Hopf bifurcation in a reaction–diffusion equation with distributed delay and Dirichlet boundary condition

Shi, Qingyan; Shi, Junping; Song, Yongli

doi:10.1016/j.jde.2017.07.024

Cited by 39 publications

(14 citation statements)

References 34 publications

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“…Combining with the spatial movement of population, the reaction-diffusion logistic equation with time delay has been proposed to consider the evolution of population distribution. Dynamics of such equation under the Neumann boundary condition was studied in [14,29,41], and the Dirichlet boundary value problem of a delayed diffusive logistic model has been considered in, for example, [2,9,34,36,40]. In general, the time delay leads to occurrence of Hopf bifurcations, and the positive steady state loses its stability and temporally oscillatory patterns arise.…”

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confidence: 99%

“…Other than the scalar equation (1), the effect of spatial heterogeneity on the competition of two species is investigated in [19-22, 25, 27]. The method we use here for Neumann boundary value problem with heterogeneous resource function is similar to earlier work for Dirichlet boundary value problem with homogeneous resource function [2,9,16,17,34,36,37,40], as the steady states in both cases are spatially non-homogeneous. That approach is powerful but also limited as it requires a precise profile of the steady state.…”

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confidence: 99%

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

Shi¹,

Shi²,

Song³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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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.

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confidence: 99%

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confidence: 99%

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

Shi¹,

Shi²,

Song³

2019

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

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“…In this way, by using the general Hopf bifurcation theorem [35][36][37][38][39][40] for functional differential equations, the results on the stability and bifurcation of system (3a)-(3c) are obtained. Theorem 2.…”

Section: Hypothesismentioning

confidence: 99%

Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

et al. 2020

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e complex chaotic dynamics and multistability of financial system are some important problems in micro-and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. e system undergoes Hopf bifurcation near the equilibrium point. e stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. e theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

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“…He found that introducing nonlocal effects into a single reaction diffusion equation would lead to the change of stability about the positive equilibrium point and the disturbance around it (it should be noted that when considering spatiotemporal patterns, the instability of the equilibrium point and the appearance of branches are often accompanied). Subsequently, scholars further considered this problem in a limited region, and studied the Neumann boundary condition [10] and Dirichlet boundary condition [27]. Recently, Guo [11,12] and Yang and Xu [32] investigated the spatiotemporal patterns for a single species reaction diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System

Han

Zhao

2021

J Nonlinear Math Phys

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This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

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Hopf bifurcation in a reaction–diffusion equation with distributed delay and Dirichlet boundary condition

Cited by 39 publications

References 34 publications

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

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

Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System

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