Proceedings of the American Mathematical Society

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



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

Authors: Xianhua Tang and Xingfu Zou
Journal: Proc. Amer. Math. Soc. 134 (2006), 2967-2974
MSC (2000): Primary 34K13; Secondary 34K20, 92D25
Published electronically: May 9, 2006
MathSciNet review: 2231621
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Abstract: By using Krasnoselskii's fixed point theorem, we prove that the following periodic $ n-$species Lotka-Volterra competition system with multiple deviating arguments

$\displaystyle (\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 $

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.

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Additional Information

Xianhua Tang
Affiliation: School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, People’s Republic of China

Xingfu Zou
Affiliation: Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7

Keywords: Positive periodic solution, Lotka-Volterra competition system
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.