具有恐惧效应和 Beddington - DeAngelis 功能反应的时空共位群内捕食模型的稳定性和分叉分析 Stability and Bifurcation Analysis of a Spatiotemporal Intraguild PredationModel with Fear Effect and Beddington-DeAngelis Functional Response

doi:10.12677/PM.2022.1212225

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PureMathematicsnØêÆ,2022,12(12),2081-2105

PublishedOnlineDecember2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.1212225

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StabilityandBifurcationAnalysisofa

SpatiotemporalIntraguildPredation

ModelwithFearEﬀectand

Beddington-DeAngelisFunctional

Response

©ÙÚ^:w.äk™êAÚBeddington-DeAngelisõU‡Až˜ +SÓ .-½5Ú©©

Û[J].nØêÆ,2022,12(12):2081-2105.DOI:10.12677/pm.2022.1212225

w

XiaoningWang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Nov.17

,2022;accepted:Dec.15

,2022;published:Dec.22

,2022

Abstract

Competitionandpredationareacommonphenomenoninecology.Whentwospecies

compete for the samelimited resources,the coexistence of predator and prey isneces-

sarytosustainthepredator-preysystem.Itisofgreatsigniﬁcancetostudywhether

predatorsandpreycancoexistinaintraguildpredationmodelwhencompetingfor

thesameresource.Inthispaper,westudythestabilityandHopfbifurcationofa

spatiotemporalintraguildpredationmodelwithafeareﬀect,theconditionsforthe

coexistenceofpredatorandpreyinaintraguildpredationmodelarederivedbydis-

cussingtheexistenceofequilibriumpoints,localandglobalasymptoticstabilityand

uniform persistence,the conditionof stability of equilibrium point is obtained by Lya-

punovmethod andHelvetz criterion.Finally,wetake thefearfactorasthebranching

parameterandobtaintheconditionsfortheexistenceofHopfbifurcationateach

equilibriumpoint.

Keywords

IntraguildPredationModel,Beddington-DeAngelis Functional Response, Fear Eﬀect,

Stability,HopfBifurcation

2022byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI:10.12677/pm.2022.12122252087nØêÆ

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= (u

∗

2.3.²ï:ÛÜ-½5

XÚ(1.3)3(u,v,w)?JacobiÝXe













,(3)

Ù¥

1−2u

1+kv

−

(v+w)(1+b

v+b

(1+au+b

v+b

=−[

ku(1−u)

(1+kv)

u(1+au)+(b

−b

)uw

(1+au+b

v+b

],J

−b

)uv−u(1+au)

(1+au+b

v+b

v(1+b

v)+vw(ac+e

)

(1+au+b

v+b

u−cw)(1+au+b

(1+au+b

v+b

−δ

=−

cv(1+b

v)+uv(ac+e

)

(1+au+b

v+b

w(1+b

v+b

w)−acεvw

(1+au+b

v+b

cεw(1+b

w)+uw(acε−b

)

(1+au+b

v+b

u+εcv)(1+au+b

(1+au+b

v+b

−δ

DOI:10.12677/pm.2022.12122252088nØêÆ

w

e¡ÏLOŽXÚ(1.3)3z‡²ï:?JacobiÝAŠ,5(½ù²ï:-½5.

½n3(i) ²…²ï:E

= (0,0,0)´Ã^‡Ø-½.

(ii)emax{<

}<1,KŒ²…²ï:E

= (1,0,0)´ÛÜìC-½;ÄKE

´Ø-½

.

y²(i) XÚ(1.3)3²ï:E

?JacobiÝ•







100

0−δ

00−δ







.(4)

Ý(4)AŠ•λ

(0)

= 1 >0,λ

(0)

= −δ

Úλ

(0)

= −δ

.Ïd,²ï:E

´Ø-½.

(ii)XÚ(1.3)3²ï:E

?JacobiÝ•







−1−

1+a

−

1+a

−δ

1+a

−δ







.(5)

Ý(5)A•§•

(λ+1)



λ−(

1+a

−δ

)



λ−(

1+a

−δ

)



= 0.

