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.

中文翻译：

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通过争夺资源实现许多物种的动态

本文关注的是物种消耗非交互资源的资源竞争的数学模型。该微分方程组是通过在平衡状态下重新对麦克阿瑟的竞争模型进行正规化而正式获得的，并且与[S. S. Mirrahimi B. Perthame和JY Wakano J. Math。生物学，64（7）：1189-1223，2012]。作为一个动力系统，会发生自组织的不同物种的生成。给出了生存的必要条件。我们通过一个优化问题证明了进化稳定分布（ESD）的存在，并提出了一种直接计算ESD的独立算法。在特定的结构条件下，系统的解决方案随着时间的发展显示出接近离散ESD的能力。系统的时间离散化被证明满足两个期望的特性：正性和能量耗散。数值例子说明了某些有趣的生物学现象。