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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(6),3955-3964
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.116424
˜aäk‡zƒŽÓ .
•`¼
äää···
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2022c527F¶¹^Fϵ2022c619F¶uÙFϵ2022c628F
Á‡
ÏLÚ\Ó ö- .¥š‚5Ó¼¼ê§ï Ä˜aäk‡zƒŽÓ .§T
.²ï:•35ÚÛÜ-½5^‡§ÏLELyapunov ¼ê5ä²ï:±ŒÛ-
½5"(½š‚5¼e)²L²ï:§¦^Pontryagin 4ŒŠn•`¼üѧ
l«3y2i)Ô«+Ø«ýœ¹e§é•’]?1|^ÚmuÓž§Ø=‰•
¬Jø•Œ²L|d§…‘±°)XÚ²ï"
'…c
2i)Ô§Ó ö- .§‡zƒŽ§-½5§•`¼üÑ
OptimalHarvestofaPredationModel
withAllelopathy
YajingYuan
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:May27
th
,2022;accepted:Jun.19
th
,2022;published:Jun.28
th
,2022
©ÙÚ^:ä·.˜aäk‡zƒŽÓ .•`¼[J].A^êÆ?Ð,2022,11(6):3955-3964.
DOI:10.12677/aam.2022.116424
ä·
Abstract
Byintroducingthenonlinearcapturefunctioninthepredator-preymodel,akindof
predator-preymodelwithplantingphaseisstudied.Theconditionsfortheexistence
andlocalstabilityoftheequilibriumpointofthemodelareobtained.Theglobal
stabilityaroundthepositiveequilibriumpointisjudgedbyconstructingLyapunov
function,theecologicalandeconomicequilibriumpointundernonlinearharvesting
isdetermined,andtheoptimalharvestingstrategyisobtainedbyusingPontryagin
maximumprinciple,whichrevealsthatwhileensuringthenonextinctionofplankton
population,theutilizationanddevelopmentoffisheryresourcesnotonlyprovides
fishermenwiththemaximumeconomicprofit, butalsomaintainsthebalanceofmarine
ecosystem.
Keywords
Plankton,Prey-PredatorModel,Allelopathy,Stability,OptimalHarvestStrategy
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
2i)Ô´)¹3°,à6ÚÑ¥¤6)Ô.ŠâE'X,§©•2i‡ÔÚ2iÄ
Ô.2i‡Ô´Y) Ôó¥•ÄE,´°‚¸¥•Ä Ô5.2iÄÔ´²LY
ÄÔ,´¥þY•¥~aÚÙ¦²LÄÔ-‡,é•’uÐäk-‡¿Â.‘XÄåÆ
XÚuÐ,Nõ)ó Šö5¿,Y)«+U)˜zÆÔŸ,ùzÆÔŸéÙ¦Y)«
+©Oke-Ú³›)•Š^[1]. <‚Ï~re-)•zÆÔŸ¡•-ƒ,r³›)•zÆ
ÔŸ¡•Óƒ,¿…r˜«ÔŸ)Óƒ³›Ù§ÔŸ)•y–‰‡zƒŽ[2].
Maynard−Smith ÄgJÑ£ãü‡Ô«m‡zƒŽŠ^êÆ.[3].¯¢þ,˜‡Ô«
êþO\ŒU¬ÏL)‡zƒŽŠ^½öe -Ô)5K•,˜‡Ô«½Ù¦A‡Ô«
)•,lK•G!üO[4]. 8c'u2i)Ô‡zƒŽ ïÄ®²˜-‡¤J[5–8].(
DOI:10.12677/aam.2022.1164243956A^êÆ?Ð
ä·
ÜMaynard−Smith [3].JÑ±e2i)Ô‡zƒŽÓ .







