Global behavior for a diffusive predator-prey system with Holling type II functional response

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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:} [email protected] 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=(du+αu) +αu( –_Ku) –_+βmuv_amu, x∈,t> ,

vt=(dv+αuv+αv) –rv+_+cβmuv_amu, 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}_∂_;_η_{is 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,αandαare

self-diffusion rates;αis 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₍_c_βr_–_ar₎,αcKm_Km(c₍β_c_β–_–ar_ar)–₎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:

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() If_K₍_c_βr_–_ar₎<m≤_K₍_c_βr_–_ar₎+_Ka andcβ>ar, a unique positive constant solution (u*,v*)is globally asymptotically stable;

() If_K₍_c_βr_–_ar₎<m<_K₍_c_βr_–_ar₎+_Ka₍_cc_ββ_–_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

p_d_t_]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),W_p()∩C∞(,T),C∞(). (.)

If

sup u(·,t)_w

p(),v(·,t)wp():  <t<T

<∞,

thenT= +∞.

ChooseM=max{K,uL∞()}. 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+αu)u,u∈L∞(QT)for the solution to the following equation:

ut=

(d+αu)u

+αu

 – u K

– βmuv

 +amu, (x,t)∈×(,T),

∂ηu= , (x,t)∈∂×(,T),

u(x, ) =u(x)≥, x∈,

where d,αare positive constants and≤u∈L(QT).Then there exists a positive

con-stant C(T),depending onuW_()anduL∞(),such that

X_W,

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Furthermore,

∇X∈V(QT), ∇u∈L

(n+)

n ₍_Q

T). (.)

Proof FromX= (d+αu)u, it is easy to ﬁnd that

Xt= (d+ αu)X+C–Cv, (.)

where C=dαu+ (αα–d_Kα)u– α_Kαu andC= _+β_amumu (d+ αu).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+Cv||X|dxdt. (.)

Using the Hölder inequality and Young inequality to estimate the right-hand side of (.), we have

C+CvL₍_Q

T)XL(QT)≤m

 +vL₍_Q

T)

X_L₍_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 onuW

() anduL∞(). So, we know∇X∈V(QT). SinceX∈

L₍_Q

T), it follows from the elliptic regularity estimate [, Lemma .] that

QT |Xxixj|

_d_x_d_t_≤_m

, i,j= , . . . ,n.

From (.), we haveXt∈L(QT). Hence,X_W,

 (QT)≤C(T). Moreover, (.) can be

ob-tained by use of the Sobolev embedding theorem.

Lemma . Assume that w∈W_P,(QT)∩C,(¯×[,T))is a bounded function satisfying

wt≤a(x,t,w)w+f(x,t) in QT

with the boundary condition ∂_∂w_v≤on∂QT,where f ∈L

P₍_Q

T).Then∇W is in Lp(QT).

The proof of the above lemma can be found in [, Proposition .].

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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)|

β_d_x_≤_C

T(∀t≤T),

there exists a positive constant M,independent of w but possibly depending on n,,p,β and CT,such that

w_Lp˜₍_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(Q_T)≤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

+ δdXX+αu( + δu).

By Lemma . withp= ,a(x,t,ξ) =d_+ δdξ,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∈

W_q,(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 some}_α_∈_{(, ).}_{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.

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Firstly, integrating the ﬁrst equation of (.) over, we have

d dt

udx=

u

α–αu

K –

βmv

 +amu

dx≤

αudx– α K

udx

≤α

udx– α K||

udx



.

Thus, for allt≥, we can obtain

udx≤M_,

whereM_=max{K||,udx}.

Furthermore,

u_L₍_Q

T)≤

T



M_dtM. (.)

Secondly, linear combination of the second and ﬁrst equations of (.) and integrating overyields

d dt

(cu+v) dx=c

αudx–cα K

udx–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



M_dtM. (.)

