Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

Abstract

In this paper, we study the Hopf bifurcation and spatiotemporal pattern formation of a delayed diffusive logistic model under Neumann boundary condition with spatial heterogeneity. It is shown that for large diffusion coefficient, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady state at a critical time delay value, and the dependence of corresponding spatiotemporal patterns on the heterogeneous resource function is demonstrated via numerical simulations. Moreover, it is proved that the heterogeneous resource supply contributes to the increase of the temporal average of total biomass of the population even though the total biomass oscillates periodically in time.

Hopf bifurcation of a delayed single population model with patch structure

“…On the other hand, we see from Remark 2.13 that the large dispersal rate d may dilute the effect of the connectivity between patches, and the first Hopf bifurcation value of model (1.3) tends to that of the "average" model (2.43) when the dispersal rate d tends to infinity. For model (1.2), we see from [32] that when d is large or near some critical value, delay r could induce Hopf bifurcation, and similar results could also be found in [8,9,11,12,17,18,19,22,35,36,41,42] for other delayed reaction-diffusion models. It is of interest to consider whether Hopf bifurcation could occur for model (10,8,16,20,24,12,18,6,14).…”

“…This method could also be used to study the Hopf bifurcation for some other population models under the homogeneous Dirichlet boundary conditions, see [9,12,17,18,19,22,35,36,41,42] and the references therein. If m(x) is spatially heterogenous, the effect of spatial heterogeneity on model (1.2) was investigated in [28] for τ = 0, and delay induced Hopf bifurcation was considered in [32]. Moreover, the advection term, which indicates movements towards better quality habitat or refers to unidirectional bias of movements, was also taken into consideration for model (1.2) by many researchers.…”

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

“…A temporally oscillatory solution emerges from the Hopf bifurcation, and this solution is spatially non-homogeneous under Dirichelt boundary condition [5,37,38,42] or with spatial heterogeneity [34]. Similar Hopf bifurcation and temporally oscillatory solution are also found when the delay is distributed one as type (b) [16,33,49].…”

“…boundary value problem, the positive steady state solution loses its stability via a Hopf bifurcation when the delay τ is large [27,34,47], while the same phenomenon is also proved for small amplitude positive steady state for Dirichlet boundary value problem [5,37,38,42].…”

Hopf bifurcation in a delayed reaction–diffusion–advection equation with ideal free dispersal

“…We also point out that there are several mathematical models formulated to describe the effect of time delay on Hopf bifurcation of the spatially nonhomogeneous positive equilibrium. These models include single population models [18][19][20]23], competition diffusion systems [24,26], predator-prey diffusion models [17,25] and nonlocal delay models [10,11,14,15].…”