dx
dt
= rx(1−
x
k
)−
β
1+αx
xy−γx
2
y,
dy
dt
= −δy+
aβxy
1+αx
,
(1.1)
Ù¥xL«2i‡Ô,´ «+—Ý,yL«2iÄÔ, ´Ó ö«+—Ý.γL«2iÄÔé
2i‡ÔÓƒ³›Ç. γx
2
´yÔ«éxÔ«—Ý˜«õU‡A.§´ÏL2iÄÔ)˜
«kÓÔŸ5{Ž2i‡Ô¦^Ó].ëêr“LSO•Ç. φ(x) =
β
1+αx
´HollingΠ .õ
U‡A¼ê[9],“LÓ öÓ Ç.δ•Ó ö«+g,kÇ. a•=zÇ.k•‚¸NBþ.
3¼Ae,·‚b.(1.1)¥2i‡ÔØäkû’-‡5,Ó öØä/¼,
¤±Â¼¼ê±•ãå¼•Œ²L|d.8c2•A^¼¼ê,̇k±en«a
.[10]:
(i)~ê¼¼ê,éӼ閱~ê¼Ç?1ÓM,¡•½•ÓMüÑ;
(ii)'~¼¼ê,¡•½ãåþ ÓMüÑ,=b½ü žmS¼þ†ÓMãåþ9«+Œ
¦È¤',˜„P•H(y,E) = qEy;
(iii)š‚5¼¼ê,=H(y,E)=
qEy
d
1
E+d
2
y
,Ù¥qL«ŒÓ¼Xê,EL«éÓ ö«+
¼ãå,d
1
,d
2
´~ê.ØJuy,~ê¼¼ê‘ÅÏéÔ,'~¼¼ê´éu
½E,H(y,E)¬‘yÃ.‚5O\.š‚5¼¼êžØþãØy¢A,¿…÷
vlim
E→+∞
qEy
d
1
E+d
2
y
=
qE
d
2
,lim
y→+∞
qEy
d
1
E+d
2
y
=
qy
d
1
. =š‚5¼¼êé¼ãåY²Ú«+´ÝþLy
ÑÚA,lL²š‚5¼•y¢.Ïd,3•3š‚5¼¼êœ¹e,.(1.1)?U
•:









dx
dt
= rx(1−
x
k
)−
β
1+αx
xy−γx
2
y,
dy
dt
= −δy+
aβ
1+αx
xy−
qEy
d
1
E+d
2
y
.
(1.2)
Nõ;[ÚÆöïÄˆ«ˆ«+ÄåXÚ¼¯K,3•’]+n!¾Á³››
+•Œþ¤J. Lv[11]ïÄ˜‡é ko«n«+.,ém˜« ÚÓ
ö?1ƒÓÓMãåþÓ¼,‰ÑÄu•Œ²LÂÃÛÉ•`››. Manna[12]ïÄ^
ƒÓÓ¼ãåþӞ¼ü«+•`¼üÑ.oäz[13]A^Pontryagin4ŒŠn?Ø
€9DÂ.•`nÜ››.
©Äk©Û.(1.2) ²ï:•35Ú-½5,,?Ø•`¼››üÑ,•)ºƒA
(Ø.
2.²ï:-½5
•{üå„,·‚Ú\ÃþjCþ:u=
1
k
x,v=
β
r
y,t=
τ
λ
,Ãþjëê•:
m= αk,η=
γk
β
,s=
δ
r
,b=
aβk
r
,h=
qEd
2
β
,d=
βEd
1
d
2
r
.
DOI:10.12677/aam.2022.1164243957A^êÆ?Ð
ä·
2g^tL«τ,.(1.2)C•Xe/ª:









