Permanence and global stability in a discrete n-species competition system with feedback controls

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Abstract

In this paper, we investigate the following discrete n-species competition system with feedback controls:xi(k+1)=xi(k)expbi(k)-j=1naij(k)xj(k)-j=1,jincij(k)xi(k)xj(k)-di(k)ui(k),Δui(k)=ri(k)-ei(k)ui(k)+fi(k)xi(k),i=1,2,,n,which describes the effect of toxic substances and age structures simultaneously. Some sufficient conditions are established on the permanence and the global stability of the system. These results are also applied to some special cases.

Introduction

The coexistence (one mathematical concept of coexistence of species is permanence) and global stability of population models are of great interest in mathematical biology. There is a rich literature on this topic. For results on models described by differential equations, see, for example, [1], [2], [3], [4], [5], [6], [14], [20], [21], [22], [24], [25], [27], [28], [29] and the references therein, while for results on models governed by difference equations, to mention a few, see [8], [11], [12], [13], [15], [16], [17], [18], [19], [23], [26], [30], [31], [32] and the references therein.

We should mention that results on models of differential equations are mostly for Lotka–Volterra systems. Though Lotka–Volterra systems are basic models on mathematical ecology, the growth rate of some competitive species does not correspond with that of Lotka–Volterra models (see Chen [7]). The reason is that the linear mathematics used neglects many important factors such as the effect of toxic substances. To incorporate the effect of toxic substances, Chattopadhyay [2] assumed that each species produces a substance toxic to each other but only when the other is present. This produces the following system on two competitive species:dN1dt=N1(K1-α1N1-β12N2-γ1N1N2),dN2dt=N2(K2-α2N2-β21N1-γ2N1N2).

Moreover, as we know, ecosystems in the real world are often disturbed by unpredictable forces which can result in changes in biological parameters such as survival rates. Of particular interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control, we call the disturbance functions as control variables. For more discussion on this direction, we refer to [9], [10], [27].

Recently, Xia et al. [28] studied the following n-species competitive system with feedback controls:x˙i(t)=xi(t)bi(t)-j=1naij(t)xj(t)-j=1,jincij(t)xi(t)xj(t)-di(t)ui(t),u˙i(t)=ri(t)-ei(t)ui(t)+fi(t)xi(t),i=1,2,,n.Some sufficient conditions were obtained for the existence of a unique almost periodic solution for model (1.2).

However, many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Therefore, lots have been done on discrete Lotka–Volterra systems.

With the incorporation of the effect of fluctuating environment, a realistic modification of (1.1) isdN1dt=N1(t)[r1(t)-a11(t)N1(t)-a12(t)N2(t)-b1(t)N1(t)N2(t)],dN2dt=N2(t)[r2(t)-a21(t)N1(t)-a22(t)N2(t)-b2(t)N1(t)N2(t)].Assuming that the average growth rates in (1.3) change at equally spaced time intervals and estimates of the population sizes are made at equally spaced time intervals, Zhang and Fang [31] obtained the following discrete analog of (1.3),N1(k+1)=N1(k)exp{r1(k)-a11(k)N1(k)-a12(k)N2(k)-b1(k)N1(k)N2(k)},N2(k+1)=N2(k)exp{r2(k)-a21(k)N1(k)-a22(k)N2(k)-b2(k)N1(k)N2(k)}for k=0,1,2,. Under the assumption that ri, aij>0, bi>0 (i,j=1,2) are ω-periodic, where ω is a fixed positive integer, some sufficient conditions are provided for the existence of multiple positive periodic solutions by employing the coincidence degree theory.

However, results on discrete models with feedback controls are rare. To the best of our knowledge, only Liu et al. [19] studied the global attractivity of the following logistic difference model with feedback control,Nn+1=Nnexprn1-Nn-mk-cμn,Δμn=-aμn+bNn-m.Therefore, in this paper, we consider the following discrete n-species competition system with feedback controls:xi(k+1)=xi(k)expbi(k)-j=1naij(k)xj(k)-j=1,jincij(k)xi(k)xj(k)-di(k)ui(k),Δui(k)=ri(k)-ei(k)ui(k)+fi(k)xi(k),i=1,2,,n,where Δ is the first-order forward difference operator, that is, Δui(k)=ui(k+1)-ui(k). Throughout this paper, we always assume that ei(·):N={0,1,2,}(0,1); ri(·), di(·), bi(·), aij(·), cij(·) and fi(·):NR+=[0,) are bounded sequences. System (1.5) can be regarded as a discrete analog of system (1.2) and hence it reflects the effect of toxic substances and age structures simultaneously.

