Abstract
In this paper, a three-species delayed food chain system with mutual interference and impulses is established. By utilizing the continuation theorem, the comparison theorem, some analysis techniques, and constructing a suitable Lyapunov functional, some sufficient conditions for the existence of positive periodic solutions, permanence, and global attractivity of the system are obtained. The conditions obtained are related to the mutual interference constant m, prey refuge constant θ, delays, and impulses. An example is given to show the feasibility of the results.
Highlights
1 Introduction Impulsive differential equations are used for the mathematical simulation of processes which are subject to impulses during their evolution
Many evolution processes in nature are characterized by the fact that, at certain moments of time, they experience some abrupt changes of state
In this paper, we are concerned with the following delayed three-species Beddington-type food chain system with mutual interference and impulsive control methods
Summary
Impulsive differential equations are used for the mathematical simulation of processes which are subject to impulses during their evolution. Do proposed and studied a constant periodic releasing for top predator or periodic impulsive immigration for top predator—a three-species food chain system with impulsive control and Beddington–DeAngelis functional response [9] Motivated by these facts, in this paper, we are concerned with the following delayed three-species Beddington-type food chain system with mutual interference and impulsive control methods. If L is a Fredholm operator of index zero, there exist continuous projectors P : X → X and Q : Y → Y such that Im P = Ker L, Ker Q = Im L = Im(I – Q) By Lemma 2.1, we know that the operator equation Lξ = Nξ has at least one solution in ∩ Dom L.
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