Stability and bifurcation of mutual system with time delay

Abstract

In this paper, we study the stability and bifurcation in a mutual model with a delay τ, where τ is regarded as a parameter. It is found that there are stability switches, and Hopf bifurcation occur when the delay τ passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions in the first bifurcation value is given using the normal form method and center manifold theorem.

Introduction

Two species cohabit a common habitat and each species enhances the average growth rate of the other, this type of ecological interaction is known as facultative mutualism. Mutual phenomenon can increase viability and make species persistently multiply. As far as we know, the research on mutual system is less than prey-predator system [1] and competitive system [2]. Moreover, the known results of mutual system mainly focus on stability and persistence [3], [4]. Research on the ecologic system stability of positive equilibrium and existence of periodic solutions is very critical, which can help us to realize the law for species quantity and predict the trend for species quantity. In this paper we study the stability of positive equilibrium and existence of periodic solutions of mutual system with a delay

$$\text{x}\text{̇}\text{(t)=r}{}_1\text{x(t)}\text{1−}\text{x(t−τ)}\text{k}{}_1\text{+ax(t)y(t),}\text{y}\text{̇}\text{(t)=r}{}_2\text{y(t)}\text{1−}\text{y(t−τ)}\text{k}{}_2\text{+bx(t)y(t),}$$

where r_ℓ, k_ℓ, a, b, (ℓ=1,2) are positive, τ⩾0. x(t) and y(t) are respectively densities of two species at time t, r_ℓ is the intrinsic growth rate of two species, k_ℓ is the ecosystem support or environmental carrying capacity for two species, a and b are the rates of transmission between two species. For Eq. (1.1), He and Gopalsamy recently study the stability and persistence by Liapunov’s second method in [5]. It is well-known that the delay is the key to give rise to difference of between the delay differential equations and ordinary differential equations. A natural question is how the delay affects the dynamics of Eq. (1.1). The purpose of this paper is to answer this question partially.

When the delay is zero, Theorem 3.1^′ in [4] indicates that the positive equilibrium of system (1.1) is global asymptotically stable. The theorem shows that system (1.1) has no non-trivial positive periodic solutions when the delay is zero. Applying Hopf bifurcation theory, we find that the system has positive periodic solutions for some τ. In addition, formulae for determining the direction and stability of bifurcating periodic solutions are presented by the center manifold theorem and normal form method. Finally, a numerical simulation example is performed for supporting the analysis results. The remainder of the paper is organized as follows: employing the method introduced by Cooke and Grossman [6] and regarding the delay τ as a parameter, we investigate the characteristic equation and get the existence of the stability switches and Hopf bifurcation in Section 2; the formulae for determining bifurcation direction and stability of the bifurcating periodic solutions are presented in Section 3; in Section 4, numerical simulation is carried out.