Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443
h ps://doi.o g/10.1186/s13662-021-03599-z
RESEARCH Open Access
Quali a i e analysis o a disc e e- ime
phy oplank on–zooplank on model wi h
Holling ype-II esponse and oxici y
Muhammad Salman Khan1, Ma ia Sam een1* , Hassen Aydi2and Manuel De la Sen3
*Co espondence:
[email p o ec ed];
msam [email protected]
1Depa men o Ma hema ics,
Quaid-I-Azam Uni e si y, 45320,
Islamabad, Pakis an
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
The in e ac ion among phy oplank on and zooplank on is one o he mos impo an
p ocesses in ecology. Disc e e- ime ma hema ical models a e commonly used o
desc ibing he dynamical p ope ies o phy oplank on and zooplank on in e ac ion
wi h nono e lapping gene a ions. In such ype o gene a ions a new age g oup
swaps he olde g oup a e egula in e als o ime. Keeping in obse a ion he
dynamical eliabili y o con inuous- ime ma hema ical models, we con e a
con inuous- ime phy oplank on–zooplank on model in o i s disc e e- ime
coun e pa by applying a dynamically consis en nons anda d diffe ence scheme.
Mo eo e , we discuss boundedness condi ions o e e y solu ion and p o e he
exis ence o a unique posi i e equilib ium poin . We discuss he local s abili y o
ob ained sys em abou all i s equilib ium poin s and show he exis ence o
Neima k–Sacke bi u ca ion abou unique posi i e equilib ium unde some
ma hema ical condi ions. To con ol he Neima k–Sacke bi u ca ion, we apply a
gene alized hyb id con ol echnique. Fo explana ion o ou heo e ical esul s and o
compa e he dynamics o ob ained disc e e- ime model wi h i s con inuous
coun e pa , we p o ide some mo i a ing nume ical examples. Mo eo e , om
nume ical s udy we can see ha he ob ained sys em and i s con inuous- ime
coun e pa a e s able o he same alues o pa ame e s, and hey a e uns able o
he same pa ame ic alues. Hence he dynamical consis ency o ou ob ained
sys em can be seen om nume ical s udy. Finally, we compa e he modified hyb id
me hod wi h old hyb id me hod a he end o he pape .
Keywo ds: Phy oplank on–zooplank on model; Boundedness; Local s abili y
analysis; Neima k–Sacke bi u ca ion; Gene alized hyb id con ol me hod
1 In oduc ion
The s udy o ma hema ical models o popula ion dynamics is conside ed as a key a ea
in abs ac ecology om he ime when he amous Lo ka–Vol e a model was p esen ed
[1]. The lea ning o o ganism mo emen and sp eading has u n ou o be a undamen al
elemen o unde s anding a chain o ecological in e oga ions associa ed wi h he spa-
io empo al s udy o dynamics o popula ions [2]. Plank ons a e eno mously flexible in
abundance, bo h empo ally and spa ially. Plank on a iabili y depends on na u al along
©The Au ho (s) 2021. This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use,
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Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 2 o 29
wi h physical p ocedu e o he spa ial s uc u e. Na u al p ocesses include, o ins ance,
de elopmen ,g azing,andbeha io ,andphysicalp ocedu esinclude, o ins ance,mixing
andla e als i ing.Nonlinea i yo ecosys emsen i elycon ibu es o hespa ialo ganiza-
ion in plank on alloca ions [3]. In ma ine ecology he wo d plank on e e s o he spon-
aneously mo ing and ain ly swimming o ganisms. Commonly, plank on is pa ed in o
wo species, he phy oplank on species and zooplank on species. Phy oplank on species
a e iny in hei size wi h a single celled s uc u e [4]. Phy oplank on a e beneficial o
aqua ic li e and p oduce hal o he oxygen in he wo ld h ough he p ocess o pho o-
syn hesis. Phy oplank on popula ion is exe ing uni e sal-scale effec on a mosphe e by
anspo ingCO2 omwa e su ace o hedep ho oceans.Mainly, hisp ocesshappens
due o hei dea h, sinking, and p ima y p oduc ion [4]. I is obse ed ha algal species
ise abundan ly in damped, we , and ma ine en i onmen s. The s ages o speedy g ow h,
slow s agna ion, and accele a ed decline in he numbe o cells collec i ely c ea e analgal
bloom. This phenomenon o accele a ed a ia ion in he densi y o phy oplank on popu-
la ion is he cen al ai in he plank on ecosys em [4]. Despi e he ac ha he sudden
eme genceand disappea ingo bloomsisno clea , heundesi ableeffec o damagingal-
gal blooms on he heal h o mankind, aqua ic li e, and fishe ies ade can be easily seen
[4].
On he incidence o blooms, phy oplank on and zooplank on in e ac wi h each o he ,
and he s udy o his in e ac ion is he poin o ocus o many scien ific in es iga ions [5].
Phy oplank on p oduces oxic ma e ials oa e p eda ionsby hei p eda o s (zooplank-
on). Fu he mo e, his is he opic o in e es o many esea che s om many decades.
Ma hema ical modeling o in e ac ions be ween plank on species p o ides us an impo -
an op ional me hod in imp o ing he knowledge o any indi idual ela ed o he biolog-
ical and physical mechanisms conce ning o he ecological s udy o plank on popula ion
[5].
The au ho s in [6] ha e conside ed a plank on–nu ien model ela ed o aqua ic en-
i onmen by conside a ion o plank onic blooms. In [7] he au ho s ha e examined he
influence o pe iodici y and seasonali y on plank onic dynamics.
In [8] he au ho s ha e p esen ed wo ma hema ical models connec ed o plank on
ecosys em along wi h a s ong ep esen a ion o i al sep ic phy oplank on and i uses.
The au ho s in [9] ha e con empla ed he effec o p eda ion on compe i o y elimina ion
and he coexis ence o compe i o y p eda o s. Mo eo e , hey p esen ed and explo ed a
one-phy oplank on wo-zooplank on model along wi h he conside a ion o ha es ing.
