A spatially and size-structured population model with unbounded birth process

Abed Boulouz

doi:10.3934/dcdsb.2022038

Article Contents

2022, Volume 27, Issue 12: 7169-7183. Doi: 10.3934/dcdsb.2022038

This issue Previous Article Chaotic motion and control of the driven-damped Double Sine-Gordon equation Next Article On the Cauchy problem for a nonlocal nonlinear Schrödinger equation

A spatially and size-structured population model with unbounded birth process

Abed Boulouz

Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Hay Dakhla, BP8106, 80000–Agadir, Morocco

Received: May 31, 2021

Revised: December 31, 2021

Early access: March 2022

Published: December 2022

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Abstract

In this paper, we consider a spatially and size structured population model with unbounded birth process. Firstly, the model is transformed into a closed-loop system, and hence the well-posedness is established by using the feedback theory of regular linear systems. Moreover, the solution to the resulting closed-loop system is given by a perturbed semigroup. Secondly, we give a condition on birth and death rates in such a way that the solution decays exponentially. To do this, we show that the semigroup solution is positive and hence we derive a characterization of exponential stability due to the technique tools of positive semigroups. We mention that our results extend a previous work in [D. Yan and X. Fu, Comm. Pure Appl. Anal. 15 (2016), 637–655] to the unbounded situation.

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Mathematics Subject Classification: Primary: 92D25, 47D06; Secondary: 35B40.

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