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

By using Krasnoselskii's fixed point theorem, we prove that the following periodic $n-$species Lotka-Volterra competition system with multiple deviating arguments

$(\ast)\quad\quad \dot{x}_i(t)=x_i(t)\left[r_i(t)-\sum_{j=1}^{n}a_{ij}(t)x_j(t-\tau_{ij}(t)) \right],\quad i=1, 2, \ldots, n,\qquad\quad$

has at least one positive $\omega-$periodic solution provided that the corresponding system of linear equations

$(\ast\ast)\qquad\qquad\qquad\qquad\quad \sum_{j=1}^{n}\bar{a}_{ij} x_j= \bar{r}_i, \quad i=1, 2, \ldots, n,\qquad\qquad\qquad\qquad\quad$

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

$\bar{r}_i=\frac{1}{\omega}\int_{0}^{\omega}r_i(s)ds >0; \ \bar... ...\frac{1}{\omega}\int_{0}^{\omega}a_{ij}(s)ds \ge 0, \quad i, j=1, 2, \ldots, n.$

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.