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

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.