Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti

Kai Wang; Hongyong Zhao; Hao Wang

doi:10.3934/dcdsb.2022122

Article Contents

2023, Volume 28, Issue 2: 1279-1299. Doi: 10.3934/dcdsb.2022122

This issue Previous Article Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation Next Article Periodic solutions in distribution of stochastic lattice differential equations

Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China
3.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

^* Corresponding author
^* Corresponding author

Received: December 31, 2021

Revised: March 31, 2022

Early access: June 2022

Published: February 2023

This work is supported by Natural Science Foundation of China (No. 11971013), the NSERC Individual Discovery Grant RGPIN-2020-03911 and NSERC Accelerator Grant RGPAS-2020-00090, the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0169) and Nanjing University of Aeronautics and Astronautics PhD short-term visiting scholar project (No. ZDGB2021026) at the University of Alberta

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Abstract

This paper is devoted to investigate the existence of traveling wave solutions for a partially degenerate Aedes aegypti model with nonlocal effects. By taking specific kernel forms and time scale transformation, we transform the nonlocal model into a singularly perturbed system with small parameters. A locally invariant manifold for wave profile system is obtained with the aid of the geometric singular perturbation theory, and then the existence of traveling wave solutions is proved provided that the basic reproduction number $ \mathcal{R}_0>1 $ through utilizing the Fredholm orthogonal method. Furthermore, we study the asymptotic behaviors of traveling wave solution with the help of asymptotic theory. The methods used in this work can help us overcome the difficulty that the solution map associated with the system is not compact. Numerically, we perform simulations to demonstrate the theoretical results.

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Mathematics Subject Classification: Primary: 35B40, 35C07, 35K57.

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Figure 1. The sketch of the distribution of the roots of $ \Sigma_1(\zeta) $

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Figure 2. Graphs of $ u_1(t, x) $ and $ u_2(t, x) $ with respect to the spatial variable $ x $ for fixed time: $ t = 3, 5, 20, 30, 35, 40, 50 $

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Figure 3. The dependence of the $ c_2^\ast $ on parameters

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Figure 4. Sensitivity of $ \mathcal{R}_0 $ to parameters of model (3)

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