ion e sions [16], p esence o bi-
u ca ions [17, 18] o esonances [19], and pe u ba ions o
he basic model. See, o ins ance [18]. On he o he hand, an
ex ended Be e on model on isola ed ime scales is analyzed
in de ail in [20]. Also, an ex ension o he Be e on–Hol
model including disc e e delays in he e olu ion dynamics
has been in es iga ed in [21]. A Be e on–Hol model ex-
ension including disc e e delays in he e olu ion dynamics
has been in es iga ed in [21]. On he o he hand, i can be
poin ed ou ha Be e on–Hol -based models a e used also
by biologis s when moni o ing shing s ock a ailabili y and
shing mig a ions o e alua e he ecommended maximum
numbe o cap u es (o ecommended ha es ing ac ion) o
a oid he en i onmen deg ada ion and species ex inc ion.
See, o ins ance [22, 23], and some o he e e ences he ein.
In his pape , we ocus on a gene alized Be e on–Hol
equa ion which assumes a quad a ic- ype penal y o he
popula ion excess desc ibing he po en ial in e nal com-
pe ence be ween he indi iduals o ood, e uge, e c. Te
ha es ing ac ion is conside ed join ly wi h e en ually
p esen independen consump ion i necessa y. I is seen ha
he p esence o such a e m can ansla e in o he p esence o
wo o he equilib ium poin s. Te pape also designs species
e olu ion con ol laws by moni o ing he ha es ing ac ion
and he in uence in he esul s o conside ing a modelling
unc ion o Allee’s e ec which makes di cul g owing o
e en can cause ex inc ion o small numbe s o ep oduc i e
indi iduals.
Te pape is o ganized as ollows. Sec ion 2 deals wi h
he equilib ium poin s in he p esence and absence o
ha es ing ac ion, conside ed oge he wi h e en ual in-
dependen consump ion, in he case when he in insic
g ow h a e elemen s exceed uni y. Te ha es ing se-
quence can ha e posi i e, nega i e o null elemen s. Te
local asymp o ic s abili y o each easible ( ha is being eal
and non-nega i e) equilib ium poin s is cha ac e ized in
he case when he pa ame e izing sequences con e ge o
limi s. Sec ion 3 de elops wo me hods o de i e con ol
laws o he ha es ing ac ions again i he in insic g ow h
a e exceeds uni y o all ime. Te s p oposed me hod is
based on he con e gence o he solu ion sequence o he
popula ion o a p esc ibed a ge ed equilib ium poin o he
popula ion alue by choosing he ha es ing sequence.
Classical c i e ia o con e gence o sequences, such as
D’Alembe , Cauchy, and Raabe c i e ia, a e in ol ed in he
espec i e moni o ing ules o he ha es ing ac ion. Te
second me hod elies on a sample- o-sample moni o ing o
he solu ion sequence o a ge a p esc ibed e olu ion
pa e n by designing he ha es ing sequence. Nex , Sec ion
4 elies on in oducing Allee’s e ec o modi y he basic
Be e on–Hol equa ion o desc ibe he si ua ion a ising
unde small numbe s o indi iduals which make di cul
he ep oduc i e ac ion and can lead o ex inc ion e en he
in insic g ow h a e exceeds uni y o all ime. Te
equilib ium poin s, hei s abili y condi ions as well as
ex inc ion condi ions a e in es iga ed i he in insic
g ow h a e exceeds uni y o all ime. Te second pa o
his sec ion p oposes a penal y e m in he Be e on–Hol
equa ion o high le els o popula ion in he absence o
ha es ing. Te esul ing equilib ium poin s and hei
s abili y issues a e also in es iga ed i he in insic g ow h
a e exceeds uni y o all ime. On he o he hand, Sec ion 5
elies on ex inc ion condi ions and he local asymp o ic
s abili y o he ex inc ion equilib ium poin i he in insic
g ow h sequence has elemen s being less han uni y in he
absence o ha es ing by conside ing he modi ed model
wi h quad a ic penal y ime o high le els o popula ion.
Te local asymp o ic s abili y o he posi i e equilib ium
poin s is also in es iga ed. Sec ion 6 is de o ed o discuss
some nume ical examples and, nally, some conclusions
end he pape . In he ollowing, he subsequen no a ion is
used:
Z0+and Z0−deno e, espec i ely, he se s o non-
nega i e and nonposi i e in ege numbe s
Z+and Z−deno e, espec i ely, he se s o posi i e and
nega i e in ege numbe s
R0+and R0−deno e, espec i ely, he se s o non-
nega i e and nonposi i e eal numbe s
R+and R−deno e, espec i ely, he se s o posi i e and
nega i e eal numbe s
2Disc e e Dynamics in Na u e and Socie y
2. Equilib ium Poin s and Thei Local
Asymp o ic S abili y unde Posi i e, Null, o
Nega i e Ha es ing
“Ha es ing” is e e ed o shing and hun ing subjec o
adminis a i e egula ion which depends on he species
popula ion. On he o he hand, “independen consump ion”
e e s o posi i e o nega i e supplies o ex a popula ions
due o mig a ions om ou side o he en i onmen unde
conside a ion [10, 13, 14]. In he sequel, we conside bo h
e ec s oge he in eg a ed in he same addi i e pe u ba ion
sequence o ha o he s anda d popula ion e olu ion se-
quence. Conside he ollowing, in gene al, ime- a ying
Be e on–Hol equa ion subjec o ha es ing ac ion e en-
ually combined wi h independen consump ion:
yn+1�anKnyn
Kn+an−1
yn−hn,∀n∈Z0+,(1)
wi h ini ial condi ion y0≥0, whe e an
∞
n�0is he in insic
g ow h a e o he species and Kn
∞
0is he en i onmen
ca ying capaci y sequence, bo h being posi i e eal se-
quences. I a cons ain yn+1≥zn+1≥0; ∀n∈Z0+is p e xed
o some sequence zn
∞
n�0⊂R0+so ha he ha es ing se-
quence hn
∞
n�0has o ul l:
hn∈−∞,anKnyn
Kn+an−1
yn−zn+1
,∀n∈Z0+.(2)
Te ha es ing sequence de nes he popula ion amoun
which is no ela ed o he dynamics e olu ion wi hin he
en i onmen because o na u al ep oduc ion and dead
conce ns. I is ela ed o an inc ease o dec ease o in-
di iduals due o popula ion ux ei he om o o he habi a
plus e en ual dec ease o popula ion due o hun ing o
shing. In his way, he sequence can ake nega i e alues a
a pa icula sampling ins an because o he sign in (1), his
si ua ion will co espond o an inc ease o he amoun s o
indi iduals), posi i e alues ( ha is, dec ease o popula ion),
o ze o ( ha is, he popula ion is jus modi ed by he na u al
ep oduc ion and dead wi hin he conside ed habi a ).
Acco ding o ha philosophy, he ha es ing is conside ed in
his pape as he e en ual combina ion o an e en ual a-
di ional ha es ing ( ha is, hun ing/ shing) and e en ual
mig a ions in bo h senses om o o he habi a unde s udy.
Also, he hun ing o shing includes, in gene al, legal o
illegal ac ions (poaching).
