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

Abstract

A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered. The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation. The direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are derived by the center manifold theory and normal form method. In particular, the global continuation of the Hopf bifurcation branches are investigated with a careful estimate of the bounds and periods of the periodic orbits, and the existence of multiple periodic orbits are shown.

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

“…By using the implicit function theorem as in [Busenberg & Huang, 1996;Su et al, 2012], we can give a profile of u λ near λ * without proof. Lemma 1.…”

“…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].…”

“…Then our main result is that: if τ i is a "dominated delay", then it has an effect on the stability of the positive equilibrium and delay induced Hopf bifurcation can occur when λ is near λ * ; and if τ i is not a "dominated delay", then it cannot affect the stability/instability of the positive equilibrium when λ is near λ * . Hence the results in [Busenberg & Huang, 1996;Su et al, 2012] can be generalized for the multiple delay case. As in [Chen & Shi, 2012], throughout the paper, we denote τ = max{τ i : i = 1, 2, .…”

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].…”

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

“…(1.2) have been considered in many papers, for example, [1,2,9,17,39] and the references therein. On the other hand, spatiotemporal pattern formation for the nonlocal Fisher-KPP type equation (again without delay effect) has been studied in [15,16,26,28].…”