Huppe e al. [10]conside edanu ien –phy oplank onmodel oexamine hedynami-
calbeha io o phy oplank onblooms.In[11] heau ho sha ep esen edazooplank on–
phy oplank on model wi h ha es ing. Fu he mo e, hey ha e explained ha he ex a
exploi a ionmayex e mina e hepopula ionwhilesui ableha es inggua an ies hecon-
solida ion o bo h popula ions. Mo eo e , nume ous s udies ha e hei poin o ocus
on phy oplank on–zooplank on models along wi h a sou ce o nu ien , he oxic con-
sequence o plank on species, he su i al o plank on species, o he ha es ing effec s
[9–16].I is con enien oin oduce he oxin c ea inglag du ing he s udy o he dynam-
ics o phy oplank on–zooplank on models. The au ho s in [17]ha ep esen edama he-
ma ical model including ime lag in oxin deli e ance by phy oplank on. The wo k done
in [18–21] mo i a ed us o s udy he dynamics o a phy oplank on–zooplank on popula-
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 3 o 29
ion model wi h oxici y. Mo eo e , he oxic subs ance is eleased by phy oplank on and
some imes by o he ex e nal sou ces.
We conside he basic phy oplank on–zooplank on model p esen ed by Cha opad-
hayay e al. [22]. Fu he mo e, his ma hema ical model is based on he ollowing con-
di ions.
•Wesuppose ha z( )and p( )a e he sizes o zooplank on and phy oplank on
popula ions, espec i ely.
• Zooplank on popula ion ea s phy oplank on popula ion and hen ecycles hem in o
hei own communi y. The unc ional esponse αp( )z( )
a+p( ) ep esen s he p eda ion a e
o zooplank on popula ion on phy oplank on species. Mo eo e , his p eda ion
inc eases he g ow h a e o zooplank on, which is ep esen ed by he e m βp( )z( )
a+p( ).
• We assume ha zooplank on popula ion becomes in ec ed by ea ing in ec ed
phy oplank on popula ion. Addi ionally, he in ec ion in phy oplank on may be
p oduced due o ex e nal oxic subs ance (see [22]).
• We assume ha he in ec ion in phy oplank on may be p oduced due o ex e nal oxic
subs ance(see[22]).
• Phy oplank on popula ion has logis ic g ow h [21] in he absence o zooplank on
popula ion, whe e is hei exponen ial a e o g ow h, and kis he maximum
ca ying capaci y o en i onmen .
Unde hese condi ions we ha e he ollowing phy oplank on–zooplank on model [22]:
⎧
⎨
⎩
dp
d = p( )(1– p( )
k)–α (p( ))z( ),
dz
d =β (p( ))z( )–δz( )–ρg(p( ))z( ). (1.1)
Kuang [23] ha e inspec ed he limi cycle beha io in Gause- ype p eda o –p ey sys ems
wi h Holling ype-II esponse [24]. In addi ion, he e ealed ha he s udy o dynamical
p ope ies o p eda o –p eymodels usingaHolling- ype esponse unc ion isbe e han
hes udyo dynamicso p eda o –p eymodelswi hou usingHolling esponse.Gene ally,
Holling ype-II esponseismodeledanddesc ibedbyusing ec angula hype bola,andi s
ma hema ical o m is gi en as
ϕ(x)= x
a+x,
whe e ais any cons an . By using Holling ype-II esponse we ge he ollowing ma he-
ma ical o m o sys em (1.1):
⎧
⎨
⎩
dp
d = p( )(1– p( )
k)–αp( )
a+p( )z( ),
dz
d =βp( )
a+p( )z( )–δz( )–ρp( )
a+p( )z( ). (1.2)
• Nex , we assume ha he ime lag o p oduc ion and media ion o oxic subs ance by
phy oplank on is ze o.
• We in oduce he ca chabili y coefficien s q1and q2 o phy oplank on and
zooplank on popula ions espec i ely. Gene ally, unc ional o m o ha es ing is
exp essed by using he hypo hesis o ca ch-pe -uni -effo [25].
• Mo eo e , we in oduce Eas he pa ame e o combined effo o ha es ing o
popula ion [25].
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 4 o 29
Unde hese modifica ions, sys em (1.2) akes he ollowing ma hema ical o m:
⎧
⎨
⎩
dp
d = p( )(1– p( )
k)–αp( )
a+p( )z( )–m1p3( )–q1Ep( ),
dz
d =βp( )
a+p( )z( )–δz( )–ρp( )
a+p( )z( )–m2z2( )–q2Ez( ), (1.3)
whe e he pa ame e s in sys em (1.3) a e nonnega i e and defined as ollows:
a: cons an o pa ial cap u ing sa u a ion.
α: maximal akeo e a e o zooplank on on phy oplank on.
β: con e sion a e o phy oplank on–zooplank on (β<α).
ρ: oxici y a e o phy oplank on pe uni biomass.
δ: na u al a e o dea h o zooplank on popula ion.
Mo eo e , he e mm1p3( )appea inginsys em(1.3) ep esen s hein ec ionp oduced
in phy oplank on popula ion due o an ex e nal oxic subs ance. In addi ion, d2
dp2(m1p3)=
6m1p>0showsanaccele a ingg ow ho oxicsubs ancepa allel ophy oplank onpopu-
la ion.Thisis due o ac ha app oxima elyeach indi idual in phy oplank onpopula ion
isinc easinglyconsuming he oxicsubs ances.Howe e , he educ iono g azingbyzoo-
plank on due oxici y effec is ep esen ed by he e m m2z2( ). Fu he mo e, he oxici y
effec on zooplank on popula ion is less han phy oplank on popula ion, whe e m1and
m2a e he oxici y coefficien s wi h 0 <m2<m1[25].
Ob iously, i is app op ia e o explo e he dynamics o any biological model by di -
e ence equa ions ins ead o diffe en ial equa ions when we a e dealing wi h nono e -
lapping popula ions. Fu he mo e, obse a ion and analysis o chaos in any biological
sys em by using diffe ence equa ions is be e han by using diffe en ial equa ions [26].
Hencei isin e es ing os udybiologicalmodelsindisc e e o m.Recen ly,Ghanba iand
Gómez-Aguila [27] discussed he dynamics o nu ien –phy oplank on–zooplank on
sys em wi h a iable-o de ac ional de i a i es. Mo eo e , he au ho s in [28]explo ed
he exis ence o chaos in a cance model using ac ional de i a i es by means o expo-
nen ial decay and he Mi ag-Leffle law. Beigi e al. [29] discussed he use o ein o ce-
men lea ning o effec i e accina ion s a egies o co ona i us disease 2019 (COVID-
19). The au ho s in [30] analyzed he ole o zooplank on dynamics o Sou he n Ocean
phy oplank onbiomassandglobalbiogeochemicalcycles.Fo mo ede ailon heanalysis
o a ious dynamical sys ems, we e e he in e es ed eade o [31–37]. The e a e a i-
ous ma hema ical echniques o con e ing he sys ems o diffe en ial equa ions o hei
co esponding disc e e coun e pa s. To achie e his goal, he usual way is applying s an-
da d diffe ence schemes such as Runge–Ku a me hods and Eule app oxima ions. How-
e e ,nume icalinconsis encyisexpe iencedwi h heapplica iono usualfini ediffe ence
me hods. Hence, o a oid his nume ical inconsis ency, we can apply he nons anda d fi-
ni e diffe ence me hod gi en by Mickens [38].
