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A dynamically consistent nonstandard finite difference scheme for a predator–prey model. (English) Zbl 1459.37088

Summary: The interaction between prey and predator is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of predator and prey interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for a class of predator – prey systems with Holling type-III functional response. Positivity, boundedness, and persistence of solutions are investigated. Analysis of existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes a Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes a Neimark-Sacker bifurcation at its interior fixed point. A numerical simulation is provided to strengthen our theoretical discussion.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
39A30 Stability theory for difference equations
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