Stability and Bifurcation in a Diffusive Logistic Population Model with Multiple Delays

Abstract

A diffusive logistic population model with multiple delays and Dirichlet boundary condition is considered in this paper. The stability/instability of the positive equilibrium and delay induced Hopf bifurcation are investigated. Moreover, we show which kind of delay could actually affect the dynamics. Int. J. Bifurcation Chaos 2015.25. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 08/18/15. For personal use only. 1550107-2 Int. J. Bifurcation Chaos 2015.25. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 08/18/15. For personal use only.

Dirichlet problem for a diffusive logistic population model with two delays

“…Depending on parameter ranges, there may exist multiple sequences of Hopf bifurcation points and stability switches may occur. This is not observed in [4,16], where only one sequence of critical values is obtained. The reason is that instantaneous density dependence is not incorporated in the model in [4], while the model in [16] does not consider multiple delays.…”

“…When Ω is bounded, various boundary conditions can be imposed, which include Dirichlet boundary condition, Neumann boundary condition, and Robin boundary condition. In this case, the study focuses on the existence of steady states and their stability (see, for example, [3,4,5,13,16,22] and references therein). In many studies of Hopf bifurcations, the diffusion rate is often chosen as the bifurcation parameter.…”

“…In many studies of Hopf bifurcations, the diffusion rate is often chosen as the bifurcation parameter. To the best of our knowledge, only Chen and Wei [4] and Su et al [16] chose time delay as the bifurcation parameter. Su et al [16] proposed the following diffusive model…”

“…They also investigated the occurrence of Hopf bifurcation from this positive steady state solution, the direction of the Hopf bifurcation, the stability of the bifurcating periodic orbits, and global continuation of the Hopf bifurcation branches. Chen and Wei [4] studied a different diffusive model…”

“…Some results on the linear stability of the positive equilibrium, oscillations of solutions, and the occurrence of limit cycles have been obtained; see, for example, [2,4,7,10,15,21] and references therein. In particular, Ruan [15] considered…”