In gene al, whene e a nons anda d fini e diffe ence scheme is p oposed, i is aimed
on hep ese a iono he ollowing p ope ieso he espec i econ inuous- ime sys em:
posi i i yo esul s,boundedness,s abili yo equilib iumpoin s,andbi u ca ions.Mo e-
o e , he o ma ion o hese ype o diffe ence schemes is no s aigh o wa d, and he e
a enousualways o hei cons uc ion,whichisp obablyconside ed asmajo downside
o nons anda d diffe ence schemes. Hence by aking in o accoun he o iginal dynamical
p ope ies o model (1.3) a disc e e- ime model om (1.3)isob ainedbyusingMickens-
ypenons anda dschemesuch ha i emainsdynamicallyconsis en [39].Implemen ing
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 5 o 29
heMickens- ypenons anda dschemeonmodel(1.3),wege he ollowingdisc e e- ime
ma hema ical model:
⎧
⎨
⎩
pn+1–pn
h= pn(1– pn+1
k)–αpn+1
a+pnzn–m1p2
npn+1 –q1Epn+1,
zn+1–zn
h=βpn
a+pnzn–δzn+1 –ρpn
a+pnzn+1 –m2znzn+1 –q2Ezn+1,(1.4)
whe e h>0 is aken as a s ep size o he nons anda d scheme. Fu he mo e, (1.4)canbe
w i en in o he ollowing ma hema ical o m:
⎧
⎪
⎨
⎪
⎩
pn+1 =(1+h )pn
1+h(
kpn+αzn
a+pn+m1p2
n+q1E),
zn+1 =(1+hβpn
a+pn)zn
1+h(ρpn
a+pn+δ+m2zn+q2E),(1.5)
whe e β>ρ. Mo eo e , ou model (1.5) loses i s biological consis ency whene e β<ρ
(see [25]), which is impossible biologically. Hence, o he es o ou pape , we assume
ha a>kand β>ρ.
2 Boundedness and exis ence o fixed poin s o sys em (1.5)
Toob ains eadys a eso sys em (1.5),weconside he ollowing wo-dimensionalsys em
o equa ions:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
p=(1+h )p
1+h(
kp+αz
a+p+m1p2+q1E),
z=(1+hβp
a+p)z
1+h(ρp
a+p+δ+m2z+q2E).(2.1)
Sol ing (2.1), we can ge he ollowing equilib ium poin s: (0,0) which is an ex inc ion
poin o bo hpopula ions,(√4k2m + 2–4Ek2mq1–
2km ,0),whichisanex inc ionequilib ium o
zooplank on popula ion, and he unique posi i e equilib ium (p,z). Addi ionally, he fi s
componen o hepoin (√4k2m + 2–4Ek2mq1–
2km ,0) emainsposi i e o >Eq1.Theexis ence
and uniqueness o (p,z) can be s udied as ollows. Suppose ha p0>0andz0>0.Then
eachsolu ion(pn,zn)o sys em(1.5)mus sa is ypn>0andzn>0 o alln≥0.Then om
he fi s equa ion o sys em (1.5) i ollows ha
pn+1 ≤(1+h )pn
1+h
kpn. (2.2)
Consequen ly, sol ing (2.2) and hen aking he limi , we ge
limsup
n→∞ pn≤k. (2.3)
In he same way, om second equa ion o sys em (1.5)wege
zn+1 =(1+hβpn
a+pn)zn
1+h(ρpn
a+pn+δ+m2zn+q2E)
≤(1+hβk
a+k)zn
1+h(ρk
a+k+m2zn).
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 6 o 29
Hence, we can ob ain he uppe bound o zooplank on popula ion:
limsup
n→∞ zn≤k(β–ρ)
m2(k+a). (2.4)
Finally, we ha e he ollowing heo em abou he boundedness o all solu ions o (1.5).
Theo em 2.1 Assume ha 0<p0≤kand0<z0≤k(β–ρ)
m2(k+a).Then o all n≥0, e e y posi-
i e solu ion (pn,zn)o sys em (1.5)is bounded and con ained in he se [0,k]×[0, k(β–ρ)
m2(k+a)]
whene e β>ρ.
Nex , we conside he equa ion sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
p=(1+h )p
1+h(
kp+αz
a+p+m1p2+q1E),
z=(1+hβp
a+p)z
1+h(ρp
a+p+δ+m2z+q2E).(2.5)
F om (2.5)wege he ollowingpai :
p=a(β–ρ)
(β–ρ)–(δ+m2z+q2E)–a,z=(a+p)( – p
k–m1p2–q1E)
α.
F om his pai we can w i e
F(p)= a(β–ρ)
(β–ρ)–(δ+m2 (p)+q2E)–a–p, (2.6)
whe e
(p)=(a+p)( – p
k–m1p2–q1E)
α, (2.7)
wi h
(0)= a( –q1E)
α>0
and
F(0)= a(δ+m2 (0)+q2E)
(β–ρ)–(δ+m2 (0)+q2E)>0.
Fu he mo e, a he uppe bound and o each λ∈(0,k], i (β–ρ)>(δ+m2 (λ)+q2E),
hen
F(λ)= a(δ+m2 (λ)+q2E)
(β–ρ)–(δ+m2 (λ)+q2E)–λ<0,
whe e
(λ)=–(a+λ)(m1 2+q1E)
α<0.
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 7 o 29
Hence F(p)=0 has a leas one posi i e eal oo in [0,k]. Fu he mo e, we can see ha
F(λ)=–1+ a(β–ρ)(m2 (λ))
((β–ρ)–(δ+m2 (λ)+q2E))2<0,
whe e
(λ)=–(a+λ)(
k+2m1λ)
α+ – λ
k–m1λ2–q1E
α<0,
whene e
–q1E<
k+m1λλ
o e e y λ∈[0,k]. Hence he equa ion F(p) =0 has a unique posi i e solu ion in [0,k].
