### AIMS Mathematics

2023, Issue 6: 13390-13410. doi: 10.3934/math.2023678
Research article

# Bifurcations and chaotic behavior of a predator-prey model with discrete time

• Received: 18 February 2023 Revised: 16 March 2023 Accepted: 20 March 2023 Published: 06 April 2023
• MSC : 34K18, 37L10, 39A28

• In this paper, the dynamical behavior of a predator-prey model with discrete time is discussed in terms of both theoretical analysis and numerical simulation. The existence and stability of four equilibria are analyzed. It is proved that the system undergoes Flip bifurcation and Hopf bifurcation around its unique positive equilibrium point using center manifold theorem and bifurcation theory. Additionally, by applying small perturbations to the bifurcation parameter, chaotic cases occur at some corresponding internal equilibria. Finally, numerical simulations are provided with the help of maximum Lyapunov exponent and phase diagrams, which reveal a complex dynamical behavior.

Citation: Binhao Hong, Chunrui Zhang. Bifurcations and chaotic behavior of a predator-prey model with discrete time[J]. AIMS Mathematics, 2023, 8(6): 13390-13410. doi: 10.3934/math.2023678

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• In this paper, the dynamical behavior of a predator-prey model with discrete time is discussed in terms of both theoretical analysis and numerical simulation. The existence and stability of four equilibria are analyzed. It is proved that the system undergoes Flip bifurcation and Hopf bifurcation around its unique positive equilibrium point using center manifold theorem and bifurcation theory. Additionally, by applying small perturbations to the bifurcation parameter, chaotic cases occur at some corresponding internal equilibria. Finally, numerical simulations are provided with the help of maximum Lyapunov exponent and phase diagrams, which reveal a complex dynamical behavior.

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