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December  2020, 25(12): 4659-4676. doi: 10.3934/dcdsb.2020118

A delayed differential equation model for mosquito population suppression with sterile mosquitoes

Yuanxian Hui 1,2, , Genghong Lin 1, , Jianshe Yu 1,, and  Jia Li 3,

1. 

Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
2. 

School of Mathematics and Statistics, Pu'er University, Pu'er 665000, China
3. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA

* Corresponding author: Jianshe Yu

Received  October 2019 Revised  December 2019 Published  December 2020 Early access  March 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11631005), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT 16R16), the China Postdoctoral Science Foundation (No.2019M660196), and the Guangzhou Postdoctoral International Training Program Funding Project

The technique of sterile mosquitoes plays an important role in the control of mosquito-borne diseases such as malaria, dengue, yellow fever, west Nile, and Zika. To explore the interactive dynamics between the wild and sterile mosquitoes, we formulate a delayed mosquito population suppression model with constant releases of sterile mosquitoes. Through the analysis of global dynamics of solutions of the model, we determine a threshold value of the release rate such that if the release threshold is exceeded, then the wild mosquito population will be eventually suppressed, whereas when the release rate is less than the threshold, the wild and sterile mosquitoes coexist and the model exhibits a complicated feature. We also obtain theoretical results including a sufficient and necessary condition for the global asymptotic stability of the zero solution. We provide numerical examples to demonstrate our results and give brief discussions about our findings.

Keywords: Global asymptotic stability, mosquito population suppression, sterile mosquitoes, delay differential equation, Hopf bifurcation.
Mathematics Subject Classification: Primary: 34K20, 34K60; Secondary: 92D25.
Citation: Yuanxian Hui, Genghong Lin, Jianshe Yu, Jia Li. A delayed differential equation model for mosquito population suppression with sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4659-4676. doi: 10.3934/dcdsb.2020118
References:
[1]

L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson, Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Diseases, 10 (2010), 295-311.  doi: 10.1089/vbz.2009.0014.  Google Scholar

[2]

R. Anguelov, Y. Dumont and J. Lubuma, Mathematical modeling of sterile insect technology for control of anopheles mosquito, Comput. Math. Appl., 64 (2012), 374-389.  doi: 10.1016/j.camwa.2012.02.068.  Google Scholar

[3]

J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109-117.  doi: 10.1016/0025-5564(75)90028-0.  Google Scholar

[4]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 2$^nd$ edition, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[6]

L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM J. Appl. Math., 74 (2014), 1786-1809.  doi: 10.1137/13094102X.  Google Scholar

[7]

L. Cai, J. Huang, X. Song and Y. Zhang, Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes, Discrete Contin. Dyn. Syst. Ser. B., 24 (2019), 6279-6295.  doi: 10.3934/dcdsb.2019139.  Google Scholar

[8]

N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar

[9]

H. Diaz, A. A. Ramirez, A. Olarte and C. Clavijo, A model for the control of malaria using genetically modified vectors, J. Theoret. Biol., 276 (2011), 57-66.  doi: 10.1016/j.jtbi.2011.01.053.  Google Scholar

[10]

V. A. Dyck, J. Hendrichs and A. S. Robinson (eds.), Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management, Springer, Dordrecht, 2005. Google Scholar

[11]

K. R. Fister, M. L. Mccarthy, S. F. Oppenheimer and C. Collins, Optimal control of insects through sterile insect release and habitat modification, Math. Biosci., 244 (2013), 201-212.  doi: 10.1016/j.mbs.2013.05.008.  Google Scholar

[12]

J. Hale, Theory of Functional Differential Equations, 2$^nd$ edition, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[13]

L. Hu, M. Tang, Z. Wu, Z. Xi and J. Yu, The threshold infection level for Wolbachia invasion in random environments, J. Differential Equations, 266 (2019), 4377-4393.  doi: 10.1016/j.jde.2018.09.035.  Google Scholar

[14]

M. Huang, J. Luo, L. Hu, B. Zheng and J. Yu, Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations, J. Theoret. Biol., 440 (2018), 1-11.  doi: 10.1016/j.jtbi.2017.12.012.  Google Scholar

[15]

M. Huang, M. Tang and J. Yu, Wolbachia infection dynamics by reaction-diffusion equations, Sci. China Math., 58 (2015), 77-96.  doi: 10.1007/s11425-014-4934-8.  Google Scholar

[16]

