Electron. J. Diff. Equ., Vol. 2012 (2012), No. 91, pp. 1-10.

Positive almost periodic solutions for state-dependent delay Lotka-Volterra competition systems

Yongkun Li, Chao Wang

Abstract:
In this article, using Mawhin's continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive almost periodic solutions for the system of equations
$$ \dot{u}_i(t)=u_i(t)\Big[r_i(t)-a_{ii}(t)u_i(t) -\sum_{j=1, j\neq i}^na_{ij}(t)u_j\big(t-\tau_j(t,u_1(t), \dots,u_n(t))\big)\Big], $$
where $r_i,a_{ii}>0$, $a_{ij}\geq0(j\neq i$, $i,j=1,2,\dots,n)$ are almost periodic functions, $\tau_i\in C(\mathbb{R}^{n+1},\mathbb{R})$, and $\tau_i(i=1,2,\dots,n)$ are almost periodic in $t$ uniformly for $(u_1,\dots,u_n)^T\in\mathbb{R}^n$. An example and its simulation figure illustrate our results.

Submitted April 12, 2012. Published June 7, 2012.
Math Subject Classifications: 34K14, 92D25.
Key Words: Lotka-Volterra competition system; almost periodic solutions; coincidence degree; state dependent delays.

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Yongkun Li
Department of Mathematics, Yunnan University
Kunming, Yunnan 650091, China
email: yklie@ynu.edu.cn
Chao Wang
Department of Mathematics, Yunnan University
Kunming, Yunnan 650091, China
email: super2003050239@163.com

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