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

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

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)].

Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 92D25.

 Citation: • • Figure 1.  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

Figure 2.  (a) An unstable limit cycle created by the subcritical Hopf bifurcation; (b) A stable limit cycle created by the supercritical Hopf bifurcation

Figure 3.  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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