R E S E A R C H
Open Access
Global behavior for a diffusive predator-prey
system with Holling type II functional
response
Yanzhong Zhao
**Correspondence: yanzhongz@yahoo.com.cn Department of Basic Research, Qinghai University, Xining, 810016, China
Abstract
A strongly coupled self- and cross-diffusion predator-prey system with Holling type II functional response is considered. Using the energy estimate, Sobolev embedding theorem and bootstrap arguments, the global existence of non-negative classical solutions to this system in which the space dimension is not more than five is obtained.
MSC: 35K50; 35K55; 35K57; 92D40
Keywords: predator-prey system; cross-diffusion; global solution
1 Introduction
In this paper, we consider the global existence of non-negative classical solutions to the following diffusion predator-prey system with Holling type II functional response:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ut=(du+αu) +αu( –Ku) –+βmuvamu, x∈,t> ,
vt=(dv+αuv+αv) –rv++cβmuvamu, x∈,t> ,
∂ηu(x,t) =∂ηv(x,t) = , x∈∂,t> ,
u(x, ) =u(x)≥,v(x, ) =v(x)≥, x∈,
(.)
whereis a bounded region inRn(n≥) with a smooth boundary∂;ηis the outward
normal on∂,∂η=∂/∂η;u(x) andv(x) are non-negative smooth functions and are not
identically zero;uandvdenote the population densities of predator and prey, respectively;
α,β,r,a,K,mandcare positive constants, andm∈(, ];d andd are the diffusion
rates of the two species;αij(i,j= , ) are given non-negative constants,αandαare
self-diffusion rates;αis the cross-diffusion rate. It means that the diffusion is from one
species of high-density areas to the other species of low-density areas. See [, ] for more details on the ecological backgrounds of this system.
Obviously, the non-negative equilibrium solutions of system (.) are (K, ) and (u*,v*) =
(m(cβr–ar),αcKmKm(c(βcβ––arar)–)αcr). For the reaction-diffusion problem of system (.),i.e.,αii= (i=
, ), the global attraction, persistence and stability of non-negative equilibrium solutions are studied in []. The main result can be summarized as follows:
() IfK(cβr–ar)<m≤K(cβr–ar)+Ka andcβ>ar, a unique positive constant solution (u*,v*)is globally asymptotically stable;
() IfK(cβr–ar)<m<K(cβr–ar)+Ka(ccββ–ar) andcβ>ar, a positive constant solution(u*,v*)is
locally asymptotically stable.
In view of the study of dynamic behavior of a predator-prey reaction-diffusion system with Holling type II functional response, a natural problem is what the global behavior for a predator-prey cross-diffusion system (.) is. To the best of our knowledge, the existing results are very few. In this paper, we consider the space dimension to be less than six, and initial functionu(x) andv(x) under some smooth conditions, using the energy estimate,
Sobolev embedding theorem and bootstrap arguments, we consider the global existence of non-negative classical solutions for system (.).
We denoteQT=×(,T).u∈Wq,(QT) means thatu,uxi,uxixj (i,j= , . . . ,n) andut are inLq(Q
T).uLp,q(Q T)= [
T
(
|u(x,t)|pdx)
q
pdt]q.u
V(QT)=sup≤t≤Tu(·,t)L()+
∇uL(Q
T).
2 Auxiliary results
Lemma . Let(u,v)be the solution of(.).There exists a positive constant M(≥)such
that
≤u≤M, ≤v, ∀t≥. (.)
Proof Firstly, the existence of local solutions for system (.) is given in [–]. Roughly speaking, ifu,v∈Wp(),p> , there exists the maximumT≤+∞such that system
(.) admits a unique non-negative solution
u,v∈C[,T),Wp()∩C∞(,T),C∞(). (.)
If
sup u(·,t)w
p(),v(·,t)wp(): <t<T
<∞,
thenT= +∞.
ChooseM=max{K,uL∞()}. By use of the maximum principle, the non-negative
solution of system (.) can be derived from the maximum principle,i.e.,u,v≥ for all
t≥. This completes the proof of Lemma ..
