Positive almost periodic solutions for a predator-prey Lotka-Volterra system with delays

• Main Author: Yuan Ye
• Format: Article
• Language: English
• Published: University of Szeged 2012-08-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: predator-prey lotka-volterra system; almost periodic solutions; coincidence degree; delays
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=1716

In this paper, by using Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost periodic solutions are obtained for the predator-prey Lotka-Volterra competition system with delays \begin{eqnarray*} \left\{\begin{array}{lll} \frac{{\mathrm{d}} u_i(t)}{{\mathrm{d}}t}&=&u_i(t)\bigg [a_i(t)-\sum\limits_{ l=1}^{n}a_{il}(t)u_l(t-\sigma_{il}(t))-\sum\limits_{j=1}^{m}b_{ij}(t)v_j(t-\tau_{ij}(t))\bigg],i=1,\ldots,n,\\ \frac{{\mathrm{d}} v_j(t)}{{\mathrm{d}} t}&=&v_j(t)\bigg[-r_j(t)+\sum\limits_{l=1}^{n}d_{jl}(t)u_l(t-\delta_{jl}(t))-\sum\limits_{h=1}^{m}e_{jh}(t)v_h(t-\theta_{jh}(t))\bigg ], j=1,\ldots,m, \end{array}\right. \end{eqnarray*} where $a_i,r_j, a_{il}, b_{ij},d_{jl},e_{jh}\in C(\mathbb{R},(0,\infty)\sigma_{il},\tau_{ij},\delta_{jl},\theta_{jh}\in C(\mathbb{R},\mathbb{R})(i,l=1,\ldots,n, j,h=1,\ldots,m)$ are almost periodic functions.