Global solutions for a general predator-prey model with cross-diffusion effects

Zhang Lina, Fu, Fu Shengmao

Description

This work investigates a general Lotka-Volterra type predator-prey model with nonlinear cross-diffusion effects representing the tendency of a predator to get closer to prey. Using the energy estimate method, Sobolev embedding theorems and the bootstrap arguments, the existence of global solutions is proved when the space dimension is less than 10. MSC: 35K57, 35B35.

Sujets

• Predation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

ZhangandFuAdvancesinDifferenceEquations2012,2012:117
http://www.advancesindifferenceequations.com/content/2012/1/117
RESEARCH OpenAccess
Globalsolutionsforageneralpredator-prey
modelwithcross-diffusioneffects
*LinaZhang andShengmaoFu
*Correspondence:
[e-mail] Abstract
DepartmentofMathematics,
ThisworkinvestigatesageneralLotka-Volterratypepredator-preymodelwithNorthwestNormalUniversity,
Lanzhou730070,P.R.China nonlinearcross-diffusioneffectsrepresentingthetendencyofapredatortogetcloser
toprey.Usingtheenergyestimatemethod,Sobolevembeddingtheoremsandthe
bootstraparguments,theexistenceofglobalsolutionsisprovedwhenthespace
dimensionislessthan10.
MSC: 35K57;35B35
Keywords: globalsolution;predator-prey;cross-diffusion
1 Introduction
Considerthefollowingpredator-preymodelwithcross-diffusioneffects:
⎧
⎪u –d (+ δu)u =ug(u)–p(u,v)v, x ∈ ,t>, t ⎪
⎪⎪ ⎪ γ⎪⎨v –d + αv+ v =v –d–sv+cp(u,v) , x ∈ ,t>,t  l+u ()
∂u ∂v⎪⎪ = =, x ∈ ∂ ,t>,⎪∂ν ∂ν⎪⎩
u(x,)=u (x) ≥( ≡), v(x,)=v (x) ≥( ≡), x ∈ , 
nwhere ⊂R (n ≥) is a bounded domain with smooth boundary ∂ , ν is the outward
unitnormalvectoroftheboundary∂ ,thegivencoefficientsd,s,c,d ,d , δ, α, γ andlare 
positive constants. The functions g ∈C ([,∞)) and p ∈C([,∞) ×[,∞)) are assumed
tosatisfythefollowingtwohypothesesthroughoutthispaper:
(H) g()>andg (u)< forallu>,andthereexistsapositiveconstantK suchthat
g(K)=.
(H) Forallu,v ≥, ≤p(u,v) ≤Ch(u)forsomepositiveconstantC andacontinuous
functionh(u).
In the model (), u and v represent the densities of the prey and predator respectively,
the constants d, s and c refer to the predator death rate, the inter-specific competition
coefficient and the conversion rate of the prey to the predator, and g(u)representsthe
growth rate of the prey in the absence of the predator. The function p(u,v) is called the
functionalresponseofthepredatortotheprey.Theconstantsd andd arethediffusion 
rates of the prey and predator respectively, δ and α are referred to as self-diffusion
preslsures,andtheterm d [γv/(+u )] inthepredatorequationiscross-diffusionpressure,
which expresses the population flux of the predator resulting from the presence of prey
© 2012 Zhang and Fu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
inanymedium,providedtheoriginalworkisproperlycited.ZhangandFuAdvancesinDifferenceEquations2012,2012:117 Page2of9
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species.Biologically,thistermrepresentsthetendencyofthepredatortochaseitsprey.It
isclearthatsuchanenvironmentofprey-predatorinteractionoftenoccursinreality.For
example, in [–], with the similar biological interpretation, the authors also introduced
thesamecross-diffusiontermasin()tothepredatorofvariouspredator-preymodels.
Mathematically,oneofthemostimportantproblemfor()istoestablishtheexistence
of global solutions. However, to our knowledge, there are only a few works. In [], the
globalexistenceresultwasshownwithoutanyrestrictionsonspacedimensionsbyH.Li,
P.PangandM.Wang.Buttheirresultsarenotvalidfor()becausesomerestrictionsare
requiredforthetermp(u,v)v/u,sothatthestandardreactiontermlikeLotka-Volterratype
isexcludedintheirworks.Thepurposeofthispaperistoestablishasufficientcondition
fortheexistenceofglobalsolutionsfor()withoutanyrestrictionsonthetermp(u,v)v/u
inthehigherdimensionalcase.Ourmainresultisasfollows.
Theorem . Let l ≥ and n<.Assumethatu ,v satisfy the zero Neumann bound- 
+λary condition and belong to C () for some< λ<.Then() possesses a unique
non+λ,(+λ)/ +λ,(+λ)/negative solution (u,v),suchthatu ∈ C ( × [,∞)) and v ∈ C ( ×
+λ,(+λ)/[,∞)).Moreover,ifl=orl ≥,thenv ∈C ( ×[,∞)).
Astothestationaryproblemfor(),therehavebeensomeworks,forexample,see[,,
–]andreferencestherein.
The paper is organized as follows. In Section , we present some known results and
pprove some preliminary results that are used in Section .InSection ,weestablish L -
∞estimates for v.InSection ,weestablish L -estimates for v and give a proof of
Theorem..
2Preliminaries
Forthetime-dependentsolutionsof(),thelocalexistenceisanimmediateconsequence
of[–].Theresultscanbesummarizedasfollows.
Theorem A Let u ,v ∈W (),wherep>n, be nonnegative functions. Then the system  p
 ∞() has a unique nonnegative solution (u,v) such that u,v ∈C([,T),W ()) ∩C ((,T),p
∞C ()),whereT ∈ (,+∞] is the maximal existence time of the solution. If the solution
(u,v)satisfiestheestimate

