Existence and asymptotic behaviors of traveling waves of a modified vector-disease model

Zengji Du,Zhaosheng Feng

doi:10.3934/cpaa.2018090

Abstract

In this paper, we are concerned with the existence and asymptotic behavior of traveling wave fronts in a modified vector-disease model. We establish the existence of traveling wave solutions for the modified vector-disease model without delay, then explore the existence of traveling fronts for the model with a special local delay convolution kernel by employing the geometric singular perturbation theory and the linear chain trick. Finally, we deal with the local stability of the steady states, the existence and asymptotic behaviors of traveling wave solutions for the model with the convolution kernel of a special non-local delay.

Highlights

In the past decades, one has seen that vector-borne diseases have become major public health problems throughout the world
Li-Zhu [21] studied limit cycles bifurcating from the limit periodic sets of the predator-prey systems with the response functions of Holling types, as well as the multiplicity of such limit cycles by applying the geometric singular perturbation theory developed by Dumortier-Roussarie etc [8, 5]
We use two methods to prove the existence of a traveling wave solution connecting two steady states: the geometric singular perturbation theory and the method of upper-lower solutions

Summary

Introduction

One has seen that vector-borne diseases have become major public health problems throughout the world. Li-Zhu [21] studied limit cycles bifurcating from the limit periodic sets of the predator-prey systems with the response functions of Holling types, as well as the multiplicity of such limit cycles by applying the geometric singular perturbation theory developed by Dumortier-Roussarie etc [8, 5] Another useful approach is to use the cross iteration method as well as the Schauder’s fixed point theorem to prove the existence of traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions [12, 25].

It becomes

To verify

Define t

It is easy to see that