Two species nonlocal diffusion systems with free boundaries

Yihong Du; Mingxin Wang; Meng Zhao

doi:10.3934/dcds.2021149

Article Contents

2022, Volume 42, Issue 3: 1127-1162. Doi: 10.3934/dcds.2021149

This issue Previous Article Zero-dimensional and symbolic extensions of topological flows Next Article Topological mild mixing of all orders along polynomials

Two species nonlocal diffusion systems with free boundaries

1.
School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
2.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
3.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
4.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

^*Corresponding author: Mingxin Wang
^*Corresponding author: Mingxin Wang

Received: May 2021

Revised: September 2021

Early access: September 2021

Published: March 2022

Y. Du was supported by the Australian Research Council. M. Wang was supported by NSFC Grant 11771110. M. Zhao was supported by NWNU-LKQN2021-16 and a scholarship from the China Scholarship Council

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Abstract

We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [7], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.

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Mathematics Subject Classification: 35K57, 35R20, 92D25.

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