The Traveling Waves For A Single Species Model With Two Age Structures

As is known to all,in the field of mathematics,population ecology is one of the most extensive and mature branches in application of ecology.Many research of population models can be come down to the study of reaction-diffusion equation,during the study of population diffusion phenomenon,when many species are affected by the time-varying environment,their behavior of dynamic have great difference,such as seasonal variation: in the different seasons,due to temperature and food supply problems,the birth rate,death rate and the rate of diffusion change correspondingly.Therefore,it is more important theretical and practical significance to study the time-varying environment.This paper is concerned with the existence of the traveling wave fronts in a nonlocal diffusion model with two age structures and delay,which are also influenced by the time varying environment.First,by using the Fourier transformation,we derive a single species model of nonlocal diffusion with two age structures and delay,we study the initial value problem and establish comparison principle,by constructing appropriate upper-lower solution,the existence of traveling waves is proved.The lattice dynamical system is used to describe processes of the growth and invasion of species in the space of discrete plaque environment,and the population living environment is generally in the two-dimensional space.So,we concerned with the existence and uniqueness of the traveling wave fronts with two age structures and delay model in a 2D lattice dynamical system,which are also influenced by the time varying environment.First,by using the two-dimension Fourier transformation,we derive a lattice dynamic population model with two age classes and maturation delay,then,we study the initial value problem and establish comparison principle,by constructing appropriate upper-lower solution,the existence and uniqueness of traveling waves is proved.