# American Institute of Mathematical Sciences

May  2021, 26(5): 2561-2580. doi: 10.3934/dcdsb.2020195

## Large-time behavior of matured population in an age-structured model

 1 School of Mathematics and Statistics, Xidian University, Xi'an, China 2 Institut de Mathématiques de Bordeaux, Université de Bordeaux, Bordeaux, France

* Corresponding author: Linlin Li

Received  October 2019 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The first author is supported by the NSF of Shaanxi Province of China (2020JQ-289) and the Program for New Century Excellent Talents in University (XJS200701)

In this paper, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. We prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. We prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation. At last, we present numerical simulations.

Citation: Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195
##### References:

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##### References:
the matured population $w(t,x)$ for $t = 0$ and $t = 0.25$ with no control
the matured population $w(t,x)$ for $t = 0.5$ and $t = 1$ with no control
the matured population $w(t,x)$ for $t = 0$ and $t = 0.25$ with control $u(a,w)$
the matured population $w(t,x)$ for $t = 0.5$ and $t = 1$ with control $u(a,w)$
the matured population $w(t,x)$ for $t = 0$, $0.25$
the matured population $w(t,x)$ for $t=0.5$, $1$
the matured population $w(t,x)$ for $1.5$, $2$
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