Ci a ion: Mondal, S.; Saman a, G.;
De la Sen, M. Dynamics o
Oxygen-Plank on Model wi h
Va iable Zooplank on Sea ch Ra e in
De e minis ic and Fluc ua ing
En i onmen s. Ma hema ics 2022,10,
1641. h ps://doi.o g/10.3390/
ma h10101641
Academic Edi o s: Sophia Jang and
Jui-Ling Yu
Recei ed: 3 Ap il 2022
Accep ed: 5 May 2022
Published: 11 May 2022
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ma hema ics
A icle
Dynamics o Oxygen-Plank on Model wi h Va iable
Zooplank on Sea ch Ra e in De e minis ic and
Fluc ua ing En i onmen s
Sudeshna Mondal 1, Gu up asad Saman a 1and Manuel De la Sen 2,*
1Depa men o Ma hema ics, Indian Ins i u e o Enginee ing Science and Technology,
Shibpu , How ah 711103, India; [email p o ec ed] (S.M.);
[email p o ec ed] o [email p o ec ed] (G.S.)
2Ins i u e o Resea ch and De elopmen o P ocesses, Uni e si y o he Basque Coun y,
48940 Leioa, Bizkaia, Spain
*Co espondence: [email p o ec ed]
Abs ac :
I is es ima ed by scien is s ha 50–80% o he oxygen p oduc ion on he plane comes om
he oceans due o he pho osyn he ic ac i i y o phy oplank on. Some o his p oduc ion is consumed
by bo h phy oplank on and zooplank on o cellula espi a ion. In his a icle, we ha e analyzed
he dynamics o he oxygen-plank on model wi h a modi ied Holling ype II unc ional esponse,
based on he p emise ha zooplank on has a a iable sea ch a e, a he han cons an , which is
ecologically meaning ul. The posi i i y and uni o m boundedness o he s udied sys em p o e
ha he model is well-beha ed. The easibili y condi ions and s abili y c i e ia o each equilib ium
poin a e discussed. Nex , he occu ence o local bi u ca ions a e exhibi ed aking each o he i al
sys em pa ame e s as a bi u ca ion pa ame e . Nume ical simula ions a e illus a ed o e i y he
analy ical ou comes. Ou indings show ha (i) he sys em dynamics change ab up ly o a low
oxygen p oduc ion a e, esul ing in deple ion o oxygen and plank on ex inc ion; (ii) he p oposed
sys em has oscilla o y beha io in an in e media e ange o oxygen p oduc ion a es; (iii) i always has
a s able coexis ence s eady s a e o a high oxygen p oduc ion a e, which is dissimila o he ou come
o he model o a coupled oxygen-plank on dynamics whe e zooplank on consumes phy oplank on
wi h classical Holling ype II unc ional esponse. Las ly, he e ec o en i onmen al s ochas ici y is
s udied nume ically, co esponding o ou p oposed sys em.
Keywo ds: oxygen-plank on model; modi ied Holling ype II; s abili y analysis; local bi u ca ions
MSC: 37M05; 92D25; 92D40
1. In oduc ion
Plank on a e he nume ous se ies o o ganisms obse ed in wa e o ai ha a e no able
o p opel hemsel es agains wa e cu en s o wind, espec i ely. The indi idual o ganisms
cons i u ing plank on a e known as plank e s. In he ocean, hey o e a i al sou ce o
meals o many small and massi e aqua ic o ganisms, including bi al es, ish and whales.
The plan ypes o he plank on communi y a e e e ed o as phy oplank on, hey acqui e
hei s eng h h ough pho osyn hesis, as do ees and di e en plan s on land. This means
phy oplank on need o ha e sola ligh , so hey li e wi hin he p ope ly-li loo laye s
o oceans and lakes. Zooplank on a e he animal componen s o he plank onic ne wo k,
and hey a e he p inciple ood supply o ish and o he aqua ic animals. Phy oplank on
a e no he bes meal sou ce o zooplank on; howe e , hey o e a massi e quan i y o
oxygen o human and di e en dwelling animals a e soaking up ca bon dioxide ia
pho osyn hesis om he en i onmen . Some o his oxygen p oduc ion is consumed by bo h
phy oplank on and zooplank on because o espi a ion [
1
,
2
]. Fu he mo e, a dec ease in he
Ma hema ics 2022,10, 1641. h ps://doi.o g/10.3390/ma h10101641 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 1641 2 o 24
oxygen p oduc ion a e by phy oplank on may ha e a disas ous e ec o li ing animals,
including humankind. The e o e, he s udy o he possible ange o oxygen p oduc ion
a es is impo an o sus ain sys em dynamics.
Ma hema ical modeling is a esea ch ool ha can e eal he dynamic p ope ies o
he oxygen-plank on model. Recen ly, esea che s ha e analyzed a ma hema ical model o
oxygen-plank on in e ac ions wi ha Holling ype II unc ional esponse [
3
–
5
], whe e he
sea ch a e o he p eda o popula ion was cons an , i.e., independen o he p ey popula-
ion [
6
–
8
]. Howe e , i seems easonable ha p eda o s can a y hei sea ch a es based
on he a ailabili y o p ey. In 1977, Hassel e al. [
9
] expe imen ally obse ed ha he
sea ch a e o a ious in e eb a e p eda o s, speci ically zooplank on, depended on he
biomass o he p ey (phy oplank on) popula ion. In 2020, Dalziel e al. [
10
] analyzed he
dynamics o a p eda o –p ey model wi h a a iable p eda o sea ch a e. In 2021, Mondal
and Saman a [
11
] s udied he dynamic na u e o a p eda o –p ey model wi h he impac
o a p eda o ’s ea , whe e he sea ch a e o he p eda o depended on he biomass o he
p ey species. Recen ly, hey also in es iga ed he dynamic beha io o a oxin-p oducing
plank on model whe e he zooplank on’s sea ch a e depended on he biomass o he phy-
oplank on popula ion, a he han being assumed cons an [
12
]. Mo i a ed om he abo e
discussions, we p oposed and analyzed he dynamic beha io o he oxygen-plank on
model wi h a a iable zooplank on sea ch a e, a he han cons an , whe e oxygen is p o-
duced by he pho osyn he ic ac i i y o phy oplank on du ing he day ime and consumed
by phy o and zooplank on o hei espi a ion.
