Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2020; 57(1): 249-281

Online first article August 19, 2019      Printed January 1, 2020

https://doi.org/10.4134/JKMS.j190036

Copyright © The Korean Mathematical Society.

Bifurcation analysis of a single species reaction-diffusion model with nonlocal delay

Jun Zhou

Southwest University

Abstract

A reaction-diffusion model with spatiotemporal delay modeling the dynamical behavior of a single species is investigated. The parameter regions for the local stability, global stability and instability of the unique positive constant steady state solution are derived. The conditions of the occurrence of Turing (diffusion-driven) instability are obtained. The existence of time-periodic solutions, the existence and nonexistence of nonconstant positive steady state solutions are proved by bifurcation method and energy method. Numerical simulations are presented to verify and illustrate the theoretical results.

Keywords: Reaction-diffusion model, nonlocal delay, spatiotemporal patterns, turing instability, Hopf bifurcation, steady state bifurcation, nonconstant positive solutions

MSC numbers: Primary 35J55, 35K55, 92C15, 92C40