Persistence And Periodic Solution For Two-species Dynamical Systems With Impulses

Biological dynamical systems described by ordinary differential equat-ions have been studied by many authors and the research results are very rich. But many phenomena exhibit impulsive effects, so it is more natural to describe the processes which characterized by the fact that at certain moments of time they experience a change of state abruptly with impulsi-ve differential equations. This is particularly true in the case of describing the growth of species and the behavior of epidemic dynamics. In [1], the persistence for Holling type ⅡLotka-Volterra predator-prey system wit-h impulsive perturbations on the predator was studied. This paper develo-ps the persistence for community defence model with impulses and gets a sufficient condition. In spite of the great number of investigations of the delayed differential equations without impulsive effect(see[2-10]), the delayed impulsive differential equations have not been elaborated. In this paper, sufficient conditions are obtained for the existence of a positive periodic solution for the ratio-dependent predator-prey food chain model with time delays. In this paper, we first consider two-species dynamical systems with fu-nctional response and impulse where x (t ), y (t )represent the densities of prey and predator at time t resp-ectively, a is the intrinsic growth rate of prey, b is the rate of intraspecif-ic competition or density dependence, cdenotes the death rate of predator, k is the rate of conversing prey into predator. Then we consider the following two-species ratio-dependent dynamical systems with time delays where x (t ), y (t ) represent the densities of prey and predator at time t r-espectively, τis a positive constant time delay, b1 ( t ) , a1 ( t ) , m1 ( t ) ,b2 ( t ), c ( t ), a 2( t ),m2 ( t )> 0are continuous T ? periodic functions, z+ = {1 ,2,L} , The initial functions are ( ) ( ( ) ( ))? t = ?1 t ,?2t, where 0 < t1 < t 2< L < tk 0 such that ck + q = ck ,d k +q = dk, t k + q = tk+T , 0 < t k +1? t k< T.The main conclusions are given by Theorem 1 There exists a constant M > 0 such that 0 < x (t ) ≤M, 0 < y (t )≤M for each solution ( x (t ), y (t ))of (1 ) with all t large enough. Theorem 2 system (1 ) is permanent if p < aταcω. Theorem 3 Assume that kα? cβ> 0 , cT ? q ln (1 + p)> 0 , max ??? H 1,ln ??? kcωα??? ? 2 aT??? ??? ???, ( )21ln 1 0qkkb T d=? ??? ∏+ ???> , a2 > b2 , T > τ( ) ( )2 21ln 1 0qlkka T τb T d=? ? + ??? ∏+ ???>, ( )1 11l u 1 ln q1 k0kc m b cT =? ??? + ??? ∏+ ??????> where ( )0g : 1Tg t dt,= T∫[ ] ( )g u =: m0,aTx y t,[ ] ( )g l =: m0,iTny t. then system ( 2 ) has a T ? periodic solution.