¤±, Ý(5) AŠ•λ

(1)

= −1,λ

(1)

1+a

−δ

= δ

−1),λ

(1)

1+a

−δ

= δ

−1).

max{<

}<1ž,²ï:E

´ÛÜìC-½;‡ƒ,E

´Ø-½.

½n4b<

>1,k>b

¤á,eλ

(2)

<0,T

(2)

<0,KÃIGÓ ö²ï:E

,0)´

ÛÜìC-½;eλ

(2)

>0½T

(2)

>0,KE

,0)´Ø-½.

y²XÚ(1.3)3²ï:E

?JacobiÝ•







00a







.(6)

Ù¥

1+kv

aδ

(1−u

)

−u

= −

(1−u

)

(1+kv

)

−

(1−u

)

(1+kv

)

−b

−u

(1+au

)

(1+au

)

= (e

−aδ

)

1−u

1+kv

= −

(1−u

)

1+kv

= −

(1+b

)+u

+ac)

(1+au

)

−δ

εc(1−u

)

1+kv

DOI:10.12677/pm.2022.12122252089nØêÆ

w

Ý(6)A•§•

(λ−a

)



−(a

)λ+a

−a



= 0

-D

(2)

= a

−a

1−u

1+kv

(k−b

)(1−u

)

(1+kv

)

1+kv

−aδ

)

(2)

= a

(

−b

)(1−u

)−u

1+kv

,λ

(2)

−δ

εc(1−u

)

1+kv

d‰ˆ½nŒλ

(2)

+λ

(2)

,λ

(2)

>0.λ

(2)

<0…T

(2)

<0ž,Ý(6)A

Šλ

(2)

,λ

(2)

äkK¢Ü,Ïd,²ï:E

´ÛÜìC-½;λ

(2)

>0½T

(2)

>0ž,²ï:

´Ø-½.

½n5b<

>1,e<

…aδ

,KÃIG ²ï:E

,0,w

)´ÛÜìC-

½;e<

,δ

−<

) >c(1−u

½e

<(aδ

−b

)(1−u

) ž,²ï:E

,0,w

)

´Ø-½.

y²XÚ(1.3)3²ï:E

?JacobiÝ•













.(7)

Ù¥

aδ

(1−u

)−u

= (1−u

)(

−ku

)−

= −

[1−b

(1−u

)]

−e

−c(1−u

),b

= (e

−aδ

)(1−u

),b

= (εc−b

)(1−u

),b

= −b

(1−u

Ý(7)A•§•

(λ−b

)



−(b

)λ+b

−b



= 0.

-D

(3)

= b

−b

= δ

(1−u

)

−b

−aδ

)

+4b

>0,

(3)

= b

= (

aδ

−b

)(1−u

)−u

,λ

(3)

−e

−c(1−u

d‰ˆ½nŒλ

(3)

+ λ

(3)

,λ

(3)

>0.λ

(3)

<0…T

(2)

<0,=<

…

aδ

ž,Ý(7)AŠλ

(3)

,λ

(3)

äkK¢Ü,Ïd,²ï:E

´ÛÜìC-½;

(3)

>0½T

(3)

>0,=<

,δ

−<

) >c(1−u

½e

<(aδ

−b

)(1−u

)ž,²ï

´Ø-½.

½n6e~ê²ï:E

∗

)÷v±e^‡µ

)εc>b

,1+kv

∗

(1−u

∗

),e

>aδ

(i= 1,2)

)1+au

∗

>a(1−u

∗

)

−aδ

) >(e

−aδ

)(c+b

)

[(1+kv

∗

)−b

(1−u

∗

)]v

∗

>(εc−b

)[(1+kv

∗

)−b

(1−u

∗

)]w

∗

DOI:10.12677/pm.2022.12122252090nØêÆ

w

K~ê²ï:E

∗

)´ÛÜìC-½.

y²XÚ(1.3)3²ï:E

∗

?JacobiÝ•

∗













.(8)

Ù¥

∗

1+kv

∗

a(1−u

∗

)

1+au

∗

−1

= −

∗

1+kv

∗

k(1−u

∗

)

1+kv

∗

(1+kv

∗

)−b

(1−u

∗

)

1+au

∗

1+kv

∗

(1−u

∗

)−(1+kv

∗

)

1+au

∗

−aδ

∗

1+au

∗

= −

∗

1+au

∗

<0,

= −

(c+b

∗

1+au

∗

<0,c

−aδ

∗

1+au

∗

(εc−b

)

1+au

∗

= −

∗

1+au

∗

<0.