Lemma . Let X= (d+αu)u,u∈L∞(QT)for the solution to the following equation:
ut=
(d+αu)u
+αu
– u K
– βmuv
+amu, (x,t)∈×(,T),
∂ηu= , (x,t)∈∂×(,T),
u(x, ) =u(x)≥, x∈,
where d,αare positive constants and≤u∈L(QT).Then there exists a positive
con-stant C(T),depending onuW()anduL∞(),such that
XW,
Furthermore,
∇X∈V(QT), ∇u∈L
(n+)
n (Q
T). (.)
Proof FromX= (d+αu)u, it is easy to find that
Xt= (d+ αu)X+C–Cv, (.)
where C=dαu+ (αα–dKα)u– αKαu andC= +βamumu (d+ αu).C andC are
bounded in QT from (.). Multiplying (.) by –X and integrating by parts over Qt
yields
∇X(x,t)dx–
∇X(x, )dx+d
Qt
|X|dxdt
≤
QT
|C+Cv||X|dxdt. (.)
Using the Hölder inequality and Young inequality to estimate the right-hand side of (.), we have
C+CvL(Q
T)XL(QT)≤m
+vL(Q
T)
XL(Q
T)
≤m( +M)
d
+d
X
L(Q
T) (.)
with somem> . Substituting (.) into (.), we obtain
sup ≤t≤T
∇X(x,t)
dx+d
Qt
|X|dxdt≤m,
where m depends onuW
() anduL∞(). So, we know∇X∈V(QT). SinceX∈
L(Q
T), it follows from the elliptic regularity estimate [, Lemma .] that
QT |Xxixj|
dxdt≤m
, i,j= , . . . ,n.
From (.), we haveXt∈L(QT). Hence,XW,
(QT)≤C(T). Moreover, (.) can be
ob-tained by use of the Sobolev embedding theorem.
Lemma . Assume that w∈WP,(QT)∩C,(¯×[,T))is a bounded function satisfying
wt≤a(x,t,w)w+f(x,t) in QT
with the boundary condition ∂∂wv≤on∂QT,where f ∈L
P(Q
T).Then∇W is in Lp(QT).
The proof of the above lemma can be found in [, Proposition .].
Lemma . Let p> ,p˜= +n(pp+).If
sup ≤t≤T
w
L
p
p+()+∇wL (Q
T)<∞,
and there exist positive constantsβ∈(, )and CTsuch that
|w(·,t)|
βdx≤C
T(∀t≤T),
there exists a positive constant M,independent of w but possibly depending on n,,p,β and CT,such that
wLp˜(Q
T)≤M
+ sup
≤t≤T
w(t)
L
p p+()
p n(p+)˜p
∇w
˜
p
L(Q
T)
.
Finally, one proposes some standard embedding results which are important to obtain theL∞andC+α,+α(Q
T) normal estimates of the solution for (.).
Lemma . There exists a constant C(T)such that∇uL(QT)≤C(T).
Proof Letδ=α/d,X= ( +δu)u. By Lemma .,u is bounded. Therefore,X is also
bounded. By Lemma ., we haveX∈W,(QT). Moreover,Xsatisfies
Xt≤d( + δu)X+αu( + δu)
=
d
+ δdXX+αu( + δu).
By Lemma . withp= ,a(x,t,ξ) =d+ δdξ,f(x,t) =αu(x,t)( + δu(x,t)), we obtain
the desired result.
Lemma . Let ⊂ Rn be a fixed bounded domain and∂⊂C. Then for all u∈
Wq,(QT)with q≥,one has
() ∇uLp(Q
T)≤CuWq,(QT),∀≤p≤
(n+)q
n+–q,q<n+ ;
() ∇uLp(QT)≤CuWq,(QT),∀≤p≤ ∞,q=n+ ;
() ∇u
Cα,α(QT)≤CuW
,
q (QT),∀ –
n+
q ≤α≤,q>n+ ,
where C is a positive constant dependent on q,n,,T.
3 The existence of classical solutions
The main result about the global existence of non-negative classical solutions for the cross-diffusion system (.) is given as follows.
Theorem . Assume that u> and v> satisfy homogeneous Neumann boundary
conditions and belong to C+α(¯)for someα∈(, ).Then system(.)has a unique
non-negative solution(u,v)∈C+α,+α(¯ ×[,∞))if the space dimension is n≤.
