Permanence and global attractivity of a discrete multispecies Lotka–Volterra competition predator–prey systems

Abstract

In this paper, we propose a discrete multispecies Lotka–Volterra competition predator–prey systems. For general non-autonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

Introduction

The aim of this paper is to investigate the dynamic behavior of the following discrete n + m-species Lotka–Volterra competition predator–prey systems

$$\begin{array}{cc}\hfill & x_i(k+1)=x_i(k)\exp \left[b_i(k)-\sum _{l=1}^n{a}_{\mathit{il}}(k)x_l(k)-\sum _{l=1}^m{c}_{\mathit{il}}(k)y_l(k)\right]\text{,}\hfill \\ \hfill & y_j(k+1)=y_j(k)\exp \left[-r_j(k)+\sum _{l=1}^n{d}_{\mathit{jl}}(k)x_l(k)-\sum _{l=1}^m{e}_{\mathit{jl}}(k)y_l(k)\right]\text{,}\hfill \end{array}$$

where i = 1, 2, … , n; j = 1, 2, … , m; x_i(k) is the density of prey species i at kth generation. y_j(k) is the density of predator species j at kth generation. a_il(k) and e_jl(k) measures the intensity of intraspecific competition or interspecific action of prey species and predator species, respectively. b_i(k) representing the intrinsic growth rate of the prey species x_i; r_j(k) representing the death rate of the predator species y_j.

Dynamic behaviors of population models governed by difference equations had been studied by a number of papers, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and the references cited therein. It has been found that the autonomous discrete systems can demonstrate quite rich and complicated dynamics, see [1], [2], [3], [4], [18], [19]. Recently, more and more scholars paid attention to the non-autonomous discrete population models, since such kind of model could be more appropriate.

Zhou and Zou [15] had studied the dynamic behavior of the following non-autonomous single species discrete model

$$x(k+1)=x(k)\exp \left[r(k)\left(1-\frac{x(k)}{K(k)}\right)\right]\text{.}$$

Sufficient conditions on the persistence and the existence of a stable periodic solution of the system (1.2) are obtained.

Recently, Chen and Zhou [14] further generalized the system (1.2) to the following two-species Lotka–Volterra competition system

$$\begin{array}{cc}\hfill & x(k+1)=x(k)\exp \left[r_1(k)\left(1-\frac{x(k)}{K_1(k)}-{\mu }_2(k)y(k)\right)\right]\text{,}\hfill \\ \hfill & y(k+1)=y(k)\exp \left[r_2(k)\left(1-{\mu }_1(k)x(k)-\frac{y(k)}{K_2(k)}\right)\right]\text{.}\hfill \end{array}$$

They obtained the sufficient conditions which guarantee the persistence of the system (1.3). Also, for the periodic case, they obtained the sufficient conditions which guarantee the existence of a globally stable periodic solution of the system.

Wang and Lu [12] proposed the following Lotka–Volterra model

$$x_i(k+1)=x_i(k)\exp \left[r_i(k)-\sum _{j=1}^n{a}_{\mathit{ij}}(k)x_j(k)\right]\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where x_i(k) is the density of population i at kth generation, r_i(k) is the growth rate of population i at kth generation, a_ij(k) measures the intensity of intraspecific competition or interspecific action of species. By constructing a suitable Lyapunov function and using the finite covering theorem of Mathematic Analysis, they obtained a set of sufficient conditions which ensure the system to be globally asymptotically stable.

As was pointed out by Berryman [20], the dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Already, there are several scholars done works on the persistence, existence of positive periodic solution etc. of two species discrete predator–prey system, see [17], [21], [22], [23] and the references cited therein. However, to the best of the author’s knowledge, to this day, still no scholar done works on multispecies competition predator–prey ecosystems.