Theo em2.2 Assume ha 0<p0≤kand0<z0≤k(β–ρ)
m2(k+a).Then o
–q1E<
k+m1λλ
and
>q1E,
he e exis s a unique posi i e cons an solu ion (p,z)o sys em (1.5)in [0,k]×[0, k(β–ρ)
m2(k+a)]i
and only i o each λ∈(0,k], we ha e
(β–ρ)>δ+m2 (λ)+q2E.
In addi ion, o λ=0,
(β–ρ)<δ+m2 (λ)+q2E.
3 S abili y analysis o sys em (1.5) abou i s fixed poin s
To discuss he s abili y o sys em (1.5) abou all i s equilib ium poin s, we compu e he
a ia ionalma ix V(p,z)o sys em(1.5)abou eacho i sfixedpoin (p,z).Thema ixV(p,z)
is gi en by
V(p,z)=j11 j12
j21 j22.
The cha ac e is ic polynomial M(ξ)o V(p,z)is
M(ξ)=ξ2–T ξ+D , (3.1)
whe e
T =(j11 +j22)
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 8 o 29
and
D =j11j22 –j12j21.
Thenex lemmadesc ibes hecondi ionspa allel o heJu ycondi ion o hes abili yo
fixed poin s; see [40].
Lemma3.1([40]) Le M(ξ)=ξ2–T ξ+D andM(1) >0.I ξ1,ξ2a e he oo so M(ξ)=0,
hen:
(a)|ξ1|<1and |ξ2|<1i and only i M(–1)>0 and D <1;
(b)|ξ1|>1and |ξ2|>1i and only i M(–1) >0 and D >1;
(c)|ξ1|<1and |ξ2|>1o (|ξ1|>1and |ξ2<|1) i and only i M(–1)<0;
(d)ξ1and ξ2 ep esen complex conjuga es wi h |ξ1|=1=|ξ2|i and only i T 2–4D <0
and D =1.
I ξ1and ξ2a e cha ac e is ic alues o (3.1), hen he poin (p,z)is sink i |ξ1|<1and
|ξ2|<1.Fu he mo e,i islocallyasymp o icallys able.Thepoin (p,z)isknownasasou ce
( epelle )i |ξ1|>1and |ξ2|>1,and i p o ides ins abili y condi ion o he gi en sys em.
The poin (p,z)is a saddle poin i |ξ1|<1and |ξ2|>1o (|ξ1|>1and |ξ2|<1).Finally,
(p,z)is nonhype bolic i condi ion (d)is sa isfied.
Fi s ly,wewills udy hes abili yo sys em(1.5)abou popula ion eeequilib iumpoin
(0,0). The a ia ional ma ix V(p,z) o sys em (1.5) e alua ed a (0,0) is
V(0,0) =1+h
1+Ehq10
01
1+hδ+Ehq2.
Fu he mo e, V(0,0) isadiagonalma ix.Hencesys em(1.5)has woeigen alues ela ed o
hepopula ion eeequilib iumpoin (0,0),ξ1=1+h
1+Ehq1andξ2=1
1+hδ+Ehq2,whe e,ξ1andξ2
a e oo so hecha ac e is icequa iono hema ixV(0,0).I isclea ha |ξ2|=|1
1+hδ+Ehq2|<
1 o all pa ame ic alues. Now by conside ing he condi ion |ξ2|< 1 we a e now able o
desc ibe s abili y condi ions o sys em (1.5)abou (0,0).
P oposi ion3.2 Le ξ1andξ2be he oo so hecha ac e is icequa iono hema ixV(0,0)
andsuppose ha |ξ2|<1 o allpa ame ic alues.Le (0,0)beapopula ion eefixedpoin
o sys em (1.5). Then (0,0) is sink o saddle i and only i <Eq1o >Eq1, espec i ely.
Nex , we will explo e he local s abili y o sys em (1.5) abou he zooplank on- ee
equilib ium (√4k2m + 2–4Ek2mq1–
2km ,0). Clea ly, he fi s componen in he pai
(√4k2m + 2–4Ek2mq1–
2km ,0) is posi i e i and only i >q1E.Le V1(√4k2m + 2–4Ek2mq1–
2km ,0)
be he a ia ional ma ix o he wo-dimensional sys em (1.5) abou he zooplank on
eeequilib ium(√4k2m + 2–4Ek2mq1–
2km ,0).ThenV1(√4k2m + 2–4Ek2mq1–
2km ,0)has he ollowing
o m:
V1(x,0)=k2(1+h )(1+Ehq1–hm1x2)
(k+Ehkq1+hx( +km1x))2–hk2(1+h )αx
(a+x)(k+Ehkq1+hx( +km1x))2
0a+x+hβx
a+ahδ+x+h(δ+ρ)x+Ehq2(a+x),
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 9 o 29
whe e x=√4k2m + 2–4Ek2mq1–
2km .Mo eo e ,V1(x,0) has he cha ac e is ic polynomial
M(ξ)=ξ2–T V1(x,0)+De V1(x,0)(3.2)
wi h
T V1(x,0)=a+x+hβx
a+ahδ+x+h(δ+ρ)x+Ehq2(a+x)
+k2(1+h )(1+Ehq1–hm1x2)
(k+Ehkq1+hx( +km1x))2
and
De V1(x,0)
=k2(1+h )(a+x+hβx)(1+Ehq1–hm1x2)
(a+ahδ+x+h(δ+ρ)x+Ehq2(a+x))(k+Ehkq1+hx( +km1x))2.
Henceweha e he ollowingp oposi ionabou helocals abili yo sys em(1.5)abou he
zooplank on- ee equilib ium (√4k2m1 + 2–4Ek2m1q1–
2km1,0).
P oposi ion 3.3 Le ξ1and ξ2be he cha ac e is ic oo s o (3.2), and le >q1E.I
(√4k2m1 + 2–4Ek2m1q1–
2km1,0)=(x,0)is a zooplank on- ee cons an solu ion o (1.5), hen:
(a)(x,0) emains inside he uni disk i and only i
1+Ehq1–hm1x2<(k+Ehkq1+hx( +km1x))2
k2(1+h )(3.3)
and
βx<aδ+(δ+ρ)x+Eq2(a+x). (3.4)
(b)(x,0)lies ou side he uni disk i and only i
1+Ehq1–hm1x2>(k+Ehkq1+hx( +km1x))2
k2(1+h )(3.5)
and
βx>aδ+(δ+ρ)x+Eq2(a+x). (3.6)
(c)(x,0)isasaddlepoin i andonlyi oneo he ollowingpai so inequali ies(3.4)–(3.5)
o (3.3)–(3.6)is sa isfied.
