ABSTRACT

For a $C^1$ map T from $C=[0,+\mathrm{\infty }{)}^N$ to C of the form $T_i\left(x\right)=x_i{f}_i\left(x\right)$, the dynamical system $x\left(n\right)=T^n\left(x\right)$ as a population model is competitive if $\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{x}_j}\le 0$ $(i\ne j)$. A well know theorem for competitive systems, presented by Hirsch [On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn. 2(2) (2008), pp. 169–179] and proved by Ruiz-Herrera [Exclusion and dominance in discrete population models via the carrying simplex, J. Differ. Equ. Appl. 19(1) (2013), pp. 96–113] with various versions by others, states that, under certain conditions, the system has a compact invariant surface $\mathrm{\Sigma }\subset C$ that is homeomorphic to ${\mathrm{\Delta }}^{N-1}=\{x\in C:x_1+\cdots +x_N=1\}$, attracting all the points of $C\setminus \left\{0\right\}$, 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 $Df=\left(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{x}_j}\right)$ 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.

摘要

为一个 $C^{\mathrm{1个}}$来自的地图T$C=[0，+\mathrm{\infty }{）}^{ñ}$到C的形式${Ť}_{\mathrm{一世}}（X）=X_{\mathrm{一世}}{F}_{\mathrm{一世}}（X）$动力系统 $X（ñ）={Ť}^{ñ}（X）$ 作为人口模型具有竞争力 $\frac{\mathrm{\partial }{F}_{\mathrm{一世}}}{\mathrm{\partial }{X}_{Ĵ}}\le 0$ $（\mathrm{一世}\ne Ĵ）$。甲众所周知定理竞争的系统，通过赫希[呈现在存在和有竞争力的动力系统的负载单形的独特性，生物化学。达因 2（2）（2008），第169-179页]，并由Ruiz-Herrera证明[通过携带的单纯形法，J。Differ。提出的离散人口模型中的排他性和优势地位。等式 应用 19（1）（2013），第96–113页]，其他版本则指出，在某些条件下，系统具有紧凑的不变表面$\mathrm{\Sigma }\subset C$ 这是同胚的 ${\mathrm{\Delta }}^{ñ-\mathrm{1个}}=\{X\in C：X_{\mathrm{1个}}+\cdots +X_{ñ}=\mathrm{1个}\}$，吸引了 $C\setminus \left\{0\right\}$，并称为携带单纯形。该定理已被大量引用所接受。在本文中，我们指出其条件之一要求所有${ñ}^{\mathrm{2个}}$ 雅可比矩阵的项 $dF=（\frac{\mathrm{\partial }{F}_{\mathrm{一世}}}{\mathrm{\partial }{X}_{Ĵ}}）$消极地表达是不必要的，过于严格。我们通过将条件简化为要求Df的每个输入均为非正负且每个条件为正，证明了修改后的携带单纯形的存在性和唯一性$F_{\mathrm{一世}}$ 严格减少 $X_{\mathrm{一世}}$。作为主要结果应用的一个例子，提供了充分的条件，以使物种消失以及一种物种相对于另一种的优势。