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Positive periodic solution for a two-species ratio-dependent predator-prey system with time delay and impulse. (English) Zbl 1111.34051

The paper deals with a predator-prey model with delay and impulses of the form

x ˙(t)=x(t)b 1 (t)-a 1 (t)x(t)-c(t)y(t) m 1 (t)y(t)+x(t),y ˙(t)=y(t)-b 2 (t)+a 2 (t)x(t-τ) m 2 (t)y(t-τ)+x(t-τ),fortt k , x(t k + )-x(t k - )=c k x(t k ),y(t k + )-y(t k - )=d k y(t k ),(x(0+),y(0+))=(x 0 ,y 0 ),(x(t),y(t))=(ϕ 1 (t),ϕ 2 (t)),for-τt0·

The authors apply the continuation fixed-point theorem of coincindence degree theory to provide sufficient conditions for the existence of a periodic solution of the problem.

MSC:
34K13Periodic solutions of functional differential equations
34K45Functional-differential equations with impulses
34K60Qualitative investigation and simulation of models
92D25Population dynamics (general)