Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes

Shangbing Ai; Jia Li; Jianshe Yu; Bo Zheng

doi:10.3934/dcdsb.2021172

Article Contents

2022, Volume 27, Issue 6: 3039-3052. Doi: 10.3934/dcdsb.2021172

This issue Previous Article Hartman and Nirenberg type results for systems of delay differential equations under

$ (\omega,Q) $

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Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes

1.
Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL 35899, USA
2.
Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China

^* Corresponding author: Jianshe Yu
^* Corresponding author: Jianshe Yu

Received: February 2021

Revised: May 2021

Published: June 2022

This work is supported in part by National Natural Science Foundation of China (12071095, 11971127, 11631005)

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Abstract

A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is derived where the wild mosquito population is composed of larvae and adult classes and only sexually active sterile mosquitoes are included as a function given in advance. The strategy of constant releases of sterile mosquitoes is considered but periodic and impulsive releases are more focused on. Local stability of the origin and the existence of a positive periodic solution are investigated. While mathematical analysis is more challenging, numerical examples demonstrate that the model dynamics, determined by thresholds of the release amount and the release waiting period, essentially match the dynamics of the alike one-dimensional models. It is also shown that richer dynamics are exhibited from the two-dimensional stage-structured model.

Keywords:

Mathematics Subject Classification: Primary: 34C25, 34D05, 92D25; Secondary: 92D30, 92D40.

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Figure 1. Schematic illustration of the phase-plane analysis in Theorem 3.3

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Figure 2. Schematic illustration of the Poincare map in Theorem 3.3

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Figure 3. Schematic illustration of the phase-plane analysis in Theorem 3.4

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Figure 4. Parameters are given in (23) and the release amount threshold $ c^* = 4.35979 $. With $ c = 3 < c^* $, the threshold waiting period threshold $ T^* $ seems between 6 and 7. When $ T = 6 $, the origin is locally, not globally, asymptotically stable and there exists a locally asymptotically stable positive periodic solution as shown in the left figure. When $ T = 7 $, there exists a globally asymptotically stable positive periodic solution as shown in the right figure. Here only the solution curves for the larvae are presented

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Figure 5. Parameters are still given in (23) with the same release amount threshold $ c^* = 4.35979 $, but now $ c = 6 > c^* $. When $ T = 5.5 $, the origin is globally asymptotically stable as shown in the left figure, and when $ T = 7 $ there exists a globally asymptotically stable positive periodic solution as shown in the right figure. Here again only the solution curves for the larvae are presented

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Figure 6. With the same parameters in (23), and also the same $ c = 6 > c^* = 4.35979 $ when $ T = 6 $, the origin is only locally asymptotically stable and there exists a locally asymptotically stable positive periodic solution

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