I u ns ou ha any equilib ium poin needs o be eal
non-nega i e in o de o be easible ( ha is, eal and pos-
i i e) as i is add essed in he subsequen esul .
Theo em 1. Assume ha an
∞
0(⊂[1, a]) ⟶a,Kn
∞
0
(⊂[0, K]) ⟶K,zn
∞
0(⊂[0, z]) ⟶zand hn
∞
0⟶h
wi h hn∈(− ∞,(anKnyn)/(Kn+(an−1)yn)− zn+1];
∀n∈Z0+. Ten, he solu ion o (1) has:
(i) A eal posi i e equilib ium poin y�K>0and
a null equilib ium poin (ex inc ion) a y0�0i
h�0.
(ii) I h≥K hen he e is no nonex inc ion easible
equilib ium poin and he nonex inc ion equilib ium
poin s a e easible i h<K.
(iii) I h≠0 hen he po en ial nonex inc ion equilib ium
poin s y1>0and y2≥y1>0a e gi en by
y1,2�(a−1)(K−h)∓�������������������������
(a−1)2(K−h)2−4(a−1)Kh
2(a−1),
(3)
which a e eal i and only i h∈((− ∞,((a+1−
2��
a
√)/(a−1))K)]∪[((a+1+2��
a
√)/(a−1))K, +
∞). Te equilib ium poin y2is easible i and only i
h∈[− ∞, K]which es ic s he abo e-gi en inequali y
o ealness. Also, he equilib ium poin y1is easible i
and only i h∈[0, K]. I h�K(a+1∓2��
a
√)/(a−1)
hen y1�y2>0. In e ms o he in insic g ow h a e,
he nonex inc ion equilib ium poin s y1and y2a e bo h
easible i h∈[0, K]and (a−1)(K−h)2≥4Kh,
equi alen ly, i a≥((K+h)/(K−h))2(a>1i
h�0). Also, y1is no easible o h<0and y2is easible
o h<0i espec i e o a(>1)and K(>0).
P oo . No e di ec ly ha he ex inc ion le el y0�0 is an
equilib ium poin . Also, by eplacing in (1) he limi s o he
a ious sequences, one ge s a single oo y�Ki h�0 and,
i h≠0, hen
(a−1)y2+(a−1)(h−K)y+Kh �0.(4)
Tus, since a>1, i h≥K hen he e is no eal non-
nega i e solu ion o (1) since (4) ails o y>0 and h≥K.
I 0 <h<K, o i h<0, hen he oo s o (2) a e
y1�(a−1)(K−h)− �������������������������
(a−1)2(K−h)2−4(a−1)Kh
2(a−1),
y2�(a−1)(K−h)+ �������������������������
(a−1)2(K−h)2−4(a−1)Kh
2(a−1).
(5)
P ope ies [(i)-(ii)] ha e been p o ed. On he o he
hand, he nonex inc ion equilib ium poin s y1and y2a e
bo h easible o h∈[0, K)i (a−1)(K−h)2≥4Kh,
equi alen ly, i a≥(K+h)/(K−h)2(a>1 i h�0). Also, y1
is no easible o h<0 om (5) and y2is easible, also om
(5), o h<0 i espec i e o a(>1)and K(>0).
Disc e e Dynamics in Na u e and Socie y 3
No e ha , in o de o he oo s o (4) o be eal, since
a>1, i is needed ha θ(h)≥0, whe e
θ(h) �(a−1)(K−h)2−4Kh �(a−1)h2+(a−1)K2−2(a−1)Kh −4Kh
�(a−1)h2+(a−1)K2−2aKh +2Kh −4Kh �(a−1)h2+(a−1)K2−2Kh(a+1).(6)
No e ha , y1,2a e eal i and only i he ze os o θ(h), ha
is, h1,2�K(a+1∓2��
a
√)/(a−1), a e non-nega i e eal since
a>1. Since θ(h)is a con ex pa abola, θ(h)≥0, and bo h
equilib ium poin s y1and y2a e eal (and hey can be
posi i e) i and only i h∈(− ∞, K(a+1−2��
a
√/
(a−1)]∪[K(a+1+2��
a
√)/(a−1),+∞). Con a ily, i
h∈(K((a+1−2��
a
√))/(a−1)), K(a+1+2��
a
√)/(a−1))
hen he ze os a e no eal and he nonex inc ion equilib ium
poin s y1,2ne e exis . Since, hei easibili y implies ha
h≤K hen he equilib ium poin y2is easible i and only i
h∈[− ∞, K]. Since (a+1−2��
a
√)/(a−1)<1 o a>1. Also,
he equilib ium poin y1is easible i and only i
h∈[0, K(a+1−2��
a
√)/(a−1)]. I is ob ious ha h�h1,2�
K(a+1∓2��
a
√)/(a−1) hen y1�y2. P ope y (iii) is p o ed.
Te use o he in e se sequence o ha o a he pop-
ula ion e olu ion sequence is o in e es o de i e easily some
in e es ing esul s conce ning he s abili y and he asymp-
o ic boundedness o he solu ion as i is add essed in he
subsequen esul : □
Theo em 2. De ne he in e se sequence o he solu ion o (1)
as xn�y−1
n;∀n∈Z0+. Te ollowing p ope ies hold:
(i) Te in e se sequence xn
∞
n�0o he solu ion yn
∞
n�0is
gi en by he disc e e equa ion:
xn+1�μnxn+]n+hI
n;∀n∈Z0+,(7)
whe e μn�a−1
n;]n� (1−μn)K−1
n� (an−1)a−1
nK−1
n,
subjec o hn∈(− ∞,(anKnyn)/(Kn+(an−1)yn)];
∀n∈Z0+, and hI
nis ze o i hn�0 o any n∈Z0+,
wi h
hn�anKnhI
n
2 1 −a−1
n
+hI
n+a−1
nxn
Kn
xn+K−1
nan+a−1
n−2
+hI
nan−1
;∀n∈Z0+,(8)
o x0�y−1
0>0. Te solu ion is equi alen ly
exp essed om gi en ini ial condi ions as ollows:
xn+1�
n
i�0
μi
⎞
⎠x0+n
i�0
n
j�i+1
μj
⎞
⎠]i+hI
i
.
⎛
⎝
⎛
⎝
(9)
(ii) xn
∞
n�0is bounded, equi alen ly, yn
∞
n�0does no
anish nei he a any sample no asymp o ically
(and hen he popula ion does no ex inguish ei-
he in ni e ime o asymp o ically) i
lim supn⟶∞ n
i�0(n
j�i+1[μj])(]i+hI
i)<∞. In
pa icula , i an
∞
n�0⊂[a, a]and
lim in n⟶∞an≥a>1 hen
xn≤−
a
−nx0+1−−
a
−n
1−asup
0≤i≤n−1
]i+hI
i
;∀n∈Z+,
(10)
is bounded o any gi en ni e x0≥0 o all n∈Z0+
i sup0≤i≤n−1|]i+hI
i|<+∞. Fo any ni e x0>0,
so ha y0�x−1
0, he sequence yn�x−1
n
∞
n�0
sa is es:
yn≥1−−
a
−
a
−n1−−
a
x0+1−−
a
−n
sup
0≤i≤n−1
]i+hI
i
>0; ∀n∈Z+.