M. Huang, M. Tang, J. Yu and B. Zheng, The impact of mating competitiveness and incomplete cytoplasmic incompatibility on Wolbachia-driven mosquito population suppression, Math. Biosci. Eng., 16 (2019), 4741-4757.  doi: 10.3934/mbe.2019238.  Google Scholar

[17]

M. Huang, M. Tang, J. Yu and B. Zheng, A stage structured model of delay differential equations for Aedes mosquito population suppression, Dicrete Contin. Dyn. Syst. doi: 10.3934/dcds.2020042.  Google Scholar

[18]

M. Huang, J. Yu, L. Hu and B. Zheng, Qualitative analysis for a Wolbachia infection model with diffusion, Sci. China Math., 59 (2016), 1249-1266.  doi: 10.1007/s11425-016-5149-y.  Google Scholar

[19]

Y. Hui, G. Lin and Q. Sun, Oscillation threshold for a mosquito population suppression model with time delay, Math. Biosci. Eng., 16 (2019), 7362-7374.  doi: 10.3934/mbe.2019367.  Google Scholar

[20]

G. E. Hutchinson, Circular causal systems in ecology, Ann. NY. Acad. Sci., 50 (1948), 221-246.  doi: 10.1111/j.1749-6632.1948.tb39854.x.  Google Scholar

[21] G. E. Hutchinson, An Introduction to Population Ecology, Yale University Press, New Haven, Conn., 1978.   Google Scholar
[22]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in Dynamics Reported: Expositions in Dynamical Systems, Vol. 1, Springer, Berlin, 1992,164–224. doi: 10.1007/978-3-642-61243-5.  Google Scholar

[23]

J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyn., 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613.  Google Scholar

[24]

J. Li, M. Han and J. Yu, Simple paratransgenic mosquitoes models and their dynamics, Math. Biosci., 306 (2018), 20-31.  doi: 10.1016/j.mbs.2018.10.005.  Google Scholar

[25]

Y. Li, F. Kamara, G. Zhou, S. Puthiyakunnon, C. Li, Y. Liu and et al., Urbanization increases Aedes albopictus larval habitats and accelerates mosquito development and survivorship, PLoS Negl. Trop. Dis., 8 (2014), E3301. doi: 10.1371/journal.pntd.0003301.  Google Scholar

[26]

F. Liu, C. Yao, P. Lin and C. Zhou, Studies on life table of the nature population of Aedes albopictus, Acta Sci. Natur. Univ. Sunyatseni., 31 (1992), 84-93.   Google Scholar

[27]

Z.-W Liu, Y.-Y Zhang and Y.-Z Yang, Population dynamics of Aedes (stegomyia) albopictus (Skuse) under laboratory conditions, Acta Entomol. Sin., 28 (1985), 274-280.   Google Scholar

[28]

E. Liz, Delayed logistic population models revisited, Publ. Mat., Vol. EXTRA (2014), 309–331.  Google Scholar

[29]

A. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.  doi: 10.1071/ZO9540009.  Google Scholar

[30]

R. S. Patterson, D. E. Weidhaass, H. R. Ford and C. S. Lofgren, Suppression and elimination of an island population of Culex pipiens quinquefasciatus with sterile males, Science, 168 (1970), 1368-1369.  doi: 10.1126/science.168.3937.1368.  Google Scholar

[31]

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.  Google Scholar

[32]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.   Google Scholar

[33]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[34]

J. Yu, Modelling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917.  Google Scholar

[35]

J. Yu and J. Li, Dynamics of interactive wild and sterile mosquitoes with time delay, J. Biol. Dyn., 13 (2019), 606-620.  doi: 10.1080/17513758.2019.1682201.  Google Scholar

[36]

J. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, J. Difference Equ. Appl., 25 (2019), 1549-1567.  doi: 10.1080/10236198.2019.1669578.  Google Scholar

[37]

D. Zhang, X. Zheng, Z. Xi, K. Bourtzis and J. R. L. Gilles, Combining the sterile insect technique with the incompatible insect technique: I-impact of Wolbachia infection on the fitness of triple- and double-infected strains of Aedes albopictus, PLoS One, 10 (2015), 1-13.  doi: 10.1371/journal.pone.0121126.  Google Scholar

[38]

B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743-770.  doi: 10.1137/13093354X.  Google Scholar

[39]

B. Zheng, M. Tang, J. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5.  Google Scholar

show all references

References:
[1]

