The stability and Hopf bifurcation for a predator–prey system with time delay

Abstract

In this paper, we consider a predator–prey system with time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. We study the impact of the time delay on the stability of the model and by choosing the delay time τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time τ passes some critical values. Using normal form theory and central manifold argument, we also establish the direction and the stability of Hopf bifurcation. Finally, we perform numerical simulations to support our theoretical results.

Introduction

There has been great interest during the last few decades in dynamical characteristics of population models and among these models, predator–prey systems play an important role in population dynamics. Many theoreticians and experimentalists have concentrated on the stability of the predator–prey systems and more specifically they have investigated on the stability of such systems when the time delays incorporated into the models. Such delayed systems has received great attention since time delay may have very complicated impact on the dynamical behavior of the system such as periodic structure, bifurcation and so on. For references see [1], [4], [5], [7], [8], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [23], [24], [25], [26], [27], [28], [29], [31].

The main purpose of this paper is to study the stability of the following ratio-dependent delayed predator–prey system

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}N(t)}{\mathrm{d}t}=r_{1}N(t)-ϵP(t)N(t)\text{,}\hfill \\ \hfill & \frac{\mathrm{d}P(t)}{\mathrm{d}t}=P(t)\left(r_2-\theta \frac{P(t-\tau )}{N(t)}\right)\text{,}\hfill \end{array}$$

where $r_1\text{,}{r}_2\text{,}ϵ$ and θ are positive constants, $N(t)$ and $P(t)$ can be interpreted as the densities of prey and predator populations at time t, respectively, and $\tau ⩾0$ denotes the delay time for the predator density. In this model, predator density is logistic with time delay and the carrying capacity proportional to prey density. In many of the studies related to stability of predator–prey models, they consider constant carrying capacity, however in this study, we focus on the carrying capacity proportional to prey density which shows really interesting behavior in terms of dynamical structure.

The model

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}N(t)}{\mathrm{d}t}=r_{1}N(t)-ϵP(t)N(t)\text{,}\hfill \\ \hfill & \frac{\mathrm{d}P(t)}{\mathrm{d}t}=P(t)\left(r_2-\theta \frac{P(t)}{N(t)}\right)\text{,}\hfill \end{array}$$

i.e., when there is no delay was considered by Zhou et al. [30] in detail. They first studied the stability conditions of the equilibrium point for the system and then by introducing so called Allee effect ([2], [3], [5], [9], [22], [30]) in different forms, they investigated the impact of this effect on the dynamics of this predator–prey system.

Our aim is to study the stability of delayed predator–prey system (0.1) and investigate how the time delay τ effects the dynamics of this system. To analyze the system, first, we investigate the local stability of the equilibrium point of the corresponding characteristic equation of the system and obtain the general stability criteria involving the delay time. Second, by choosing the delay τ as bifurcation parameter, we show that the positive equilibrium loses its stability and the system exhibits Hopf bifurcation. Then based on the normal form approach and the center manifold theory introduced by Hassard and Kazarinoff [6], we derive the formula for determining the properties of Hopf bifurcation of the model. More specifically, it is proven that the Hopf bifurcation is subcritical and the bifurcating periodic solutions are unstable under certain conditions. Finally, to support these theoretical results, we illustrate them by numerical simulations.

This paper is organized as follows: In Section 1, we first focus on the stability and Hopf bifurcation of the positive equilibrium. In Section 2, we derive the direction and stability of Hopf bifurcation by using normal form and central manifold theory. Finally in Section 3, numerical simulations are performed to support the stability results.