# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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## On positive periodic solutions of Lotka-Volterra competition systems with deviating argumentsHTML articles powered by AMS MathViewer

by Xianhua Tang and Xingfu Zou
Proc. Amer. Math. Soc. 134 (2006), 2967-2974 Request permission

## Abstract:

By using Krasnoselskii’s fixed point theorem, we prove that the following periodic $n-$species Lotka-Volterra competition system with multiple deviating arguments \begin{equation*} (\ast )\quad \quad \dot {x}_i(t)=x_i(t)\left [r_i(t)-\sum _{j=1}^{n}a_{ij}(t)x_j(t-\tau _{ij}(t)) \right ],\quad i=1, 2, \ldots , n,\qquad \quad \end{equation*} has at least one positive $\omega -$periodic solution provided that the corresponding system of linear equations \begin{equation*} (\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 \end{equation*} 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 $\bar {r}_i=\frac {1}{\omega }\int _{0}^{\omega }r_i(s)ds >0;\ \ \ \bar {a}_{ij}=\frac {1}{\omega }\int _{0}^{\omega }a_{ij}(s)ds \ge 0, \quad i, j=1, 2, \ldots , n.$ 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.
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• Xianhua Tang
• Affiliation: School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, People’s Republic of China
• Email: tangxh@mail.csu.edu.cn
• Xingfu Zou
• Affiliation: Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
• MR Author ID: 618360
• Email: xzou@uwo.ca
• Received by editor(s): August 13, 2004
• Received by editor(s) in revised form: April 29, 2005
• Published electronically: May 9, 2006
• Additional Notes: The first author was supported in part by NNSF of China (No. 10471153), and the second author was supported in part by the NSERC of Canada and by a Faculty of Science Dean’s Start-Up Grant at the University of Western Ontario.
• Communicated by: Carmen C. Chicone