Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

Dynamics of many species through competition for resources

Pages: 737 – 760



Wenli Cai (Department of Mathematics, China University of Mining and Technology, Beijing, China)

Hailiang Liu (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)


This paper is concerned with a mathematical model of competition for resource where species consume noninteracting resources. This system of differential equations is formally obtained by renormalizing the MacArthur’s competition model at equilibrium, and agrees with the trait-continuous model studied by [S. Mirrahimi, B. Perthame and J.Y. Wakano, J. Math. Biol., 64(7):1189–1223, 2012]. As a dynamical system, self-organized generation of distinct species occurs. The necessary conditions for survival are given. We prove the existence of the evolutionary stable distribution (ESD) through an optimization problem and present an independent algorithm to compute the ESD directly. Under certain structural conditions, solutions of the system are shown to approach the discrete ESD as time evolves.The time discretization of the system is proven to satisfy two desired properties: positivity and energy dissipation. Numerical examples are given to illustrate certain interesting biological phenomena.


competition for resources, steady states, evolutionary stable distribution, relative entropy

2010 Mathematics Subject Classification

37N25, 65M08, 92D15

Cai was supported partially by National Natural Science Foundation of China (No.: 11701557, 11701556), National foreign special projects (No.: BG20190001019), and the Fundamental Research Funds for the Central Universities (No.: 2020YQLX05). Liu was partially supported by the National Science Foundation under Grant DMS1812666.

Received 17 March 2020

Accepted 30 October 2020

Published 5 May 2021