This a icle is o ganized as ollows: we ha e ocused on he cons uc ion o he basic
model in Sec ion 2. The de i a ion o posi i i y and uni o m boundedness is shown in Sec-
ion 3. Sec ion 4desc ibes he easibili y c i e ia and s abili y condi ions o all he equilib ia.
Fu he mo e, he occu ence o local bi u ca ions a e exhibi ed in Sec ion 5. In Sec ion 6,
we conduc nume ical simula ions using MATLAB o alida e he analy ical indings. The
impac o he oxygen p oduc ion a e on he exis ence o he in e io equilib ium poin
as well as he main quali a i e di e ence be ween he p oposed model and he sys em
analyzed by Seke ci and Pe o skii [
3
] a e discussed. This sec ion also consis s o he e ec
o en i onmen al s ochas ici y on he p oposed oxygen-plank on model by pe u bing
some pa ame e s o he sys em wi h Gaussian whi e noise e ms. This wo k ends wi h a
discussion and he ou comes o he analy ical consequences.
2. Cons uc ion o Basic Model
A ma ine ecosys em is a complica ed sys em wi h many nonlinea ly in e ac ing species,
o ganic subs ances, and ino ganic chemical componen s. Co espondingly, a " ealis ic”
ecosys em model can consis o many equa ions. In his a icle, we a e mos ly in e es ed in
he dynamics o he oxygen-plank on model, whe e oxygen is p oduced by he pho osyn-
he ic ac i i y o phy oplank on.
Re isi ing an oxygen-plank on model sys em gi en in [
3
,
5
] and aking a modi ied
Holling ype II unc ional esponse, whe e he sea ch a e o he p eda o (zooplank on)
depends on he biomass o he p ey (phy oplank on), a he han being cons an ( o de ails,
see [10–12]), we conside he ollowing model ( o de ails see Figu e 1):
dc
d =Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc
dp
d =Bc
c+c1−γpp−ap2z
ahp2+p+g−σp(1)
dz
d = ηc2
c2+c2
4!.ap2z
ahp2+p+g−µz
wi h ini ial condi ions:
c(0)>0, p(0)>0, z(0)>0. (2)
Ma hema ics 2022,10, 1641 3 o 24
He e,
c
is he amoun o oxygen, and
p
and
z
a e he biomass o phy oplank on and
zooplank on, espec i ely. All he pa ame e s a e posi i e due o hei biological meaning
and a e desc ibed in Table 1:
Table 1. Desc ip ion o biologically meaning ul pa ame e s.
Pa ame e s Desc ip ions
Ae ec o en i onmen al ac o s on he a e o oxygen p oduc ion due o he pho osyn hesis o phy oplank on
δmaximum pe capi a phy oplank on espi a ion a e
νmaximum pe capi a zooplank on espi a ion a e
m a e o oxygen loss due o he biochemical eac ion in a ma ine ecosys em
Bmaximum phy oplank on pe capi a g ow h a e in he high oxygen limi
ci,i=0, 1, 2, 3, 4 hal sa u a ion cons an o he co esponding p ocesses
γmo ali y a e due o in aspeci ic compe i ion among indi idual phy oplank on
amaximally achie able sea ch a e o zooplank on
hhandling ime o zooplank on
ghal sa u a ion cons an
σna u al mo ali y a e o phy oplank on. I is assumed ha B>σ
η∈(0, 1)maximum eeding e iciency
µmo ali y a e o zooplank on
μz
ap2z
ahp2+p+g
νcz
c+c3
δcp
c+c2
mc
γp2+σ p
Ac0p
c+c0
zooplank on
phy oplank on oxygen
Figu e 1.
G aphical scheme ep esen ing he in e ac ions among oxygen, phy oplank on, and zoo-
plank on, whe e phy oplank on p oduce oxygen h ough pho osyn he ic ac i i y in sunligh and
consume i du ing he nigh o hei espi a ion; zooplank on depend on phy oplank on o hei
g ow h and consume oxygen o hei espi a ion.
Desc ip ion o sys em (1):
•
The e m
Ac0
c+c0
desc ibes he a e o oxygen p oduc ion pe uni o phy oplank on
biomass du ing he day ime by pho osyn he ic ac i i y;
δcp
c+c2
and
νcz
c+c3
indica e he
espi a ion o phy oplank on and zooplank on, espec i ely, and
mc
is he loss o
oxygen due o na u al deple ion in a ma ine ecosys em.
•
The e m
Bcp
c+c1
desc ibes he g ow h o phy oplank on depending on he amoun o
a ailable oxygen. The unc ion
ap2
ahp2+p+g
is named as a modi ied Holling ype II unc-
ional esponse, based on he p emise ha he zooplank on’s sea ch a e is dependen
on he biomass o phy oplank on, a he han being cons an ( o de ails,
see [10,11]
).
Again, he consumed phy oplank on biomass is ans o med in o zooplank on biomass
Ma hema ics 2022,10, 1641 4 o 24
wi h an e iciency o
ηc2
c2+c2
4
, which depends on he oxygen concen a ion (zooplank on
die due o insu icien oxygen).
The ollowing a e p ope ies o a modi ied Holling ype II unc ional esponse
H(p) = ap2
ahp2+p+g
1. H(p)is a smoo h unc ion, and H(p) = 0 o p=0.
2. H0(p) = ap(p+2g)
(ahp2+p+g)2>
0, i.e.,
H
inc eases wi h
p
and
lim
p→∞H(p) = 1
h
, i.e.,
H(p)
sa u a es a 1
h o a la ge p ey popula ion.
