# American Institute of Mathematical Sciences

November  2019, 24(11): 6279-6295. doi: 10.3934/dcdsb.2019139

## Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes

 1 School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000 2 China, College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China 3 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

* Corresponding author: Yuyue Zhang and Jicai Huang. The second author is supported by NSFC (No.11471133, No.11871235)

Received  August 2018 Revised  January 2019 Published  November 2019 Early access  July 2019

Fund Project: The first author is supported by NSFC (No.11871415, No.11371305, No.11671346) and Nanhu Scholars Program for Young Scholars XYNU.

To reduce or eradicate mosquito-borne diseases, one of effective methods is to control the wild mosquito populations by using the sterile insect technique. Dynamical models with different releasing strategies of sterile mosquitoes have been proposed and investigated in the recent work by Cai et al. [SIAM. J. Appl. Math. 75(2014)], where some basic analysis on the dynamics are given and some complicated dynamical behaviors are found by numerical simulations. While their findings seem exciting and promising, yet the models could exhibit much more complex dynamics than it has been observed. In this paper, to further study the impact of the sterile insect technique on controlling the wild mosquito populations, we systematically study bifurcations and dynamics of the model with a proportional release rate of sterile mosquitoes by bifurcation method. We show that the model undergoes saddle-node bifurcation, subcritical and supercritical Hopf bifurcations, and Bogdanov-Takens bifurcation as the values of parameters vary. Some numerical simulations, including the bifurcation diagram and phase portraits, are also presented to illustrate the theoretical conclusions. These rich and complicated bifurcation phenomena can be regarded as a complement to the work by Cai et al. [SIAM. J. Appl. Math. 75(2014)].

Citation: Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139
##### References:
 [1] P. L. Alonso, G. Brown, M. Arevalo-Herrera, F. Binka, C. Chitnis and F. Collins, et al., A research agenda to underpin malaria eradication, PLoS Med., 8 (2011), e1000406. doi: 10.1371/journal.pmed.1000406. [2] H. J. Barclay, Pest population stability under sterile releases, Res. Popul. Ecol., 24 (1982), 405-416.  doi: 10.1007/BF02515585. [3] K. W. Blayneh and J. Mohammed-Awel, Insecticide-resistant mosquitoes and malaria control, Math. Biosc., 252 (2014), 14-26.  doi: 10.1016/j.mbs.2014.03.007. [4] L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215.  doi: 10.1007/s00285-012-0546-5. [5] 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. [6] Centers for Disease Control and Prevention (CDC), Malaria's impact worldwide., Available from: https://www.cdc.gov/malaria/malaria_worldwide/impact.html. [7] S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Reprint of the 1994 original. Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511665639. [8] F. Dumortier, R. Roussarie, J. Sotomayor and H. Żoładek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math., 1480, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353. [9] Y. Dumont and J. M. Tchuenche, Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), 809-854.  doi: 10.1007/s00285-011-0477-6. [10] L. Esteva and H. Mo Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132-147.  doi: 10.1016/j.mbs.2005.06.004. [11] J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24pp. doi: 10.1142/S0218127413501642. [12] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101. [13] J. Huang, S. Ruan and J. Song, Bifurcaitons in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Diff. Equat., 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024. [14] J. Ito, A. Ghosh, L. A. Moreira, E. A. Wimmer and M. Jacobs-Lorena, Transgenic anopheline mosquitoes impaired in transmission of a malaria parasite, Nature, 417 (2002), 452-455.  doi: 10.1038/417452a. [15] W. Jiang, X. Li and X. Zou, On a reaction-diffusion model for sterile insect release method on a bounded domain, Internat. J. Biomath., 7 (2014), 1450030, 17pp. doi: 10.1142/S1793524514500302. [16] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Economic Entomolgy, 48 (1955), 459-462. [17] Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer-Verlag, New York Heidelberg, 1998. [18] R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis, Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Diff. Equat., 245 (2008), 442-467.  doi: 10.1016/j.jde.2007.10.034. [19] J. Li, Simple mathematical models for interacting wild and transgenic mosquito populations, Math. Biosci., 189 (2004), 39-59.  doi: 10.1016/j.mbs.2004.01.001. [20] J. Li, Discrete-time models with mosquitoes carrying genetically-modified bacteria, Math. Biosci., 240 (2012), 35-44.  doi: 10.1016/j.mbs.2012.05.012. [21] X. Li, J. Ren, S. A. Campbell, G. S. K. Wolkowicz and H. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 785-807.  doi: 10.3934/dcdsb.2018043. [22] L. Perko, Differential Equations and Dynamical Systems, 3rd edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8. [23] J. Ren and L. Yu, Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model, Journal of Nonlinear Science, 26 (2016), 1895-1931.  doi: 10.1007/s00332-016-9323-8. [24] J. Ren, L. Yu and S. Siegmund, Bifurcations and chaos in a discrete predator-prey model with Crowley-Martin functional response, Nonlinear Dynamics, 90 (2017), 19-41.  doi: 10.1007/s11071-017-3643-6. [25] S. Ruan and W. Wang, Dynamical behavior of an epidmeic model with a nonlinear incidence rate, J. Diff. Equat., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [26] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896. [27] S. M. White, P. Rohani and S. M. Sait, Modelling pulsed releases for sterile insect techniques: fitness costs of sterile and transgenic males and the effects on mosquito dynamics, J. Appl. Ecology, 47 (2010), 1329-1339.  doi: 10.1111/j.1365-2664.2010.01880.x. [28] World Health Organization (WHO), 10 facts on malaria, December, 2016. Available from: http://www.who.int/features/factfiles/malaria/en/. [29] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. [30] Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equation, Transl. Math. Monogr., 101, Amer. Math. Soc., Providence, RI, 1992. [31] 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.