(d)(x,0)is nonhype bolic i and only i one o he ollowing condi ions is sa isfied:
1+Ehq1–hm1x2=(k+Ehkq1+hx( +km1x))2
k2(1+h )
o
aδ+(δ+ρ–β)x+Eq2(a+x)=0.
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 16 o 29
Finally, o w i e he linea pa o (4.3) in he canonical ma ix o m a ˘
h=0, we conside
he simila i y ans o ma ion
P
Z= 12 0
– 11 –℘X
Y,, (4.6)
whe e
=T (0)
2,
and
℘=4D (0)–T 2(0)
2.
Then om (4.6)weha e
X
Y=1
12 0
– 11
℘ 12 –1
℘P
Z.. (4.7)
By using ans o ma ion (4.6)weha e henex au ho i a i e o mo sys em(4.3):
X
Y→–℘
℘
X
Y+˘
F(X,Y)
˘
G(X,Y), (4.8)
whe e
˘
F(X,Y)= 16P3
12 + 17P2Z
12 + 13P2
12 + 18PZ2
12 + 14PZ
12 +Z3 19
12 +Z2 15
12
+O|X|+|Y|4
and
˘
G(X,Y)=(– 11) 16
12℘– 26
℘P3+(– 11) 17
12℘– 27
℘P2Z
+(– 11) 13
12℘– 23
℘P2+(– 11) 18
12℘– 28
℘PZ2
+(– 11) 14
12℘– 24
℘PZ +(– 11) 19
12℘– 29
℘Z3
+(– 11) 15
12℘– 25
℘Z2+O|X|+|Y|4
wi hP= 12XandZ=(– 11)X–℘Y.Hencebyusing hes anda d heo yo no mal o m
o analysis o bi u ca ion we can calcula e he fi s Lyapuno exponen a (X,Y)=(0,0)
as ollows:
=–Re(1–2ξ1)ξ2
2
1–ξ1θ20θ11–1
2|θ11|2–|θ02|2+Re(ξ2θ21)˘
h=0,
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 17 o 29
whe e
θ20 =1
8˘
FXX –˘
FYY +2˘
GXY +i(˘
GXX –˘
GYY –2˘
FXY),
θ11 =1
4˘
FXX +˘
FYY +i(˘
GXX +˘
GYY),
θ02 =1
8˘
FXX –˘
FYY –2˘
GXY +i(˘
GXX –˘
GYY +2˘
FXY),
θ21 =1
16(˘
FXXX +˘
FXYY +˘
GXXY +˘
GYYY)
+i
16(˘
GXXX +˘
GXYY –˘
FXXY –˘
FYYY).
Due o a o emen ioned analysis, we ha e he ollowing heo em (see [45–50]).
Theo em4.1 Assume ha (3.7), (3.8), and (3.9)a e sa isfied and =0.Then he unique
posi i e fixed poin (p,z)o sys em (1.5)unde goes Neima k–Sacke bi u ca ion.Addi ion-
ally,i <0, hen o h >˘
h,an a ac ing in a ian closed cu e bi u ca es om he fixed
poin (p,z), and i >0, hen o h <˘
h,a epelling in a ian closed cu e bi u ca es om
he fixed poin (p,z).
5 Modified hyb id con ol s a egy o con olling bi u ca ion and chaos
Gene ally, disc e e- ime sys ems a e mo e complex o analyze as compa ed o a
con inuous- ime one. Fo su i al o li e in any en i onmen , i is necessa y ha he pop-
ula ion does no expe ience any i egula si ua ion. Hence, o con olling acciden al un-
e enanduns ablebeha io inanyma hema icalsys em,chaoscon olisconside ed obe
an applied ool o e ading his complex and chao ic beha io [51–53]. In his pa o he
pape ,wes udya eedbackcon olme hodwi hpa ame e pe u ba ion omo euns able
and i egula ajec o ies owa d he s able ajec o ies. The mos use ul and well-known
me hod in he field o chaos is gi en by O e al. [51] o con ol pe iod-doubling bi u -
ca ion, which is known as OGY me hod. La e on, nume ous s a egic con ol me hods
a e de eloped (see [53]). He e we conside a modified hyb id con ol me hod o con ol
he Neima k–Sacke bi u ca ion and chaos. Fu he mo e, his ma hema ical me hod is
well applicable o e e y disc e e- ime sys em expe iencing he pe iod-doubling bi u ca-
ion and chaos. O iginally, a hyb id me hod was p oposed by Liu e al. [52]. Mo eo e , i
wasde eloped ocon ol hepe iod-doublingbi u ca ion(see[54,55]).He ewe e o med
heexis inghyb id con ol echnique[52] ocon ol heNeima k–Sacke bi u ca ionand
chaos. Fu he mo e, he newly de eloped echnique has shown be e esul s o almos
e e y disc e e dynamical sys em. Conside he ollowing n-dimensional disc e e dynami-
cal sys em:
Zn+1 =g(Zn,μ) (5.1)
wi h Zn∈n,n∈Z, and he pa ame e μ∈ o which sys em (5.1) expe iences he bi-
u ca ion. The pu pose o p oposing he e o med me hod o con olling he bi u ca ion
is ecap u ing he ex eme ange o s able egion in (5.1) by lessening he leng h o uns a-
ble egion.Hencewep esen he ollowinggene alizedhyb idcon olme hodbyapplying
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 18 o 29
s a e eedback along wi h pa ame e pe u ba ion;
Zn+k=L3g()(Zn,μ)+1–L3Zn, (5.2)
whe e∈Z,and0<L<1isapa ame e o con olling hebi u ca ionappea ingin(5.2).
In addi ion, g()is he k h alue o g(·). By applica ion o (5.2) osys em(1.5)wege he
ollowing sys em:
⎧
⎪
⎨
⎪
⎩
pn+1 =L3((1+h )pn
1+h(
kpn+αzn
a+pn+m1p2
n+q1E))+(1–L3)pn,
zn+1 =L3((1+hβpn
a+pn)zn
1+h(ρpn
a+pn+δ+m2zn+q2E))+(1–L3)zn.(5.3)
Fu he mo e, sys ems (5.3)and(1.5) ha e he same cons an solu ions. Addi ionally, he
Jacobian ma ix o (5.3)abou (p,z) is gi en as ollows:
⎛
⎝1–hL3p((a–k) +ekq1+p(2 +km1(2a+3p)))
k(1+h )(a+p)–hL3αp
(1+h )(a+p)
hL3z(–β+δ+ρ+eq2+m2z)2
a(β+hβδ–ρ+ehβq2+hβm2z)
β+hβδ–ρ+ehβq2+h(β–L3β+L3ρ)m2z
β+hβδ–ρ+ehβq2+hβm2z⎞
⎠. (5.4)
The ollowing heo em desc ibes a necessa y and sufficien condi ion o local s abili y o
sys em (5.3)abou (p,z).