du
dt
= u(1−u)−
uv
1+mu
−ηu
2
v,
dv
dt
−sv+
buv
1+mu
−
hv
d+v
,
(2.1)
d.(2.1)²ï:k²…²ï:E
0
=(0,0),š²…²ï:E
1
=(1,0),²ï:
E
∗
= (u
∗
,v
∗
),v
∗
=
(1−u
∗
)(1+mu
∗
)
1+mu
∗
+ηu
∗
,Ù¥u
∗
÷v˜ng•§
Au
3
∗
+Bu
2
∗
+Cu
∗
+D = 0,(2.2)
A = m(ms−b) >0,B = m(2s+b+bd−ηds−ηh)−m
2
(s+ds+h)+b(ηd−1) >0,
C = bd+b+s−(4ms+2mh+ηds+ηh) >0,D = −(s+sd+h) <0.
3ùp-f(u)=Au
3
+Bu
2
+Cu+ D.Ï•f(0)=D=−(s+ sd+ h)<0,¤±˜ng•§
(2.2)ª–k˜‡Š.2|^(kÎÒ{K,(H
0
) :3ms+3m
2
s+hbd>m
2
+ηhm+b÷
vž,§k•˜ŠE
∗
= (u
∗
,v
∗
).
2.1.²ï:ÛÜ-½5
½n1.(2.1)3z‡²ï:±ŒÛÜ-½5
(i)²…²ï:E
0
= (0,0)o´Ø-½;
(ii)
b
1+m
<s+
hd
d
2
ž,š²…²ï:E
1
= (1,0)´ìC-½;
(iii)
hv
∗
(d+v
∗
)
2
<
b(d+v
∗
)
2
h(1+mu
∗
)
2
(
u
∗
1+mu
∗
+ηu
2
∗
)ž,²ï:E
∗
= (u
∗
,v
∗
)´ìC-½,‡ƒ´Ø-
½.
y².(2.1)3?¿:(u,v) JacobiÝ•
J =
1−2u−
v
(1+mu)
2
−2ηuv−
u
1+mu
−ηu
2
bv
(1+mu)
2
−s+
bu
1+mu
−
hd
(d+v)
2
!
.(2.3)
(i).(2.1)3²…²ï:E
0
= (0,0)?JacobiÝ•
J
E
0
=




10
0−s−
hd
d
2




,(2.4)
DOI:10.12677/aam.2022.1164243958A^êÆ?Ð
ä·
Ý(2.4)AŠ•1 Ú−(s+
hd
d
2
) <0, Ïd²…²ï:E
0
o´Ø-½.
(ii).(2.1)3š²…²ï:E
1
= (1,0)?JacobiÝ•
J
E
1
=





−1−
1
m+1
−η
0−s+
b
1+m
−
hd
d
2





,(2.5)
Ý(2.5)AŠ•−1 Ú−s+
b
1+m
−
hd
d
2
.
b
1+m
<s+
hd
d
2
ž,š²…²ï:E
1
= (1,0)´ìC
-½.
(iii).(2.1)3²ï:E
∗
= (u
∗
,v
∗
)?JacobiÝ•
J
E
∗
=





a
11
a
12
a
21
a
22





,(2.6)
a
11
= 1−2u
∗
−
v
∗
(1+mu
∗
)
2
−2ηu
∗
v
∗
,a
12
= −
u
∗
1+mu
∗
−ηu
2
∗
,
a
21
=
bv
∗
(1+mu
∗
)
2
,a
22
= −s+
bu
∗
1+mu
∗
−
hd
(d+v
∗
)
2
.
Ý(2.6)A•§•
λ
2
−Θλ+Λ = 0,(2.7)
Ù¥
Θ = tr[J(E
∗
)] = a
11
+a
22
,
Λ = det[J(E
∗
)] = a
11
a
22
−a
12
a
21
,
l,²ï:-½5dΘ ÚΛ ÎÒ¤û½.Ïd,
hv
∗
(d+v
∗
)
2
<
b(d+v
∗
)
2
h(1+mu
∗
)
2
(
u
∗
1+mu
∗
+ηu
2
∗
),.
(2.1)²ï:E
∗
´ÛÜìC-½.‡ƒ, ²ï:E
∗
´Ø-½.
2.2.²ï:Û-½5
3ù˜!¥,·‚ÏLEÜ·Lyapunov ¼ê5ïÄ.(2.1)²ï:E
∗
Û-½
5.
½n2b(H
0
)¤á,mv
∗
<1+mu
∗
ž,.(2.1)²ï:E
∗
´ÛìC-½.
y²ELyapunov ¼ê
V(u,v) = [(u−u
∗
)−u
∗
ln
u
u
∗
]−ϕ[(v−v
∗
)−v
∗
ln
v
v
∗
],
DOI:10.12677/aam.2022.1164243959A^êÆ?Ð
ä·
Ù¥ϕ•–½~ê.÷X.(2.1))éV(u,v) ¦k
dV(u,v)
dt
=
u−u
∗
u
du
dt
−ϕ
v−v
∗
v
dv
dt
=(u−u
∗
)[1−u−
v
1+mu
−ηuv]−ϕ(v−v
∗
)[−s+
bu
1+mu
−
h
d+v
].(2.8)
éu²ï:E
∗
= (u
∗
,v
∗
),·‚k˜|²ï•§









