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

Linlin Li; Bedreddine Ainseba

doi:10.3934/dcdsb.2020195

Article Contents

2021, Volume 26, Issue 5: 2561-2580. Doi: 10.3934/dcdsb.2020195

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

Linlin Li^1, , and
Bedreddine Ainseba^2,

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
^* Corresponding author: Linlin Li

Received: September 30, 2019

Revised: March 31, 2020

Early access: June 2020

Published: May 2021

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)

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Abstract

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.

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Mathematics Subject Classification: Primary:35Q92, 35B40;Secondary:35K10.

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Figure 1. the matured population $ w(t,x) $ for $ t = 0 $ and $ t = 0.25 $ with no control

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Figure 2. the matured population $ w(t,x) $ for $ t = 0.5 $ and $ t = 1 $ with no control

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Figure 3. the matured population $ w(t,x) $ for $ t = 0 $ and $ t = 0.25 $ with control $ u(a,w) $

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Figure 4. the matured population $ w(t,x) $ for $ t = 0.5 $ and $ t = 1 $ with control $ u(a,w) $

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Figure 5. the matured population $ w(t,x) $ for $ t = 0 $, $ 0.25 $

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Figure 6. the matured population $ w(t,x) $ for $ t=0.5 $, $ 1 $

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Figure 7. the matured population $ w(t,x) $ for $ 1.5 $, $ 2 $

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