  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
Mirrahimi S, Perthame B, Wakano JY [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.
