# 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 & Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139
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##### References:
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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