(11)
(iii) De ne bn�1−μn� (an−1)/anby assuming ha
an
∞
n�0⊂(1,∞]subjec o ∞
n�0bn�∞
n�0(an−1)/
an�∞. Assume ha |hI
n+(an−1)/(Knan)|
≤ε(an−1)/an o any in ege n≥n0, some n0∈Z0+
and some ε∈R+. Ten, lim supn⟶∞xn
≤ε⇔lim in n⟶∞yn≥ε−1. Te cons ain
|hI
n+(an−1)/Knan|≤ε(an−1)/anis sa is ed unde
any o he subsequen s ipula ions o each n∈Z0+:
(1) 0≤hI
n≤(ε−K−1
n)(an−1)/an equi ing ha he
ha es ing sequence hn≥0and he ca ying ca-
paci y Kn≥1/ε
(2) hI
n<0and (an−1)/anK−1
n≥|hI
n|≥(K−1
n−ε)
(an−1)/an≤hI
n<0 equi ing ha hn<0and
Kn<1/ε
(3) hI
n<0and (an−1)/Knan<|hI
n|≤(an−1)/an
(ε+K−1
n) equi ing ha hn<0
(i ) Te ex inc ion equilib ium poin y0�0is uns able.
Te wo posi i e equilib ium poin s y1,2in (5) a ising
4Disc e e Dynamics in Na u e and Socie y
when he pa ame ical sequences an
∞
n�0⟶a,
Kn
∞
n�0⟶K,hn
∞
n�0⟶h ul l he ollowing
heo ies:
(a) y2(>y1)is join ly easible and locally asymp-
o ically s able i −∞<h<(( ��
a
√−1)/(��
a
√+
1))K
(b) I y1(<y2)is easible hen i is no locally as-
ymp o ically s able
(c) I h� (( ��
a
√−1)/(��
a
√+1))K hen he unique
nonex inc ion equilib ium poin y1�y2� (K
−h)/2�K/(��
a
√+1)is join ly easible and lo-
cally asymp o ically s able
P oo . One ob ains (7) di ec ly om he ollowing equi -
alen exp ession o (1):
xn+1�1
yn+1�1
anKnyn/Kn+an−1
yn
−hn�Kn+an−1
yn
anKnyn−hnKn+an−1
yn
�a−1
nxn−a−1
n−1
K−1
n+hI
n;∀n∈Z0+,
(12)
subjec o hn∈(− ∞, anKnyn/(Kn+(an−1)yn)];∀n∈Z0+
o keeping he sample- o-sample non-nega i i y o yn
∞
n�0,
whe e xn�y−1
n;∀n∈Z0+, and
hI
n�xn+1−a−1
nxn+a−1
n−1
K−1
n�Knxn+an−1
anKn−gn−a−1
nxn+a−1
n−1
K−1
n;∀n∈Z0+,(13)
wi h gn�hn(Knxn+an−1);∀n∈Z0+. One ge s om (13)
ha
K−1
n+hI
n−a−1
nK−1
n−xn
gn�anKnhI
n;∀n∈Z0+,
(14)
which leads o
gn�anKnhI
n
K−1
n+hI
n−a−1
nK−1
n−xn
�hnKnxn+an−1
;∀n∈Z0+,
(15)
so ha
hn�anKnhI
n
K−1
n+hI
n−a−1
nK−1
n−xn
Knxn+an−1
�anKnhI
n
2xn+K−1
nan+a−1
n−2
+hI
nKnxn+an−1
−2a−1
nxn+a−1
nKnx2
n
�anKnhI
n
2 1 −a−1
n
+hI
n+a−1
nxn
Kn
xn+K−1
nan+a−1
n−2
+hI
nan−1
;∀n∈Z0+,
(16)
which leads di ec ly o (8). Equa ion (9) ollows di ec ly
om ecu si e calcula ions wi h (7). P ope y (i) has been
p o ed. P ope y (ii) is a di ec consequence o P ope y (i)
since n+1
i�0[μi]<n
i�0[μi]<1; ∀n∈Z0+and limn⟶ ∞n
i�0
[μi] � 0 since lim in n⟶∞an≥a>1.
To p o e P ope y (iii), we ew i e an uppe -bounding
exp ession o (7) as
xn+1�1−bn
xn+bn
Kn+hI
n≤1−bn
xn+bn
Kn+hI
n
≤1−bn
xn+εnbn;∀n∈Z0+,(17)
Disc e e Dynamics in Na u e and Socie y 5
since an
∞
n�0⊂[1,∞]implies ha bn�1−μn� (an−1)/
an∈[0,1];∀n∈Z0+subjec o ∞
n�0bn�∞
n�0(an−1)/an�
∞. Since i is also assumed ha |hI
n+bn/Kn|≤εbn�
ε(an−1)/an;∀n∈Z0+, his cons ain is achie ed i
|hI
n+bn/Kn| � |hI
n+(an−1)/Knan|≤εnbn≤ε(an−1)/an;∀n
(≥n0)∈Z0+, and some n0∈Z0+, whe e εn
∞
n�0⊂[0,ε] o
any in ege n≥n0, some n0∈Z0+and some ε∈R+ hen i
ollows om [Lemma 1.2 (i), [24]] ha 0 ≤lim sup
n⟶∞ xn≤ε.
Te s ipula ion 1 is go om he cons ain s hI
n≥0 (implying
ha hn≥0) and hen |hI
n+(an−1)/Knan| � |hI
n+(an−1)/
Knan|≤ε(an−1)/an,∀n∈Z0+.
Te s ipula ion 2 ollows o hI
n<0 (implying ha hn<0)
and |hi
n|≤bnK−1
nso ha he subsequen cons ain holds
|hI
n+(an−1)/Knan| � (an−1)/Knan− |hI
n|;∀n∈Z0+.
Te s ipula ion 3 ollows o hI
n<0 and |hi
n|≤bnK−1
nso
ha he ollowing cons ain holds |hI
n+(an−1)/Knan|
� |hI
n|− (an−1)/Knan;∀n∈Z0+.
P ope y (iii) has been p o ed. P ope y (i ) ollows
om a local pe u ba ion analysis. I is assumed ha he
in a ian equa ion (1) pe u bed om any equilib ium poin
yas yn�y+δyn;∀n∈Z0+. Te linea ized pe u ba ion
ansmi ed o he nex sample is y+δyn+1�aK/K/((y+
δyn)+a−1)− h;∀n∈Z0+which leads o |δyn+1|≤K2a/
(K+y(a−1))2|δyn|+o(|δyn|);∀n∈Z0+and, o su -
cien ly small |δyn|,|δyn+1/δyn|<1, ∀n∈Z0+, so ha yis
locally asymp o ically s able, i and only i K��
a
√/(K+
y(a−1))<1. Equi alen ly, i and only i y>K(��
a
√
−1)/(a−1) � K/(��
a
√+1). Con a ily, he equilib ium
poin yis no locally asymp o ically s able ( ha is, ei he
c i ically s able o uns able) i and only i
K��
a
√/(K+y(a−1))≥1, ha is, i and only i
y≤K/(��
a
√+1). In pa icula , i is uns able i
K��
a
√/(K+y(a−1))>1, ha is, i y<K/(��
a
√+1). Te
local asymp o ic s abili y cons ain ails and he ins abili y
cons ain holds i y�y0�0 (ex inc ion equilib ium poin )
since a>1.
