Stability of bifurcating periodic solutions in a delayed reaction–diffusion population model

Abstract

A delayed reaction-diffusion model of the Fisher type with a single discrete delay and zero-Dirichlet boundary conditions on a general bounded open spatial domain with a smooth boundary is considered. The stability of a spatially heterogeneous positive steady state solution and the existence of Hopf bifurcation about this positive steady state solution are investigated. In particular, by using the normal form theory and the centre manifold reduction for partial functional differential equations, the stability of bifurcating periodic solutions occurring through Hopf bifurcations is investigated. It is demonstrated that the bifurcating periodic solution occurring at the first bifurcation point is orbitally asymptotically stable while those occurring at the other bifurcation points are unstable.

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

“…We proved that when λ > dπ 2 / 2 , the model has a unique positive steady state solution u λ , and for a fixed λ satisfying 0 < λ − dπ 2 / 2 1, there exists a sequence of the delay values {τ n } ∞ n=0 so that a forward Hopf bifurcation occurs at each τ = τ n from the positive steady state u λ . For (1.6), the stability of the bifurcating periodic solutions were studied by Yan and Li [46]. The result in [42] generalized earlier result of Busenberg and Huang [2], in which a diffusive Hutchinson equation with Dirichlet boundary condition was considered.…”

“…For the local stability and Hopf bifurcation around the positive steady state solution of (1.4), we analyze the characteristic equation using the approach in Busenberg and Huang [2], which has been utilized in many other studies of stability of non-constant steady state solution [1,42,46]. The first part of analysis here is conducted under a similar framework as in [2,42], but the analysis here is more difficult with the presence of both delayed and instantaneous effect on the growth rate.…”

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

“…(1.1) unstable through a Hopf bifurcation. Related work can also be found in References [11,12,18,19,22].…”

Dynamical analysis of a logistic equation with spatio-temporal delay

“…For a long time, it has been recognized that delays not only can cause the loss of stability but also induce various oscillations and periodic solutions, see [13][14][15][16]. For example, for the Neumann boundary value problem, (1.2) has been considered in [17,18] and they considered the stability and related Hopf bifurcation from the homogeneous equilibrium.…”