Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect

Abstract

A reaction-diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel.

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

“…In Proposition 2.9 of [6], the authors delt with the stability of a positive equilibrium in a reaction-diffusion equation with a nonlocal reaction term. And we find that the method can also be used here to deal with the stability of a positive equilibrium in a reaction-diffusion equation with spatiotemporal delay.…”

Stability and bifurcation in a reaction–diffusion model with nonlocal delay effect

“…Here, we emphasize the importance of nonlocal growth rate per capita in (1), since in the reality individuals sometimes compete for resource not only in their immediate neighborhood but also in a more board domain. Though such nonlocal delay equations in mathematical biology have been studied extensively in the literature [1,2,4,5,7,10,12], to the best of our knowledge, the qualitative study of the nonlocal delay effect has not been found. As we shall see, model (1) allows an indepth analysis and then to gain insight into possible mechanisms behind the observed behavior of the nonlocal delay effect.…”

“…Very recently, Chen and Shi [5] considered the stability/instability of the positive spatially nonhomogeneous steady-state solutions and the associated Hopf bifurcation for the following diffusive logistic population model with nonlocal delay effect:…”

Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

“…It is recognized that the stability and bifurcation analysis for a non-constant steady state solution (which is natural for Dirichlet boundary condition) is more difficult than the one for a constant steady state solution (which is natural for Neumann boundary condition) [2,7,18,40,42], as the spatial profile of the non-constant steady state solution is usually not known, which makes the characteristic equation analysis much harder. Analysis in [2,42] has also been extended to a diffusive logistic equation with nonlocal delay effect [3].…”