Dirichlet problem of a delay differential equation with spatial non-locality on a half plane. (February 2018)

• Record Type: Journal Article
• Authors: Hu, Wenjie; Duan, Yueliang; Zhou, Yinggao
• Is Part Of: Nonlinear analysis. Volume 39(2018)
• Page Start: 300
• Page End: 320
• Publication Date: 2018-02
• Subjects: Dirichlet problem -- A priori estimate -- Heterogeneous steady state -- Global attractivity -- Nonlocal delay differential equation; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.07.009
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we are concerned with modeling and analyzing the dynamics for a two-stage species that lives on a half plane. We first derive a spatially nonlocal and temporally delayed differential equation that describes the mature population on a semi-infinite environment with a homogeneous Dirichlet condition. For the derived model, we are able to show that the solutions induce a k -set contraction semiflow with respect to the compact open topology on a bounded positive invariant set attracting every solution of the equation. To describe the global dynamics, we first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect, which enables us to show the repellency of the trivial equilibrium. Using the estimate, k -set contracting property as well as Schauder fixed point theorem, we then establish the existence of a positive spatially heterogeneous steady state. At last, we show global attractivity of the nontrivial steady state by employing dynamical system approaches.