Stability and Hopf bifurcation analysis for the diffusive delay logistic population model with spatially heterogeneous environment

Abstract

In this work, we have studied stability and Hopf bifurcation analysis for use in a delayed diffusive logistic population equation in spatially heterogeneous environments. The solutions of the 1-D reaction-diffusion equation are considered using the Galerkin technique. Full maps of Hopf bifurcation are determined for the parameters of maturation time, diffusion coefficient and growth rate. In addition, the effects of the free parameters in this model have been examined with the consequence that they can destabilize or stabilize the solution. The Hopf bifurcations for proliferation rate decreased as the maturation time increased while the diffusion coefficient grew. Furthermore, bifurcation diagrams and examples of periodic limited cycle solutions and 2D phase plane maps have been constructed. The comparisons between the numerical simulations with the analytical solutions provided confirmatory evidence and the validation of the technique used, with an excellent agreement compared for all the examples shown. Lindstedt–Poincaré in perturbation theory was applied to calculate the asymptotic results around the Hopf bifurcation point for both the one and two-term analytical systems.