Proof Whenn= , the proof is similar to the methods of [–]. So, we just give the proof of Theorem . forn= , , , . The proof is divided into three parts.
Firstly, integrating the first equation of (.) over, we have
d dt
udx=
u
α–αu
K –
βmv
+amu
dx≤
αudx– α K
udx
≤α
udx– α K||
udx
.
Thus, for allt≥, we can obtain
udx≤M,
whereM=max{K||,udx}.
Furthermore,
uL(Q
T)≤
T
MdtM. (.)
Secondly, linear combination of the second and first equations of (.) and integrating overyields
d dt
(cu+v) dx=c
αudx–cα K
udx–r
vdx
≤c(r+α)
udx– cα K||
udx
–r
(cu+v) dx
≤c(r+α)M–r
(cu+v) dx.
So, we get
(cu+v) dx≤max
c(r+α)M
r ,
cu(x) +v(x)
dx
M, ∀t≥. (.)
Further,
vL(Q
T)≤
T
MdtM. (.)
Then multiplying both sides of the second equation of system (.) byvand integrating over, we obtain
d dt
vdx= –
∇v∇dv+αuv+αv
dx–r
vdx+cmβ
uv
+amudx
≤–d
|∇v|dx–α
v∇v∇udx– α
v|∇v|dx+cβ
a
vdx.
Integrating the above expression in [, ] yields
v(x,t) dx–
v(x) dx+
Qt
(d+αu+ αv)|∇v|dxdt+r
Qt vdxdt
≤–α
Qt
∇u·v· ∇vdxdt+cβ a
Qt
Estimating the first term on the right-hand side of (.),
Qt
∇u·v· ∇vdxdt=
Qt
∇u·v· ∇v·vdxdt
≤ε
Qt
v|∇u|dxdt+
ε
Qt
v· |∇v|dxdt
≤ε
Qt
vdxdt+
ε
Qt
|∇u|dxdt
+
ε
Qt
v· |∇v|dxdt. (.)
Substituting (.) into (.) yields
v(x,t) dx–
v(x) dx+
(d+αu+ αv)|∇v|dxdt+r
Qt vdxdt
≤
αε+
cβ
a Qt
vdxdt+α ε
Qt
|∇u|dxdt+α
ε
Qt
v|∇v|dxdt. (.)
Selectε>ααand denoteC= α–αε. Notice the positive equilibrium point of (.)
exists under conditioncβ>ar, then
v(x,t) dx+ d
Qt
|∇v|dxdt+ α u∞
Qt
|∇v|dxdt+C
Qt
v|∇v|dxdt
≤
v(x) dx+
αε–r+
cβ
a Qt
vdxdt+α ε
Qt
|∇u|dxdt. (.)
By Lemma .,∇uL(QT)≤C(T). Integrating the above inequality and using the
Gronwall inequality, we get
sup <t<T
vdx≤C(T).
Hence, there exists a positive constantMsuch that vdx≤M. Furthermore, we have
vL(Q
T)≤
T
MdtM. (.)
Secondly, multiplying both sides of the second equation of system (.) byqvq–(q> )
and integrating over, we have
d dt
vqdx≤–(q– )d q
∇
vqdx–q(q– )α
(q+ )
∇
vq+ dx
–q(q– )α
vq–∇u· ∇vdx+q
vq
–r+ cβmu +amu
Integrating the above equation over [,t] (t≤T), it is clear that
vq(x,t) dx+(q– )d q
Qt
∇
vqdxdt+q(q– )α
(q+ )
Qt
∇
vq+ dxdt
≤
vq(x) dx–q(q– )α
Qt
vq–∇u· ∇vdxdt+cβq a
Qt
vqdxdt. (.)
By Lemma ., it can be found that∇u∈L(nn+)(QT). According to the Hölder inequality and Young inequality, we get
–q(q– )α
Qt
vq–∇u· ∇vdxdt
≤q(q– )α
q+
Q
t
vq– ∇v
q+
· ∇udxdt
≤q(q– )α
q+ ∇uL(nn+)(QT)v q–
Ln+(Q
T)∇
vq+
L(Q
T)
≤C∇
vq+
L(Q
T)v q–
Ln+(Q
T)
≤Cε
∇
vq+
L(Q
T)+ C
εv q–
Ln+(Q
T). (.)