From the point of view of biology, in the sequel, we only consider solutions of system (1.5) with xi(0)>0 and ui(0)>0, i=1,2,,n. Then it is easy to see that such a solution denoted by {(X(k),U(k))}{(x1(k),,xn(k),u1(k),,un(k))T} of system (1.5) is positive, that is, xi(k)>0 and ui(k)>0 for i=1,2,,n and all kN.

The main purpose of this paper is to investigate the permanence and global stability of system (1.5). Our study was also motivated by that of Chen and Zhou [8], where they studied the following two-species Lotka–Volterra competition model:x(n+1)=x(n)expr1(n)1-x(n)K1(n)-μ2(n)y(n),y(n+1)=y(n)expr2(n)1-μ1(n)x(n)-y(n)K2(n).Some sufficient conditions on the persistence and existence of a globally stable periodic solution for system (1.6) were established in [8].

Definition 1.1

System (1.5) is said to be permanent if there are positive constants Mi, Li, mi, li, i=1,2,,n, such that any positive solution {(X(k),U(k))} of system (1.5) satisfiesmiliminfkxi(k)limsupkxi(k)Mi,liliminfkui(k)limsupkui(k)Lifor all i=1,2,,n.

Definition 1.2

System (1.5) is said to be globally stable if limk|xi(k)-xi*(k)|=0andlimk|ui(k)-ui*(k)|=0,i=1,,nfor any two positive solutions {(X(k),U(k))} and {(X*(k),U*(k))} of system (1.5).

For the simplicity and convenience of exposition, for any bounded sequence {u(k)}, we introduce the following two notations: u¯=supkNu(k)andu̲=infkNu(k).

The organization of this paper is as follows: In the next two sections, we give sufficient conditions on permanence and global stability of system (1.5), respectively. To conclude this paper, in Section 4, the main results are applied to a discrete time analog of a differential system studied by Xiao et al. [29].

Section snippets

Permanence

To establish the main result of this section, Theorem 2.5 on permanence of system (1.5), we need some preparations.

Lemma 2.1

Yang [30]

Assume that {x(k)} satisfies x(k)>0 and x(k+1)x(k)exp{r(k)(1-ax(k))}for k[k1,), where a is a positive constant. Then limsupkx(k)1ar¯exp(r¯-1).

Lemma 2.2

Yang [30]

Assume that {x(k)} satisfies x(k+1)x(k)exp{r(k)(1-ax(k))},kK0,limsupkx(k)K and x(K0)>0, where a is a constant such that aK>1 and K0N. Then liminfkx(k)1aexp{r¯(1-aK)}.

Proposition 2.3

For any positive solution {(X(k),U(k))} of system (1.5), we

Global stability

On the basis of permanence, we further investigate the stability of system (1.5) and provide the following sufficient conditions that guarantee the global stability of system (1.5).

Theorem 3.1

Assume that (H) holds. Moreover, supposeλi=max1-a¯iiMi-j=1,jinc¯ijMiMj,1-a̲iimi-j=1,jinc̲ijmimj+j=1,jina¯ijMj+j=1,jinc¯ijMiMj+d¯i<1,δi=(1-e̲i)+f¯iMi<1for i=1,2,,n. Then system (1.5) is globally stable.

Proof

Let {(X(k),U(k))} and {(X*(k),U*(k))} be any two positive solutions of system (1.5). We need to show that

Applications

The objective of this section is to apply Theorems 2.5 and 3.1 to the following discrete two-species Lotka–Volterra competitive system with feedback controls:x1(k+1)=x1(k)exp{b1(k)-a11(k)x1(k)-a12(k)x2(k)-d1(k)u1(k)},x2(k+1)=x2(k)exp{b2(k)-a21(k)x1(k)-a22(k)x2(k)-d2(k)u2(k)},Δu1(k)=r1(k)-e1(k)u1(k)+c1(k)x1(k),Δu2(k)=r2(k)-e2(k)u2(k)+c2(k)x2(k),where {ri(k)}, {bi(k)}, {aij(k)}, {ci(k)}, {ei(k)}(0,1) and {di(k)} are bounded nonnegative sequences.

System (4.1) can be regarded as a discrete time

References (32)

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The research of Liao and Zhou was supported by the National Natural Science Foundation of China (No. 10471086) and Educational Committee Foundation of Hunan Province (No. 05C494), while the research of Chen was partially supported by the Natural Science and Engineering Research Council of Canada (NSERC) and the Early Researcher Award Program (ERA) of Ontario.

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