Qualitative analysis of a predator–prey model with rapid evolution and piecewise constant arguments

Abstract

The qualitative analysis of a predator–prey model with rapid evolution and piecewise constant arguments is investigated in this work. The discrete model, which determines the dynamical behavior of the corresponding differential model, is achieved by calculation. First, the sufficient conditions for the existence and local stability of the equilibriums are concluded from the linearized stability theorem and latent root method. Second, the global stability of the equilibriums is discussed through the Poincaré–Bendixson theorem. Furthermore, it is proved that the system has at most one limit cycle. Third, by using the bifurcation theory it is found that the model can undergo the saddle-node bifurcation; the flip bifurcation; and the Neimark–Sacker bifurcation. From the qualitative analysis it can be found that the exponential growth rate and the ratio between the fast and slow timescales have profound influence on the dynamic behavior of the model. Finally, numerical examples carry out to justify the main results in this work.