This paper deals with global asymptotic behaviour of the dynamics for $N$-dimensional competitive Kolmogorov differential systems of equations $\frac{dx_i}{dt} =x_if_i(x), 1\leq i\leq N, x\in \R^N_+$. A theory based on monotone dynamical systems was well established by Morris W Hirsch (Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51--71). One of his theorems is outstanding and states the existence of a co-dimension 1 compact invariant submanifold $\Sigma$ that attracts all the nontrivial orbits under certain assumptions and, in practice, under the condition that the system is totally competitive (all $N^2$ entries of the Jacobian matrix $Df$ are negative). The submanifold $\Sigma$ has been called carrying simplex since then and the theorem has been well accepted with many hundreds of citations. In this paper, we point out that the requirement of total competition is too restrictive and too strong; we prove the existence and uniqueness of a carry simplex under the assumption of strong internal competition only (i.e. $N$ diagonal entries of $Df$ are negative), a much weaker condition than total competition. Thus, we improve the theorem significantly by dramatic cost reduction from requiring $N^2$ to $N$ negative entries of $Df$. As an example of applications of the main result, the existence and global attraction (repulsion) of a heteroclinic limit cycle for three-dimensional systems is discussed and two concrete examples are given to demonstrate the existence of such heteroclinic cycles.

中文翻译：

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柯尔莫哥洛夫微分系统中携带单纯形的存在唯一性

本文讨论了 $N$ 维竞争 Kolmogorov 微分方程组 $\frac{dx_i}{dt} =x_if_i(x), 1\leq i\leq N, x\in \R 动力学的全局渐近行为^N_+$。Morris W Hirsch 很好地建立了基于单调动力系统的理论（竞争或合作的微分方程系统：III.竞争物种，非线性，1（1988），51--71）。他的一个定理是杰出的，它指出存在一个共维 1 紧致不变子流形 $\Sigma$，它在某些假设下吸引了所有非平凡轨道，并且在实践中，在系统完全竞争的条件下（所有 $N雅可比矩阵 $Df$ 的 ^2$ 个条目为负）。从那时起，子流形 $\Sigma$ 就被称为携带单纯形，并且该定理已被数百次引用而广为接受。在本文中，我们指出全面竞争的要求过于严格和过于强烈；我们仅在强内部竞争的假设下证明了进位单纯形的存在性和唯一性（即 $Df$ 的 $N$ 对角线条目为负），这是比完全竞争弱得多的条件。因此，我们通过将 $N^2$ 的成本大幅降低到 $N$ 的 $Df$ 负条目来显着改进定理。作为主要结果的应用实例，讨论了三维系统异宿极限环的存在和全局吸引（排斥），并给出了两个具体的例子来证明这种异宿环的存在。