Almost Periodic Solution for an Epidemic Prey-Predator System with Impulsive Effects and Multiple Delays

Abstract

A nonautonomous epidemic prey-predator system with impulsive effects and multiple delays is considered; further, there is an epidemic disease in the predator. By the mean-value theorem of multiple variables, integral inequalities, differential inequalities, and other mathematical analysis skills, sufficient conditions which guarantee the permanence of the system are obtained. Furthermore, by constructing a series of Lyapunov functionals it is proved that there exists a unique uniformly asymptotically stable almost periodic solution of the system.

Factorization of Scalar Piecewise Continuous Almost Periodic Functions

“…In order to obtain p.c.a.p. solutions to (1.4), [8,9,15,16,20,21] make the assumption (A1) The functions {φ i } d i=1 are bounded and p.c.a.p.…”

Factorization of Scalar Piecewise Continuous Almost Periodic Functions

“…has a bounded solution v with inf n∈Z |v(n)| > 0, then there exists a unique f ∈ AP (R, R) and u ∈ AP (Z, R), up to a nonzero multiplicative constant, such that As for impulsive differential equations, Theorems 1.1 and 1.2 indicate the direction of looking for solutions in the factorization form and provide an answer to the almost periodicity of the functions in an assumption (see (A1) below) of [8,9,14,15,19,20]. Changing variables is an important tool in dynamics.…”

“…In order to obtain p.c.a.p. solutions to (4), [8,9,14,15,19,20] make the assumption (A1) the functions {ϕ i } d i=1 are bounded and p.c.a.p. However, there exists no theoretical reason for the almost periodicity of the functions in (5) in literature.…”