Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting

Ming Liu, Linyi Ma and Dongpo Hu ()
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Ming Liu: School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Linyi Ma: School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Dongpo Hu: School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Mathematics, 2024, vol. 12, issue 18, 1-44

Abstract: This paper delves into the dynamics of a discrete-time predator–prey system. Initially, it presents the existence and stability conditions of the fixed points. Subsequently, by employing the center manifold theorem and bifurcation theory, the conditions for the occurrence of four types of codimension 1 bifurcations (transcritical bifurcation, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation) are examined. Then, through several variable substitutions and the introduction of new parameters, the conditions for the existence of codimension 2 bifurcations (fold–flip bifurcation, 1:2 and 1:4 strong resonances) are derived. Finally, some numerical analyses of two-parameter planes are provided. The two-parameter plane plots showcase interesting dynamical behaviors of the discrete system as the integral step size and other parameters vary. These results unveil much richer dynamics of the discrete-time model in comparison to the continuous model.

Keywords: predator–prey model; discrete-time model; bifurcation; resonance; two-parameter plane (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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