Ý(8)A•§•

λ+P

= 0.(9)

Ù¥

= −(c

)

∗

1+kv

∗



1−

a(1−u

∗

)

1+au

∗



∗

1+au

∗

= c

−c

∗

)

(1+kv

∗

)(1+au

∗

)

k(e

−aδ

)(1−u

∗

(1+kv

∗

)

(1+au

∗

)

∗

{(1+kv

∗

)[(e

−aδ

∗

+(e

−aδ

∗

]−(1−u

∗

)(b

∗

)}

(1+kv

∗

)(1+au

∗

)

+(cε−b

)(c+b

)]v

∗

(1+au

∗

)

= c

−c

)+c

−c

)+c

−c

)

= −

∗

1+kv

∗



a(1−u

∗

)

1+au

∗

−1



[(c+b

)(cε−b

)+b

∗

(1+au

∗

)

−

∗

1+kv

∗



k(1−u

∗

)

1+kv

∗

(1+kv

∗

)−b

(1−u

∗

)

1+au

∗



[(e

−aδ

)(c+b

)−b

−aδ

)]v

∗

(1+au

∗

)

−

∗

1+kv

∗



(1−u

∗

)−(1+kv

∗

)

1+au

∗



−aδ

)+(e

−aδ

)(εc−b

)]v

∗

(1+au

∗

)

DOI:10.12677/pm.2022.12122252091nØêÆ

w

d^‡(H

)Œ•,

sgn(J

∗

) =







−−−

+−−

++−







.(10)

d^‡(H

)(H

)Œ•,P

>0,P

>0.

−P

= c

(−c

−c

)+c

(−c

−c

)+c

(−c

−c

)

)+c

−c

)

−c

)+c

(11)

d^‡(H

)†(10)ÎÒŒ•,

(εc−b

)[b

(1−u

∗

)−(1+kv

∗

)]u

∗

(1+kv

∗

)(1+au

∗

)

∗

(1+kv

∗

)(1+au

∗

)



k(1−u

∗

)

1+kv

∗

(1+kv

∗

)−b

(1−u

∗

)

1+au

∗



∗

[(1+kv

∗

)−b

(1−u

∗

)]v

∗

−(εc−b

)[(1+kv

∗

)−b

(1−u

∗

)]w

∗

}

(1+kv

∗

)(1+au

∗

)

(1−u

∗

(1+kv

∗

)

(1+au

∗

)

lP

−P

>0.

Ïd,^‡(H

) (H

)¤áž, P

>0, P

−P

>0,dRouth-Hurwitz

â[42]Œ•,A•§(9)ŠþäkK¢Ü,?XÚ(1.3)~ê²ï:E

∗

´ÛÜìC-

½.‡ƒ,A•§(9)ŠØkK¢Ü…¢ÜØ•",Ïd~ê²ï:E

∗

´Ø-½.

2.4.~ê²ï:Û-½5

ù˜Ü©·‚Ì‡ïÄ¦~ê²ï:E

∗

)3R

SÛìC-½¿©^‡.