3. H00(p) = −2a2hp3−6a2ghp2+2ag2
(ahp2+p+g)3
, and
H00(p)p=0=2a
g>
0. The e o e,
H00(p)
has a
unique posi i e oo , and i changes sign om posi i e o nega i e a he unique in-
lec ion poin . A g aphical ep esen a ion o
H(p)
and
H00(p)
is p esen ed in Figu e 2.
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
p
H(p)
(a)p e ses H(p)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−4
−2
0
2
4
6
8
10
p
H"(p)
(b)p e ses H00(p)
Figu e 2.
G aphical ep esen a ions o (
a
)
H(p)
and (
b
)
H00(p)
o he pa ame ic se
{a=
3,
h=
1.2,
g=0.3}.
3. Posi i i y and Uni o m Boundedness
Theo em 1. Solu ions o (1) wi h (2) exis uniquely and a e posi i e o all ≥0.
P oo .
Since he igh hand sides o (1) a e comple ely con inuous unc ions and locally
Lipschi zian in he domain
R3
+
, solu ions o (1) wi h (2) exis uniquely in
[
0,
ξ)
, whe e
0<ξ≤∞[13].
F om he i s equa ion o (1), we ha e:
c( ) = c(0)exp−Z
0δp(θ)
c(θ) + c2
+νz(θ)
c(θ) + c3
+md(θ)
+Z
0
Ac0p(u)
c(u) + c0expZu
δp(θ)
c(θ) + c2
+νz(θ)
c(θ) + c3
+md(θ)du >0,
since c(0)>0.
F om he second equa ion o sys em (1), we ha e:
p( ) = p(0)expZ
0Bc(θ)
c(θ) + c1−γp(θ)−ap(θ)z(θ)
ahp2(θ) + p(θ) + g−σdθ>0,
since p(0)>0.
F om he las equa ion o sys em (1), we ha e:
z( ) = z(0)exp"Z
0( ηc2(θ)
c2(θ) + c2
4!.ap2(θ)
ahp2(θ) + p(θ) + g−µ)dθ#>0, since z(0)>0.
Ma hema ics 2022,10, 1641 5 o 24
The e o e, c( )>0, p( )>0 and z( )>0 o all ≥0.
Hence, he heo em is p o ed.
Theo em 2. Solu ions o (1) wi h (2) a e uni o mly bounded.
P oo . F om he second equa ion o sys em (1), we ob ain:
dp
d ≤Bp −γp2−σp
= (B−σ)p(1−p
B−σ
γ)
⇒lim sup
→∞
p( )≤B−σ
γ.
Le
Ω=c+p+z.
Then,
dΩ
d =dc
d +dp
d +dz
d
=Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc +Bc
c+c1−γpp−ap2z
ahp2+p+g−σp
+ ηc2
c2+c2
4!ap2z
ahp2+p+g−µz
≤Ac0p
c+c0
+Bcp
c+c1
+ap2z
ahp2+p+g ηc2
c2+c2
4−1!−γp2−{mc +σp+µz}
≤Ac0p
c+c0
+Bcp
c+c1−γp2−{mc +σp+µz}, since 0 <η<1
≤(A+B)p−γp2−{mc +σp+µz}
≤(A+B)2
4γ−{mc +σp+µz}. (3)
Le
κ=min{m,σ,µ}.
Then, om (3), we ob ain:
dΩ
d +κΩ≤(A+B)2
4γ.
Using he di e en ial inequali y:
0<Ω(c( ),p( ),z( ))≤(A+B)2
4γκ 1−e−κ +e−κ Ω(c(0),p(0),z(0)).
∴0<Ω(c( ),p( ),z( ))≤(A+B)2
4γκ +e, o any e>0, as →∞.
Hence, e e y solu ion o (1) en e s in o he egion:
W=(c,p,z)∈R3
+: 0 <p( )≤B−σ
γ; 0 <c( ) + p( ) + z( )≤(A+B)2
4γκ +e,e>0.
Ma hema ics 2022,10, 1641 6 o 24
4. Exis ence o Equilib ia o (1) wi h S abili y Analysis
4.1. Equilib ium Poin s
Sys em (1) has he ollowing equilib ium poin s (s eady s a es):
1.
T i ial equilib ium poin
E0(
0, 0, 0
)
co esponding o deple ion o oxygen and he
ex inc ion o plank on;
2.
Plane equilib ium poin
E1(e
c,e
p, 0)
(zooplank on ee), whe e
e
p=1
γBe
c
e
c+c1−σ
, and
e
cis a posi i e oo o he ollowing equa ion:
X1c4+X2c3+X3c2+X4c+X5=0.
He e,
X1=−mγ
,
X2=−(c1+c2)−c0+ (B−γ)δ
,
X3=−c1c2−c0(c1+c2) + (B−
γ)(A−δ)c0+δγ,X4=−c0c1c2+ (B−γ)Ac0c2−γc0(A−δ),X5=−γAc0c1c2.
3.
In e io (coexis ence) equilib ium
b
E(b
c
,
b
p
,
bz)
, whe e
b
c
,
b
p
, and
bz
can be ob ained by
sol ing he ollowing sys em o equa ions using he so wa e MATHEMATICA:
Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc =0,
Bc
c+c1−γp−apz
ahp2+p+g−σ=0,
ηc2
c2+c2
4!.ap2
ahp2+p+g−µ=0.
4.2. Local S abili y
Now, we will de e mine he s abili y beha io o he biologically easible equilib ium
poin s o sys em (1).
The Jacobian ma ix J0a E0(0, 0, 0)is gi en by:
J0=
−m A 0
0−σ0
0 0 −µ
.
