Bifurcation Analysis of a Mosquito Population Model with a Saturated Release Rate of Sterile Mosquitoes

Releasing sterile mosquitoes is a method of mosquito control that uses area-wide inundative releases of sterile male mosquitoes to reduce reproduction in a field population of wild mosquitoes. In this paper, we consider a mosquito population model with a nonlinear saturated release rate of sterile mosquitoes and study the complex dynamics and bifurcations of the model. It is shown that there are a weak focus of multiplicity 3 and a nilpotent cusp of codimension 4 for various parameter values and the model exhibits Hopf bifurcation of codimension 3 and Bogdanov-Takens bifurcation of codimension 2 as the parameter values vary. Our analysis also shows that there exists a critical release rate coefficient of sterile mosquitoes, above which the mosquito population can be eliminated and below which the interacting sterile and wild mosquitoes coexist in the form of multiple periodic oscillations and steady states for some initial populations. Numerical simulations are presented to demonstrate the coexistence of a homoclinic loop and a limit cycle, the existence of two limit cycles, and the existence of three limit cycles, respectively.