Bifurcations for a predator–prey system with two delays

Abstract

In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.

On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay

“…If a 11 a 22 − a 12 a 21 < 0, then Theorem 7 can be used to show that when one of the delays varies, the positive equilibrium E * loses its stability and a Hopf bifurcation can occur. In fact, Faria [32] proved the following result (see also Song et al [79] …”

Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment

“…Recently, researchers are using more than one delay to study the effect of past history of the system populations [14][15][16][17][18], as in reality time delays occur in almost every biological situation [19] and are assumed to be one of the reasons of regular fluctuations in population density [20]. Delay is frequently introduced in a biologically realistic predator-prey model.…”

“…Let me now assume that the system size expansion is valid such that the correlations ( = 1, 2) given by (17) decrease with the increase of the population size and they are assumed to be of the order of the inverse of the population size [24,27,28]:…”

Bifurcation analysis for a ratio-dependent predator-prey system with multiple delays

“…So the aim of this paper is to study the dynamical behavior of system (1.4), for which we investigate the stability of delayed Leslie-Gower predator-prey system (1.4) and analyse how the time delay τ effects the dynamics of this system. We would like to mention that the bifurcation in a predator-prey system with a single or multiple delays had been investigated by many researchers [9,10,19,21,22]. However, to the best of our knowledge, for this case, few results for system (1.4) have been obtained.…”