Qualitative properties and bifurcations of discrete-Time Bazykin-Berezovskaya predator-prey model

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin-Berezovskaya predator-prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-Time Bazykin-Berezovskaya predator-prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge-Kutta method (IRK) with very small step size. The comparison yields that discrete-Time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-Time model exhibits flip and Niemark-Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.