# American Institute of Mathematical Sciences

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

 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  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.

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, 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
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)
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)
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
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
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
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
 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] Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042 [3] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [4] Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727 [5] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [6] Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689 [7] Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395 [8] Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451 [9] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [10] Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541 [11] Hongyan Yin, Cuihong Yang, Xin'an Zhang, Jia Li. The impact of releasing sterile mosquitoes on malaria transmission. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3837-3853. doi: 10.3934/dcdsb.2018113 [12] Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268 [13] Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 [14] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [15] Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259 [16] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [17] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [18] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [19] Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 [20] Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (20)
• HTML views (44)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]