He e, he eigen alues a e
λ1=−m<
0,
λ2=−σ<
0, and
λ3=−µ<
0. Since all
eigen alues a e nega i e, so, E0(0, 0, 0)is always locally asymp o ically s able (LAS).
The Jacobian ma ix J1a E1(e
c,e
p, 0)is gi en by:
J1=
−Ac0e
p
(e
c+c0)2−δc2e
p
(e
c+c2)2−mme
c
e
p−νe
c
e
c+c3
Bc1e
p
(e
c+c1)2−γe
p−ae
p2
ahe
p2+e
p+g
0 0 ηe
c2
e
c2+c2
4ae
p2
ahe
p2+e
p+g−µ
.
He e, one eigen alue is
λ1=ηe
c2
e
c2+c2
4ae
p2
ahe
p2+e
p+g−µ
, and he o he eigen alues can be
ob ained by sol ing he equa ion:
λ2−Q1λ+Q2=0, (4)
whe e
Q1=−Ac0e
p
(e
c+c0)2−δc2e
p
(e
c+c2)2−m−γe
p<
0 and
Q2=γe
phAc0e
p
(e
c+c0)2+δc2e
p
(e
c+c2)2+mi−
Bc1me
c
(e
c+c1)2>0.
Hence, we ha e he ollowing heo em:
Theo em 3. E1(e
c,e
p, 0)is LAS i ηe
c2
e
c2+c2
4ae
p2
ahe
p2+e
p2+g−µ<0.
Ma hema ics 2022,10, 1641 7 o 24
The Jacobian ma ix bJa b
E(b
c,b
p,bz)is gi en by:
bJ=
a11 a12 a13
a21 a22 a23
a31 a32 a33
whe e
a11 =−Ac0b
p
(b
c+c0)2−δc2b
p
(b
c+c2)2−νc3bz
(b
c+c3)2−m<
0,
a12 =−δb
c
b
c+c2+Ac0
b
c+c0=b
c
b
pnνbz
b
c+c3+mo>
0,
a13 =−νb
c
b
c+c3
<
0,
a21 =Bc1b
p
(b
c+c1)2>
0,
a22 =Bb
c
b
c+c1−
2
γb
p−ab
pbz(b
p+2g)
(ahb
p2+b
p+g)2−σ=−γb
p−
ab
pbz(g−ahb
p2)
(ahb
p2+b
p+g)2
,
a23 =−ab
p2
ahb
p2+b
p+g<
0,
a31 =2ηc2
4b
c
(b
c2+c2
4)2
ab
p2bz
ahb
p2+b
p+g>
0,
a32 =ηb
c2
(b
c2+c2
4)
ab
pbz(b
p+2g)
(ahb
p2+b
p+g)2>0
and a33 =0.
The cha ac e is ic equa ion co esponding o b
E(b
c,b
p,bz)is
λ3+C1λ2+C2λ+C3=0
whe e
C1=−(a11 +a22)
,
C2=−a23a32 −a13a31 +a11a22 −a12a21
, and
C3=−{−a11a23a32 +
a12a23a31 +a13(a21a32 −a22a31)}.
By Rou h-Hu wi z’s c i e ia [
14
],
b
E(b
c
,
b
p
,
bz)
has h ee eigen alues wi h nega i e eal
pa s i
C1>
0,
C3>
0, and
C1C2>C3
. So, he local s abili y condi ion o
b
E(b
c
,
b
p
,
bz)
is
desc ibed in he ollowing heo em:
Theo em 4. b
E(b
c,b
p,bz)is LAS i a22 <0and a11a22 >a12a21.
5. Local Bi u ca ions
A local bi u ca ion occu s when a pa ame e change causes he s abili y (o ins abili y)
o an equilib ium (o ixed poin ) o change. In con inuous sys ems, his co esponds o he
eal pa o an eigen alue o an equilib ium passing h ough ze o.
5.1. T ansc i ical Bi u ca ion
Theo em 5. Sys em (1) unde goes a ansc i ical bi u ca ion i µ[ c]=ηe
c2
e
c2+c2
4ae
p2
ahe
p2+e
p+g.
P oo .
To p o e a ansc i ical bi u ca ion, we apply So omayo ’s heo em [
14
] by consid-
e ing
µ
as he bi u ca ion pa ame e . Acco ding o his heo em, one eigen alue o
J1
a he
bi u ca ion poin mus be ze o.
The eigen ec o s o
J1= [pij]
and
(J1)T
co esponding o he ze o eigen alue a e
ob ained as:
V=(0, 2, 1)T
and
W=(0, 0, 1)T
, espec i ely, whe e
2=−p13
p12
and
p11 =−Ac0e
p
(e
c+c0)2−δc2e
p
(e
c+c2)2−m
,
p12 =me
c
e
p
,
p13 =−νe
c
e
c+c3
,
p21 =Bc1e
p
(e
c+c1)2
,
p22 =−γe
p
,
p23 =−ae
p2
ahe
p2+e
p+g, and p31 =p32 =p33 =0.
Compu e ∆1,∆2, and ∆3as ollows:
∆1=WT·Fµe
c,e
p, 0; µ[ c]= (0, 0, 1)·
∂F1
∂µ
∂F2
∂µ
∂F3
∂µ
(E1(e
c,e
p,0);µ[ c])
⇒∆1= (0, 0, 1)·
0
0
−z
(E1(e
c,e
p,0);µ[ c])
=0,
whe e F=(F1,F2,F3)T, and F1,F2, and F3a e gi en by:
F1=Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc,
F2=Bc
c+c1−γpp−ap2z
ahp2+p+g−σp,
F3=ηc2
c2+c2
4·ap2z
ahp2+p+g−µz.