sup u(·,t) , v(·,t) :t ∈(,T) < ∞, W () W ()p p
 thenT=+∞.If,inaddition,u ,v ∈W (),thenu,v ∈C([,∞),W ()).  p p
Let
Q = ×[,T),T
q

/qT pp p p,p p,qu = u(x,t) dx dt , L (Q )=L (Q ),L (Q ) T TT


 p p pu , = u + u + ∇ u + ∇ u ,L (Q ) t L (Q ) L (Q ) pW (Q ) T T T L (Q )p T T

u = sup u(·,t) + ∇u(x,t) .V (Q )  T L () L (Q )T≤t≤TZhangandFuAdvancesinDifferenceEquations2012,2012:117 Page3of9
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Lemma. (i)Let(u,v)beasolutionof()in[,T).ThereexistsapositiveconstantM =
∞max{K,u }suchthat ≤u ≤M andv ≥. L () 
(ii)ForanyT>,thereexisttwopositiveconstantsC (T)andC (T)suchthat 

sup v(·,t) ≤C (T), v  ≤C (T).  L (Q ) L () T
≤t≤T
(iii)ForanyT>,thereexistsapositiveconstantC (T)suchthat ∇ u  ≤C (T). L (Q ) T
Proof (i)Applyingthemaximumprincipleto(),wehavetheresult(i).
(ii) Integrating the second equation of () over the domain and by (i) and (H), we
have

d
vdx = v –d–sv+cp(u,v) dx
dt

≤(d+cM) vdx–s v dx


s
≤(d+cM) vdx– vdx .
||
Then we have sup v(·,t)  ≤ C (T). Integrating the above inequality from  to≤t≤T L ()
T,wehave
sv ≤(d+cM)TC (T)+ v  .   L ()L (Q )T
Therefore, v  ≤C (T).L (Q )T
(iii)Letw =(+ δu)u,thenw satisfiestheequation 
w =d (+δu)w +c +c v,t    
where c =(+δu)ug(u)and c =–(+δu)p(u,v). Since u, c , c are bounded and v ∈   
,L (Q ),wecanprove w ∈W (Q ) inthesamewayasLemma.in[].Moreover, wT  T 
satisfies

w ≤d (+δu)w +(+δu)ug(u)=d +δw w +(+δu)ug(u).t     
Applying Proposition . in [], we obtain ∇w ∈ L (Q ), which in turn implies T
∇ u  ≤C (T). L (Q )T
pInordertoestablishL -estimatesforsolutionsof(),weneedthefollowingresultwhich
canbefoundin[].
Lemma. Letq>andq=+q/[n(q+)].Supposethatwsatisfies
sup w q/(q+) + ∇ w  < ∞,L (Q )L () T
≤t≤T
βand there exist positive constants β ∈(,) and C > such that |w(·,t)| dx ≤C forT T
all t ∈[,T]. Then there exists a positive constant M independent of w but depending onZhangandFuAdvancesinDifferenceEquations2012,2012:117 Page4of9
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n, ,q, β