½n7e(H

)†±e^‡

)aδ

(1+kv

∗

)−b

(1−u

∗

) >0

)2b

(1+kv

∗

)

<k(1−u

∗

)[aδ

(1+kv

∗

)−b

(1−u

∗

)]

)

>[e

(cε−b

)−e

(c+b

)]

(c+b

) >e

(cε−b

)

[aδ

(1+kv

∗

)−b

(1−u

∗

)] >[e

(c+b

)−e

(εc−b

)][b

(1−u

∗

)−aδ

(1+kv

∗

)]

DOI:10.12677/pm.2022.12122252092nØêÆ

w

¤á,K~ê²ï:E

∗

)3R

={(u,v,w) : u(t) >0,v(t) >0,w(t) >0}S´Ûì

C-½.

y²3R

= {(u,v,w) : u(t) >0,v(t) >0,w(t) >0}S½ÂLyapunov¼ê•

V= (u−u

∗

−u

∗

(v−v

∗

−v

∗

(w−w

∗

−w

∗

)

(12)

éª(12)÷XXÚ(1.3))'užmt¦Œ

= (

u−u

∗

)

v−v

∗

)

w−w

∗

)

duE

∗

´XÚ(1.3)~ê²ï:,k

∗

1+au

∗

1−u

∗

1+kv

∗

−cw

∗

1+au

∗

= δ

∗

+εcv

∗

1+au

∗

= δ

,,uv

∗

−uv

∗

= v

∗

(u−u

∗

)−u

∗

(v−v

∗

), u

∗

v−uv

∗

= u

∗

(v−v

∗

)−v

∗

(u−u

∗

)

∗

−v

∗

w= w

∗

(v−v

∗

)−v

∗

(w−w

∗

), v

∗

w−vw

∗

= v

∗

(w−w

∗

)−w

∗

(v−v

∗

)

∗

−u

∗

w= w

∗

(u−u

∗

)−u

∗

(w−w

∗

), u

∗

w−uw

∗

= u

∗

(w−w

∗

)−w

∗

(u−u

∗

²L{üOŽ,

(

u−u

∗

)

= −



1+kv

−

a(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b



(u−u

∗

)

−



k(1−u

∗

)

(1+kv)(1+kv

∗

)

(1+kv

∗

)−b

(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b



(u−u

∗

)(v−v

∗

)

−



(1+kv

∗

)−b

(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b



(u−u

∗

)(w−w

∗

)

(

v−v

∗

)

−aδ

(1+au+b

v+b

(u−u

∗

)(v−v

∗

)−

(1+au+b

v+b

(v−v

∗

)

−

c+b

(1+au+b

v+b

(v−v

∗

)(w−w

∗

)

(

w−w

∗

)

−aδ

(1+au+b

v+b

(u−u

∗

)(w−w

∗

)−

(1+au+b

v+b

(w−w

∗

)

cε−b

(1+au+b

v+b

(v−v

∗

)(w−w

∗

)

DOI:10.12677/pm.2022.12122252093nØêÆ

w

l

= −



1+kv

−

a(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b



(u−u

∗

)

−

(1+au+b

v+b

(v−v

∗

)

−

(1+au+b

v+b

(w−w

∗

)



−aδ

(1+au+b

v+b



(u−u

∗

)(v−v

∗

)



−

k(1−u

∗

)

(1+kv)(1+kv

∗

)

−

(1+kv

∗

)−b

(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b



(u−u

∗

)(v−v

∗

)



−aδ

(1+au+b

v+b

−

(1+kv

∗

)−b

(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b



(u−u

∗

)(w−w

∗

)



cε−b

(1+au+b

v+b

−

c+b

(1+au+b

v+b



(v−v

∗

)(w−w

∗

)

-l

= −

1+kv

−

a(1−u

∗

)

(1+kv

∗

)(1+au+b

v+b

= l

(1−u

∗

)−aδ

(1+kv

∗

)

(1+kv

∗

)(1+au+b

v+b

−

k(1−u

∗

)

(1+kv)(1+kv

∗

)

= −

(1+au+b

v+b

= l

(1−u

∗

)−aδ

(1+kv

∗

)

(1+kv

∗

)(1+au+b

v+b

= −

(1+au+b

v+b

= l

(cε−b

)−e

(c+b

)

(1+au+b

v+b













d(H

)(H

)Œ•,



<0.