Ma hema ics 2022,10, 1641 8 o 24
∆2=WT·hDFµe
c,e
p, 0; µ[ c]Vi= (0, 0, 1)·
∂2F1
∂c∂µ
∂2F1
∂p∂µ
∂2F1
∂z∂µ
∂2F2
∂c∂µ
∂2F2
∂p∂µ
∂2F2
∂z∂µ
∂2F3
∂c∂µ
∂2F3
∂p∂µ
∂2F3
∂z∂µ
(E1(e
c,e
p,0);µ[ c])
·
0
2
1
⇒∆2= (0, 0, 1)·
0 0 0
0 0 0
0 0 −1
(E1(e
c,e
p,0);µ[ c])
.
0
2
1
=−16=0.
∆3=WT·hD2Fe
c,e
p, 0; µ[ c](V,V)i= (0, 0, 1)·D
∂F1
∂c 1+∂F1
∂p 2+∂F1
∂z 3
∂F2
∂c 1+∂F2
∂p 2+∂F2
∂z 3
∂F3
∂c 1+∂F3
∂p 2+∂F3
∂z 3
(E1(e
c,e
p,0);µ[ c])
.
1
2
3
⇒∆3= (0, 0, 1)·
∂2F1
∂2c 2
1+∂2F1
∂2p 2
2+∂2F1
∂2z 2
3+2∂2F1
∂c∂p 1 2+2∂2F1
∂c∂z 1 3+2∂2F1
∂p∂z 2 3
∂2F2
∂2x 2
1+∂2F2
∂2y 2
2+∂2F2
∂2z 2
3+2∂2F2
∂x∂y 1 2+2∂2F2
∂x∂z 1 3+2∂2F2
∂y∂z 2 3
∂2F3
∂2x 2
1+∂2F3
∂2y 2
2+∂2F3
∂2z 2
3+2∂2F3
∂x∂y 1 2+2∂2F3
∂x∂z 1 3+2∂2F3
∂y∂z 2 3
(E1(e
c,e
p,0);µ[ c])
⇒∆3=2ae
p(e
p+2g)
(ahe
p2+e
p+g)2×ηe
c2
(e
c2+c2
4) 26=0.
Thus, by So omayo ’s heo em [
14
], sys em (1) exhibi s a anc i ical bi u ca ion a
µ=µ[ c].
Rema k 1.
Simila ly, i can be p o ed ha sys em (1) exhibi s ansc i ical bi u ca ions aking any
one o he pa ame e s h, σ, m, η, a, and γas a bi u ca ion pa ame e .
5.2. Hop -Bi u ca ion
The cha ac e is ic equa ion o sys em (1) a b
E(b
c,b
p,bz)is gi en by
λ3+C1(A)λ2+C2(A)λ+C3(A) = 0, (5)
whe e Ci(A) o i=1, 2, 3 we e de ined ea lie .
To de e mine he Hop -bi u ca ion a ound
b
E(b
c
,
b
p
,
bz)
o sys em (1), le us conside
A
as
he bi u ca ion pa ame e . Fo his pu pose, le us i s s a e he ollowing Theo em:
Theo em 6
(Hop -Bi u ca ion Theo em [
15
])
.
I
C1(A)
,
C2(A)
, and
C3(A)
a e con inuously
di e en iable unc ions o A in a small neighbou hood o A[H]∈Rsuch ha Equa ion (5) has:
(i) a pai o imagina y eigen alues
λ=p1(A)±ip2(A)
wi h
p1(A)∈R
,
p2(A)∈R
, so
ha hey become pu ely imagina y a A =A[H]and dp1
dA |A=A[H]6=0,
(ii) he o he eigen alue is nega i e a
A=A[H]
, hen a Hop -bi u ca ion occu s a ound
b
E(b
c
,
b
p
,
bz)
a
A=A[H]
(i.e., a s abili y change o
b
E(b
c
,
b
p
,
bz)
accompanied by he c ea ion o a limi
cycle a A =A[H]).
Theo em 7.
Sys em (1) possesses a Hop -bi u ca ion a ound
b
E(b
c
,
b
p
,
bz)
when
A
passes h ough
A[H], p o ided C1(A[H])>0, C3(A[H])>0, and C1(A[H])C2(A[H]) = C3(A[H]).
P oo . A A=A[H], he oo s o he equa ion:
λ2+C2(λ+C1)=0
a e
λ1=i√C2
,
λ2=−i√C2
, and
λ3=−C1
, whe e
C1
,
C2
and
C3
a e di e en ial unc ions
o
A
. Fu he mo e, in he dele ed neighbo hood o
A[H]
, he oo s (eigen alues) a e
λ1(A) =
p1(A) + ip2(A)
,
λ2(A) = p1(A)−ip2(A)
, and
λ3=p3(A)
(
p3(A) = −C1
), whe e
pi(A)
a e eal o i=1, 2, 3.
Ma hema ics 2022,10, 1641 9 o 24
Now, we will e i y he ans e sali y condi ion:
d
dA (Re λi(A))A=A[H]6=0, i=1, 2.
Subs i u ing λ(A) = p1(A) + ip2(A)in o he cha ac e is ic Equa ion (5), we ha e:
(p1+ip2)3+C1(A)(p1+ip2)2+C2(A)(p1+ip2)+C3(A) = 0 (6)
Di e en ia ing wi h ega d o A, we ha e:
3(p1+ip2)2(˙
p1+i˙
p2)+2C1(p1+ip2)( ˙
p1+i˙
p2)+˙
C1(p1+ip2)2
+C2(˙
p1+i˙
p2)+˙
C2(p1+ip2)+˙
C3=0 (7)
Compa ing he eal and imagina y pa s, we ob ain:
X1˙
p1−X2˙
p2+X3=0 (8)
and
X2˙
p1+X1˙
p2+X4=0 (9)
whe e
X1=3p2
1−p2
2+2C1p1+C2
X2=6p1p2+2C1p2
X3=˙
C1p2
1−p2
2+˙
C2˙
p1+˙
C3
X4=2˙
C1p1p2+˙
C2p2.
