The diffusive model for Aedes aegypti mosquito on a periodically evolving domain

Mengyun Zhang; Zhigui Lin

doi:10.3934/dcdsb.2018330

Article Contents

2019, Volume 24, Issue 9: 4703-4720. Doi: 10.3934/dcdsb.2018330

This issue Previous Article Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system Next Article Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping

The diffusive model for Aedes aegypti mosquito on a periodically evolving domain

Mengyun Zhang^1, and
Zhigui Lin^1,2, ,

1.
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2.
Key Laboratory of PCBHFASQ, Yangzhou University, Yangzhou 225002, China

* Corresponding author
* Corresponding author

Received: May 31, 2018

Revised: July 31, 2018

Early access: January 2019

Published: July 2019

The work is partially supported by the NNSF of China (Grant No. 11771381, 61877052)

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Abstract

This paper deals with a reaction-diffusion model on a periodically and isotropically evolving domain in order to explore the diffusive dynamics of Aedes aegypti mosquito, where we divide it into two sub-populations: the winged population and an aquatic form. The spatial-temporal risk index $ R_0(\rho) $ depending on the domain evolution rate $ \rho(t) $ as well as its analytical properties is investigated. The long-time behaviors of the periodic solutions under the condition $ R_0(\rho)>1 $ and $ R_0(\rho)\leq1 $ are explored, respectively. Moreover, we consider the specific case where $ \rho(t)\equiv1 $ to better understand the impact of the periodic evolution rate on the persistence and extinction of Aedes aegypti mosquito. Numerical simulations further verify our analytical results that the periodic domain evolution has a significant impact on the dispersal of Aedes aegypti mosquito.

Keywords:

Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 92D25.

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Figure 1. $ \rho(t) = 1 $. For fixed domain, we have $ R_0(1.1)<1 $. Graphs $ (a)-(c) $ show that $ M $ tends to 0

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Figure 2. $ \rho(t) = e^{0.1(1-\cos(4t))} $. For big evolution rate $ \rho(t) $ with $ \overline{\rho^{-2}}<1 $, we have $ R_0(\rho)>1 $. Graph $ (a) $ shows that $ M $ stabilizes to a positive periodic steady state. Graphs $ (b) $ and $ (c) $, which are the cross-sectional view and contour map respectively, present the periodic evolution of the domain

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Figure 3. $ \rho(t) = 1 $. For fixed domain, we have $ R_0(1.1)>1 $. Graphs $ (a)-(c) $ show that $ M $ tends to a positive steady state

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Figure 4. $ \rho(t) = e^{0.1(\cos(4t)-1)} $. For small evolution rate $ \rho(t) $ with $ \overline{\rho^{-2}}>1 $, we couldn't figure out the value of $ R_0(\rho) $. But we can see from graphs $ (a)-(c) $ that mosquitoes become extinct eventually

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