Asynchronous exponential growth for a two-phase size-structured population model and comparison with the corresponding one-phase model

Meng Bai,Shihe Xu

doi:10.1080/17513758.2018.1501104

Abstract

ABSTRACTIn this paper we study a two-phase size-structured population model with distributed delay in the birth process. This model distinguishes individuals by ‘active’ or ‘resting’ status. The individuals in the two life-stages have different growth rates. Only individuals in the ‘active’ stage give birth to the individuals in the ‘active’ stage or the ‘resting’ stage. The size of all the newborns is 0. By using the method of semigroups, we obtain that the model is globally well-posed and its solution possesses the property of asynchronous exponential growth. Moreover, we give a comparison between this two-phase model with the corresponding one-phase model and show that the asymptotic behaviours of the sum of the densities of individuals in the ‘active’ stage and the ‘resting’ stage and the solution of the corresponding one-phase model are different.

Highlights

In this paper we study a two-phase size-structured population model with distributed delay in the birth process
By using the method of semigroups, we obtain that the model is globally well-posed and its solution possesses the property of asynchronous exponential growth
Unlike the non-distributed delay case, the time lag considered here can change from 0 to the maximal value, i.e. it is distributed in an interval

Summary

Introduction

We study a two-phase size-structured population model with distributed delay in the birth process. We shall consider the special case where γ1(x) = γ2(x) = γ (x) and shows that the asymptotic behaviours of the sum of the densities of individuals in the ‘active’ stage and the ‘resting’ stage and the solution of the corresponding one-phase model are different. This result shows that the asymptotic behaviours of the sum of the densities of individuals in the ‘active’ stage and the ‘resting’ stage and the solution of the one-phase model are different and the research on the model with two stages is meaningful.

Reduction and well-posedness

Asynchronous exponential growth

Relation with the one-phase model