   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
DOI: https://doi.org/10.1090/S0002-9939-06-08320-1
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 \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

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