Theo em5.1 The posi i e cons an solu ion (p,z)o sys em (5.3)is locally asymp o ically
s able i and only i
|T |<1+D <2,
whe e T and D a e he ace and de e minan o (5.4), espec i ely.
Fo he unde s anding o limi a ion o modified hyb id con ol echnique, we ha e he
ollowing ema k.
Rema k Like he hyb id me hod [52], he modified hyb id me hod (5.2)is easibleand
efficien o hose disc e e- ime ma hema ical models o which he s epsize pa ame e is
aken as a bi u ca ion pa ame e .
6 Hyb id con ol o Neima k–Sacke bi u ca ion
In hissec ion,weapply hehyb id echnique[52] osys em(1.5) ocon ol heNeima k–
Sacke bi u ca ion.Mo eo e , hisme hodisusedascon ols a egybymany esea che s
o con olling he pe iod-doubling bi u ca ion, Neima k–Sacke bi u ca ion, and chaos
unde he effec s o pe iod-doubling bi u ca ion (see [54,55]). By applica ion o a hyb id
me hod [52] osys em(1.5)wege he ollowingsys em:
⎧
⎪
⎨
⎪
⎩
pn+1 =S1((1+h )pn
1+h(
kpn+αzn
a+pn+m1p2
n+q1E))+(1–S1)pn,
zn+1 =S1((1+hβpn
a+pn)zn
1+h(ρpn
a+pn+δ+m2zn+q2E))+(1–S1)zn,(6.1)
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 19 o 29
whe e0<S1<1isacon olpa ame e .Fu he mo e,sys ems(6.1)and(1.5)ha e hesame
cons an solu ions. Addi ionally, he Jacobian ma ix o (6.1)abou (p,z)is
⎛
⎝1–hS1p((a–k) +ekq1+p(2 +km1(2a+3p)))
k(1+h )(a+p)–hS1αp
(1+h )(a+p)
hS1z(–β+δ+ρ+eq2+m2z)2
a(β+hβδ–ρ+ehβq2+hβm2z)
β+hβδ–ρ+ehβq2+h(β–S1β+S1ρ)m2z
β+hβδ–ρ+ehβq2+hβm2z⎞
⎠. (6.2)
7 Nume ical simula ion
In his sec ion, we nume ically s udy he dynamics o (1.5). This s udy is a di ec e i-
fica ion o ou heo e ical analysis and analy ic esul s we ha e p o ed in he p e ious
sec ions. Pa icula ly, in his sec ion, we s udy he exis ence and di ec ion o Neima k–
Sacke bi u ca ion by using nume ic alues o he pa ame e s. In addi ion, in his sec ion,
we ake he ini ial condi ions in he leas neighbo hood o he equilib ium poin (p,z) o
each case s udy.
Example 7.1 Le a= 2.0099,q1= 0.0189,q2= 1.2994, = 10.5923,E= 0.9959,k= 1.3997,
β= 98.499,α= 2.9999,δ= 0.0384,ρ= 10.5842,m1= 0.6222,m2= 0.4422,p0= 0.105348,
z0= 6.8884251, and h∈(0,1)]. In his case he ex inc ion equilib ium and nonex-
inc ion equilib ium o zooplank on popula ion a e (x,0) = (1.26553,0) and (p,z)=
(0.1053484,6.888425), espec i ely. Then om sys em (1.5)weha e
limsup
n→∞ pn≤k=1.3997.
Then by using he alue k= 1.3997 in he second equa ion o sys em (1.5)wege
limsup
n→∞ zn≤k(β–ρ)
m2(k+a)=81.61590194902372.
Hence we ha e (0.1053484,6.888425) ∈[0,k]×[0, k(β–ρ)
m2(k+a)] o β>ρ, which e ifies Theo-
em 2.1.
Addi ionally, we ha e
(0)= a( –q1E)
α=7.084113606170539>0
and
F(0)= a(δ+m2 (0)+q2E)
(β–ρ)–(δ+m2 (0)+q2E)=0.10754185654404144 >0.
Fu he mo e, o λ= 1.3997, we ha e (β–ρ) = 88.34599 and (δ+m2 (λ)+q2E)=
4.465067496648613. Then (β–ρ)>(δ+m2 (λ)+q2E), and we ge
F(λ)= a(δ+m2 (λ)+q2E)
(β–ρ)–(δ+m2 (λ)+q2E)–λ=–1.2921581434559586<0,
whe e
(λ)=–(a+λ)(m1 2+q1E)
α=–79.36413532682512<0.
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 20 o 29
Mo eo e , we ha e
F(λ)=–1+ a(β–ρ)(m2 (λ))
((β–ρ)–(δ+m2 (λ)+q2E))2=–1.0580185510185083<0,
whe e
(λ)=–(a+λ)(
k+2m1λ)
α+ – λ
k–m1λ2–q1E
α=–10.993342080620321<0,
which e ifies Theo em 2.2.
Example 7.2 Le a= 2.0099,q1= 0.0189,q2= 1.2994, = 10.5923,E= 0.9959,k= 1.3997,
β= 98.499,α= 2.9999,δ= 0.0384,ρ= 10.5842,m1= 0.6222,m2= 0.4422,p0= 0.105348,
z0=6.8884251, and h∈(0,1)]. Then sys em (1.5) akes he o m
⎧
⎪
⎨
⎪
⎩
pn+1 =(1+10.5923h)pn
1+h(10.5923
1.3997 pn+2.9999zn
2.0099+pn+0.6222p2
n+0.0188),
zn+1 =(1+h98.499pn
2.0099+pn)zn
1+h(10.5842pn
2.0099+pn+0.0384+0.4422zn+1.2941).(7.1)
Addi ionally, in his case he ex inc ion equilib ium and nonex inc ion equilib ium o
zooplank onpopula iona e(x,0)=(1.26553,0)and(p,z)=(0.1053484,6.888425), espec-
i ely. In his case he g aphical beha io o bo h popula ion a iables is shown in Fig. 2.
