Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms

Li Ma; Youquan Luo

doi:10.3934/dcdsb.2020022

Article Contents

2020, Volume 25, Issue 7: 2555-2582. Doi: 10.3934/dcdsb.2020022

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Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms

Li Ma^, and
Youquan Luo

Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, College of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi 341000, China

^* Corresponding author: Li Ma
^* Corresponding author: Li Ma

Received: December 2018

Revised: September 2019

Early access: April 2020

Published: July 2020

The first author is supported by Jiangxi Science and Technology Project (Grant No. GJJ170566) and NSF of China (Grants No. 11801089).

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Abstract

In this paper, we investigate a class of nonlocal dispersal logistic equations with nonlocal terms

$\left\{ {\begin{array}{*{20}{l}}{{u_t} = Du + {u^q}\left( {\lambda + a(x)\int_\Omega b (x){u^p}} \right),}&{{\rm{\quad\quad in\quad\quad}}\Omega \times (0, + \infty ),}\\{u(x,0) = {u_0}(x) \ge 0}&{{\rm{\quad\quad\quad\quad\quad\quad\quad in\quad\quad}}\Omega ,}\\{u = 0,}&{{\rm{ on\quad\quad}}{{\mathbb{R}}^N}\backslash \Omega \times (0, + \infty ),}\end{array}} \right.$

where $ \Omega\subset \mathbb{R}^N(N\geq1) $ is a bounded domain, $ \lambda\in \mathbb{R} $, $ 0<q\leq1 $, $ p>0 $, $ a,b\in C(\overline{\Omega}) $, $ b\geq0 $, $ b\neq0 $ and $ a $ verifies either $ a>0 $ or $ a<0 $. $ Du = \int_\Omega J(x-y)u(y,t){\rm{d}}y-u(x,t) $ represents the nonlocal dispersal operators, which is continuous and nonpositive. Under some suitable assumptions we establish the existence, uniqueness or multiplicity and stability of positive stationary solution with nonlocal reaction term by using sub-supersolution methods, Lerray-Schauder degree theory and Lyapunov-Schmidt reduction and so on.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35K57, 92D25.

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