Existence and global attractivity of positive periodic solutions of competitor–competitor–mutualist Lotka–Volterra systems with deviating arguments

In this paper, we study the existence and global attractivity of periodic solutions of competitor–competitor–mutualist Lotka–Volterra systems with deviating arguments ( ∗ ) { x 1 ′ ( t ) = x 1 ( t ) ( r 1 ( t ) − a 11 ( t ) x 1 ( t − τ 11 ( t ) ) − a 12 ( t ) x 2 ( t − τ 12 ( t ) ) + a 13 ( t ) x 3 ( t − τ 13 ( t ) ) ) x 2 ′ ( t ) = x 2 ( t ) ( r 2 ( t ) − a 21 ( t ) x 1 ( t − τ 21 ( t ) ) − a 22 ( t ) x 2 ( t − τ 22 ( t ) ) + a 23 ( t ) x 3 ( t − τ 23 ( t ) ) ) x 3 ′ ( t ) = x 3 ( t ) ( r 3 ( t ) + a 31 ( t ) x 1 ( t − τ 31 ( t ) ) + a 32 ( t ) x 2 ( t − τ 32 ( t ) ) − a 33 ( t ) x 3 ( t − τ 33 ( t ) ) ) , where x 1 ( t ) and x 2 ( t ) denote the densities of competing species at time t , x 3 ( t ) denotes the density of cooperating species at time t , r i , a i j ∈ C ( R , [ 0 , ∞ ) ) and τ i j ∈ C ( R , R ) are w -periodic functions ( ω > 0 ) with r ̄ i = 1 w ∫ 0 w r i ( s ) d s > 0 ; a ̄ i j = 1 w ∫ 0 w a i j ( s ) ≥ 0 , i , j = 1 , 2 , 3 . We obtain sufficient conditions for the existence and global attractivity of positive periodic solutions of (∗) by Krasnoselskii’s fixed point theorem and the construction of Lyapunov functions.