离散Kolmogorov系统修正携带单形的存在唯一性
On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems
摘要：
对于一个从$C =[0, +\infty)^N$到$C$的$C^1$映射$T$，如果$\frac{\partial f_i}{\partial x_j}\leq 0$ $(i\not= j)$，则作为种群模型的动力系统$x(n) =T^n(x)$是竞争的。由Hirsch（J. Bio.动力学2（2008）169- 179），并由Ruiz-Herrera（J. Differ.当量ag Appl.19（2013）96- 113）与其他人的各种版本指出，在某些条件下，系统具有与$Δ^{N-1} =\{x\in C: x_1+ \cdots + x_N=1\}$同胚的紧凑不变曲面$Σ\subset C$，吸引$C\setminus\{0\}$的所有点，并且称为携带单形。该定理已被广泛接受，并被大量引用。在本文中，我们指出，它的一个条件，要求所有的$N^2$项目的雅可比矩阵$Df = (\frac{\partial f_i}{\partial x_j})$是负的是不必要的强大和过于限制。我们证明了修改携带单纯形的存在性和唯一性，通过减少该条件，要求每一个条目的$Df$是非正的，每个$f_i$是严格减少$x_i$。作为主要结果应用的一个例子，提供了物种消失和一个物种对其他物种的优势的充分条件。
英文摘要：
For a $C^1$ map $T$ from $C =[0, +\infty)^N$ to $C$ of the form $T_i(x) = x_if_i(x)$, the dynamical system $x(n) =T^n(x)$ as a population model is competitive if $\frac{\partial f_i}{\partial x_j}\leq 0$ $(i\not= j)$. A well know theorem for competitive systems, presented by Hirsch (J. Bio. Dyn. 2 (2008) 169--179) and proved by Ruiz-Herrera (J. Differ. Equ. Appl. 19 (2013) 96--113) with various versions by others, states that, under certain conditions, the system has a compact invariant surface $Σ\subset C$ that is homeomorphic to $Δ^{N-1} =\{x\in C: x_1+ \cdots + x_N=1\}$, attracting all the points of $C\setminus\{0\}$, and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the $N^2$ entries of the Jacobian matrix $Df = (\frac{\partial f_i}{\partial x_j})$ to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of $Df$ to be nonpositive and each $f_i$ is strictly decreasing in $x_i$. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.
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