L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson, Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Diseases, 10 (2010), 295-311.  doi: 10.1089/vbz.2009.0014.  Google Scholar

[2]

R. Anguelov, Y. Dumont and J. Lubuma, Mathematical modeling of sterile insect technology for control of anopheles mosquito, Comput. Math. Appl., 64 (2012), 374-389.  doi: 10.1016/j.camwa.2012.02.068.  Google Scholar

[3]

J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109-117.  doi: 10.1016/0025-5564(75)90028-0.  Google Scholar

[4]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 2$^nd$ edition, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[6]

L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM J. Appl. Math., 74 (2014), 1786-1809.  doi: 10.1137/13094102X.  Google Scholar

[7]

L. Cai, J. Huang, X. Song and Y. Zhang, Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes, Discrete Contin. Dyn. Syst. Ser. B., 24 (2019), 6279-6295.  doi: 10.3934/dcdsb.2019139.  Google Scholar

[8]

N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar

[9]

H. Diaz, A. A. Ramirez, A. Olarte and C. Clavijo, A model for the control of malaria using genetically modified vectors, J. Theoret. Biol., 276 (2011), 57-66.  doi: 10.1016/j.jtbi.2011.01.053.  Google Scholar

[10]

V. A. Dyck, J. Hendrichs and A. S. Robinson (eds.), Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management, Springer, Dordrecht, 2005. Google Scholar

[11]

K. R. Fister, M. L. Mccarthy, S. F. Oppenheimer and C. Collins, Optimal control of insects through sterile insect release and habitat modification, Math. Biosci., 244 (2013), 201-212.  doi: 10.1016/j.mbs.2013.05.008.  Google Scholar

[12]

J. Hale, Theory of Functional Differential Equations, 2$^nd$ edition, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[13]

L. Hu, M. Tang, Z. Wu, Z. Xi and J. Yu, The threshold infection level for Wolbachia invasion in random environments, J. Differential Equations, 266 (2019), 4377-4393.  doi: 10.1016/j.jde.2018.09.035.  Google Scholar

[14]

M. Huang, J. Luo, L. Hu, B. Zheng and J. Yu, Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations, J. Theoret. Biol., 440 (2018), 1-11.  doi: 10.1016/j.jtbi.2017.12.012.  Google Scholar

[15]

M. Huang, M. Tang and J. Yu, Wolbachia infection dynamics by reaction-diffusion equations, Sci. China Math., 58 (2015), 77-96.  doi: 10.1007/s11425-014-4934-8.  Google Scholar

[16]

M. Huang, M. Tang, J. Yu and B. Zheng, The impact of mating competitiveness and incomplete cytoplasmic incompatibility on Wolbachia-driven mosquito population suppression, Math. Biosci. Eng., 16 (2019), 4741-4757.  doi: 10.3934/mbe.2019238.  Google Scholar

[17]

M. Huang, M. Tang, J. Yu and B. Zheng, A stage structured model of delay differential equations for Aedes mosquito population suppression, Dicrete Contin. Dyn. Syst. doi: 10.3934/dcds.2020042.  Google Scholar

[18]

M. Huang, J. Yu, L. Hu and B. Zheng, Qualitative analysis for a Wolbachia infection model with diffusion, Sci. China Math., 59 (2016), 1249-1266.  doi: 10.1007/s11425-016-5149-y.  Google Scholar

[19]

Y. Hui, G. Lin and Q. Sun, Oscillation threshold for a mosquito population suppression model with time delay, Math. Biosci. Eng., 16 (2019), 7362-7374.  doi: 10.3934/mbe.2019367.  Google Scholar

[20]

G. E. Hutchinson, Circular causal systems in ecology, Ann. NY. Acad. Sci., 50 (1948), 221-246.  doi: 10.1111/j.1749-6632.1948.tb39854.x.  Google Scholar

[21] G. E. Hutchinson, An Introduction to Population Ecology, Yale University Press, New Haven, Conn., 1978.   Google Scholar
[22]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in Dynamics Reported: Expositions in Dynamical Systems, Vol. 1, Springer, Berlin, 1992,164–224. doi: 10.1007/978-3-642-61243-5.  Google Scholar

[23]

J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyn., 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613.  Google Scholar

[24]

J. Li, M. Han and J. Yu, Simple paratransgenic mosquitoes models and their dynamics, Math. Biosci., 306 (2018), 20-31.  doi: 10.1016/j.mbs.2018.10.005.  Google Scholar

[25]