1−u
∗
−
v
∗
1+mu
∗
−ηu
∗
v
∗
= 0,
−s+
bu
∗
1+mu
∗
−
h
d+v
∗
= 0,
(2.9)
·‚ò(2.8)ªÚ(2.9)ª(Ü3˜åk
dV(u,v)
dt
=(u−u
∗
)[−u−
v
1+mu
−ηuv+u
∗
+
v
∗
1+mu
∗
+ηu
∗
v
∗
)]
−ϕ(v−v
∗
)[
bu
1+mu
−
h
d+v
−
bu
∗
1+mu
∗
+
h
d+v
∗
],
ϕ=
(1+mu
∗
)(ηu
∗
(1+mu)+1)
b
,·‚
dV(u,v)
dt
=(u−u
∗
)
2
[−1+
mv
∗
(1+mu)(1+mu
∗
)
−ηv]−
(1+mu
∗
)(ηu
∗
(1+mu)+1)
b
(v−v
∗
)
2
=−[(u−u
∗
)
2
(1−
mv
∗
(1+mu)(1+mu
∗
)
+ηv)+
(1+mu
∗
)(ηu
∗
(1+mu)+1)
b
(v−v
∗
)
2
] ≤0.
Ïd,ŠâLyapunov−Lasalle ØCn[14],²ï:E
∗
´ÛìC-½.
3.)²L²ï:•35
ÏLÑÈÓ¼Ó öÚ ¼oÂ\ÚÝ\ÓM¤^¤Ä±²ž,Œ±ˆ
¤¢)Ô²ï.3.(1.1)¥,ÏLÓ ö(=2iÄÔ)3û’þäk-‡¿Â,B\š‚
5¼¼ê.Ïd,·‚Œ±ˆ¤¢)Ô²ï,ù¿›)²ïÚ²L²ï,)Ô²ïd
dx
dt
=
dy
dt
= 0 ‰Ñ.3ùp,½Âc•zü ÓMrÝð½¤, P•zü )ÔþÑÈd‚,¤
±ÀÂ\´Ñȼ)Ôþ¤¼oÂ\~¼ãåo¤.ÀÂ\•
π(x,y,E) =
PEqy
d
1
E+d
2
y
−cE = (
Pqy
d
1
E+d
2
y
−c)E,
DOI:10.12677/aam.2022.1164243960A^êÆ?Ð
ä·
@o)²ï:A
∞
= (x
∞
,y
∞
,E
∞
),de•§‰Ñ















