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 equation(∗∗){x1′(t)=x1(t)(r1(t)−a11(t)x1(t−τ11(t))−a12(t)x2(t−τ12(t))+a13(t)x3(t−τ13(t)))x2′(t)=x2(t)(r2(t)−a21(t)x1(t−τ21(t))−a22(t)x2(t−τ22(t))+a23(t)x3(t−τ23(t)))x3′(t)=x3(t)(r3(t)+a31(t)x1(t−τ31(t))+a32(t)x2(t−τ32(t))−a33(t)x3(t−τ33(t))), where x1(t)x1(t) and x2(t)x2(t) denote the densities of competing species at time tt, x3(t)x3(t) denotes the density of cooperating species at time tt, ri,aij∈C(R,[0,∞))ri,aij∈C(R,[0,∞)) and τij∈C(R,R)τij∈C(R,R) are ww-periodic functions (ω>0ω>0) with r̄i=1w∫0wri(s)ds>0;āij=1w∫0waij(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.