Ten, he ex inc ion equilib ium poin is uns able.
Fo add essing he local asymp o ic s abili y o he o he
wo equilib ium poin s y�y1and y�y2, p o ided hey a e
easible and dis inc ( ha is, he adicand o (5) is eal
posi i e), no e ha equilib ium poin s a e locally asymp-
o ically s able i and only i
y1,2�(a−1)(K−h)∓�������������������������
(a−1)2(K−h)2−4(a−1)Kh
2(a−1)>K
��
a
√+1.(18)
Conside ing y2unde he abo e cons ain , ha one
becomes equi alen o
�������������������������
(a−1)2(K−h)2−4(a−1)Kh
>(��
a
√−1)[2K− ( ��
a
√+1)(K−h)].(19)
Te abo e-gi en cons ain (19) holds i u> , whe e
u�(a−1) (a−1)(K−h)2−4Kh
�(a−1)a(K−h)2− (K+h)2
�α1h2+β1h+c1,
(20)
whe e
α1�(a−1)2β1� − 2K a2−1
;c1�(a−1)2K2,(21)
and
�(( ��
a
√−1)[2K− ( ��
a
√+1)(K−h)])2
�(a+1−2��
a
√)4K2+(a+1+2��
a
√)K2+h2−2Kh
−4K(��
a
√+1)(K−h)
�α2h2+β2h+c2,
(22)
whe e
α2�α1�(a−1)2;β2�2K(a−1)[2��
a
√−a−1];c2�K2a2+6a−4a��
a
√−4��
a
√+1
.(23)
6Disc e e Dynamics in Na u e and Socie y
Tus, u> , as necessa y condi ion a≥((K+h)/(K−
h))2, equi alen ly, h≤(( ��
a
√−1)/(��
a
√+1))Kin o de o
he adicand in he de ni ion o y1,2 o be non-nega i e
(equilib ium easibili y condi ion), wi h a>((K+h)/
(K−h))2i h≠K(��
a
√−1)/(��
a
√+1)and h≠0. Fo h�0,
y2�Kis locally asymp o ically s able om u> since ��
a
√+
1>��
a
√−1 holds i ially. Now (20)–(23), one ge s ha y2is
locally asymp o ically s able, ha is, u> , i and only i :
h<c1−c2
β2−β1�(a−1)2−a2+6a−4a��
a
√−4��
a
√+1
2(a−1)[2��
a
√−a−1]+2a2−1
K�(a+1)��
a
√−2a
(a−1)��
a
√K���
a
√−1
��
a
√+1K, (24)
which coincides wi h he easibili y condi ion. Tus, y2is
bo h easible and locally asymp o ically s able i and only i
−∞<h<(( ��
a
√−1)/(��
a
√+1))K.
Fo he local asymp o ic s abili y o y1, equa ion (19)
becomes modi ed as ollows:
−�������������������������
(a−1)2(K−h)2−4(a−1)Kh
>(��
a
√−1)[2K− ( ��
a
√+1)(K−h)],(25)
so ha , one ge s he wo associa ed cons ain s:
(1) (��
a
√−1)[2K− ( ��
a
√+1)(K−h)]≤0
(2) ��
u
√��������������������������
(a−1)2(K−h)2−4(a−1)Kh
<�
√� −
(��
a
√−1)[2K− ( ��
a
√+1)(K−h)]⇔u<
Te abo e-gi en s condi ion is equi alen o
h≤(( ��
a
√−1)/(��
a
√+1))Ksince a>1. Tis cons ain is he
easibili y cons ain o ha es ing also al eady needed o
y
→2. Te abo e-gi en second condi ion is equi alen o jus
o e e se he equali y in he cons ain (24), ha is,
h>(a+1)��
a
√−2a
(a−1)��
a
√K���
a
√−1
��
a
√+1K, (26)
so ha y1≠y2is bo h easible and locally asymp o ically
s able i and only i
��
a
√−1
��
a
√+1K≥h>��
a
√−1
��
a
√+1K, (27)
which is a con adic ion. Tus, y1is uns able i easible and
dis inc o y2.
I y1�y2is easible, ha is, he adicand o (5) is null so
ha h� (( ��
a
√−1)/(��
a
√+1))K, hen he equilib ium poin
is gi en by y1�y2� (K−h)/2 �K/(��
a
√+1)which sa -
is es i ially he abo e gi en local asymp o ic s abili y
condi ion y1�y2≥K/(��
a
√+1)so ha he con uen
nonex inc ion equilib ium poin
y1�y2� (K−h)/2 �K/(��
a
√+1) esul ing wi h
h� (( ��
a
√−1)/(��
a
√+1))Kis locally asymp o ically s able.
P ope y (i ) has been p o ed.
Conce ning Teo em 2(i), no e ha he denomina o in
he igh -hand-side o (8) canno be ze o a any sample since
he he alue o he sequence hn
∞
n�0is bounded by
hypo hesis. □
Rema k 1. No e ha , he admissible ha es ing sequence o
Teo em 2(iii) can be gene a ed om (8) by gene a ing
hI
n
∞
n�0as ollows by ul lling one o he s ipula ions 1–3 o
each n∈Z0+:
(a) T ough he s ipula ion 1 in he p oo o Teo em 2:
hI
n� (ε−K−1
n)(an−1)/an−σn≥0; ∀n∈Z0+, whe e
ε∈R+is chosen such ha Kn≥ε−1;∀n∈Z0+and he
sequence σn
∞
n�0is gene a ed subjec o 0 ≤σn
≤(ε−K−1
n)(an−1)/an;∀n∈Z0+. No e ha , hI
n≥0
and hn≥0; ∀n∈Z0+.
(b) T ough he s ipula ion 2 in he p oo o Teo em 2:
hI
n� (1−an)/anKn+σn, whe e 0 ≤σn≤((an−1)
/an) (2K−1
n−ε);∀n∈Z0+and ε∈R+is chosen such
ha Kn<ε−1. No e ha , hI
n<0 and hn<0.
(c) T ough he s ipula ion 3 in he p oo o Teo em 2:
hI
n� ((1−an)/an)(1/Kn+ε)+σn, whe e
0≤σn≤((an−1)/an)ε;∀n∈Z0+and ε∈R+. No e
ha , hI
n<0 and hn<0.
Rema k 2. Te local asymp o ic s abili y o he equilib ium
poin s add essed in Teo em 2 (i ) elies o he cases o
absence o ha es ing in he s eady s a e dynamics (h�0)
o in he cases o s a iona y shing/hun ing (h>0)o
s a iona y epopula ion ac ions (h<0). Tose cases co -
espond o cons an alues o he ha es ing sequence in
ni e ime o asymp o ically. In he pape , he dynamics is
globally s able i he popula ion solu ion sequence is
bounded o any gi en ni e ini ial condi ion. Tis ci -
cums ance migh be compa ible wi h he e en ha some o
he equilib ium poin s be locally uns able, s able, o c i -
ically s able i he e a e mo e han one equilib ium poin s.
An equilib ium poin is said o be globally asymp o ically
s able i i globally s able and all solu ion con e ges as-
ymp o ically o such a poin o any gi en ni e ini ial
condi ions.
