Positive Almost Periodic Solutions for a Delayed Predator–Prey Model with Hassell-Varley Type Functional Response

Tianwei Zhang,Yongzhi Liao,Liyan Pang

doi:10.3390/mca21020010

Abstract

By means of a fixed point theorem of coincidence degree theory, sufficient conditions are established for the existence of a positive almost periodic solution to a kind of delayed predator–prey model with Hassell-Varley type functional response. The method used in this paper offers a possible means to study the existence of positive almost periodic solutions to the models in biological populations. Finally, an example as well as numerical simulations are given to illustrate the feasibility and effectiveness of our results.

Highlights

It is well-known that the theoretical study of predator–prey systems in mathematical ecology has a long history starting with the pioneering work of Lotka and Volterra [1,2]
It is important to study the existence of almost periodic solutions to Motivated by the above reason and considering that a delay may occur in the functional response of System (2), in this paper, we consider the following almost periodic predator–prey model with Hassell-Varley type functional response and time-varying delays:
By using a fixed point theorem of coincidence degree theory, some criterions for the existence of positive almost periodic solution to a kind of delayed predator–prey model with Hassell-Varley type functional response are obtained

Summary

Introduction

It is well-known that the theoretical study of predator–prey systems in mathematical ecology has a long history starting with the pioneering work of Lotka and Volterra [1,2]. In [17], Wang considered the following periodic predator–prey model with Hassell-Varley type functional response and time-varying delay:. Motivated by the above reason and considering that a delay may occur in the functional response of System (2), in this paper, we consider the following almost periodic predator–prey model with Hassell-Varley type functional response and time-varying delays: c(t) N2 (t−τ (t)) mN2θ (t−τ(t))+N1(t). Motivated by the above reason, the main purpose of this paper is to establish some new sufficient conditions based on the existence of positive almost periodic solutions to System (4) by using Mawhin’s continuous theorem of coincidence degree theory.

Preliminaries

Results

Two Examples and Numerical Simulations

Conclusions