Choose an appropriate numberεsatisfyingCε
≤
q(q–)α
(q+) . Substituting (.) into (.)
and takingv¯=vq+ , we have
¯ v
q
q+(x,t) dx+
Qt
|∇¯v|dxdt
≤
vqdx+
C
ε¯v
(q–)
q+
L
(q–)(n+)
q+ (Q
T) +cβq
a ¯v
q q+
L
q q+(Q
T)
≤C
+¯v
(q–)
q+
L
(q–)(n+)
q+ (Q
T) +¯v
q q+
L
q q+(Q
T)
. (.)
Let
E≡ sup <t<T
¯ v
q
q+(x,t) dx+
QT
|∇¯v|dxdt.
From (.), we know
E≤C
+¯v
(q–)
q+
L
(q–)(n+)
q+ (Q
T) +¯v
q q+
L
q q+(Q
T)
.
Whenq<nn(n–+), it is easy to find that
q
q+< <q˜and
(q–)(n+)
q+ <q˜= + q n(q+). So,
E≤C
+¯v
(q–)
q+
Lq˜(Q T)+¯v
q q+
L˜q(Q T)
Setβ=q+ ∈(, ). It follows from theL()-estimate forvthat
¯vLβ()=
¯v(·,t)β
dx β =v β
L()≤M
β
, ∀t≤T.
By Lemma . and (.), we know
E≤C
+
M+M sup <t<T
v¯(·,t)
q n(q+)˜q
L
q q+()
∇¯vq˜
L(Q
T)
(q–)
q+
+
M+M sup <t<T
v¯(·,t)
q n(q+)q˜
L
q q+()
∇¯vq˜
L(Q
T)
q q+
≤C
+sup <t<T
¯v(·,t)
q q+
L
q q+()
(q–)
n(q+)q˜
∇¯vL(Q
T)
(q–) (q+)q˜
+
sup <t<T
v¯(·,t)
q q+
L
q q+()
(q–)
n(q+)˜q ∇¯v
L(Q
T)
q
(q+)q˜
≤C
+E
(q–)
n(q+)q˜E
(q–) (q+)q˜ +E
q n(q+)q˜E
q
(q+)˜q. (.)
Obviously, n((qq+)–)q˜ +((qq+)–)q˜ ∈(, ) and n(q+)q q˜ +(q+)qq˜ ∈(, ). It is easy to know thatEis bounded by use of reduction to absurdity. Sinceq<nn(n+)–,
(q+)q˜
∈(, (n+)
n– ). So,¯vLq˜(Q T) is bounded,i.e.,v∈L(q+) q˜(QT). Denote(q+)q˜
still asq. So,
v∈Lq(QT), ∀q∈
,(n+ ) n–
. (.)
Finally, whenn= , , , , (n– )q<n+ nwithq= . Forn≤, takingq= in (.), it
follows from (.) that there exists a positive constantMsuch that
vV(QT)≤M. (.)
By embedding theorem, we get
v
L(nn+)(QT)≤ M.
(ii)L∞-estimate forv.
The second equation of system (.) can be written as the following divergence form:
∂v ∂t=
n
i,j=
∂ ∂xi
aij(x,t)
∂v ∂xj
+ n i= ∂ ∂xi
ai(x,t)v
+v
–r+ cβmu +amu
, (.)
whereaij(x,t) = (d+αu+ αv)δij,ai(x,t) =αuxjandδijis the Kronecker sign. In order to apply the maximum principle [] to (.), we need to prove the following conditions:
() vV(QT)is bounded; () ni,j=aij(x,t)ξiξj≥ν
() ni=a
i(x,t),v(–r+ cβmu
+amu)Lp,r(QT)≤μ, whereν,μare positive constants and
r+
n
p= –χ, <χ< ,p∈
n ( –χ), +∞
,r∈
–χ, +∞
,n≥. (.)
