A food chain model for two resources in un-stirred chemostat

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Abstract

This paper deals with a multiple food chain model in un-stirred chemostat, where the prey feeds on two growth-limiting, nonreproducing resources. The conditions for existence of positive steady states are established. The global bifurcation of positive solutions is considered also. Furthermore, we obtain the conditions under which the predator possesses uniformly strong persistence or goes to extinction. The techniques used in this paper include the degree theory, the global bifurcation theory, the semigroup theory, and the maximum principle.

Introduction

In this paper, we consider the following reaction–diffusion system:St=dSxx-m1uf(S,R),Rt=dRxx-m2ug(S,R),ut=duxx+u(m1f(S,R)+cm2g(S,R))-m3vh(u),vt=dvxx+m3vh(u),Sx(t,0)=Rx(t,0)=-1,ux(t,0)=vx(t,0)=0,Sx(t,1)+γS(t,1)=0,Rx(t,1)+γR(t,1)=0,ux(t,1)+γu(t,1)=0,vx(t,1)+γv(t,1)=0subject to nonnegative initial data (S0(x),R0(x),u0(x),v0(x)) with u0,v00. System (1.1) describes a multiple food chain model in an un-stirred chemostat with two growth-limiting, nonreproducing resources. More precisely, v is a predator with the prey u, and u feeds on two substitutable nutrients S and R. The response functions f(S,R),g(S,R),h(u) take the formsf(S,R)=S1+aS+bR,g(S,R)=R1+aS+bR,h(u)=ua1+u,where a,b,a1 are positive constants.

Mathematical analysis for chemostat models involving two limiting resources can be found in [2], [4], [12], [14], [15], [21], [22], where [21], [22] are for un-stirred ones. As for the study of un-stirred food chain chemostat models with single nutrient, refer to, e.g. [16], [24].

If v0, then (1.1) is reduced toSt=dSxx-m1uf(S,R),Rt=dRxx-m2ug(S,R),ut=duxx+u(m1f(S,R)+cm2g(S,R)),Sx(t,0)=Rx(t,0)=-1,Sx(t,1)+γS(t,1)=0,Rx(t,1)+γR(t,1)=0,ux(t,0)=0,ux(t,1)+γu(t,1)=0.By using of the comparison principle and the bifurcation theory, Wu and Wolkowicz [21] studied the coexistence and nonexistence to steady states of (1.2), as well as the long-time behavior of solutions for the related limiting system.

One goal of our study is to discuss coexistence steady states to (1.1), namely, positive solutions of the systemdSxx-m1uf(S,R)=0,dRxx-m2ug(S,R)=0,duxx+u(m1f(S,R)+cm2g(S,R))-m3vh(u)=0,dvxx+m3vh(u)=0,Sx(0)=Rx(0)=-1,ux(0)=vx(0)=0,Sx(1)+γS(1)=0,Rx(1)+γR(1)=0,ux(1)+γu(1)=0,vx(1)+γv(1)=0.We call (S,R,u,v) a positive solution of (1.3) if every component of (S,R,u,v) is positive in (0, 1). We will establish sufficient conditions for the existence of positive solutions of (1.3) by means of the fixed point index theory. We also deal with the global structure of the positive solutions in the case of m1=m2. Another goal of the paper is to study the dynamics of (1.1). We will investigate extinction-persistence of populations, namely, persistence of u and v, extinction for both u and v, and extinction for v only.

This paper is organized as follows. In Section 2, we propose sufficient conditions for positive steady states of (1.1). A global bifurcation of positive solutions of (1.3) is investigated in Section 3. Finally, in Section 4, we give the extinction-persistence conditions to (1.1).

Section snippets

Positive steady states

In this section, we will investigate the existence and nonexistence of positive solutions of (1.3) by using the fixed point index theory.

Set w=S+cR+u+v. Then w satisfiesdwxx=0,x(0,1),wx(0)=-(1+c),wx(1)+γwx(1)=0.It follows that w=(1+c)z with z=1+γγ-x, and (1.3) is equivalent todSxx-m1((1+c)z-S-cR-v)f(S,R)=0,dRxx-m2((1+c)z-S-cR-v)g(S,R)=0,dvxx+m3vh((1+c)z-S-cR-v)=0,Sx(0)=Rx(0)=-1,vx(0)=0,Sx(1)+γS(1)=0,Rx(1)+γR(1)=0,vx(1)+γv(1)=0.By the maximum principle, we can easily get the following lemma:

Lemma 2.1

Let

Global bifurcation

In this section, we discuss the global structure of positive solutions in the case of m1=m2. That is to say, S and R share the same consumption rate, although the maximum growth rates of u with respect to R and S are different when c1. We regard m3 as a bifurcation parameter and suppose that all other constants are fixed. When v0 (1.3) is reduced to the systemdSxx-m1uf(S,R)=0,dRxx-m2ug(S,R)=0,duxx+u(m1f(S,R)+cm2g(S,R))=0,Sx(0)=Rx(0)=-1,ux(0)=0,Sx(1)+γS(1)=0,Rx(1)+γR(1)=0,ux(1)+γu(1)=0with u=(1

Extinction and persistence

In this section, we discuss extinction and persistence conditions to system (1.1). The techniques by Dung [8] will be used here.

By a similar argument as that for Lemma 2.1 in [13], one can easily show that the solution (S,R,u,v) of (1.1) exists globally, nonnegative, bounded, and satisfiesS+cR+u+v-(1+c)z=O(e-ϑt)for some ϑ>0 as t+. Moreover, it follows from the maximum principle that S,R,u,v>0.

In order to derive the desired extinction result, we deal with (1.1) in Y=[Lp(0,1)]4,p>1. Set s=z-S

Discussion

Now discuss the obtained conclusions for the un-stirred chemostat model of multiple food chain with two resources. Biologically, we are interested in the persistence of the predator v. The conditions of positive steady states of the model are established in Theorem 2.1, for which, besides the conditions (i) or (ii) required by the existence of semitrivial solution (S¯,R¯,u¯,0), we need naturally in addition m3>κ0(mˆ), where κ0 is the principal eigenvalue ofdϕxx+κh(u)ϕ=0,x(0,1),ϕx(0)=0,ϕx(1)+γϕ(

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