rx(1−
x
k
)−
βxy
1+αx
−γx
2
y= 0,
−δy+
aβxy
1+αx
−
qEy
d
1
E+d
2
y
= 0,
(
Pqy
d
1
E+d
2
y
−c)E = 0,
(3.1)
•yA
∞
•3,7L±
Pqy
d
1
E+d
2
y
>c.²OŽ•§(3.1))²ï:A
∞
αLǥXe
/ª:
x
∞
=
P(δd
1
−q)+cd
2
P(aβd
1
−δαd
2
)+cd
2
α
,
y
∞
=
r(k−x
∞
)(1+αx
∞
)
βk+γx
∞
k(1+αx
∞
)
,
E
∞
=
d
2
y
∞
[δ(1+αx
∞
)−aβx
∞
]
aβx
∞
d
1
−(1+αx
∞
)(δd
1
+q)
.
½n3
cd
2
p
<q<δd
1
,aβd
1
<δαd
1
+αqž,)²ï:A
∞
= (x
∞
,y
∞
,E
∞
)•3.
5: XJE>E
∞
, @oÓ¼Ôo¤ò‡Ll•’¼oÂ\,˜•¬¬;É›”,
¦‚g,¬òÑ•’,E>E
∞
ØUÕ‘±.‡ƒ, E<E
∞
,K•’•\k|Œã,Ïd,3m
˜¼•’¥,§¬áÚ5õ•¬,ùòé¼óŠ)5ŒK•,ƒA邸
Ú•’]muÚ|^Ò¬kBŠ^,¤±,E <E
∞
•ØUÕ/±.
4.•`¼üÑ
•(½˜‡•`¼üÑ,Ú\˜‡ëYÂ\žm6JcŠ:
J(E) =
Z
∞
0
e
−εt
π(x,y,E)dt=
Z
∞
0
e
−εt
(
Pqy
d
1
E+d
2
y
−c)Edt,
Ù¥, ε•byÇ, E(t) •Ó ö«+Ó¼ãå þ ,÷v››•V=[0,E
max
]. E
max
´ ÚÓ
ö¼Œ1þ•, E
ε
L«• `››,¤éAG•x
ε
,y
ε
. ·‚A
ε
=(x
ε
,y
ε
) ••Z²ï
:.·‚8I´3y4XÚ«+Œ±YuÐcJe,3››•V þ(½#N››E(t),¦
.(1.2)²LЊ(x(0),y(0))=(x
0
,y
0
) )4ëYÂ\žm6J cŠˆ•ŒŠ.=
•`››E
ε
÷vJ(E
ε
) = maxJ(E).
y3½Â•`››M—î¼ê
H = e
−εt
(
PqEy
d
1
E+d
2
y
−cE)+λ
1
[rx(1−
x
k
)−
βxy
1+αx
−γx
2
y]+λ
2
[−δy+
aβxy
1+αx
−
qEy
d
1
E+d
2
y
],
Ù¥λ
1
,λ
2
´Š‘Cþ,Óžk
∂H
∂E
=: σ(t),Œ„σ(t)¦E(t)30†E
max
þ5£ƒ†,¡σ(t)•ƒ
DOI:10.12677/aam.2022.1164243961A^êÆ?Ð
ä·
†¼ê.duM—î¼êH 3››Cþ¥´‚5,¤±•`››9ÛÉ››Úbang−bang
››(3§þ.½e.)(Ü.Ïd,3ù«œ¹e,éA•`üÑXe:
E(t) =