show all references

##### References:
 [1] P. L. Alonso, G. Brown, M. Arevalo-Herrera, F. Binka, C. Chitnis and F. Collins, et al., A research agenda to underpin malaria eradication, PLoS Med., 8 (2011), e1000406. doi: 10.1371/journal.pmed.1000406. [2] H. J. Barclay, Pest population stability under sterile releases, Res. Popul. Ecol., 24 (1982), 405-416.  doi: 10.1007/BF02515585. [3] K. W. Blayneh and J. Mohammed-Awel, Insecticide-resistant mosquitoes and malaria control, Math. Biosc., 252 (2014), 14-26.  doi: 10.1016/j.mbs.2014.03.007. [4] L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215.  doi: 10.1007/s00285-012-0546-5. [5] 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. [6] Centers for Disease Control and Prevention (CDC), Malaria's impact worldwide., Available from: https://www.cdc.gov/malaria/malaria_worldwide/impact.html. [7] S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Reprint of the 1994 original. Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511665639. [8] F. Dumortier, R. Roussarie, J. Sotomayor and H. Żoładek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math., 1480, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353. [9] Y. Dumont and J. M. Tchuenche, Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), 809-854.  doi: 10.1007/s00285-011-0477-6. [10] L. Esteva and H. Mo Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132-147.  doi: 10.1016/j.mbs.2005.06.004. [11] J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24pp. doi: 10.1142/S0218127413501642. [12] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101. [13] J. Huang, S. Ruan and J. Song, Bifurcaitons in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Diff. Equat., 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024. [14] J. Ito, A. Ghosh, L. A. Moreira, E. A. Wimmer and M. Jacobs-Lorena, Transgenic anopheline mosquitoes impaired in transmission of a malaria parasite, Nature, 417 (2002), 452-455.  doi: 10.1038/417452a. [15] W. Jiang, X. Li and X. Zou, On a reaction-diffusion model for sterile insect release method on a bounded domain, Internat. J. Biomath., 7 (2014), 1450030, 17pp. doi: 10.1142/S1793524514500302. [16] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Economic Entomolgy, 48 (1955), 459-462. [17] Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer-Verlag, New York Heidelberg, 1998. [18] R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis, Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Diff. Equat., 245 (2008), 442-467.  doi: 10.1016/j.jde.2007.10.034. [19] J. Li, Simple mathematical models for interacting wild and transgenic mosquito populations, Math. Biosci., 189 (2004), 39-59.  doi: 10.1016/j.mbs.2004.01.001. [20] J. Li, Discrete-time models with mosquitoes carrying genetically-modified bacteria, Math. Biosci., 240 (2012), 35-44.  doi: 10.1016/j.mbs.2012.05.012. [21] X. Li, J. Ren, S. A. Campbell, G. S. K. Wolkowicz and H. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 785-807.  doi: 10.3934/dcdsb.2018043. [22] L. Perko, Differential Equations and Dynamical Systems, 3rd edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8. [23] J. Ren and L. Yu, Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model, Journal of Nonlinear Science, 26 (2016), 1895-1931.  doi: 10.1007/s00332-016-9323-8. [24] J. Ren, L. Yu and S. Siegmund, Bifurcations and chaos in a discrete predator-prey model with Crowley-Martin functional response, Nonlinear Dynamics, 90 (2017), 19-41.  doi: 10.1007/s11071-017-3643-6. [25] S. Ruan and W. Wang, Dynamical behavior of an epidmeic model with a nonlinear incidence rate, J. Diff. Equat., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [26] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896. [27] S. M. White, P. Rohani and S. M. Sait, Modelling pulsed releases for sterile insect techniques: fitness costs of sterile and transgenic males and the effects on mosquito dynamics, J. Appl. Ecology, 47 (2010), 1329-1339.  doi: 10.1111/j.1365-2664.2010.01880.x. [28] World Health Organization (WHO), 10 facts on malaria, December, 2016. Available from: http://www.who.int/features/factfiles/malaria/en/. [29] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. [30] Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equation, Transl. Math. Monogr., 101, Amer. Math. Soc., Providence, RI, 1992. [31] 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.
The phase portraits of system (1) with $a = 1, \mu _{1} = \frac{37}{312}, \xi _{1} = \frac{41}{312}, \mu _{2} = \frac{703}{6240}, \xi _{2} = \frac{19}{3120}$. (a) No positive equilibrium when $b = \frac{49}{400}$; (b) A cusp when $b = \frac{19}{160}$; (c) Two positive equilibria when $b = \frac{9}{80}$, $E_1^*$ is a saddle, $E_2^*$ is a stable focus
(a) An unstable limit cycle created by the subcritical Hopf bifurcation; (b) A stable limit cycle created by the supercritical Hopf bifurcation
The bifurcation diagram and phase portraits of system (23) with $b = \frac{95}{800}$. (a) Bifurcation diagram; (b) No equilibria when $(\lambda_1, \lambda_2) = (0.01, -0.008)$ lies in the region Ⅰ; (c) An unstable focus when $(\lambda_1, \lambda_2) = (0.01, -0.011)$ lies in the region Ⅱ; (d) An unstable limit cycle when $(\lambda_1, \lambda_2) = (0.01, -0.012)$ lies in the region Ⅲ; (e) An unstable homoclinic loop when $(\lambda_1, \lambda_2) = (0.01, -0.01253)$ lies on the curve HL; (f) A stable focus when $(\lambda_1, \lambda_2) = (0.01, -0.013)$ lies in the region Ⅳ
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