= l

−l

(1+kv)(1+au+b

v+b

{

(1+kv

∗

)

−k(1−u

∗

)[aδ

(1+kv

∗

)−b

(1−u

∗

)]

}

(1+kv)

(1+kv

∗

)(1+au+b

v+b

−

4ab

(1−u

∗

)(1+kv

∗

)(1+kv)

(1+kv)

(1+kv

∗

)(1+au+b

v+b

−

(1+kv)

(1−u

∗

)−aδ

(1+kv

∗

)]

(1+kv)

(1+kv

∗

)(1+au+b

v+b

−

(1−u

∗

)

(1+au+b

v+b

(1+kv)

(1+kv

∗

)(1+au+b

v+b

d(H

)(H

)Œ•,



= l

+2l

−l

{

(cε−b

)−e

(c+b

)]

−4b

}

(1+kv)(1+kv

∗

)[(1+kv

∗

)(1+au+b

v+b

w)−a(1−u

∗

)(1+kv)]

(1+kv)

(1+kv

∗

)

(1+au+b

v+b

(cε−b

)−e

)][b

(1−u

∗

)−aδ

(1+kv

∗

)](1+kv)

(1−u

∗

)−aδ

(1+kv

∗

)]

(1+kv)

(1+kv

∗

)

(1+au+b

v+b

DOI:10.12677/pm.2022.12122252094nØêÆ

w

(1+kv)

(1−u

∗

)−aδ

(1+kv

∗

)]

(1+kv)

(1+kv

∗

)

(1+au+b

v+b

(1+kv)[b

(1−u

∗

)−aδ

(1+kv

∗

)]

(1+kv)

(1+kv

∗

)

(1+au+b

v+b

−

(1+kv)(1−u

∗

)[e

(cε−b

)−e

)][b

(1−u

∗

)−aδ

(1+kv

∗

)](1+au+b

v+b

(1+kv)

(1+kv

∗

)

(1+au+b

v+b

kδ

(1−u

∗

)(1+au+b

v+b

(1+kv)

(1+kv

∗

)

(1+au+b

v+b

<0.

P˜u=u−u

∗

,˜v=v−v

∗

,˜w=w−w

∗

,l

=(˜u,˜u,˜w)

L(˜u,˜u,˜w)≤0.…=u=u

∗

v= v

∗

,w= w

∗

ž,

= 0,dLaSalleC©Ún[43],·‚E

∗

S´ÛìC-½.

2.5.Hopf©|

ù˜Ü©·‚±™êÏfkŠ•©|ëê,$^Sotomayor½n[44]?Øˆ²ï:?u)Hopf

©|^‡.

½n8©|ëêkL.Šk= k

ž,XÚ(1.3)Œ7ÃIGÓ ö²ï:E

²{ ¿Ê

Å©|,ùpk

÷vµQ

)=a

) +a

)=0,λ

)−a

)

det(J(E

))|

k=k

= s

) >0,

dλ

k=k

ϕ−2σ

−4σ

6= 0.

y²λ(k)=λ

(k)+ iλ

(k)•(λ−a

)[λ

−(a

)λ+a

−a

]=0A

Š,òλ(k)“\¿©l¢ÜÚJÜ,











−3λ

(λ

−λ

)+s

−s

= 0,

3λ

−λ

−2s

= 0,

(13)

s

= a

(k)+a

(k),

= a

(k)a

(k)−a

(k)a

(k)+a

(k)[a

(k)+a

(k)],

= a

(k)[a

(k)a

(k)−a

(k)a

(k)].

AŠ¢Ü•"ž,TAŠ´Ý,ÃIGÓ ö²ï:E

Ï)Hopf©|”

,Òkλ

) = 0…λ

) 6= 0,“\(13)ªXe

•§|:











−s

= 0

−λ

= 0

Q

= a

)+a

) =0ž,ks

= a

),Kλ

)−a

),l

det(J(E

))|

k=k

= s

) >0.é(13)ªü>'uëêk¦











(3λ

−3λ

−2s

)

dλ

+(2s

−6λ

)

dλ

= (λ

−λ

)

−λ

(3λ

−3λ

−2s

)

dλ

+(6λ

−2s

)

dλ

= 2λ

−λ

duλ

= 0,¿-σ

= s

3λ

,σ

= 2s

,ϕ=

−λ

,ψ=−λ

.Kþª•§|C

DOI:10.12677/pm.2022.12122252095nØêÆ

w

•λ

) 6= 0,“\(13)ªXe•§|:











dλ

+σ

dλ

= ϕ

dλ

−σ

dλ

= ψ

)

dλ

k=k

ϕ−2σ

−4σ

6= 0,y..