E
max
,σ(t) >0 ⇔λ
2
e
εt
<P−
c(d
1
E+d
2
y)
2
qd
2
y
2
,
0,σ(t) <0 ⇔λ
2
e
εt
>P−
c(d
1
E+d
2
y)
2
qd
2
y
2
.
λ
2
e
εt
´b½d‚, λ
2
e
εt
<P−
c(d
1
E+d
2
y)
2
qd
2
y
2
´Â¼Ó öÀÂ\.l²LÆÝw,1˜‡^
‡)º•’ÓMÂ\Œub½d‚ž,•¬3¼|œ¹,ùòy¦‚ už•õÓMã
å.1‡^‡)º•’ÓMÂ\$uÓM¤ž,•¬ò¬‘›”, —•’ÓM¹ÄÊ
Ž.σ(t)=0ž,ù¿›X ^rzü ãå¼¤u3-Y²eTãå™5>S|
dbyŠ,lk
λ
2
qd
2
y
2
(d
1
E+d
2
y)
2
= e
−εt
(
Pqd
2
y
2
(d
1
E+d
2
y)
2
−c) =
∂π
∂E
e
−εt
.(4.1)
|^Pontryagin4ŒŠn[15],Š‘Cþ7L÷v¤‰ÑŠ‘•§
∂λ
1
∂t
= −
∂H
∂x
= −[λ
1
(r−
2rx
k
−
βy
(1+αx)
2
−2γxy)+λ
2
aβy
(1+αx)
2
],(4.2)
∂λ
2
∂t
= −
∂H
∂y
= −[e
−εt
Pqd
1
E
2
(d
1
E+d
2
y)
2
+λ
1
(−
βx
1+αx
−γx
2
)+λ
2
(−δ+
aβx
1+αx
−
qd
1
E
2
(d
1
E+d
2
y)
2
)].
(4.3)
(Ü(4.1)Ú(4.2)ªŒ±
∂λ
1
∂t
= −A
2
λ
1
−A
3
λ
2
,(4.4)
Ù¥
A
1
= e
−εt
P−
c(d
1
E+d
2
y)
2
qd
2
y
2
,A
2
= r−
2rx
k
−
βy
(1+αx)
2
−2γxy,A
3
=
aβy
(1+αx)
2
,
¦)‚5•§(4.4)Ï),
λ
1
(t) =
A
1
A
3
e
−εt
A
2
−ε
.(4.5)
Šâ(4.3)ªŒ±
∂λ
2
∂t
= −B
1
e
−εt
+B
2
λ
2
,(4.6)
Ù¥
B
1
=
Pqd
1
E
2
(d
1
E+d
2
y)
2
+
A
1
A
3
A
2
−ε
(−
βx
1+αx
−γx
2
),B
2
= δ−
aβx
1+αx
+
qd
1
E
2
(d
1
E+d
2
y)
2
.
¦)‚5•§(4.6)Ï),
λ
2
(t) =
B
1
e
−εt
B
2
+ε
,(4.7)
DOI:10.12677/aam.2022.1164243962A^êÆ?Ð
ä·
w,λ
1
(t),λ
2
(t)÷vî5^‡.d(4.1)Ú(4.7)ªŒ±
c= (P−
B
1
B
2
+ε
)
qd
1
E
2
(d
1
E+d
2
y)
2
,(4.8)
ddŒ••`²ï)A
ε
= (x
ε
,y
ε
,E
ε
),
x
ε
=
P(δd
1
−q)+cd
2
P(aβd
1
−δαd
2
)+cd
2
α
,
y
ε
=
r(k−x
ε
)(1+αx
ε
)
βk+γx
ε
k(1+αx
ε
)
,
E
ε
=
d
2
y
ε
[δ(1+αx
ε
)−aβx
ε
]
aβx
ε
d
1
−(1+αx
ε
)(δd
1
+q)
.
d(4.8)ª•,ε→∞ž,
Pqd
1
E
2
(d
1
E+d
2
y)
2
−c= 0,
¤±
∂π
∂E
e
−εt
= 0,
ù¿›XÕbyǬ—™5ü ãå|d~.Ïd,ε→0 ž,ÂÈ•Œ.
5.(Ø
T©ïÄ˜a2i)Ôäk‡zƒŽŠ^š‚5Ó öÓ¼Ó ..3ù˜a.
¥,æ^š‚5¼¼ê,òû’•’¥•3ÓMÚ«+SÓ ƒ(Ü,ïÄäk‡zƒŽŠ^
Ó .ÄåÆÚ• `¼. •‘±°)XÚ²ïÚŒ±YuÐ,é2i)Ô«+¦^
•`¼››üÑ,A^Pontryagin4ŒŠn,&?XÚS•ŒŒ±Yþ.(JL²,æÂ
žb½d‚÷vî5^‡,"byÇ)•ŒÂÃ,l?˜Ú`²•`¼üÑQ Œ±Ü
n/muÚ¦^],qŒ±¢y²LÃ,3°)XÚ¥äk-‡¿Â.
ë•©z
[1]Mukhopadhyay,A.,Chattopadhyay,J.andTapaswi,P.K.(1998)ADelayDifferentialEqua-
tionsModelofPlanktonAllelopathy.MathematicalBiosciences,149,167-189.
https://doi.org/10.1016/S0025-5564(98)00005-4
[2]•=¬,y#‰,º˜.êÆ)Æ.†ïÄ•{[M].®:®‰Æч,1998.
[3]Smith,M.(1974)ModelsinEcology.UniversityPress.