In he hi d case, he la ges posi i e equilib ium poin is
la ge unde nega i e s a iona y ha es ing (ha ing
a meaning o s a iona y epopula ion and/o immig a ion o
he habi a om ou side), han he equilib ium poin K
a ising in he absence o ha es ing. In he second case, he
global s abili y condi ion leads o he conclusion ha he
la ge equilib ium poin y2is locally asymp o ically s able
and he smalle one y1is no locally asymp o ically s able
unless hey a e coinciden o a s a iona y ha es ing e o
h� (( ��
a
√−1)/(��
a
√+1))K.
Disc e e Dynamics in Na u e and Socie y 7
Te ollowing esul p o es ha , i he ha es ing ac ion
sequence has a limi h, hen a limi poin o he solu ion
canno exceed he amoun K−h.
P oposi ion 1. I an
∞
0⊂(1, a)⟶a,Kn
∞
0(⊂[0,
K]) ⟶K,hn
∞
n�0⟶hand yn
∞
n�0⟶y hen
h�0⟺(y�0)∨(y�K)
h>0⟺y<K−h
h<0⟺y>K+|h|.(28)
P oo . On ge s om (1) ha
hn�anKnyn
Kn+an−1
yn−yn+1;∀n∈Z0+.(29)
I yn
∞
n�0⟶y, one ge s by aking limi s in (29) as
n⟶ ∞ ha
h�aK
K+(a−1)y−1
y�(a−1)(K−y)
K+(a−1)yy, (30)
and, equi alen ly,
hK �(a−1)(K−y−h)y, (31)
|h|K� − hK �(a−1)(y− |h|− K)i h<0.(32)
Ten, he gi en p ope ies ollow di ec ly om (31) and .
Te ollowing esul es ablishes he boundedness o he
solu ion sequence unde bounded non-nega i e
ha es ing. □
P oposi ion 2. Assume ha an
∞
0⊂[1, a],Kn
∞
0⊂[0, K],
hn
∞
n�0(⊂R0+)I hn
∞
n�0⊂([0, anKnyn/(Kn+(an−
1)yn)]);∀n∈Z0+ hen yn
∞
0(⊂R0+)is bounded.
P oo . Assume on he con a y ha yn
∞
n�0⟶+∞.Ten,
om L´Hopi al ule o quo ien s wi h nume a o and
denomina o ending o in ni y, since he ha es ing se-
quence is non-nega i e,
lim
yn⟶ ∞n⟶ ∞ sup yn+1−anKnyn
Kn+an−1
yn
�lim
n⟶ ∞ sup yn+1−anKn
an−1
≤0,
(33)
so ha
lim sup
n⟶∞ yn+1�lim
n⟶ ∞ yn+1≤lim in
n⟶∞
anKn
an−1<+∞.
(34)
A con adic ion o he sequence yn
∞
n�0 o di e ge,
which comple es he p oo . □
3. Con ol Laws o Moni o ing he
Ha es ing Ac ion
Te s pa o his sec ion is add essed o de i e ha es ing
con ol laws based on Teo em 2(i ) gua an eeing he
con e gence o a p esc ibed equilib ium poin x∗unde he
assump ion ∞
n�0(an−1)/an�∞on he in insic g ow h
sequence an
∞
n�0. Known c i e ia o absolu e con e gence
o se ies o o con e gence o se ies o non-nega i e ele-
men s o a p esc ibed limi can be used o calcula e he
ha es ing con ol sequence hn
∞
n�0based on he p e ious
calcula ion o hI
n
∞
n�0, which e ec s he ha es ing e ec in
he in e se o he solu ion sequence, so as o sa is y Teo em
2(i ). Te las pa o he sec ion p oposes ha es ing con ol
laws which make he solu ion sequence o he popula ion
e olu ion o sample- o-sample, a he han asymp o ically,
beha e acco ding o a p esc ibed sui able pa e n.
Now, ew i e he popula ion solu ion sequence
xn�y−1
n
∞
n�0as an equilib ium pe u ba ion in he o m
xn+x∗
∞
n�0, whe e x∗is he sui ed equilib ium poin and
xn�xn−x∗;∀n∈Z0+. Tus, one can ew i e om (6) he
one-s ep ahead e olu ion o he inc emen al sequence
xnin
he o m o Lemma 1.2 (iii) o [24] as ollows:
xn+1�1−bn
xn−bnx∗+]n+hI
n
�1−bn
xn+ωn+cn;∀n∈Z0+,(35)
whe e o each n∈Z0+,
bn�1−μn�1−a−1
n�an−1
an
,(36)
ωn�]n+βn�an−1
anKn+βn,(37)
cn�hI
n−bnx∗−βn,(38)
o any βn
∞
0⊂Rchosen such ha hI
n−bnx∗≥
βn≥(1−an)/anKn;∀n∈Z0+which gua an ees ha ωn≥0
and cn≥0; ∀n∈Z0+. No e ha , bn∈[0,1];∀n∈Z0+. Su -
cien condi ions o con e gence
xn
∞
0(⊂R0+)⟶0⟺
xn
∞
0⟶x∗a e x0≥0, and
∞
n�0bn�∞
n�0
an−1
an�∞,(39)
ωn�o bn
⇔lim
n⟶ ∞
βn
an−1+1
anKn
�0,(40)
∞
n�0cn<∞.(41)
Te condi ion (40) is equi alen o limn⟶ ∞(βn− (1−
an)/anKn) � 0 and he condi ion (41) is gua an eed i
8Disc e e Dynamics in Na u e and Socie y
∞
n�0hI
n+1−an
an
x∗−K−1
n
<∞,(42)
since, om (38) and he condi ion hI
n−bnx∗≥
βn≥((1−an)/an)Kn;∀n∈Z0+, one ge s:
∞
n�0
cn�
∞
n�0
hI
n−bnx∗−βn
≤
∞
n�0
hI
n−an−1
an
x∗+an−1
anKn
�
∞
n�0
hI
n+1−an
an
x∗−K−1
n
<∞,
(43)
which induces also he u he necessa y condi ion
lim
n⟶ ∞(hI
n+((1−an)/an)(x∗−K−1
n)) � 0, since cn⟶0 as
n⟶ ∞, gua an eed in u n i
hI
n
∞
n�0⟶0 (equi alen ly, i hn
∞
n�0⟶0) and
K−1
n
∞
n�0⟶x∗, equi alen ly, i Kn
∞
n�0⟶y∗, o i
hI
n
∞
n�0⟶0 and an
∞
n�0⟶1
3.1. Ha es ing Con ol Law Based on d´ Alembe Con e -
gence C i e ion. a) ∞
n�0cn<∞ wi h cn≥0; ∀n∈Z0+is
gua an eed unde d´ Alembe con e gence c i e ion in
o de o xn
∞
n�0⟶x∗�1/y∗>0, equi alen ly,
yn
∞
n�0⟶y∗>0, i
hI
n+1−bn+1x∗−βn+1
hI
n−bnx∗−βn�cn≤c<1; ∀n∈Z0+,(44)
i hI
n≠bnx∗+βnand hI
n≥bnx∗+βn;∀n∈Z0+. Ten, since
bn� (an−1)/an;∀n∈Z0+, equa ion (44) leads o
bn+1x∗+βn+1�an+1−1
an+1
x∗+βn+1
≤hI
n+1�bn+1x∗+βn+1+cnhI
n−bnx∗−βn
�an+1−1
an+1−cn
an−1
an
x∗+βn+1+cnhI
n−βn
;∀n∈Z0+,
(45)
wi h limn⟶ ∞(βn− ((1−an)/anKn) � 0. One ge s om
(12), by using xn�y−1
n;∀n∈Z0+, ha
yn+1�x−1
n+1�anKnx−1
n−hnKn+an−1
x−1
n
Kn+an−1
x−1
n�anKn−hnKnxn+an−1
Knxn+an−1;∀n∈Z0+,(46)
so ha , equi alen ly,
xn+1�Knxn+an−1
anKn−hnKnxn+an−1
;∀n∈Z0+,(47)
which equalized o (7) in Teo em 2 (i) leads o:
hI
n�hI
nhn
�Knxn+an−1
anKn−hnKnxn+an−1
+1
an
1−an
Kn−xn
;∀n∈Z0+.