ž,XÚ(1.3)Œ7ÃIG ²ï:E

²{¿Ê

Å©|,ùpk

÷vµQ

)=b

) + b

)=0,λ

)−b

)

det(J(E

))|

k=k

= s

) >0,

dλ

k=k

ϕ−2σ

−4σ

6= 0.

½n9e±e^‡¤á

)

(i)P

∗

) >0,P

∗

) >0

(ii)Q(k

∗

) = P

∗

)−P

∗

) = 0

(iii)

dQ(k)

k=k

∗

6= 0

…©|ëêk²L.Šk

∗

ž,XÚ(1.3)Œ7~ê²ï:E

∗

)Hopf©|.

y²d^‡(H

)¥(i)(ii)Ñ,k= k

∗

ž,(9)ªk˜‡KŠÚü‡XJŠ,K(9)ªÒ˜

½Œ±¤(λ

)(λ+P

) = 0/ª,¿Ñn‡Š•λ

= −P

,λ

= i

√

,λ

= −i

√

.é

¤k©|ëêŠk,AŠλ

/ª•λ

= α(k)+iβ(k),λ

= α(k)−iβ(k),α(k)•¢Ü,ò

λ(k) = α(k)+iβ(k)“\(9)ª¥



α(k)+iβ(k)



(k)



α(k)+iβ(k)





α(k)+iβ(k)



= 0

(14)











dα(k)

−l

dβ(k)

(k) = 0

dα(k)

dβ(k)

(k) = 0

Ù¥

(k) = P

(k)+2P

(k)α(k)+3



α(k)

−β(k)



(k) = 2



(k)+3α(k)



β(k),

(k) =



α(k)

−β(k)



(k)+α(k)P

(k)+P

(k),l

(k) =



2α(k)P

(k)+P

(k)



β(k).

)

dα(k)

k=k

∗

= Re



dλ(k)



k=k

∗

= −



(k)l

(k)+l

(k)l

(k)



k=k

∗

DOI:10.12677/pm.2022.12122252096nØêÆ

w

(k)P

(k)+P

(k)P

(k)−P

(k)



(k)



k=k

∗

= −







dQ(k)



(k)









k=k

∗

6= 0.

3.1.±È5Úk.5

•ïÄØCáÚ••35,XÚ(1.4))k.5Ú±È5¦^±eÚn[45].

Ún1

f(s)és≥0´C

a¼ê,d>0, η≥0´~ê,?,4T∈[0,∞),

Φ ∈C

2,1

(Ω×(T,∞))∩C

1,0

(Ω×[T,∞))´¼ê.

(i)eΦ÷v











−d∆Φ ≤Φ

1+η

f(Φ)(θ−Φ),(x,t) ∈Ω×(T,∞)

= 0,(x,t) ∈∂Ω×(T,∞)

~êθ>0,Klim

t→∞

supmax

Ω

Φ(.,t) ≤θ.

(ii)eΦ÷v











−d∆Φ ≥Φ

1+η

f(Φ)(θ−Φ),(x,t) ∈Ω×(T,∞)

= 0,(x,t) ∈∂Ω×(T,∞)

~êθ>0,Klim

t→∞

infmin

Ω

Φ(.,t) ≥θ.

½n10 XÚ(1.4)3R

S¤k)´˜—k.¿•ª?\ØCáÚ•

=[0,1] ×

εce

y²·‚7Ly²é∀t>0,



u(x,0),v(x,0),w(x,0)



∈



u(x,t),v(x,t),w(x,t)



∈

†Œ±wu(x,t)>0,v(x,t)>0,w(x,t)>0,Ïd§ ÐŠu(x,0),v(x,0)Úw(x,0)´šK,

ylXÚ(1.4)1˜‡•§¥,·‚

−∆u≤u(1−u),

dÚn1¥(i)•,

lim

t→∞

supmax

Ω

u(x,t) ≤1.

