Permanence in nonautonomous Lotka–Volterra system with predator–prey

Abstract

The nonautonomous Lotka–Volterra system with m-predators and n-preys is considered in this paper. Average conditions are obtained for permanence and global attractivity in the system. It is shown that our results are generalization of those of the periodic system.

Introduction

We consider the nonautonomous predator–prey Lotka–Volterra system of the differential equations

$$\text{x}\text{̇}{}_{\mathrm{i}}\text{(t)=x}{}_{\mathrm{i}}\text{(t)}\text{b}{}_{\mathrm{i}}\text{(t)−∑}{}_{\mathrm{k=1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{ik}}\text{(t)x}{}_{\mathrm{k}}\text{(t)−∑}{}_{\mathrm{k=1}}{}^{\mathrm{m}}\text{c}{}_{\mathrm{ik}}\text{(t)y}{}_{\mathrm{k}}\text{(t)}\text{,}\text{i=1,…,n,}\text{y}\text{̇}{}_{\mathrm{j}}\text{(t)=y}{}_{\mathrm{j}}\text{(t)}\text{−γ}{}_{\mathrm{j}}\text{(t)+∑}{}_{\mathrm{k=1}}{}^{\mathrm{n}}\text{d}{}_{\mathrm{jk}}\text{(t)x}{}_{\mathrm{k}}\text{(t)−∑}{}_{\mathrm{k=1}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{jk}}\text{(t)y}{}_{\mathrm{k}}\text{(t)}\text{,}\text{j=1,…,m,}$$

where x_i(t) and y_j(t) denote the density of prey species X_i and predator species Y_j, respectively; b_i(t), γ_j(t), a_ik(t), c_il(t), d_jk(t) and e_jl(t), (i,k=1,…,n; j,l=1,…,m), i.e., the coefficients of system (1.1) are bounded and continuous functions defined on [c,+∞); γ_j(t), a_ik(t), c_il(t), d_jk(t) and e_jl(t), (i,k=1,…,n; j,l=1,…,m) are nonnegative on [c,+∞). In addition to predator–prey relation, there are also competition among predator species Y_j and among prey species X_i in system (1.1).

The competitive Lotka–Volterra system has been studied in many papers, for example papers [1], [2], [3], [4], [5], [6], [7], [8] and [11]. Some sufficient conditions are obtained for permanence, global attractivity and extinction in this system. But there are a fewer papers on system (1.1). Yang and Xu [9] studied τ-periodic system (1.1), i.e., the coefficients of system (1.1) are continuous periodic functions with a common period τ>0. Under the assumptions that the coefficients are nonnegative and some of them are strictly positive, they obtained sufficient conditions for existence, uniqueness and global attractivity of the strictly positive τ-periodic solution of the τ-periodic system (1.1). Zhao and Chen [10] considered τ-periodic system (1.1) and obtained the same results under certain sufficient conditions and the assumptions that $\text{∫}{}_0{}^{\mathrm{\tau }}\text{b}{}_{\mathrm{i}}\text{(t)}\text{d}\text{t>0}$, other coefficients are as in [9]. It is shown that paper [10] is a generalization of [9] in some sense.

In this paper, we shall study the general nonautonomous predator–prey Lotka–Volterra system (1.1) and give sufficient conditions for permanence and global attractivity in system (1.1).

First we give some notations and definitions. Given a function g(t) which is defined on [c,+∞), we set

$$\text{g}{}^{\mathrm{u}}\text{=}\text{sup}\text{{g(t)}\text{|}\text{c⩽t<+∞},}\text{g}{}^{\mathrm{l}}\text{=}\text{inf}\text{{g(t)}\text{|}\text{c⩽t<+∞}.}$$

According to Ahmad and Lazer [3], we define the lower and upper averages of a function g(t) which is continuous and bounded on [c,+∞). If c⩽t₁<t₂ we set

The lower and upper averages of g(t), denoted by m[g] and M[g], respectively, are defined by

$$\text{m[g]=}\text{lim}{}_{\mathrm{s\to +\infty }}\text{inf}\text{{A[g,t}{}_1\text{,t}{}_2\text{]}\text{|}\text{t}{}_2\text{−t}{}_1\text{⩾s}}$$

and

$$\text{M[g]=}\text{lim}{}_{\mathrm{s\to +\infty }}\text{sup}\text{{A[g,t}{}_1\text{,t}{}_2\text{]}\text{|}\text{t}{}_2\text{−t}{}_1\text{⩾s}.}$$

Since the set $\text{{A[g,t}{}_1\text{,t}{}_2\text{]}\text{|}\text{t}{}_2\text{−t}{}_1\text{⩾s}}$ gets smaller as s increases, the limits exist; and since g^l⩽A[g,t₁,t₂]⩽g^u, it follows that

$$\text{g}{}^{\mathrm{u}}\text{⩾M[g]⩾m[g]⩾g}{}^{\mathrm{l}}\text{.}$$

Assume that

$$\text{(}\text{A1}\text{)}\text{M[b}{}_{\mathrm{i}}\text{]>0,}\text{i=1,…,n,}$$$$\text{(}\text{A2}\text{)}\text{m[a}{}_{\mathrm{ii}}\text{]>0,}\text{i=1,…,n,}$$$$\text{(}\text{A3}\text{)}\text{m[e}{}_{\mathrm{jj}}\text{]>0,}\text{j=1,…,m.}$$

System (1.1) is called permanent, if there exist strictly positive reals δ, Δ and T₀>c such that δ⩽x_i(t)⩽Δ, i=1,…,n, and δ⩽y_j(t)⩽Δ, j=1,…,m, for any strictly positive solution col(x₁(t),…,x_n(t),y₁(t),…,y_m(t)) of (1.1) as t>T₀. System (1.1) is said to be globally attractive, if there exists a strictly positive solution col(x₁(t),…,x_n(t),y₁(t),…,y_m(t)) of (1.1) such that lim_t→+∞(u_i(t)−x_i(t))=0 for i=1,…,n, and lim_t→+∞(v_j(t)−y_j(t))=0 for j=1,…,m, where col(u₁(t),…,u_n(t),v₁(t),…,v_m(t)) is any other strictly positive solution of (1.1).

In the following we shall give average conditions for permanence and global attractivity in system (1.1). In Section 2, some basic results on logistic equation are obtained. In Section 3, we give our main results on system (1.1). It is show that our results are generalization of those of the τ-periodic logistic equation and τ-periodic system (1.1).