F om (8) and (9), we ob ain:
˙
p1=−(X1X3+X2X4)
X2
1+X2
2
.
Now,
X3=˙
C1p2
1−p2
2+˙
C2p1+˙
C36=˙
C1p2
1−p2
2+˙
C2p1+˙
C1C2+C1˙
C2
[since C36=C1C2in a dele ed neighbo hood o A[H]]
A A=A[H],
•Case 1: p1=0, p2=√C2
X1=−2C2,X2=2C1√C2,X36=C1˙
C2,X4=√C2˙
C2
The e o e, X2X4+X1X36=2C1C2˙
C2−2C1C2˙
C2=0
So, X2X4+X1X36=0 a A=A[H], when p1=0, p2=√C2.
•Case 2: p1=0, p2=−√C2
X1=−2C2,X2=−2C1√C2,X36=C1˙
C2,X4=−√C2˙
C2
So, X2X4+x1X36=2C1C2˙
C2−2C1C2˙
C2=0
So, X2X4+X1X36=0 a A=A[H], when p1=0, p2=−√C2.
∴d
dA (Re λi(A))|A=A[H]6=0, o i=1, 2 and p3(A[H]) = −C1(A[H])<0.
Hence, Theo em 7is p o ed using Theo em 6.
No e:
Imagina y eigen alues a e connec ed wi h any molecula p ocess (e.g., collisions) and
he e e se o ha p ocess [16].
Ma hema ics 2022,10, 1641 16 o 24
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5
1
1.5
2
2.5
3
3.5
4
4.5
µ
Oxygen
(a) Bi u ca ion diag am o c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
µ
Phy oplank on
(b) Bi u ca ion diag am o p
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
µ
Zooplank on
(c) Bi u ca ion diag am o z
Figu e 13.
Bi u ca ion diag am o sys em (1) aking
µ
as he bi u ca ion pa ame e , while he
o he s emain unchanged, as in Figu e 6.
b
E(b
c
,
b
p
,
bz)
is uns able wi h a pe iodic solu ion when
µ∈(µ[H]1=0.131580, µ[H]2=
0.354676
)
and s able when
µ∈(
0, 0.131580
)∪(
0.354676, 0.474246
)
.
Again, b
E(b
c,b
p,bz)goes o s able zooplank on ee equilib ium E1(e
c,e
p, 0)when µ>µ[ c]=0.474246.
0.5 1 1.5 2 2.5
−0.5
0
0.5
1
1.5
2
2.5
m
Oxygen
(a) Bi u ca ion diag am o c
0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
m
Phy oplank on
(b) Bi u ca ion diag am o p
0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
m
Zooplank on
(c) Bi u ca ion diag am o z
Figu e 14.
Bi u ca ion diag ams o sys em (1) aking
m
as he bi u ca ion pa ame e , while he
o he s emain unchanged, as in Figu e 6. He e,
b
E(b
c
,
b
p
,
bz)
is s able when
m∈(
0.0,
m[H]=
0.6533
)
and uns able wi h a pe iodic solu ion when
m∈(
0.6533, 2.287
)
. When
m>m[ c]=
2.287, i ial
equilib ium
E0= (
0, 0, 0
)
exis s co esponding o he deple ion o oxygen and he ex inc ion o
plank on.
Ma hema ics 2022,10, 1641 17 o 24
0.2 0.4 0.6 0.8
0
1
2
3
4
5
g
c
H
(a) Bi u ca ion diag am o c
0.2 0.4 0.6 0.8
0
0.5
1
1.5
2
g
p
H
(b) Bi u ca ion diag am o p
0.2 0.4 0.6 0.8
0
1
2
3
g
z
H
(c) Bi u ca ion diag am o z
Figu e 15.
Bi u ca ion diag ams o sys em (1) while
g
a ies om
[
0.09, 1
]
and he o he s emain
unchanged, as in Figu e 6. He e, he in e io equilib ium
b
E(b
c
,
b
p
,
bz)
is uns able wi h a pe iodic solu ion
when g∈[0.09, g[H]=0.226067)and s able when g>g[H]=0.226067.
0 0.5 1
0
2
4
6
η
c
H H H
H
BP
(a) Bi u ca ion diag am o c
0 0.5 1
0
1
2
3
4
η
p
H H
BP H
H
(b) Bi u ca ion diag am o p
0 0.5 1
0
1
2
3
η
z
BP
H
H
(c) Bi u ca ion diag am o z
Figu e 16.
Bi u ca ion diag ams o sys em (1) aking
η
as he bi u ca ion pa ame e , while he
o he s emain unchanged, as in Figu e 6. He e, he zooplank on ee equilib ium
E1(e
c
,
e
p
, 0
)
is
s able when 0
<η<η[ c]=
0.147603, and he in e io equilib ium
b
E(b
c
,
b
p
,
bz)
is s able when
η∈
(
0.147603,
η[H]1=0.198177)∪(η[H]2=
0.530864, 1
)
and uns able wi h a pe iodic solu ion when
η∈(0.198177, 0.530864). He e, ‘BP’ s ands o he ansc i ical bi u ca ion poin .
Ma hema ics 2022,10, 1641 18 o 24
0 5 10
0
2
4
a
c
H H
H
BP
(a) Bi u ca ion diag am o c
0 5 10
0
1
2
3
a
p
BP
H
(b) Bi u ca ion diag am o p
0 5 10
0
1
2
3
4
a
z
H H
H
BP
(c) Bi u ca ion diag am o z
Figu e 17.
Bi u ca ion diag ams o sys em (1) aking
a
as he bi u ca ion pa ame e , while he o he s
emain unchanged, as in Figu e 6. He e, he zooplank on ee equilib ium
E1(e
c
,
e
p
, 0
)
is s able when
0
<a<a[ c]=
0.109643, and he in e io equilib ium
b
E(b
c
,
b
p
,
bz)
is s able when
a∈(
0.109643,
a[H]=5.325675)and uns able wi h pe iodic solu ion when a>a[H]=5.325675.
