Complex dynamics and chaos control in a nonlinear discrete prey–predator model

The dynamics of prey–predator interactions are often modeled using differential or difference equations.  In this paper, we investigate the dynamical behavior of a two-dimensional discrete prey–predator system.  The model is formulated in terms of difference equations and derived by using a nonstandard finite difference scheme (NSFD), which takes into consideration the non-overlapping generations.  The existence of fixed points as well as their local asymptotic stability are proved.  Further, it is shown that the model experiences Neimark–Sacker bifurcation (NSB for short) and period-doubling bifurcation (PDB) in a small neighborhood of the unique positive fixed point under certain parametric conditions.  This analysis utilizes bifurcation theory and the center manifold theorem.  The chaos produced by NSB and PDB is stabilized.  Finally, we use numerical simulations and computer analysis to check our theories and show more complex behaviors.

difference equations
asymptotic stability
bifurcation analysis
chaos control
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