(48)
Ten, combining (45) and (48)
hI
n+1�an+1−1
an+1−cn
an−1
an
x∗+cn
Knxn+an−1
anKn−hnKnxn+an−1
+1
an
1−an
Kn−xn
+βn+1−cnβn;∀n∈Z0+.(49)
Equi alen ly,
Disc e e Dynamics in Na u e and Socie y 9
In he same way, in o de o y2 o be locally as-
ymp o ically s able, |2−a− (1−a)/Ky2|<awhich
is ul lled unde wo condi ions, namely,
(c) 0 ≤2−a− ((1−a)/K)y2<awhich holds i and
only i y2∈(2K,(2−a)/(1−a)K];
(d) 0 ≤((1−a)/K)y2− (2−a)<awhich holds i and
only i y2∈[(2−a)K/(1−a),2K/(1−a))
which a e equi alen ly combined in o he local s abili y
condi ion y2∈(2K, 2K/1 −a). Taking in o accoun ha
y2�1−a/c(1+�����������
1−4cK/1 −a
√), he condi ion y2∈[(2−
a)K/(1−a),2K/(1−a)) becomes equi alen o
2c(2−a)
(1−a)2−1
2≤1−4cK
1−a<4Kc
(1−a)2−1
2
,(87)
subjec o 0 ≤c<(1−a)/4Kand de ne pa ame e s
λc�c/a>0 (no e ha λc�0 does no need o be conside ed
since hen y2�y1�Kis s able) and λK�K/a>0. Ten, he
cons ain c<(1−a)/4K akes he o m 4λcλKa2+a−1<0
and he abo e local s abili y cons ain (87) o y2 akes he
o m (76). □
Rema k 4. No e om Teo em 4 (ii) ha any nonze o
equilib ium poin is less han he ca ying capaci y i c>0
con a ily o he case when c�0 whe e y�Kis an
equilib ium poin .
Rema k 5. No e ha , con a ily o Teo em 4(i ), i a≥1and
c�0, hen, in he absence o ha es ing, yn
∞
n�0canno be
s ic ly dec easing con e ging o ze o since, o he limi so-
lu ion sequence o be s ic ly dec easing, i is necessa y ha
(yn+1/yn) � a/(1+((a−1)/K)yn)<1 wha implies ha
yn>Kso ha yn
∞
n�0⟶0 is impossible o any gi en
posi i e ni e ini ial condi ion i an≥1; ∀n∈Z0+. Tus, he
ex inc ion equilib ium poin is uns able i an≥1; ∀n∈Z0+.
6. Nume ical Simula ions
6.1. Example 1. Tis example illus a es he esul s o Teo em
1. Te sequences an
∞
0,Kn
∞
0and zn
∞
0a e espec i ely
gene a ed by means o he ollowing di e ence equa ionsc:
an+1�ε1an+ρ1,
Kn+1�ε2Kn+ρ2,
zn+1�ε3zn+ρ3,(88)
wi h he ollowing alues o he pa ame e s:
ε1�0.9,ρ1�0.4,ε2�0.8,ρ2�200,ε3�0.75 and ρ3�100,(89)
and he ollowing ini ial condi ions:
a0�1.5, K0�500 and z0�10,(90)
In his way he condi ions o Teo em 1 abou he se-
quences an
∞
0and Kn
∞
0a e ul lled since a�limn⟶ ∞
an
�ρ1/(1−ε1) � 4 and K�limn⟶ ∞ Kn
�ρ2/(1−ε2)
�1000.
In a s simula ion, he ha es ing sequence hn
∞
0wi h
hn�5∗(0.7)nis conside ed so ha h�limn⟶ ∞ hn
�0
and hen he condi ions o Teo em 1(i) a e sa is ed. Fig-
u e 1 shows he e olu ion o he species popula ion yn
and
ha o he en i onmen ca ying capaci y Kn
i he
popula ion is ini ially y0�200. One can see ha
y�limn⟶ ∞ yn
�limn⟶ ∞ Kn
�K�1000 as Teo-
em 1(i) poin s ou .
On he o he hand, Figu e 2 displays he e olu ion o he
species popula ion and ha o he en i onmen ca ying
capaci y i he popula ion is ini ially y0�10. Again, one can
see ha y�limn⟶ ∞ yn
�limn⟶ ∞ Kn
�K�1000
al hough he ini ial popula ion is close o he equilib ium
poin y�0. Te esul s displayed in Figu es 1 and 2 illus a e
he ac ha he equilib ium poin y�0 is globally as-
ymp o ically uns able while he equilib ium poin y�Kis
globally asymp o ically s able.
In a second simula ion he same alues o (89) and (90)
a e main ained bu he sequence hn
∞
0con e ges o
a nonze o alue h. Conc e ely, he ime e olu ion o hn
∞
0is
displayed in Figu e 3 while ha o yn
and Kn
i he
popula ion is ini ially y0�145 is shown in Figu e 4.
In his example he ac ha
0≠h�limn⟶ ∞ hn
<limn⟶ ∞ Kn
�Kis obse ed so
ha he condi ions o Teo em 1(ii) and Teo em 1(iii) a e
sa is ed. In ac , he Be e on–Hol equa ion (1) possesses
wo equilib ium poin s gi en by (1). Conc e ely, such
equilib ium poin s a e y1�142.855 and y2�600. One can
see ha limn⟶ ∞ yn
�y2�600 al hough he ini ial
condi ion y0�145 is close o he equilib ium poin y1.
Such a ac illus a es ha y1is uns able while y2is globally
asymp o ically s able in he sense ha all solu ions gen-
e a ed by ni e ini ial condi ions con e ge o such an
equilib ium.
6.2. Example 2. Tis example illus a es he esul s o
Teo em 3 ela ed wi h he Allee e ec o small numbe o
indi iduals in he species. Te in insic g ow h a e and
ca ying capaci y sequences, an
∞
0and Kn
∞
0, a e gi en by
(88) wi h he same alues o he pa ame e s ε1,ε2,ρ1, and ρ2
han hose poin ed ou in (89). Mo eo e , he ha es ing
sequence hn
∞
0is ze o o all nεZ0+and he unc ion (yn)
appea ing in he modi ed Be e on-Hol equa ion (58) is
gi en by he ollowing equa ion:
yn
�yn+α1
anyn+α2
,(91)
wi h α1�0.1 and α2�1. In his way, (yn)<a−1
n o all
nεZ0+and he condi ions o Teo em 3(i) a e ul lled.
