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Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments. (English) Zbl 1186.34118

Summary: The model discussed in this paper is described by the following periodic 3-species Lotka-Volterra predator-prey system with several 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 2 (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) denotes the density of prey species at time t, x 2 (t) and x 3 (t) denote the density of predator species at time t, r i ,a ij C(,[0,)) and τ ij C(,) are w-periodic functions with

r ¯ i =1 w 0 w r i (s)ds>0;a ¯ ij =1 w 0 w a ij (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 (*).

34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
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