Global qualitative analysis for a predator–prey system with delay

Abstract

In this paper, the dynamics of a predator–prey system with a finite delay is considered. The conditions for the global stability and the existence of Hopf bifurcation at the positive equilibrium of the system are obtained. Explicit algorithms for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are derived, using the normal form theory and center manifold argument [Nussbaum RD. Periodic solutions of some nonlinear autonomous functional equations. Ann Mat Pura Appl 1974;10:263–306]. Numerical simulations supporting the theoretical analysis are also given. Global existence of periodic solutions is established by using a global Hopf bifurcation result of Wu [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981].

Introduction

The stability and periodic solutions of ecological models are of great interest in mathematical biology. The literature [1], [2], [3] on this is very extensive. In [4], criteria are established that guarantee an equilibrium to be absolutely stable. It is also shown in [5], [6], [2] that stability switches can happen when delays increase and for certain values of a delay, periodic solutions may occur through Hopf bifurcation.

In this paper we focus on a predator–prey system with delay modeled by

$$\left\{\begin{array}{c}\frac{\mathrm{d}x(t)}{\mathrm{d}t}=x(t)[r_1-a_{11}x(t)-a_{12}y(t)]\text{,}\hfill \\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}=y(t)[-r_2+a_{21}x(t)-a_{22}y(t-\tau )]\text{,}\hfill \end{array}\right.$$

with initial conditions

$$\left\{\begin{array}{c}x(t)={\Phi }_1(t)⩾0\text{,}t\in [-\tau \text{,}0)\text{,}{\Phi }_1(0)>0\text{,}\hfill \\ y(t)={\Phi }_2(t)⩾0\text{,}t\in [-\tau \text{,}0)\text{,}{\Phi }_2(0)>0\text{,}\hfill \end{array}\right.$$

where x(t) and y(t) denote the density (per square unit of the habitat) of prey and predator population, respectively; τ, r_i, a_ij (i, j = 1, 2) are positive constants by biological terms; τ is duration time of predator species to mature. System (1.1) always has a unique positive equilibrium E_∗ = (x_∗, y_∗) provided that the condition

$$(\mathrm{H})r_1{a}_{21}-r_2{a}_{11}>0$$

holds, where

$$x_{\ast }=\frac{r_1{a}_{22}+r_2{a}_{12}}{a_{11}{a}_{22}+a_{12}{a}_{21}}\text{,}{y}_{\ast }=\frac{r_1{a}_{21}-r_2{a}_{11}}{a_{11}{a}_{22}+a_{12}{a}_{21}}\text{,}$$

when the delay τ = 0, the system (1.1) simplifies to an autonomous system of ordinary differential equation of the form

$$\left\{\begin{array}{c}\frac{\mathrm{d}x(t)}{\mathrm{d}t}=x(t)[r_1-a_{11}x(t)-a_{12}y(t)]\text{,}\hfill \\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}=y(t)[-r_2+a_{21}x(t)-a_{22}y(t)]\text{.}\hfill \end{array}\right.$$

Chen [7] proved that under the condition (H), the positive equilibrium E_∗ = (x_∗, y_∗) of system (1.4) is global asymptotically stable. Recently, Wang and Ma [8] showed that the solutions of system (1.1) are bounded, uniformly persistent and the delay is harmless for the uniform persistence of system (1.1).

For a long time, the global existence of a periodic solution to the mathematical models of population dynamics has attracted much attention due to its theoretical and practical significance. It is well known that periodic solutions can arise through the Hopf bifurcation in delayed differential equations. However, these periodic solutions bifurcating from Hopf bifurcation are generally local. Therefore, it is an important mathematical subject to investigate if these nontrivial periodic solutions exist globally. Recently, a great deal of research has been devoted to the topics. One of the methods used in them is the ejective fixed point argument developed by [9], which has been successfully used to obtain the global existence of periodic solutions bifurcating from the Hopf bifurcation by many researchers (see, for example, [10], [11], [12]). The other is the global Hopf bifurcation theorem due to Erbe et al. [13], which was established using a purely topological argument. Krawcewicz et al. [14] first applied this global Hopf bifurcation theorem to a neural functional differential equation.

In this paper, we reconsider system (1.1), showing that the positive equilibrium E_∗ = (x_∗, y_∗) is globally asymptotically stable from a different aspect, and devote our attention to the global existence of periodic solutions to system (1.1).

This paper is organized as follows. In Section 2, the conditions for the global stability are obtained. In Section 3, we investigate the Hopf bifurcation at the positive equilibrium E_∗ of system (1.1). In Section 4, we use the norm form method and the center manifold theory introduced by Hassard et al. [15] to analyze the direction, the stability and the period of the bifurcating periodic solutions at critical values of τ, and perform some numerical simulations to illustrate the analytical results found. The global Hopf bifurcation will be considered in Section 5.