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<p>In this paper, we investigate a discrete-time Kolmogorov-type predator–prey model with a ratio-dependent functional response that incorporates both predator interference and handling time. In contrast to its continuous-time counterpart, the discrete model exhibits a wider variety of dynamical behaviors. We first establish the positivity and boundedness of solutions, determine all equilibrium points, and study their local stability properties. Our analysis shows that the coexistence equilibrium can undergo transcritical, period-doubling, and Neimark–Sacker bifurcations, whereas the boundary equilibria may experience transcritical and period-doubling bifurcations. In particular, the intersection of the period-doubling and Neimark–Sacker bifurcation curves gives rise to a codimension-two <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1:2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>:</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> resonance point, which plays a central role in the emergence of quasi-periodic oscillations, higher-period orbits, and chaotic dynamics. The onset of chaos is further characterized by means of the maximum Lyapunov exponent. To support the theoretical results, numerical simulations are presented in the form of bifurcation diagrams and phase portraits. In addition, a state-feedback control strategy is proposed to stabilize the coexistence equilibrium and suppress chaotic oscillations, showing that the chaotic behavior of the system can be effectively controlled.</p>

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Multi-Parameter Bifurcation Analysis of a Discrete Ratio-Dependent Predator–Prey Model: Codimension-Two Resonance, Chaos, and Control

  • 期刊论文

  • 发表日期：

    2026年6月25日

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  • Lahcen Koujane,
  • Hajar Mouhsine,
  • Karima Mokni,
  • Mohamed Ch-Chaoui

摘要

In this paper, we investigate a discrete-time Kolmogorov-type predator–prey model with a ratio-dependent functional response that incorporates both predator interference and handling time. In contrast to its continuous-time counterpart, the discrete model exhibits a wider variety of dynamical behaviors. We first establish the positivity and boundedness of solutions, determine all equilibrium points, and study their local stability properties. Our analysis shows that the coexistence equilibrium can undergo transcritical, period-doubling, and Neimark–Sacker bifurcations, whereas the boundary equilibria may experience transcritical and period-doubling bifurcations. In particular, the intersection of the period-doubling and Neimark–Sacker bifurcation curves gives rise to a codimension-two \(1:2\) 1 : 2 resonance point, which plays a central role in the emergence of quasi-periodic oscillations, higher-period orbits, and chaotic dynamics. The onset of chaos is further characterized by means of the maximum Lyapunov exponent. To support the theoretical results, numerical simulations are presented in the form of bifurcation diagrams and phase portraits. In addition, a state-feedback control strategy is proposed to stabilize the coexistence equilibrium and suppress chaotic oscillations, showing that the chaotic behavior of the system can be effectively controlled.