# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021194
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## Invasive speed for a competition-diffusion system with three species

 1 School of Mathematics and Physics, University of South China, Hengyang 421001, China 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, A1C 5S7, Canada

* Corresponding author: Chunhua Ou

Received  December 2020 Revised  May 2021 Early access July 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China grant (11626129 and 11801263), the Natural Science Foundation of Hunan Province grant (2018JJ3418); The second author is supported by the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20B512); The third author is supported by the Canada NSERC discovery grant (RGPIN-2016-04709)

Competition stems from the fact that resources are limited. When multiple competitive species are involved with spatial diffusion, the dynamics becomes even complex and challenging. In this paper, we investigate the invasive speed to a diffusive three species competition system of Lotka-Volterra type. We first show that multiple species share a common spreading speed when initial data are compactly supported. By transforming the competitive system into a cooperative system, the determinacy of the invasive speed is studied by the upper-lower solution method. In our work, for linearly predicting the invasive speed, we concentrate on finding upper solutions only, and don't care about the existence of lower solutions. Similarly, for nonlinear selection of the spreading speed, we focus only on the construction of lower solutions with fast decay rate. This greatly develops and simplifies the ideas of past references in this topic.

Citation: Chaohong Pan, Hongyong Wang, Chunhua Ou. Invasive speed for a competition-diffusion system with three species. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021194
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##### References:
The solution $u(x, t)$ at $t = 16, 36, 56$ for two sets of parameters
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