Abstract
We considered a nonautonomous two dimensional predator-prey system with impulsive effect. Conditions for the permanence of the system and for the existence of a unique stable periodic solution are obtained.
Highlights
In biomathematics, many mathematical models have been established to describe the relationships between species and the outer environment, and the connections between different species
We considered a nonautonomous two dimensional predator-prey system with impulsive effect
Many mathematical models have been established to describe the relationships between species and the outer environment, and the connections between different species
Summary
Many mathematical models have been established to describe the relationships between species and the outer environment, and the connections between different species. We consider the nonautonomous ratio-dependent predator-prey system with Holling type III functional response and impulsive effect x (t) = x(t). Where α(t), βi(t) (1 ≤ i ≤ l) are continuous ω-periodic functions; βi(t) ≥ 0 (1 ≤ i ≤ l); γi is a positive constant; there exists an integer q > 0 such that hk+q = hk, tk+q = tk + ω, and hk > −1 for all k ∈ Z+. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q : Z → Z such that Im P = KerL, Ker Q = Im P It follows that L|DomL∩KerP → Im L is invertible. From the stability property of x∗(t), one knows that the positive periodic solution of (2.1) is unique This completes the sufficient part of Lemma 2.2(2). Is a unique positive, globally asymptotically stable ω-periodic solution of (2.28)
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