On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments

Authors:	Xianhua Tang Xingfu Zou

Institution:	School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, People's Republic of China ; Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7

Abstract:	By using Krasnoselskii's fixed point theorem, we prove that the following periodic species Lotka-Volterra competition system with multiple deviating arguments $\displaystyle (\ast)\quad\quad \dot{x}_i(t)=x_i(t)\leftr_i(t)-\sum_{j=1}^{n}a_{ij}(t)x_j(t-\tau_{ij}(t)) \right],\quad i=1, 2, \ldots, n,\qquad\quad$ has at least one positive $\omega-$ periodic solution provided that the corresponding system of linear equations $\displaystyle (\ast\ast)\qquad\qquad\qquad\qquad\quad \sum_{j=1}^{n}\bar{a}_{ij} x_j= \bar{r}_i, \quad i=1, 2, \ldots, n,\qquad\qquad\qquad\qquad\quad$ has a positive solution, where $r_i, a_{ij}\in C({\mathbf{R}}, 0, \infty))$ and $\tau_{ij}\in C({\mathbf{R}}, {\mathbf{R}})$ are $\omega-$ periodic functions with $\displaystyle \bar{r}_i=\frac{1}{\omega}\int_{0}^{\omega}r_i(s)ds >0; \ \bar... ...\frac{1}{\omega}\int_{0}^{\omega}a_{ij}(s)ds \ge 0, \quad i, j=1, 2, \ldots, n.$ Furthermore, when $a_{ij}(t)\equiv a_{ij}$ and $\tau_{ij}(t)\equiv \tau_{ij}$ , $i,j =1,\ldots,n$ , are constants but $r_i(t), i=1, \ldots,n$ , remain $\omega$ -periodic, we show that the condition on $(\ast\ast)$ is also necessary for $(\ast)$ to have at least one positive $\omega-$ periodic solution.

Keywords:	Positive periodic solution Lotka-Volterra competition system

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