Chaos Detection and Bounded Control in an Allee Predator–Prey Map: Local Bifurcations, Global Crises, and Validation

• Naseema Aslam ^[1] ; Qamar Din ^[1]
  1. [1] University of Poonch Rawalakot
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We study a discrete-time predator–prey model in which the prey population is subject to a strong Allee effect and the interaction is governed by a positivity-preserving Rickertype updating rule. The system admits the extinction equilibrium, two predator-free boundary equilibria, and, under explicit parameter restrictions, a biologically meaningful coexistence equilibrium. We first establish forward invariance of the nonnegative quadrant and derive a parameter-dependent absorbing region, which ensures that the dynamics evolve in a compact invariant set. We then carry out the local bifurcation analysis of the model. In particular, we obtain explicit conditions for transcritical and flip bifurcations on the boundary and at the coexistence equilibrium, and we characterize the Neimark–Sacker bifurcation in terms of the trace and determinant of the Jacobian matrix. To describe the structure of resonant dynamics near the Neimark– Sacker curve, we identify the codimension-two resonance points associated with the strong resonances 1 : 2, 1 : 3, and 1 : 4, and derive the corresponding normal-form reductions together with the relevant transversality and nondegeneracy conditions. The analysis is complemented by a study of global dynamics. Two-parameter diagrams and numerical simulations reveal multistability, changes in basin geometry, and crisis-type transitions leading to sudden attractor enlargement and long transient irregular behavior.

    Chaotic regimes are examined through numerical diagnostics–including maximal Lyapunov exponents, long-orbit behavior, basin plots, and a numerically detected snapback- repeller configuration–while clearly distinguishing these computations from a fully interval-verified Marotto proof.We also propose a bounded bio-parameter feedback control strategy for stabilizing the coexistence state through small biologically meaningful parameter perturbations subject to saturation constraints, and numerical experiments show that the method can suppress irregular oscillations within its local guaranteed regime. Finally, we examine the empirical relevance of the model by calibrating it to predator–prey time-series data using robust parameter estimation and uncertainty quantification based on moving-block bootstrap confidence intervals. The fitted dynamics capture the long-run scale and qualitative predator–prey covariation, while abrupt demographic shocks are only partially reproduced by the minimal twodimensional structure. Taken together, the results provide a unified description of local bifurcations, resonant dynamics, global numerical transitions, and bounded local control in an Allee-regulated predator–prey system.

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