2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
γ
c
s able in e io equilib ium
BP
s able zooplank on ee equilib ium
Figu e 18.
Bi u ca ion diag am o sys em (1) aking
γ
as he bi u ca ion pa ame e , while he o he s
emain unchanged, as in Figu e 6. He e, ‘BP’ appea s a γ=γ[ c]=4.479066.
E ec o En i onmen al Noise on Sys em (1)
In a ma ine ecosys em, he oxygen-plank on model is a ec ed by he en i onmen al
noise due o he inhe en s ochas ici y o he wea he condi ions. Fo en i onmen al noise,
some o he pa ame e s o sys em (1) change andomly o e ime. In his s udy, we ha e
assumed ha he s ochas ici y a ec s he oxygen p oduc ion e m h ough pa ame e
A
,
Ma hema ics 2022,10, 1641 19 o 24
he phy oplank on g ow h e m h ough pa ame e
B
, and he zooplank on mo ali y a e
µby u ning A,B, and µin o andom a iables as ollows:
A→A+γ1( )
B→B+γ2( )(11)
µ→µ+γ3( )
whe e
γ1
,
γ2
, and
γ3
a e independen Gaussian whi e noise e ms and sa is y he ollow-
ing condi ions:
<γj( )>=0 and <γj( 1),γj( 2)>=α2
jδj( 1− 2), o j=1, 2, 3
whe e
αj
a e he in ensi ies o s eng hs o he andom pe u ba ions,
δ
is he Di ac del a
unc ion de ined by: (δ(x) = 0, o x6=0
R∞
−∞δ(x)dx =1
and <·>is he ensemble a e age o he conside ed s ochas ic p ocess.
In oducing Gaussian whi e noises, sys em (1) can be o mula ed as:
dc
d =(A+γ1( ))c0p
c+c0−δcp
c+c2−νcz
c+c3−mc
dp
d =(B+γ2( ))cp
c+c1−γp2−ap2z
ahp2+p+g−σp
dz
d = ηc2
c2+c2
4!.ap2z
ahp2+p+g−(µ+γ3( ))z
i.e., dc
d =Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc +γ1( )c0p
c+c0
dp
d =Bcp
c+c1−γp2−ap2z
ahp2+p+g−σp+γ2( )cp
c+c1
dz
d = ηc2
c2+c2
4!.ap2z
ahp2+p+g−µz−γ3( )z
dc
d =Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc +c0p
c+c0·α1
dw1
d
dp
d =Bcp
c+c1−γp2−ap2z
ahp2+p+g−σp+cp
c+c1·α2
dw2
d
dz
d = ηc2
c2+c2
4!·ap2z
ahp2+p+g−µz−α3zdw3
d
whe e
γ1=α1dw1
d
,
γ2=α2dw2
d
, and
γ3=α3dw3
d
. He e,
w=w1( ),w2( ),w3( ) ≥0
ep esen s h ee-dimensional s anda d B ownian mo ion.
Ma hema ics 2022,10, 1641 20 o 24
Hence, ou p oposed s ochas ic sys em is:
dc =Ac0p
c+c0−δcp
c+c2−νcz
c+c3−mc +c0
c+c0α1pdw1
dp =Bcp
c+c1−γp2−ap2z
ahp2+p+g−σp+p
c+c1α2cdw2(12)
dz = ηc2
c2+c2
4!.ap2z
ahp2+p+g−µz−α3zdw3.
The e ec o en i onmen al noise on he dynamics o sys em (12) is analyzed nu-
me ically by he Eule Ma uyama me hod in MATLAB. Fo his pu pose, we chose he
pa ame ic se as ollows:
{c0=1, δ=1, c2=1, m=0.5, B=1.8, c1=1.0, γ=0.7, σ=0.1, ν=0.01, µ=0.1,
h=1.2, a=3.0, η=0.7, c4=1, g=0.3, c3=1, α1=α2=α3=0.001}, (13)
bu a ied Ain a b oad ange.
When we ook
A=
10, while he o he pa ame e s emained he same as in se (13),
hen he e ec o he Gaussian whi e noises on he s ochas ic sys em (12) we e as depic ed in
Figu e 19. Fu he mo e,
Figu e 19
shows ha he oxygen, phy oplank on, and zooplank on
a ied a ound he de e minis ic coexis ence s eady-s a e alues 1.48635, 0.218551, and
0.866809, espec i ely. Hence, sys em (12) is pe sis en . In his con ex , we epea ed he
s ochas ic simula ions 20000 imes, and he nume ical esul s a e depic ed in Figu e 20,
which shows he s a iona y dis ibu ion o
c( )
,
p( )
, and
z( )
a ime
=
600. Mo eo e ,
when we chose
A=
1.8, while he emaining pa ame e s emained he same as in se (13),
hen sys em (12) was also pe sis en (see Figu e 21).
0 20 40 60 80 100 120 140 160 180 200
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Time
Oxygen
0 20 40 60 80 100 120 140 160 180 200
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
Time
Phy oplank on
0 20 40 60 80 100 120 140 160 180 200
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Time
Zooplank on
Figu e 19.
S ochas ic ajec o ies o sys em (12) when
A=
10 and he emaining pa ame e s a e same
as in se (13). Ini ial condi ions a e c(0) = 1, p(0) = 0.3 and z(0) = 1.
Ma hema ics 2022,10, 1641 21 o 24
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
0
500
1000
1500
c
F equency
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
0
200
400
600
800
1000
1200
p
F equency
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1
0
100
200
300
400
500
600
700
800
z
F equency
Figu e 20. His og ams o sys em (12) wi h he pa ame e s chosen om Figu e 19.
0 100 200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
c, p, z
c
p
z
Figu e 21. Pe sis ence o sys em (12) when A=1.8 and he emaining pa ame e s s ay unal e ed as
in Figu e 19.
