Stability and bifurcation of a time-varying delay predator–prey model with Allee effect, prey refuge and B-D function

Abstract

This paper mainly considers a time-varying delay predator–prey model with Allee effect, prey refuge and B-D functional response. First, the positivity and boundedness of solutions of such model without delay are proved. Next, the conditions for the existence and local stability of equilibriums are provided. The global stability of the system near the positive equilibrium is also obtained by constructing appropriate Lyapunov function. Then, the existence of Hopf bifurcation and the stability of periodic solutions of bifurcation are proved. Furthermore, when the time-varying delay is a periodic function, periodic perturbation is applied to the delay to suppress oscillations by the method of multiple scales, which calculates the amplitude of the perturbation to control oscillations. Finally, the results show that oscillations can be weakened, and numerical simulations are showed to verify our analytical results.