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Periodicity and attractivity of a ratio-dependent Leslie system with impulses. (English) Zbl 1216.34045

The authors study the ratio-dependent Leslie predator-pray model with impulses

x ˙ 1 =x 1 (t)b (t) - a (t) x 1 (t) - c(t)x 1 (t)x 2 (t) h 2 x 2 2 (t)+x 1 2 (t),x ˙ 2 (t)=x 2 (t)e (t) - f (t) x 2 (t) x 1 (t),tt k ,x i (t k + )=(1+h k i )x i (t k ),x i (0)>0,i=1,2,(1)

where x i (t), i=1,2 denote the density of prey and predator at time t, respectively. b, a, c, d, e, f, p, α i ,β i ,γ i C(, + ), i=1,2, are ω-periodic functions. Sufficient conditions for the existence of positive periodic solutions of (1) are derived.

Sufficient conditions for the existence of a unique positive ω-periodic solution which is globally attractive are found.

MSC:
34C60Qualitative investigation and simulation of models (ODE)
34A37Differential equations with impulses
34C25Periodic solutions of ODE
34D20Stability of ODE
92D25Population dynamics (general)
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