Hopf bifurcation analysis of a pine wilt disease model with both time delay and an alternative food source

Delayed pregnancy of predators and the Beverton-Holt-like alternative food source are key factors in controlling population density. To control the population density of Monochamus alternatus, the vector of pine wilt disease, this paper proposes a control system integrating the Holling II functional response function, Beverton-Holt-like alternative food source, and pregnancy delay. The conditions for the existence of the Hopf bifurcation were analyzed and we derived the normal form of Hopf bifurcation of the system with pregnancy delay and the Beverton-Holt-like alternative food source by using the multiple time scale method. Considering its biological significance, we selected a set of appropriate parameters for numerical simulation. Moreover, we also obtained that Hopf bifurcation can be induced under the effect of pregnancy delay. Finally, we put forward several biological elucidations that are useful for the prevention and treatment of pine wilt disease.