Figu e 5 shows he e olu ion o he species popula ion yn
i
he popula ion is ini ially y0�15. Figu e 6 displays he
e olu ion o he unc ion (yn)and he in e se o he se-
quence an
∞
0. In Figu e 5, one can see ha he species
popula ion con e ges o he ex inc ion as Teo em 3(i)
es ablishes since (yn)<a−1
n o all nεZ0+as i is shown in
Figu e 6.
16 Disc e e Dynamics in Na u e and Socie y
Now, he unc ion (yn)appea ing in he modi ed
Be e on–Hol equa ion (58) is gi en by he ollowing
equa ion:
yn
�αnyp
n+βn,(92)wi h αn� (0.7(an−1)/anKn)y1−p
nand βn�0.8a−1
n o all
nεZ0+. In his way, he condi ions o Teo em 3(ii) a e
ul lled. Figu e 7 shows he e olu ion o he species
5 1015202530354045500
n (days)
0
200
400
600
800
1000
yn
Kn
Figu e 1: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�200.
5 1015202530354045500
n (days)
yn
Kn
0
200
400
600
800
1000
Figu e 2: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�10.
5 1015202530354045500
n (days)
-200
-100
0
100
200
300
hn
Figu e 3: Time e olu ion o he ha es ing sequence.
10 20 30 40 500
n (days)
0
200
400
600
800
1000
yn
Kn
Figu e 4: E olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�145.
yn
0
2
4
6
8
10
12
14
16
10 20 30 40 50 60 700
n (days)
Figu e 5: Time e olu ion o he species popula ion i y0�15 and
(yn)<a−1
n.
10 20 30 40 50 60 700
n (days)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1/an
(yn)
Figu e 6: Time e olu ion o he in e se o an
∞
0and (yn)i
y0�15.
Disc e e Dynamics in Na u e and Socie y 17
popula ion yn
i he popula ion is ini ially y0�15. One
can see ha he species popula ion con e ges o he ex-
inc ion as Teo em 3(ii) es ablishes.
6.3. Example 3. Te ollowing wo examples illus a e he
esul s o P oposi ion 5 abou he modi ed Be e on–Hol
equa ion (63).
(i) Te sequences an
∞
0,Kn
∞
0and cn
∞
0a e e-
spec i ely de ned as
an�a01+0.5 sin 2π an
,(93)
Kn�K01+0.02 sin 2π Kn
,(94)
cn�c01+0.3 sin 2π cn
,(95)
o illus a ed he esul (i) o such a p oposi ion wi h
a0�1.8, a�0.01, K0�500, K�0.02, c0�0.03
and c�0.03. Figu e 8 shows he e olu ion o he
species popula ion yn
i he popula ion is ini ially
y0�100. One can see ha he species popula ion
nei he ex inguishes no inc eases in an un-
boundedness way as P oposi ion 5 (i) es ablishes.
(ii) Te sequences an
∞
0,Kn
∞
0, and cn
∞
0a e, e-
spec i ely, gene a ed by means o he ollowing
di e ence equa ions:
an+1�ε1an+ρ1,
Kn+1�ε2Kn+ρ2,
cn+1�ε3cn+ρ3,(96)
wi h he ollowing alues o he pa ame e s:
ε1�09,ρ1�0.4,ε2�0.8,ρ2�200,
ε3�0.75 and ρ3�0.005,(97)
and he ollowing ini ial condi ions:
a0�1.5, K0�500 and c0�0.01.(98)
In his way he condi ions o P oposi ion 5(ii) abou he
sequences an
∞
0,Kn
∞
0and cn
∞
0a e ul lled since
a�limn⟶ ∞ an
�ρ1/(1−ε1) � 4, K�limn⟶ ∞ Kn
�
ρ2/(1−ε2) � 1000 and c�limn⟶ ∞ cn
�
ρ3/(1−ε3) � 0.02. Figu e 9 shows he e olu ion o he
species popula ion yn
and ha o he en i onmen ca -
ying capaci y Kn
i he popula ion is ini ially y0�200.
One can see ha y�limn⟶ ∞
yn
� ((a−1)/2c)( �������������
1+4Kc/(a−1)
−1)≈319 <
min(K, ���������
K(a−1)/c
) � min(1000,���������
K(a−1)/c
)≈387 as
P oposi ion 1(ii) poin s ou .
On he o he hand, Figu e 10 displays he e olu ion o he
species popula ion and ha o he en i onmen ca ying ca-
paci y i he popula ion is ini ially y0�10. Again, one can see
ha y�limn⟶ ∞ yn
≈319 al hough he ini ial popula ion
is close o he equilib ium poin y�0. Te esul s displayed in
Figu es 9 and 10 illus a e he ac ha he equilib ium poin
y�0is globally asymp o ically uns able while he equilib ium
poin y�Kis globally asymp o ically s able o c>0.
6.4. Example 4. Te ollowing examples illus a e he esul s
o Sec ion 5 abou he modi ed Be e on–Hol equa ion
(63). Te sequences an
∞
0and Kn
∞
0a e, espec i ely,
gene a ed by means o he ollowing di e ence equa ions:
an+1�ε1an+ρ1,
Kn+1�ε2Kn+ρ2,(99)
wi h he ollowing alues o he pa ame e s:
ε1�0.8,ρ1�0.1,ε2�0.9 and ρ2�50,(100)
and he ollowing ini ial condi ions:
a0�1.5 and K0�300.(101)
In his way he condi ions o Sec ion 5 abou he se-
quences an
∞
0and Kn
∞
0a e ul lled since a�limn⟶ ∞
an
�ρ1/(1−ε1) � 0.5 and K�limn⟶ ∞ Kn
�ρ2/(1−
ε2) � 500. Se e al choices o he sequence cn
∞
0a e con-
side ed o illus a e he esul s o Sec ion 5:
(i) In he s case cn
∞
0is gi en by he di e ence
equa ion:
cn+1�ε3cn+ρ3,(102)
wi h he alues o he pa ame e s ε3�0.75 and
ρ3�0 and he ini ial condi ion c0�0.01. In his
way, c�limn⟶ ∞ cn
�ρ3/(1−ε3) � 0 and hen,
he condi ions ha a<1, K>0 and c�0 a e ul lled
so ha he modi ed Be e on–Hol equa ion has
wo equilib ium poin s, namely, he s able poin
y1�0 (ex inc ion) and he uns able one y2�K.
Figu e 11 shows he e olu ion o he species pop-
ula ion yn
and ha o he en i onmen ca ying
capaci y Kn
i he popula ion is ini ially y0�250.
5 101520253035400
n (days)
0
5
10
15
y
n
Figu e 7: Time e olu ion o he species popula ion i y0�15 and
(yn) � αnyp
n+βn.
18 Disc e e Dynamics in Na u e and Socie y
Te esul s displayed in Figu e 11 illus a es he ac
ha he equilib ium poin y1�0 is globally as-
ymp o ically s able while he equilib ium poin y2�
500 is uns able o c�0 as Sec ion 5 poin s ou .