In addi ion, Fig. 2(c) ep esen s he maximum Lyapuno exponen o sys em (7.1). In
Fig. 3,somephasepo ai sa egi en,whe eh a ies in ]0,1[. We can easily see ha he e
exis s he Neima k–Sacke bi u ca ion when hce ainly passes h ough h= 0.38022 (see
Fig. 3(b)). Fo he a o emen ioned alues o pa ame e s, he Jacobian ma ix V2(p,z) o
sys em (7.1)is
V2(0.1053484433,6.88842511)
=0.9751639697174742 –0.01143564868241927
36.869495419357726 0.5871690414953413 .
Mo eo e , he cha ac e is ic equa ion M(ξ)=0 o V2(0.1053484433,6.88842511) is
ξ2–1.8038899298174336ξ+1 =0. (7.2)
Sol ing (7.2), we ge ξ1= 0.7811665056064 + 0.6196705420078iand ξ2=
0.7811665056064–0.6196705420078iwi h |ξ1|=|ξ1|=1. In addi ion, we ha e
M(–1)=3.556545701326458 >0
and
M(1)=0.43187967890082724 >0.
Now om (4.3)weha e
˘
(P,Z)=0.105348+0.974755P–0.0116237Z–0.256573P2–0.099269PZ
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 21 o 29
+0.001282Z2–0.142822P3+0.10288P2Z+0.010039PZ2
–0.000141506Z3+O|P|+|Z|4
and
˘
g(P,Z)=6.88842+0.574993Z+37.95688P–43.12450P2+3.450296PZ
–0.0354763Z2+48.99565P3–2.366456P2Z–0.1893441PZ2
+0.00218884Z3+O|P|+|Z|4.
Finally,whensys em(4.3)iscon e edin o hecanonical o m (4.6),weob ain hema ix
1
12 0
– 11
℘ 12 –1
℘=–0.01143564869 0
–0.1939974644 –0.6196705420078615
wi h
1
12 0
– 11
℘ 12 –1
℘–1
=–87.4458482512197577 0.0
27.3762776879404903 –1.61376075222132043.
Fu he mo e, om (4.8)weha e
˘
F(X,Y)=22.073323320819P2+8.5402951060818PZ –0.1103358266003Z2
+12.2871550765P3–8.85102538857P2Z–0.8636988319698PZ2
+0.012173994624Z3+O|X|+|Y|4
and
˘
G(X,Y)=61.114564266701P2–8.141785570023PZ +0.0908219975961Z2
–81.22561954155P3+6.52880078485P2Z+0.571452207625PZ2
–0.0072969596695Z3+O|X|+|Y|4.
Addi ionally,plo s o ˘
F(X,Y)and˘
G(X,Y)wi hsolu iona (0,0)a ep esen edinFigs.1(a)
and 1(b), espec i ely, whe e P= (–0.1162370758)Xand Z= (–0.1998810610)X–
(0.6334408575)Y. Finally, we ge
θ20 =1
8˘
FXX –˘
FYY +2˘
GXY +i(˘
GXX –˘
GYY –2˘
FXY)=0.005831–0.019066i,
θ11 =1
4˘
FXX +˘
FYY +i(˘
GXX +˘
GYY)=–0.01195+0.013959i,
θ02 =1
8˘
FXX –˘
FYY –2˘
GXY +i(˘
GXX –˘
GYY +2˘
FXY)=0.02389–0.00182i,
θ21 =1
16(˘
FXXX +˘
FXYY +˘
GXXY +˘
GYYY)
+i
16(˘
GXXX +˘
GXYY –˘
FXXY –˘
FYYY)= 0.00755+0.00574i,
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 22 o 29
Figu e 1 Plo s o ˘
F(X,Y)and˘
G(X,Y) o a= 2.0099, q1= 0.0189, q2= 1.2994, = 10.5923, E= 0.9959,
k= 1.3997, β= 98.499, α= 2.9999, δ= 0.0384, ρ= 10.5842, m1= 0.6222, m2= 0.4422, and h∈(0,1)
and
=–Re(1–2ξ1)ξ2
2
1–ξ1θ20θ11–1
2|θ11|2–|θ02|2+Re(ξ2θ21)ˆ
h=0
=–0.00138464412<0.
Hence he condi ion o he exis ence o Neima k–Sacke bi u ca ion is sa isfied (see
Theo em 4.1).
Example 7.3 This example is ela ed o he s udy o con ol o Neima k–Sacke bi u ca-
ion by using gene alized hyb id echnique (5.2). To show he effec i eness o gene alized
echnique, we ha e used he same alues o pa ame e s as in Example 7.1. Conside he
ollowing sys em o diffe ence equa ions:
⎧
⎪
⎨
⎪
⎩
pn+1 =L3((1+10.5923h)pn
1+h(10.5923
1.3997 pn+2.9999zn
2.0099+pn+0.6222p2
n+0.0188))+(1–L3)pn,
zn+1 =L3((1+h98.499pn
2.0099+pn)zn
1+h(10.5842pn
2.0099+pn+0.0384+0.4422zn+1.2941))+(1–L3)zn,(7.3)
whe e a=2.0099,q1= 0.0189,q2= 1.2994, =10.5923,E=0.9959,k=1.3997,β= 98.499,
α= 2.9999,δ= 0.0384,ρ= 10.5842,m1= 0.6222,m2= 0.4422,h= 0.699909. In addi-
ion, 0 < L< 1 is he con ol pa ame e . Fu he mo e, o sys em (7.3), we ha e (p,z)=
(0.1053484433,6.88842511), which is a unique posi i e cons an solu ion o he o iginal
sys em (1.5). Addi ionally, con olled diag ams o zooplank on and phy oplank on pop-
ula ions by using models (7.3)a eshowninFigs.4(b) and 4(a), espec i ely. Finally, we
can see ha he s abili y o ini ial sys em (1.5) is ic o iously egained o la ge ange o
con ol pa ame e by using he gene alized hyb id con ol me hod (see Fig. 4).