[4]Rice,E.L.(1984)ManipulatedEcosystems:RolesofAllelopathyinAgriculture.In:Allelopa-
thy,2ndEdition,AcademicPress,Cambridge,MA,8-73.
https://doi.org/10.1016/B978-0-08-092539-4.50006-X
DOI:10.12677/aam.2022.1164243963A^êÆ?Ð
ä·
[5]Gupta,R.P.,Banerjee,M.andChandra,P.(2012)TheDynamicofTwo-SpeciesAllelopathic
CompetitionwithOptimalHarvesting.JournalofBiologicalDynamics,6,674-694.
https://doi.org/10.1080/17513758.2012.677484
[6]Abbas,S.,Banerjee,M.andHungerbjhler,N.(2010)Existence,UniquenessandStability
AnalysisofAllelopathicStimulatoryPhytoplanktonModel.JournalofMathematicalAnalysis
andApplications,367,249-259.https://doi.org/10.1016/j.jmaa.2010.01.024
[7]Tian,C.,Lai,Z.andLin,Z.(2011)PatternFormationforaModelofPlanktonAllelopathy
withCross-Diffusion.JournaloftheFranklinInstitute,348,1947-1964.
https://doi.org/10.1016/j.jfranklin.2011.05.013
[8]Tian, C. and Ruan, S. (2019) Pattern Formation and Synchronism in anAllelopathic Plankton
ModelwithDelayinaNetwork.SIAMJournalonAppliedDynamicalSystems,18,531-557.
https://doi.org/10.1137/18M1204966
[9]Holling, C.S. (1959) Some Characteristics of Simple Types of Predation and Parasitism. Cana-
dianEntomologist,91,385-398.https://doi.org/10.4039/Ent91385-7
[10]Gupta,R.P.,Chandra,P.andBanerjee,M.(2015)DynamicalComplexityofaPrey-Predator
ModelwithNonlinearPredatorHarvesting.DiscreteandContinuousDynamicalSystems—
SeriesB(DCDS-B),20,423-443.https://doi.org/10.3934/dcdsb.2015.20.423
[11]Lv,Y.,Rong,Y.andPei,Y.(2013)APrey-PredatorModelwithHarvestingforFishery
ResourcewithReserveArea.AppliedMathematicalModelling,37,3048-3062.
https://doi.org/10.1016/j.apm.2012.07.030
[12]Manna,D.,Maiti,A.andSamanta,G.P.(2018)AnalysisofaPredator-PreyModelforEx-
ploitedFishPopulationswithSchoolingBehavior.AppliedMathematicsandComputation,
317,35-48.https://doi.org/10.1016/j.amc.2017.08.052
[13]oäz,4||.€9DÂ.•`nÜ››ïÄ[J].A^êÆÚåÆ,2022,43(4):
445-452.
[14]Hale,J.K.(1969)OrdinaryDifferentialEquations.AmericanMathematicalMonthly,23,82-
122.
[15]0UŒ,ÜS`,MöL.•`››üÑA^Ä:[M].®:‰Æч,2003.
DOI:10.12677/aam.2022.1164243964A^êÆ?Ð

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