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Bifurcations in a discrete time Lotka-Volterra predator-prey system. (English) Zbl 1100.37054
Summary: A discrete-time system, derived from a predator-prey system by Euler’s method with step one, is investigated in the closed first quadrant $$\mathbb{R}^2_+$$. It is shown that the discrete-time system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and the discrete-time system has a stable invariant cycle in the interior of $$\mathbb{R}^2_+$$ for some parameter values. Numerical simulations are provided to verify the theoretical analysis and show the complicated dynamical behavior. These results reveal far richer dynamics of the discrete model compared with the continuous model of the same type.

##### MSC:
 37N25 Dynamical systems in biology 39A10 Additive difference equations 39A12 Discrete version of topics in analysis 92D25 Population dynamics (general) 37G99 Local and nonlocal bifurcation theory for dynamical systems
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