Then multiplying both sides of the second equation of system (.) byvand integrating over, we obtain

  d dt

vdx= –

∇v∇dv+αuv+αv

dx–r

vdx+cmβ

uv

 +amudx

≤–d

|∇v|_d_x_–_α 

v∇v∇udx– α

v|∇v|_d_x₊cβ

a

vdx.

Integrating the above expression in [, ] yields

v(x,t) dx–

v_(x) dx+ 

Qt

(d+αu+ αv)|∇v|dxdt+r

Qt vdxdt

≤–α

Qt

∇u·v· ∇vdxdt+cβ a

Qt

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Estimating the ﬁrst term on the right-hand side of (.),

Qt

∇u·v· ∇vdxdt=

Qt

∇u·v· ∇v·_v_d_x_d_t

≤ε

Qt

v|∇u|_d_x_d_t₊ 

ε

Qt

v· |∇v|dxdt

≤ε

Qt

v_d_x_d_t₊ 

ε

Qt

|∇u|_d_x_d_t

+ 

ε

Qt

v· |∇v|dxdt. (.)

Substituting (.) into (.) yields

v(x,t) dx–

v_(x) dx+ 

(d+αu+ αv)|∇v|dxdt+r

Qt vdxdt

≤

αε+

cβ

a Qt

vdxdt+α ε

Qt

|∇u|_d_x_d_t₊α

ε

Qt

v|∇v|_d_x_d_t_. _(.)

Selectε>_α_α_and denoteC= α–α__ε_. Notice the positive equilibrium point of (.)

exists under conditioncβ>ar, then

v(x,t) dx+ d

Qt

|∇v|_d_x_d_t_{+ }_α u∞

Qt

|∇v|_d_x_d_t₊_C 

Qt

v|∇v|_d_x_d_t

≤

v_(x) dx+

αε–r+

cβ

a Qt

vdxdt+α ε

Qt

|∇u|_d_x_d_t_. _(.)

By Lemma .,∇uL(Q_T)≤C_(T). Integrating the above inequality and using the

Gronwall inequality, we get

sup <t<T

vdx≤C(T).

Hence, there exists a positive constantM_such that vdx≤M_. Furthermore, we have

vL₍_Q

T)≤

T



M_dtM. (.)

Secondly, multiplying both sides of the second equation of system (.) byqvq–₍_q_{> )}

and integrating over, we have

d dt

vqdx≤–(q– )d q

_∇

vq_d_x_–q(q– )α

(q+ )

_∇

vq+ _d_x

–q(q– )α

vq–∇u· ∇vdx+q

vq

–r+ cβmu  +amu

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Integrating the above equation over [,t] (t≤T), it is clear that

vq(x,t) dx+(q– )d q

Qt

_∇

vq_d_x_d_t₊q(q– )α

(q+ )

Qt

_∇

vq+ _d_x_d_t

≤

vq_(x) dx–q(q– )α

Qt

vq–∇u· ∇vdxdt+cβq a

Qt

vqdxdt. (.)

By Lemma ., it can be found that∇u∈L(nn+)₍_Q_T_{). 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+

 · ∇u_d_x_d_t

≤q(q– )α

q+  ∇uL(nn+)₍_Q_T₎v q–



Ln+₍_Q

T)∇

vq+

L₍_Q

T)

≤C∇

vq+

L₍_Q

T)v q–



Ln+₍_Q

T)

≤Cε

 ∇

vq+ 

L₍_Q

T)+ C

εv q–

 

Ln+₍_Q

T). (.)

Choose an appropriate numberεsatisfyingCε

 ≤

q(q–)α

(q+) . Substituting (.) into (.)

and takingv¯=vq+ , we have

¯ v

q

q+₍_x_,_t_{) d}_x₊

Qt

|∇¯v|_d_x_d_t

≤

vqdx+

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_{) d}_x₊

QT

|∇¯v|_d_x_d_t_.

From (.), we know

E≤C

 +¯v

(q–)

q+

L

(q–)(n+)

q+ ₍_Q

T) +¯v

q q+

L

q q+₍_Q

T)

.