=é∀>0,∃t

>0,t>t

,(x,t) ∈Ω×[t

,∞]ž,¦u(x,t) ≤1+.

DOI:10.12677/pm.2022.12122252097nØêÆ

w

Ïd,3(x,t) ∈Ω×[t

,∞]þ,v•§÷v

−d

∆v≤v



(1+)

1+a(1+)+b

−δ



≤



(1+)

−δ



= δ



(1+)

−v



dÚn1¥(i)•,

lim

t→∞

supmax

Ω

v(x,t) ≤

=é∀

,∃t

,t>t

,(x,t) ∈Ω×[t

,∞]ž,¦v(x,t) ≤

+

Ïd,3(x,t) ∈Ω×[t

,∞]þ,w•§÷v

−d

∆w≤w



(1+)+cε(

+

)

−δ



≤δ







(1+)+cε



+

i

−w







du,

´?¿,2A^Ún1(i),

lim

t→∞

supmax

Ω

w(x,t) ≤

εce

½Âéuäk šKÐŠ…(u(x,0),v(x,0),w(x,0))6=0XÚ(1.4),e∃σ

,σ

þ•~

ê,¦XÚ(1.4))u(x,t),v(x,t),w(x,t)÷v

lim

t→∞

infmin

Ω

(u,t) ≥σ

,lim

t→∞

infmin

Ω

(v,t) ≥σ

,lim

t→∞

infmin

Ω

(w,t) ≥σ

,K¡XÚ(1.4)´±È.

½n11el

=1−

)(b

−ke

)

>0,l

−(c+b

)[b

(δ

+aδ

)+εce

]

(c+b

)

>0,

+εcδ

−δ

(1+a)−e

>0,KXÚ(1.4)´±È.

Ï•

−d

∆u= u



1−u

1+kv

−

v+w

1+au+b

v+b



≥u(

1−u

1+kv

−

)

≥u



(1−u)

+ke

−



≥

+ke



1−

)(b

+ke

)

−u



…-l

= 1−

)(b

+ke

)

>0,KdÚn1(ii)•,

lim

t→∞

infmin

Ω

(u,t) ≥1−

)(b

+ke

)

=é∀0 <

(0)

,∃t

,t>t

,(x,t) ∈Ω×[t

,∞]ž,¦u(x,t) ≥l

−

(0)

3(x,t) ∈Ω×[t

,∞]þ,XÚ(1.4)1‡'uv•§÷v:

−d

∆v≥v



−

(0)

)

(1+a)+(b

+εce

)+b

−

c+b



≥

(c+b

(1+a)+(b

+εce

)+b



−(c+b

)[b

(δ

+aδ

)+εce

]

(c+b

)

−v



DOI:10.12677/pm.2022.12122252098nØêÆ

w

-l

−(c+b

)[b

(δ

+aδ

)+εce

]

(c+b

)

>0,dÚn1(ii)•,

lim

t→∞

infmin

Ω

(v,t) ≥

−(c+b

)[b

(δ

+aδ

)+εce

]

(c+b

)

=é∀

(1)

>0,∃t

(1)

,t>t

(1)

,(x,t) ∈Ω×



(1)

,∞



ž,¦v(x,t) ≥l

−

(1)

aq/,XÚ(1.4)1n‡'uw•§÷v:

−d

∆w≥w



−

(0)

)+εcδ

−

(1)

)

(1+a)+e

−δ



≥

(1+a)+e



−

(0)

)+εcδ

−

(1)

)−δ

(1+a)−e

−w



-l

+εcδ

−δ

(1+a)−e

>0,dÚn1(ii)•,

lim

t→∞

infmin

Ω

(w,t) ≥

+εcδ

−δ

(1+a)−e

y..

