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

Abstract

In this paper, we concentrate on the study of a reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition. It is shown that a positive spatially nonhomogeneous equilibrium can bifurcate from the trivial equilibrium. Moreover, the stability of the bifurcated positive equilibrium is investigated. And we prove that, for the given spatiotemporal delay, the bifurcated equilibrium is stable under some conditions, and Hopf bifurcation cannot occur.

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

“…Especially, Busenberg and Huang [1996] firstly studied the Hopf bifurcation of the nonhomogeneous equilibrium for Dirichlet boundary condition, see also [Chen & Shi, 2012;Chen & Yu, 2014;Hu & Yuan, 2011;Su et al, 2009Su et al, , 2012Yan & Li, 2010] for other related delayed reaction-diffusion equations with Dirichlet boundary condition. For delayed differential equations, the effect of multiple delays has been widely investigated, see [Bélair & Campbell, 1994;Braddock & van den Driessche, 1993;Faria, 2011;Gopalsamy, 1990;Hale & Huang, 1993;Lenhart & Travis, 1986;Li et al, 1999;Mahaffy et al, 1995].…”

Stability and bifurcation on predator-prey systems with nonlocal prey competition

“…Since diffusive predator-prey and competing systems have much significant roles in population dynamics, they have been investigated extensively in the literature. We refer to [5,6,7,8,10,13,14,30,31,33,34,39,41,42,43] on the aspect of existence and nonexistence of nonconstant steady state solutions, periodic solutions and traveling wave solutions. These results could be used to explain the complex pattern formation in ecology.…”

Existence and stability of steady state solutions of reaction-diffusion equations with nonlocal delay effect

“…For the diffusion kernel defined in (1.4) (C) and Gamma distribution function (b), it is found under Dirichlet boundary condition that the small amplitude positive steady state does not undergo Hopf bifurcation and it remains stable for τ > 0 [9]. Same result holds for Neumann boundary condition and weak kernel, but Hopf bifurcation occurs for Neumann boundary condition and strong kernel [50].…”

“…Our results can be compared to a vast body of previous work on [5, 20, 36-38, 42, 46] [23, 29, 33] (A) [27, 34, 43-45, 47] [14, 16, 49] (A) (B) [8,10,21,22,46] (B) [28] (B) [3] (C) [9,18] (C) [18,39,50] (C) [4] Table 1: References on dynamics of (1.1) with different combinations of G, g and boundary conditions.…”