Whenq<n_n(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)

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Setβ=_q_+ ∈(, ). It follows from theL₍_{)-estimate for}_v_that

¯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+₍₎

∇¯vq˜

L₍_Q

T)

(q–)

q+

+

M+M sup <t<T

v_¯(·,t)

q n(q+)q˜

L

q q+₍₎

∇¯vq˜

L₍_Q

T)

q q+

≤C

 +sup <t<T

_¯v(·,t)

q q+

L

q q+₍₎

(q–)

n(q+)q˜

∇¯v_L₍_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_+)q_q_˜ ∈(, ). It is easy to know thatEis bounded by use of reduction to absurdity. Sinceq<n_n(n+)_–,

(q+)q˜

 ∈(, (n+)

n– ). So,¯vLq˜₍_Q T) is bounded,i.e.,v∈L(q+) q˜₍_Q_T_{). Denote}(q+)q˜

 still asq. So,

v∈Lq(QT), ∀q∈

,(n+ ) n– 

. (.)

Finally, whenn= , , , , (n_{– )}_q_<_n_{+ }_n_with_q_{= . For}_n_≤_{, taking}_q_{=  in (.), it}

follows from (.) that there exists a positive constantMsuch 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+αu+ αv)δij,ai(x,t) =αuxjandδijis the Kronecker sign. In order to apply the maximum principle [] to (.), we need to prove the following conditions:

() vV(QT)is bounded; () n_i_,_j_=aij(x,t)ξiξj≥ν

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() n_i_=a

i(x,t),v(–r+ cβmu

+amu)Lp,r(QT)≤μ, whereν,μare 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 (.). Sincen_i_,_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+ αu)∇u

+αu

 – u K

– βmuv

 +amu. (.)

Whenn= , , , , n+_ <(_nn_–+), it is clear thatd+ αuhas an upper bound overQTby

Lemma .. Set

q∈

n+ 

 ,

(n+ ) n– 

.

From (.), we haveαu( –_Ku) –_+βmuv_amu ∈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–Cv∈Lq(QT),∀q∈(n+_ ,(_nn_–+)). From (.), we obtaind+αu∈

Cβ,β₍_Q

T). Thus, according to the parabolic regularity result of [, pp.-,

Theo-rem .], we can conclude that

X∈W_q,(QT), ∀q∈

n+ 

 ,

(n+ ) n– 

, (.)

which implies that∇X∈L

(n+)q n+–q₍_Q

T) by Lemma ..

SinceX= (d+ αu)u, we have∇u= (d+ αu)–∇X,i.e.,∇u∈L

(n+)q n+–q₍_Q

T). It means

that |∇u|_,_|∇_v_|_∈_L((nn+)+–qq)₍_Q

T). So, ni=ai(x,t)∈L

(n+)q

(n+–q)₍_Q

T). From (.) and (.),

v(–r+_+cβ_amumu)∈Lq₍_Q T).

Then the condition () and (.) are satisﬁed by choosingp=r=_((_nn+)_+–p_p₎. According to the maximum principle [, p., Theorem .], we can conclude thatv∈L∞(QT). From

(.), there exists a positive constantMsuch 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+αu)u, we haveu=

–d√d+αX

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So,

u∈C+α,+α₍_Q

T), ∀α∈(, ). (.)

Rewrite the second equation of system (.) as

vt=∇ ·

(d+αu+ αv)∇v+α∇uv

+v

–r+ cβmu

 +amu

.

Therefore, we can conclude that v(–r+ _+cβ_amumu)∈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+αu+αv)v. So,v¯satisﬁes

¯

vt= (d+αu+ αv)v¯+f(x,t), (.)

wheref(x,t) = (d+αu+ αv)v(–r+_+cβ_amumu) +αutv. According to (.) to (.), we

haved+αu+ αv,f(x,t)∈Cσ, σ

(Q_T). Applying the Schauder estimate to (.), we

know

¯

v∈C+σ,+σ

₍_Q

T).

Fromv=–(d+αu)+

√

(d+αu)+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*_{, namely}_σ_<_α_{, we have}_C+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

(11)

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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