Propagation and heterogeneous steady states in a delayed nonlocal R-D equation without spatial translation-invariance

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Yi, Taishan¹  Zou, Xingfu² 

[1] Sun Yat Sen Univ, Sch Math Zhuhai, Zhuhai 519082, Guangdong, Peoples R China

[2] Univ Western Ontario, Dept Appl Math, London, ON N6A 5B7, Canada

关键词: Reaction-diffusion equation;    Spatial nonlocality;    Delay;    Heterogeneous steady state;    Non-translation-invariant;    Spreading speed;   

DOI  :  10.1016/j.jde.2019.09.004

来源: Elsevier

【 摘 要 】

We consider a delayed reaction-diffusion equation that models the population dynamics of a single species with the mature population living in the 1-D whole space R while the immature population only living in the half space R+, with homogeneous Dirichlet condition for the immatures at the boundary point. One of the important features of this model system is that it does have the translational-invariance. By linking the non-translational-invariant solution map for this equation to travelling wave maps for another related 1-D spatial homogeneous delay reaction-diffusion equation, we obtain some traveling-like a priori estimates for nontrivial solutions. We then establish the existence, uniqueness, and attractivity of heterogeneous steady states. As a result, we are able to describe the traveling-like asymptotic behaviours of nontrivial solutions in space-time region. These enable us to develop a new method for exploring the spreading speeds and asymptotic propagation phenomena for a class of non-translation-invariant delay reaction-diffusion equations on R. As a corollary, we also recover some results on the asymptotic spreading speeds and traveling waves for monostable and spatial homogeneous delay reaction-diffusion equations in R. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free