3.2.ÛÜ-½5

:†XÚ(1.3)˜—.yL«E=



u(x,t),v(x,t),w(x,t)



,F(E)=



(E),f

(E)



D= diag(d

),XÚ(1.4)3E

∗

?‚5z/ªXe:











= LE,x∈Ω,

n·∇E= 0,x∈∂Ω.

L= D∆+J

∗

0=µ

<µ

<...<µ

<...,´ΩþäkàgNeumann>.^‡ −∆ŽfA

Š,E(µ

)•H

(Ω)þµ

(i=0,1,2,...)éAA˜m, X•[H

(Ω)]þ



(

Ω)



4•,

= {cφ

|c∈R

},{φ

: j= 1,2,3,...,dimE(µ

)}´X

˜|IOÄ,K

∞

i=1

dim[E(µ

)]

j=1

éz˜‡i≥1,X

3ŽfLe´ØC,…é,‡i≥15`,λ´ŽfLAŠ…=§

´Ý−µ

D+J

∗

AŠž,TAŠ¿éAX

¥˜‡A•þ.−µ

D+J

∗

A•§•

(λ) = λ

λ+P

= 0

(15)

DOI:10.12677/pm.2022.12122252099nØêÆ

w



= (d

)µ

= (d

)µ

−[c

)+c

)]µ

= d

−(d

)µ

+[d

−c

)+d

−c

)+d

−c

)]µ

= q

= P

−P

= m



= (d

)(d

)−d

= −2(d

)(c

)−d

)

= d



2(c

)−c

−c





2(c

)−c

−c





2(c

)−c

−c



= P

−P

d(10)¥ÎÒŒ•m

þŒu",kQ

>0,Ïd,P

>0,dRouth-Hurwitz

OKŒ••§(15)n‡ŠÑäkK¢Ü,éz‡i≥1.

-λ= µ

ζ,K

(λ) = µ

ζ+P

(ζ)

4µ

−→∞,i−→∞ž,

lim

t→∞

(ζ)

= ζ

+(d

)ζ

+(d

)ζ+d

Ψ(ζ)

w,,

Ψ(ζ) = 0kn‡Š−d

,−d

, dëY5•∃i

, ¦

(ζ) = 0n‡Šζ

,ζ

÷v

Re{ζ

},Re{ζ

}≤−

,∀i≥i

,ùpd= min{d

},KRe{λ

},Re{λ

}≤

−µ

≤−

,∀i≥i

−

d= max



Re{λ

},Re{λ

}≤−µ

≤−



DOI:10.12677/pm.2022.12122252100nØêÆ

w

d>0,…Re{λ

},Re{λ

}≤−µ

≤−

≤−d= min





<0,∀i≥1dLŽ

fAŠ|¤Ì uŒ²¡Re{λ≤−d}S.Ïd,dDanHenry[46]½n5.1.1,·‚k±

∗

-½5(J.

½n12e^‡(H

)-(H

)¤á,Kg*ÑXÚ(1.4)~ê²ï:E

∗

´ÛÜìC-½.

5:þã½n L²g*ÑXÚŒUØ¬U CXÚ-½5,•Ò´`§g*ÑØU)Turing

Ø-½5.

4.(Ø

,IG ,IGÓ ö•65,•Ä3àgNewman>.^‡eäk™êAÚBedington-

DeAngelisõU‡Až˜ +SÓ ..éuù‡IGP.§·‚Ñ¤kŒ1²ï:

•35Ú(ÛÜ/Û)-½5^‡,•)²…²ï:!>.²ï:Ú~ê²ï:.d,„

Ú(1.4)Œ1²ï:.,,·‚l½n6Ú9Œ±w,ÃØ*Ñ´Ä•3,~ê²ï:Û

Ü-½5^‡¿vkÉ*ÑK•.

½n10ž˜XÚ(1.4)˜—±È5•3•ý«XÉ™êK••],IG ,IGÓ

önÔ«Œ±•.

êŠ©Û.

Ä7‘8

I[g,‰ÆÄ7(Nos.11761063)"

ë•©z

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