Y. Li, F. Kamara, G. Zhou, S. Puthiyakunnon, C. Li, Y. Liu and et al., Urbanization increases Aedes albopictus larval habitats and accelerates mosquito development and survivorship, PLoS Negl. Trop. Dis., 8 (2014), E3301. doi: 10.1371/journal.pntd.0003301.  Google Scholar

[26]

F. Liu, C. Yao, P. Lin and C. Zhou, Studies on life table of the nature population of Aedes albopictus, Acta Sci. Natur. Univ. Sunyatseni., 31 (1992), 84-93.   Google Scholar

[27]

Z.-W Liu, Y.-Y Zhang and Y.-Z Yang, Population dynamics of Aedes (stegomyia) albopictus (Skuse) under laboratory conditions, Acta Entomol. Sin., 28 (1985), 274-280.   Google Scholar

[28]

E. Liz, Delayed logistic population models revisited, Publ. Mat., Vol. EXTRA (2014), 309–331.  Google Scholar

[29]

A. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.  doi: 10.1071/ZO9540009.  Google Scholar

[30]

R. S. Patterson, D. E. Weidhaass, H. R. Ford and C. S. Lofgren, Suppression and elimination of an island population of Culex pipiens quinquefasciatus with sterile males, Science, 168 (1970), 1368-1369.  doi: 10.1126/science.168.3937.1368.  Google Scholar

[31]

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.  Google Scholar

[32]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.   Google Scholar

[33]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[34]

J. Yu, Modelling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917.  Google Scholar

[35]

J. Yu and J. Li, Dynamics of interactive wild and sterile mosquitoes with time delay, J. Biol. Dyn., 13 (2019), 606-620.  doi: 10.1080/17513758.2019.1682201.  Google Scholar

[36]

J. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, J. Difference Equ. Appl., 25 (2019), 1549-1567.  doi: 10.1080/10236198.2019.1669578.  Google Scholar

[37]

D. Zhang, X. Zheng, Z. Xi, K. Bourtzis and J. R. L. Gilles, Combining the sterile insect technique with the incompatible insect technique: I-impact of Wolbachia infection on the fitness of triple- and double-infected strains of Aedes albopictus, PLoS One, 10 (2015), 1-13.  doi: 10.1371/journal.pone.0121126.  Google Scholar

[38]

B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743-770.  doi: 10.1137/13093354X.  Google Scholar

[39]

B. Zheng, M. Tang, J. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.  doi: 10.1007/s00285-017-1142-5.  Google Scholar

