Global qualitative analysis for a predator–prey system with delay☆
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 bywith initial conditionswhere x(t) and y(t) denote the density (per square unit of the habitat) of prey and predator population, respectively; τ, ri, aij (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 conditionholds, wherewhen the delay τ = 0, the system (1.1) simplifies to an autonomous system of ordinary differential equation of the formChen [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.
Section snippets
Analysis of global stability
In this section, we focus on investigating the global stability of the positive equilibrium of the system (1.1), by constructing suitable Lyapunov functionals. We first give two lemmas which will be used in the proof of the main results of this section. Lemma 2.1 Every solution (x(t), y(t)) of system (1.1) with initial conditions (1.2) is positive for all t > 0. Proof From system (1.1) we know thatThis completes the proof. □ Lemma 2.2
The existence of Hopf bifurcation
In this section, we focus on investigating the existence of Hopf bifurcation at the positive equilibrium E∗ = (x∗, y∗) of the system (1.1).
we first consider the linearization of system (1.1) at E∗The characteristic equation for system (3.1) takes the formwhereThe second degree transcendental polynomial Eq. (3.2) has been extensively studied by many researchers [4], [6],
Direction and stability of Hopf bifurcation
In this section, we focus on investigating the direction, stability and period of the periodic solution bifurcating from the positive equilibrium E∗. Following the ideas of Hassard et al. [15], we derive the explicit formulae for determining the properties of the Hopf bifurcation at the critical value of by using the normal form and the center manifold theory. Without loss of generality, we denote any one of these critical values by , at which Eq. (3.2) has a pair of
Global existence of periodic solutions
In this section, we will investigate the global existence of periodic solutions of system (1.1) by using the global Hopf bifurcation theorem due to Wu [21]. For simplification of notations, setting zt = (xt, yt)T, we may rewrite system (1.1) as the following functional differential equation:where zt(θ) = z(t + θ) ∈ C([−τ, 0], R2). System (5.1) has the only positive equilibrium z∗ = E∗(x∗, y∗) under the condition (H). Following the work of Wu [21], we define
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This work is supported by Shanghai Leading Academic Discipline Project (T0401) and National Natural Science Foundation of China (10561004, 10472100).