Again, i we ake
µ=
0.5, while he o he pa ame e s emain he same as in se (13),
hen, i is no ed om Figu e 22 ha he zooplank on popula ion can no pe sis in sys-
em (12) o any o he ollowing choices: (a) A=1.8 and (b) A=10.
Fu he mo e, i is obse ed om Figu e 23 ha sys em (12) becomes ex inc o any o
he ollowing choices: (a)
A=
1.5, (b)
σ=
1.0, and (c)
m=
2.9, while he o he pa ame e s
emain he same as in se (13).
Ma hema ics 2022,10, 1641 22 o 24
0 20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
c, p, z
c
p
z
(a)A=1.8 and µ=0.5
0 20 40 60 80 100 120 140 160 180 200
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time
c, p, z
c
p
z
(b)A=10.0 and µ=0.5
Figu e 22.
Ex inc ion o he zooplank on in sys em (12) when (
a
)
A=
1.8 and
µ=
0.5, (
b
)
A=
10.0
and µ=0.5 and emaining pa ame e s a e chosen om se (13).
0 20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
c, p, z
c
p
z
(a)A=1.5
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
c, p, z
c
p
z
(b)σ=1.0
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
c, p, z
c
p
z
(c)m=2.9
Figu e 23.
Deple ion o oxygen and ex inc ion o plank on co esponding o sys em (12) when
(a)A=1.5, (b)σ=1.0, (c)m=2.9 and he emaining pa ame e s a e chosen om se (13).
7. Discussion and Conclusions
A Holling ype II unc ional esponse [
6
–
8
] is p edica ed on he assump ion ha he
sea ch a e o a p eda o is cons an , i.e., independen o he p ey popula ion. Howe e , i
seems easonable ha he p eda o can a y hei sea ch a e based on he a ailabili y o
p ey. In pa icula , i is es ima ed ha 50–80% o he oxygen p oduc ion on Ea h comes om
he oceans due o he pho osyn he ic ac i i y o phy oplank on. Some o his p oduc ion is
Ma hema ics 2022,10, 1641 23 o 24
consumed by bo h phy oplank on and zooplank on o cellula espi a ion. Fu he mo e,
zooplank on consume phy oplank on wi h a modi ied Holling ype II unc ional esponse,
based on he p emise ha he zooplank on sea ch a e is dependen on phy oplank on
( o de ails, see [
10
,
11
]). The goal o his a icle was o in es iga e he beha io o he
oxygen-plank on model wi h a modi ied Holling ype II unc ional esponse. The ollowing
summa izes ou indings:
•
The coexis ence s eady s a e is s able when 1.8
≤A<
1.966532, and i loses i s s able
na u e h ough Hop -bi u ca ion when 1.966532
<A<
7.258206 (see Figu es 7and 8).
•
The dynamic beha io o sys em (1) changes ab up ly o a low oxygen p oduc ion
a e (0
<A<
1.8), esul ing in he deple ion o oxygen and plank on ex inc ion (see
Figu e 3). This deple ion o oxygen p oduc ion will be a consequence o he global
ecological disas e .
•
Sys em (1) always has a s able coexis ence s eady s a e o a high oxygen p oduc ion
a e (see Figu e 6), i.e., he sus ainabili y o oxygen p oduc ion is possible when
A
is
la ge (
A>
7.258206). This esul is opposi e o he ou come shown by Seke ci and
Pe o skii [
3
] because hey obse ed ha he sys em dynamics we e no sus ainable
o a high oxygen p oduc ion a e. This is he main quali a i e di e ence be ween he
modi ied Holling ype II ( a iable sea ch a e, as men ioned in he p oposed model)
and he Holling ype II unc ional esponses. The e o e, he s udy o he modi ied
Holling ype II unc ional esponse is ecologically meaning ul o he sus ainabili y o
he dynamics o sys em (1), i he ne oxygen p oduc ion a e is abo e a ce ain c i ical
al e (A≥1.8).
Mo eo e , he e ec o en i onmen al noise has a s ong impac due o he inhe en
s ochas ici y o wea he condi ions. So, ou p oposed de e minis ic sys em (1) was com-
pa ed wi h a co esponding s ochas ic model (12) inco po a ing Gaussian whi e noises in
he sys em pa ame e s A,B, and µ, as men ioned in (11).
In he u u e, a ealis ic model can be p oposed o explo e he e ec s o spa ial di usion
in he pa e n o ma ion h ough di usion-d i en ins abili y.
Au ho Con ibu ions:
Concep ualiza ion, S.M., G.S. and M.D.l.S.; Me hodology, S.M., G.S. and
M.D.l.S.; In es iga ion, S.M., G.S. and M.D.l.S.; Fo mal analysis, S.M., G.S. and M.D.l.S.; W i ing—
o iginal d a p epa a ion, S.M., G.S. and M.D.l.S.; W i ing— e iew and edi ing, S.M., G.S. and
M.D.l.S. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This esea ch was unded by he Spanish Go e nmen and Eu opean Commission o i s
suppo h ough g an RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and o he Basque Go e n-
men o i s suppo h ough g an IT1207-19.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
The da a used o suppo he indings o he s udy a e a ailable wi hin
he a icle.
Acknowledgmen s:
The au ho s a e g a e ul o he anonymous e e ees, o hei ca e ul eading,
aluable commen s, and help ul sugges ions, which ha e helped hem o imp o e he p esen a ion o
his wo k signi ican ly. The hi d au ho (Manuel De la Sen) is g a e ul o he Spanish Go e nmen
and Eu opean Commission o i s suppo h ough g an RTI2018-094336-B-I00 (MCIU/AEI/FEDER,
UE) and o he Basque Go e nmen o i s suppo h ough g an IT1207-19.
Con lic s o In e es : The au ho s decla e ha hey ha e no con lic o in e es ega ding his wo k.
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