FIGURE 1 (A) is the graph of stability switch in terms of time delays for model (6) for $\tau\in(0, \overline{\tau})$, such that equilibrium $S^{(1)}_\nu$ is asymptotically stable for $\tau\in (0, \tau_1)\cup(\tau_4, \overline{\tau})$, and unstable for $\tau\in(\tau_1, \tau_4)$. Given that $\hat{\tau}\approx 25.2573 > 2\sqrt{3}\pi/(9\mu_1)\approx 20.1533$, we find a bifurcation at $\tau^{**} = \tau_4\approx 25.9708\in (\hat{\tau}, \overline{\tau})$ such that $S^{(1)}_\nu$ is asymptotically stable for $\tau\in (\hat{\tau}, \tau^{**})$, and unstable for $\tau\in (r_0^{**}, \overline{\tau})$, as shown in FIGURE 1 (B)">Figure 1.  Suppose $t_0 = 0$ and model parameters of (6) given in (15). FIGURE 1 (A) is the graph of stability switch in terms of time delays for model (6) for $\tau\in(0, \overline{\tau})$, such that equilibrium $S^{(1)}_\nu$ is asymptotically stable for $\tau\in (0, \tau_1)\cup(\tau_4, \overline{\tau})$, and unstable for $\tau\in(\tau_1, \tau_4)$. Given that $\hat{\tau}\approx 25.2573 > 2\sqrt{3}\pi/(9\mu_1)\approx 20.1533$, we find a bifurcation at $\tau^{**} = \tau_4\approx 25.9708\in (\hat{\tau}, \overline{\tau})$ such that $S^{(1)}_\nu$ is asymptotically stable for $\tau\in (\hat{\tau}, \tau^{**})$, and unstable for $\tau\in (r_0^{**}, \overline{\tau})$, as shown in FIGURE 1 (B)
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FIGURE 2 (A). When $\tau\in(\tau^{**}, \overline{\tau})$, equilibrium $S^{(1)}_\nu$ is asymptotically stable as shown in FIGURE 2 (B)">Figure 2.  Stability of the positive equilibrium $S^{(1)}_\nu$ in model (6) when $\hat{\tau} < \tau < \overline{\tau}$. Suppose $t_0 = 0$ and model parameters are given in (15). Then $\hat{\tau}\approx 25.2573$, and $\overline{\tau}\approx 28.1341$. We find a bifurcation point $\tau^{**}\approx 25.9708\in (\hat{\tau}, \overline{\tau})$. When $\tau\in(\hat{\tau}, \tau^{**})$, equilibrium $S^{(1)}_\nu$ is unstable as shown in FIGURE 2 (A). When $\tau\in(\tau^{**}, \overline{\tau})$, equilibrium $S^{(1)}_\nu$ is asymptotically stable as shown in FIGURE 2 (B)
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FIGURE 3 (A) is the phase diagram of system (3), and FIGURE 3 (B) shows the dynamics of the wild mosquito population of system (3) where solutions approach either the locally asymptotically stable zero solution or $S_v^+$ depending on their initial values">Figure 3.  Simulations for the case of $0 < b < b^*$. Parameters are given in (28) such that the release threshold $b^*\approx 28.81$ and we set $b = 20$. There exist two positive equilibria $S_v^- = 65.5632$ and $S_v^{+} = 227.77012$ where $S_v^{-}$ is unstable and $S_v^{+}$ is locally asymptotically stable. System (3) exhibits a bi-stability phenomenon followed from Theorem 4.4. FIGURE 3 (A) is the phase diagram of system (3), and FIGURE 3 (B) shows the dynamics of the wild mosquito population of system (3) where solutions approach either the locally asymptotically stable zero solution or $S_v^+$ depending on their initial values
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FIGURE 4 (A) is the phase diagram of system (3), and FIGURE 4 (B) illustrates the dynamics of the wild mosquito population of system (3) where all solutions eventually approach zero">Figure 4.  Simulations for the case of $b > b^*$. Parameters are given in (28) such that the release threshold $b^*\approx 28.81$ and we set $b = 35$ greater than $b^*$. Then the zero equilibrium $N_0$ is globally asymptotically stable, which indicates that the wild mosquito population is eventually eradicated. FIGURE 4 (A) is the phase diagram of system (3), and FIGURE 4 (B) illustrates the dynamics of the wild mosquito population of system (3) where all solutions eventually approach zero
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Figure 5.  Comparisons of the effects of $b$ on the suppression efficiency when $b > b^*$. Here parameters are given in (28). We choose four different $b$: 50, 70, 90,130, all of which are larger than the threshold release rate $b^*\approx 28.81$. It is clear that the efficiency is improved as $b$ increased
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Table 1.  Parameter values for Aedes albopictus and sterile mosquitoes taken from earlier measurements in Guangzhou
Para. Definition Range Reference
$ a $ Number of offsprings produced per [0.9043, 6.4594] [27,37]
individual, per unit of time
$ \tau $ Average maturation period of wild [22.6, 54.6] [25,26]
mosquitoes (day)
$ e^{-\mu_0\tau} $ Survival rate of the immature 0.05 [26]
mosquitoes $ (\text{day}^{-1}) $
$ \mu_1 $ Death rate of wild mosquitoes $ (\text{day}^{-1}) $ [0.0198, 0.1368] [27]
$ \mu_2 $ Death rate of sterile mosquitoes $ (\text{day}^{-1}) $ 1/7 [2,8]
$ \xi_\nu $ Carrying capacity parameter of wild 0.0025 Given
mosquitoes
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Para. Definition Range Reference
$ a $ Number of offsprings produced per [0.9043, 6.4594] [27,37]
individual, per unit of time
$ \tau $ Average maturation period of wild [22.6, 54.6] [25,26]
mosquitoes (day)
$ e^{-\mu_0\tau} $ Survival rate of the immature 0.05 [26]
mosquitoes $ (\text{day}^{-1}) $
$ \mu_1 $ Death rate of wild mosquitoes $ (\text{day}^{-1}) $ [0.0198, 0.1368] [27]
$ \mu_2 $ Death rate of sterile mosquitoes $ (\text{day}^{-1}) $ 1/7 [2,8]
$ \xi_\nu $ Carrying capacity parameter of wild 0.0025 Given
mosquitoes
[1]

Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139
[2]

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