Indeed, as far as the continuous multispecies ecosystem is concerned, Yang and Xu [24] considered the following periodic n-prey and m-predator Lotka–Volterra system of differential equations

$$\begin{array}{cc}\hfill & {\dot{x}}_i(t)=x_i(t)\left[b_i(t)-\sum _{k=1}^n{a}_{\mathit{ik}}(t)x_k(t)-\sum _{k=1}^m{c}_{\mathit{ik}}(t)y_k(t)\right]\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{;}\hfill \\ \hfill & {\dot{y}}_j(t)=y_j(t)\left[-r_j(t)+\sum _{k=1}^n{d}_{\mathit{jk}}(t)x_k(t)-\sum _{k=1}^m{e}_{\mathit{jk}}(t)y_k(t)\right]\text{,}j=1\text{,}2\text{,}\dots \text{,}m\text{,}\hfill \end{array}$$

where x_i(t) denotes the density of prey species X_i at time t, y_j(t) denotes the density of predator species Y_j at time t; b_i(t), r_j(t), a_ik(t), c_il(t), d_jk(t), and e_jl(t) (i, k = 1, … , n;  j, l = 1, … , m) are continuous periodic functions defined on [0, +∞) with a common periodic T > 0; r_j(t), a_ik(t), c_il(t), d_jk(t) and e_jl(t) are non-negative; a_ii(t), e_jj(t) are strictly positive. Under the assumption that b_i(t) are positive periodic functions they obtained a set of sufficient conditions for the existence and global attractivity of the periodic solution of system (1.5). For more works on this direction, one could refer to [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36] and the references cited therein.

Obviously, system (1.1) is the counterpart of continuous Lotka–Volterra competition predator–prey system (1.5). To the best of the author’s knowledge, this is the first time the multispecies discrete competition predator–prey system is considered. The aim of this paper is, by developing the analysis technique of Huo and Li [16] and Chen and Zhou [14], to investigate the persistence and global stability property of the system (1.1).

We say that system (1.1) is permanent if there are positive constants M and m such that for each positive solution (x₁(k), … , x_n(k),  y₁(k), … , y_m(k)) of system (1.1) satisfies

$$\begin{array}{cc}\hfill & m⩽\underset{k\to +\infty }{\mathrm{liminf}}{x}_i(k)⩽\underset{k\to +\infty }{\mathrm{limsup}}{x}_i(k)⩽M\text{,}\hfill \\ \hfill & m⩽\underset{k\to +\infty }{\mathrm{liminf}}{y}_j(k)⩽\underset{k\to +\infty }{\mathrm{limsup}}{y}_j(k)⩽M\hfill \end{array}$$

for all i = 1, 2, … , n; j = 1, 2, … , m.

Throughout this paper, we assume that b_i(k), a_il(k), c_il(n), r_j(n), d_jl(n), e_jl(n) are all bounded non-negative sequence, and use the following notations for any bounded sequence {x(n)},

$$x^u=\underset{n\in N}{\sup }x(n)\text{,}{x}^l=\underset{n\in N}{\inf }x(n)\text{.}$$

For biological reasons, we only consider solution (x₁(k), … , x_n(k),  y₁(k), … , y_m(k)) with

$$x_i(0)>0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{;}{y}_j(0)>0\text{,}j=1\text{,}2\text{,}\dots \text{,}m\text{.}$$

Then system (1.1) has a positive solution $(x_1(k)\text{,}\dots \text{,}{x}_n(k)\text{,}{y}_1(k)\text{,}\dots \text{,}{y}_m(k){)}_{k=0}^{\infty }$ passing through (x₁(0), … , x_n(0), y₁(0), … , y_m(0)).

The organization of this paper is as follow: In Section 2, we obtain sufficient conditions which guarantee the permanence of the system (1.1). In Section 3, we obtain sufficient conditions which guarantee the global stability of the positive solution of system (1.1). As a consequence, for periodic case, we obtain sufficient conditions which ensure the existence of a globally stable positive solution of system (1.1).