Codimension One and Codimension Two Bifurcations in a Discrete Kolmogorov Type Predator–prey Model

As is widely known, discrete-time models may be the suitable choice for identifying the evolutionary behavior of a species or a taxonomic group of organisms with clearly separated developmental stages and no overlapping generations. As a consequence, we propose a discrete version of Kolmogorov predator–prey model. Firstly, the parametric conditions of fixed points stability for the present model are derived. Then, we used the normal form method of discrete-time models and bifurcation theory to investigate the bifurcation behaviors of codimensions one and two for the present model. It is not necessary to transition into Jordan form and compute the center manifold approximation of the current model with this procedure. The presence of different bifurcation forms can be checked by just calculating the critical non-degeneracy coefficients. Finally, certain indicators such as bifurcation diagrams, maximum Lyapunov exponents, and phase portraits are achieved by the numerical simulation process to clarify our theoretical results and display further model characteristics with parameter values change.