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 23 o 29
Figu e 2 Plo s o sys em (1.5) o a= 2.0099, q1= 0.0189, q2= 1.2994, = 10.5923, E= 0.9959, k= 1.3997,
β= 98.499, α= 2.9999, δ= 0.0384, ρ= 10.5842, m1= 0.6222, m2= 0.4422, and h∈(0,1)
Example7.4 Thisexampleis ela ed o hes udyo con olo Neima k–Sacke bi u ca ion
by using a hyb id echnique [52] o con ol. Conside he sys em o diffe ence equa ions
⎧
⎪
⎨
⎪
⎩
pn+1 =S1((1+10.5923h)pn
1+h(10.5923
1.3997 pn+2.9999zn
2.0099+pn+0.6222p2
n+0.0188))+(1–S1)pn,
zn+1 =S1((1+h98.499pn
2.0099+pn)zn
1+h(10.5842pn
2.0099+pn+0.0384+0.4422zn+1.2941))+(1–S1)zn,(7.4)
whe e a=2.0099,q1= 0.0189,q2= 1.2994, =10.5923,E=0.9959,k=1.3997,β= 98.499,
α= 2.9999,δ= 0.0384,ρ= 10.5842,m1= 0.6222,m2= 0.4422,h= 0.699909. In addi ion,
0<S1< 1 is he con ol pa ame e . Fu he mo e, o sys em (7.4), we ha e (p,z)=
(0.1053484433,6.88842511), which is a unique posi i e cons an solu ion o he o iginal
sys em (1.5). Addi ionally, con olled diag ams o zooplank on and phy oplank on pop-
ula ions o sys em (7.4) a e espec i ely shown in Figs. 5(b)and5(a).
Example 7.5 In his example, we compa e he gene alized hyb id me hod and hy-
b id me hod [52]. F om Examples 7.3 and 7.4 we conside wo disc e e- ime mod-
els (7.3)and(7.4), espec i ely. Mo eo e , in his case, we ha e aken L,S1∈]0,1[ and
a= 2.0099,q1= 0.0189,q2= 1.2994, = 10.5923,E= 0.9959,k= 1.3997,β= 98.499,α=
2.9999,δ=0.0384,ρ=10.5842,m1=0.6222,m2=0.4422, and h∈(0,1).
Fo m bo h sys ems (7.3)and(7.4)wege (p,z)=(0.1053484433,6.88842511) as a nique
posi i efixedpoin .Addi ionally, omTable1wecanobse e ha |I1|>|I2| o each a i-
a iono hepa ame e h∈]0,1[,whe eI1andI2a e hecon olledin e alsco esponding
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 24 o 29
Figu e 3 Phase po ai s o sys em (1.5) o a= 2.0099, q1= 0.0189, q2= 1.2994, = 10.5923, E= 0.9959,
k= 1.3997, β= 98.499, α= 2.9999, δ= 0.0384, ρ= 10.5842, m1= 0.6222, m2= 0.4422, and h∈(0,1)
o he con olled sys ems (7.3)and(7.4), espec i ely.Hencewecansee om Table1 ha
he gene alized hyb id me hod (5.2) is much be e han he old hyb id me hod [52].
Example 7.6 In his example, we compa e he dynamics o sys ems (1.3)and(1.5).
Fo case(i),we akea=2.1,q1= 0.09,q2=0.3, =1.5,c= 0.14,k= 100,β=0.5,α=
0.69,δ= 0.001,ρ=0.1,m2= 0.021, and m1= 0.06. Then we ge he fixed poin (p,z)=
(2.3059,7.38854),whichisauniqueposi i econs an solu iono (1.3)and(1.5).Mo eo e ,
o he ini ial condi ions p0= 2.3059 and z0= 7.38854, Figs. 6(a) and 7(b) a e plo ed o
sys ems(1.5)and(1.3), espec i ely. Consequen ly, we cansee ha sys ems(1.5)and(1.3)
a es ablea (p,z) = (2.3059,7.38854) o m1=0.06(seeFigs.6(a) and 7(b)). In addi ion,
Khan e al. Ad ances in Diffe ence Equa ions (2021) 2021:443 Page 25 o 29
Figu e 4 Con olled diag ams o sys em (7.1) o a= 2.0099, q1= 0.0189, q2= 1.2994, = 10.5923, E= 0.9959,
k= 1.3997, β= 98.499, α= 2.9999, δ= 0.0384, ρ= 10.5842, m1= 0.6222, m2= 0.4422, h= 0.699909 and
L∈(0,1)
Figu e 5 Con olled diag ams o sys em (7.4) o a= 2.0099, q1= 0.0189, q2= 1.2994, = 10.5923, E= 0.9959,
k= 1.3997, β= 98.499, α= 2.9999, δ= 0.0384, ρ= 10.5842, m1= 0.6222, m2= 0.4422, h= 0.699909, and
S1∈(0,1)
Table 1 Compa ison o he modified hyb id me hod (5.2) and hyb id me hod [52] o L,S1∈]0,1[ and
a= 2.0099, q1= 0.0189, q2= 1.2994, = 10.5923, E= 0.9959, k= 1.3997, β= 98.499, α= 2.9999,
δ= 0.0384, ρ= 10.5842, m1= 0.6222, m2= 0.4422, h∈(0,1)
h∈]0,1[ Con olled in e al I1 o (7.3) Con olled in e al I2 o (7.4)
0.48889569 0 < L< 0.99288325491279 0 < S1< 0.97880134847096
0.58889569 0 < L< 0.983282763587931 0 < S1< 0.95068201684448
0.68889569 0 < L< 0.97635403882826 0 < S1< 0.93072628972275
0.78889569 0 < L< 0.97111704728074 0 < S1< 0.91582972183557
0.88889569 0 < L< 0.96701917628990 0 < S1< 0.90428485868004
0.98889569 0 < L< 0.96372500134675 0 < S1< 0.89507489723917
o case (ii), we ake m1= 0.025, a=2.1,q1= 0.09,q2=0.3, =1.5,c= 0.14,k= 100,β=
0.5,α=0.69,δ=0.001,ρ=0.1,m2=0.021. We ge he fixed poin (p,z)=(2.8059,8.8854),
whichis a unique posi i e cons an solu iono (1.3)and(1.5).Hencewecansee ha bo h
sys ems (1.3)and(1.5)a euns ablea (p,z)=(2.8059,8.8854) (see Figs. 6(b)and 7(a)). Fi-
nally, Fig. 6(c), (d) shows he exis ence o Neima k–Sacke bi u ca ion in sys em (1.5) o
lowe alues o s epsize h,andFig.7(c), (d) shows ha bo h a iables p( )andz( ) om
sys em (1.3)a euns ablea (p,z)=(2.8059,8.8854) (see [25]).
