Existence and stability of periodic solutions for a mosquito suppression model with incomplete cytoplasmic incompatibility

Rong Yan; Bo Zheng; Jianshe Yu

doi:10.3934/dcdsb.2022208

Article Contents

2023, Volume 28, Issue 5: 3172-3192. Doi: 10.3934/dcdsb.2022208

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Existence and stability of periodic solutions for a mosquito suppression model with incomplete cytoplasmic incompatibility

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou Center for Applied Mathematics, Guangzhou, 510006, China

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

Received: August 1, 2022

Revised: October 1, 2022

Early access: October 28, 2022

Published online: May 1, 2023

This work is supported by National Natural Science Foundation of China (12071095, 11971127), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16).

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Abstract

Cytoplasmic incompatibility (CI) caused by endosymbiotic bacteria Wolbachia has been an effective way to suppress and even eradicate wild vector mosquitoes. By assuming that CI is driven by impulsive and periodic releases of Wolbachia-infected males, we develop a time-switching ordinary differential equation model to characterize the impact of the release amount, the release period and the incomplete CI on the suppression efficiency. Sufficient conditions to guarantee the existence and uniqueness, or the existence of exactly two periodic solutions are obtained, together with their stability analyses.

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Mathematics Subject Classification: 34C25, 34D20, 34D23, 92D25, 93D20.

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Figure 1. Division of the parameter space $ \{(s_h, c):s_h\in(0, 1), \; c\in(0, \frac{1}{\xi})\} $ according to the number of positive equilibria $ \mathcal{N} $ in model (2). There are no positive equilibria in $ \Lambda_0 = \Lambda_{01}\cup\Lambda_{02}, $ where $ \Lambda_{01} = \{(s_h, c):s_h\in(0, s_h^*], \; c\in[c_0, \frac{1}{\xi})\} \; \mathit{\mbox{and}}\; \Lambda_{02} = \{(s_h, c):s_h\in(s_h^*, 1), \; c\in(c_1, \frac{1}{\xi})\} $. Model (2) has a unique positive equilibrium on $ \Lambda_1 = \Lambda_{11}\cup\Lambda_{12} \cup\Lambda_{13}\cup\Lambda_{14} $, where $ \Lambda_{11} = \{(s_h, c):s_h\in(0, s_h^*], \; c\in(0, c_0)\} , \; \Lambda_{12} = \{(s_h, c):s_h\in(s_h^*, s_h^{**}), \; c\in(0, c_0)\} , \; \Lambda_{13} = \{(s_h, c):s_h\in(s_h^*, s_h^{**}), \; c = c_0\} \; \mathit{\mbox{and}}\; \Lambda_{14} = \{(s_h, c):s_h\in(s_h^*, 1), \; c = c_1\} $. When $ (s_h, c)\in\Lambda_2 = \Lambda_{21}\cup\Lambda_{22} $ with $ \Lambda_{21} = \{(s_h, c):s_h\in(s_h^*, s_h^{**}), \; c\in(c_0, c_1)\} \; \mathit{\mbox{and}}\; \Lambda_{22} = \{(s_h, c):s_h\in[s_h^{**}, 1), \; c\in(0, c_1)\} $, model (2) has two positive equilibria

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Figure 2. Solutions of model (5)-(6) with $ (s_h, c)\in\Lambda_{11} $ in panel (a), $ (s_h, c)\in\Lambda_{12} $ in panel (b) and $ (s_h, c)\in\Lambda_{13} $ in panel (c)

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Figure 3. Solutions of model (5)-(6) with $ (s_h, c)\in\Lambda_{21} $ in panel (a) and (d), $ (s_h, c)\in\Lambda_{22} $ in panel (b) and (e), and $ (s_h, c)\in\Lambda_{14} $ in panel (c) and (f)

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Figure 4. Schematic diagram for the dynamic behaviors of wild mosquitoes under different release strategies infected with Wolbachia-infected mosquitoes. The following abbreviations are used. GAS: globally asymptotically stable, LAS: locally asymptotically stable, US: unstable, and $ T $-ps: $ T $-periodic solution

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Figure 5. Comparison of mosquito suppression effects in $ \Lambda_{01} $ and $ \Lambda_{02}^{(1)} $

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Figure 6. We plot the magnitude of $ u^- $ against $ c $ in blue circles, and the suppression efficiency $ R $ against $ c $ in red stars. Panel (a) is for $ (s_h, c)\in\Lambda_{02}^{(2)} $ and panel (b) is for $ (s_h, c)\in\Lambda_{14}\cup\Lambda_{2} $

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Figure 7. The initial value $ u^* $ against $ c $ in (a), and against $ T $ in (b)

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Table 1. Parameter combinations of $ (s_h, c, T) $ meet conditions of Theorem 1.3(1) and (2)

$ s_h $	0.95	0.98	0.90
$ c_0 $	500	-250	750
$ c_1 $	746.0760	723.2370	787.5247
$ c $	720	720	787.5247
$ (s_h, c) $	$ \Lambda_{21} $	$ \Lambda_{22} $	$ \Lambda_{14} $
$ T^* $	14.1579	14.2786	$ 14.0539 $
$ T $ in Theorem 1.3(1)	14.2	14.3	14.3
$ T $ in Theorem 1.3(2)	14.1	14.1	14.01

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