Next, we will show that the above conditions () to () are satisfied for (.). Whenn≤, it is easy to find that the condition () is satisfied by use of (.). Sinceni,j=aij(x,t)ξiξj≥
d|ξ|for allξ ∈Rn, the condition () is verified. In view of the condition (), we take
appropriateqandr. Rewrite the first equation of system (.) as
ut=∇ ·
(d+ αu)∇u
+αu
– u K
– βmuv
+amu. (.)
Whenn= , , , , n+ <(nn–+), it is clear thatd+ αuhas an upper bound overQTby
Lemma .. Set
q∈
n+
,
(n+ ) n–
.
From (.), we haveαu( –Ku) –+βmuvamu ∈Lq(QT). Therefore, all conditions of the Hölder
continuity theorem [, Theorem .] hold for (.). Hence,
u∈Cβ,β(Q
T), β∈(, ). (.)
We will discuss (.) which is the corresponding form of (.). It follows from (.) and (.) thatC–Cv∈Lq(QT),∀q∈(n+ ,(nn–+)). From (.), we obtaind+αu∈
Cβ,β(Q
T). Thus, according to the parabolic regularity result of [, pp.-,
Theo-rem .], we can conclude that
X∈Wq,(QT), ∀q∈
n+
,
(n+ ) n–
, (.)
which implies that∇X∈L
(n+)q n+–q(Q
T) by Lemma ..
SinceX= (d+ αu)u, we have∇u= (d+ αu)–∇X,i.e.,∇u∈L
(n+)q n+–q(Q
T). It means
that |∇u|,|∇v|∈L((nn+)+–qq)(Q
T). So, ni=ai(x,t)∈L
(n+)q
(n+–q)(Q
T). From (.) and (.),
v(–r++cβamumu)∈Lq(Q T).
Then the condition () and (.) are satisfied by choosingp=r=((nn+)+–pp). According to the maximum principle [, p., Theorem .], we can conclude thatv∈L∞(QT). From
(.), there exists a positive constantMsuch that
uL∞(QT),vL∞(QT)≤M, ∀T> . (.)
Therefore, the global solution to the problem (.) exists. (iii) The existence of classical solutions.
Under the conditions of Theorem ., we consider above global solutions (u,v) to be classical. By (.) and Lemma ., we know∇X∈Cα,α(Q
T),∀α∈(, ). It follows from
Lemma . in [] thatX∈C+α,+α(Q
T). SinceX= (d+αu)u, we haveu=
–d√d+αX
So,
u∈C+α,+α(Q
T), ∀α∈(, ). (.)
Rewrite the second equation of system (.) as
vt=∇ ·
(d+αu+ αv)∇v+α∇uv
+v
–r+ cβmu
+amu
.
Therefore, we can conclude that v(–r+ +cβamumu)∈L∞(QT), u, v, ∇u and ∇v are all
bounded. By the Schauder estimate [], there existsα*∈(, ) such that
v∈Cα*,α
* (Q
T). (.)
Furthermore, by the Schauder estimate, we obtain
u∈C+σ,+σ(Q
T), σ=min α,α*
. (.)
Next, the regularity ofvwill be discussed. Setv¯= (d+αu+αv)v. So,v¯satisfies
¯
vt= (d+αu+ αv)v¯+f(x,t), (.)
wheref(x,t) = (d+αu+ αv)v(–r++cβamumu) +αutv. According to (.) to (.), we
haved+αu+ αv,f(x,t)∈Cσ, σ
(QT). Applying the Schauder estimate to (.), we
know
¯
v∈C+σ,+σ
(Q
T).
Fromv=–(d+αu)+
√
(d+αu)+α¯v
α , we can see
v∈C+σ,+σ(Q
T), σ=min α,α*
. (.)
Combining (.) and (.), we get
u,v∈C+σ,+σ(Q
T).
Therefore, the result of Theorem . can be obtained forα<α*, namelyσ=α. When
α>α*, namelyσ<α, we haveC+σ,+σ
(Q
T)→Cα,
α
(Q
T). (.) and (.) are obtained
by repeating the above bootstrap argument and the Schauder estimate. This completes
the proof of Theorem ..
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
Acknowledgements
The author thanks the anonymous referee for a careful review and constructive comments. This work was supported by the Qinghai Provincial Natural Science Foundation (2011-Z-915).
Received: 6 April 2012 Accepted: 25 September 2012 Published: 9 October 2012 References
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