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 − n- species Lotka-Volterra competition system with multiple deviating arguments ( ∗ ) x ˙ i ( t ) = x i ( t ) [ r i ( t ) − ∑ j = 1 n a i j ( t ) x j ( t − τ i j ( t ) ) ] , i = 1 , 2 , … , n , \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 ω − \omega - periodic solution provided that the corresponding system of linear equations ( ∗ ∗ ) ∑ j = 1 n a ¯ i j   x j = r ¯ i , i = 1 , 2 , … , n , \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 i j ∈ C ( R , [ 0 , ∞ ) ) r_i, a_{ij}\in C({\mathbf {R}}, [0, \infty )) and τ i j ∈ C ( R , R ) \tau _{ij}\in C({\mathbf {R}}, {\mathbf {R}}) are ω − \omega - periodic functions with \[ r ¯ i = 1 ω ∫ 0 ω r i ( s ) d s > 0 ;       a ¯ i j = 1 ω ∫ 0 ω a i j ( s ) d s ≥ 0 , i , j = 1 , 2 , … , n . \bar {r}_i=\frac {1}{\omega }\int _{0}^{\omega }r_i(s)ds >0;\ \ \ \bar {a}_{ij}=\frac {1}{\omega }\int _{0}^{\omega }a_{ij}(s)ds \ge 0, \quad i, j=1, 2, \ldots , n. \] Furthermore, when a i j ( t ) ≡ a i j a_{ij}(t)\equiv a_{ij} and τ i j ( t ) ≡ τ i j \tau _{ij}(t)\equiv \tau _{ij} , i , j = 1 , … , n i,j =1,\ldots ,n , are constants but r i ( t ) ,   i = 1 , … , n r_i(t),\ i=1, \ldots ,n , remain ω \omega -periodic, we show that the condition on ( ∗ ∗ ) (\ast \ast ) is also necessary for ( ∗ ) (\ast ) to have at least one positive ω − \omega - periodic solution.