(ii) In he second case cn
∞
0is gi en by (102) wi h he
alues o he pa ame e s ε3�0.75 and ρ3� (1−
ε3)(1−a)/4K�6.25x10−5and he ini ial condi ion
c0�0.01. In his way, c�limn⟶ ∞ cn
�ρ3/(1−
ε3) � 2.5x10−4and hen, he condi ions ha a<1,
K>0 and c� (1−a)/4Ka e ul lled so ha he
modi ed Be e on–Hol equa ion has wo equi-
lib ium poin s, namely, he s able poin y1�0
(ex inc ion) and he uns able one
y2� (1−a)/2c�1000. Figu e 12 shows he e o-
lu ion o he species popula ion yn
and ha o he
en i onmen ca ying capaci y Kn
i he pop-
ula ion is ini ially y0�1050. Te esul s displayed
in Figu e 12 illus a e he ac ha he equilib ium
poin y1�0 is globally asymp o ically s able while
he equilib ium poin y2�1000 is uns able as
Teo em 4 (i) es ablishes.
(iii) In he hi d case cn
∞
0is gi en by (102) wi h he
alues o he pa ame e s ε3�0.75 and
ρ3�0.001 >(1−ε3)(1−a)/4K�6.25x10−5and
he ini ial condi ion c0�0.01. In his way,
c�limn⟶ ∞ cn
�ρ3/(1−ε3) � 0.004 and hen,
he condi ions ha a<1, K>0 and c>(1−a)/4K�
2.5x10−4a e ul lled so ha he modi ed Be -
e on–Hol equa ion has only one equilib ium
poin , namely, he s able poin y1�0 (ex inc ion).
Figu e 13 shows he e olu ion o he species pop-
ula ion yn
and ha o he en i onmen ca ying
capaci y Kn
i he popula ion is ini ially y0�250.
Te esul s displayed in Figu e 13 illus a e he ac
ha he unique equilib ium poin y1�0 is globally
y
n
0
50
100
150
200
50 100 150 200 250 3000
n (days)
Figu e 8: Time e olu ion o he species popula ion i y0�100 and
he species e olu ion is gi en by (64).
0
200
400
600
800
1000
10 20 30 40 50 60 700
n (days)
yn
Kn
Figu e 9: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�200 and he species e olu-
ion is gi en by (63).
0
200
400
600
800
1000
10 20 30 40 50 60 700
n (days)
yn
Kn
Figu e 10: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�10 and he species e olu ion
is gi en by (63).
0
100
200
300
400
500
10 20 30 40 50 60 700
n (days)
yn
Kn
Figu e 11: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�250 and he species e olu-
ion is gi en by (64) wi h a<1, K>0, and c�0.
Disc e e Dynamics in Na u e and Socie y 19
asymp o ically s able, ha is asymp o ically s able
o any ni e ini ial condi ion, as Teo em 4 (ii)
es ablishes.
(i ) In he ou h case cn
∞
0is gi en by (102) wi h he
alues o he pa ame e s ε3�0.75 and
ρ3�3x10−5<(1−ε3)(1−a)/4K�6.25x10−5and
he ini ial condi ion c0�0.01. In his way,
c�limn⟶ ∞ cn
�ρ3/(1−ε3) � 1.2x10−4and
hen, he condi ions ha a<1, K>0 and
0<c<(1−a)/4K�2.5x10−4a e ul lled so ha
he modi ed Be e on–Hol equa ion has h ee
equilib ium poin s, namely, he s able poin y1�0
(ex inc ion) and he uns able ones y2� (1−a−
������������������
(1−a)2−4ck(1−a)
)/2c≈581 >K�500 and y3
� (1−a−������������������
(1−a)2−4ck(1−a)
)/2c≈3586 <
(1−a)/c≈4167. Figu e 14 shows he e olu ion o
he species popula ion yn
and ha o he en i-
onmen ca ying capaci y Kn
i he popula ion is
ini ially y0�700. Te esul s displayed in Figu e 14
illus a e he ac ha he equilib ium poin y1�0 is
globally asymp o ically s able while he o he ones
a e uns able as Teo em 4 (ii) es ablishes.
7. Conclusions
Tis pape has discussed a gene alized ime- a ying Be -
e on–Hol equa ion which conside s he p esence o pos-
i i e o nega i e ha es ing and, e en ually, a quad a ic- ype
penal y o he popula ion excess. Such a e m akes accoun
o he po en ial in e nal compe ence be ween he coho
indi iduals o ood, e uge, e c. Te ha es ing ac ion
(desc ibing hun ing/ shing ac ions) is conside ed join ly
wi h e en ually p esen independen consump ion (de-
sc ibing mig a ions om ou side o he habi a o inside o
ice- e sa). I is seen ha he p esence o he penal y e m
can ansla e in o he p esence o wo o he posi i e equi-
lib ium poin s. Some pa icula s abili y esul s ha e been
also de i ed o he s a iona y equa ion, which a ises when
i s pa ame e izing sequences con e ge, o he case o small
le els o popula ion by in oducing a e m aking accoun o
he Allee e ec . Te pape has also designed some species
e olu ion con ol laws by moni o ing he ha es ing ac ion
and has discussed he in uence in he s abili y esul s o
conside ing a modelling unc ion o Allee e ec which
makes di cul g owing o e en can cause ex inc ion o
small numbe s o ep oduc i e indi iduals.
Te equilib ium poin s o he s a iona y solu ion in he
p esence and absence o ha es ing ac ion ha e been
cha ac e ized and hei local asymp o ic s abili y p ope ies
ha e been in es iga ed in he case o in insic g ow h a e
exceeding uni y and e en ual execu ion o ha es ing ac ions
0
100
200
300
400
500
600
700
10 20 30 40 50 60 700
n (days)
yn
Kn
Figu e 14: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�700 and he species e olu-
ion is gi en by (63) wi h a<1, K>0, and 0 <c<(1−a)/4K.
10 20 30 40 50 60 700
n (days)
0
200
400
600
800
1000
1200
yn
Kn
Figu e 12: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�1050 and he species e o-
lu ion is gi en by (63) wi h a<1, K>0, and c� (1−a)/4K.
10 20 30 40 50 60 700
n (days)
yn
Kn
0
100
200
300
400
500
Figu e 13: Time e olu ion o he species popula ion and ha o he
en i onmen ca ying capaci y i y0�250 and he species e olu-
ion is gi en by (63) wi h a<1, K>0, and c>(1−a)/4K.
20 Disc e e Dynamics in Na u e and Socie y
and in he case o he in insic g ow h a e being less han
uni y. Some nume ical examples ha e been also discussed.
Da a A ailabili y
No unde lying da a we e collec ed o p oduced in his s udy.
Con lic s o In e es
Te au ho s decla e ha hey ha e no con ic s o in e es .
Acknowledgmen s
Te au ho s a e g a e ul o he Basque Go e nmen o i s
suppo h ough G an no. IT1555-22 and o MCIN/AEI
269.10.13039/501100011033 o G an no. PID2021-
1235430B-C21/C22. Te au ho s a e also g a e ul o he
e e ees by hei use ul commen s.
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