Existence and global attractivity of positive periodic solutions of Lotka–Volterra predator–prey systems with deviating arguments

The model discussed in this paper is described by the following periodic 3-species Lotka–Volterra predator–prey system with several 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) denotes the density of prey species at time tt, x2(t)x2(t) and x3(t)x3(t) denote the density of predator 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 with r̄i=1w∫0wri(s)ds>0;āij=1w∫0waij(s)>0,i,j=1,2,3. By using Krasnoselskii’s fixed point theorem and the construction of Lyapunov function, a set of easily verifiable sufficient conditions are derived for the existence and global attractivity of positive periodic solutions of (*).