Control and bifurcation analysis of a discrete predator-prey model with Holling type-IV and Leslie-Gower dynamics

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Abstract

This paper investigates the dynamical behavior of a Euler-generated discrete predator-prey map incorporating a Holling type-IV functional response and Leslie-Gower predator dynamics. Since the Leslie-Gower term is singular at zero prey density, the biologically feasible state space is restricted to x > 0 and y ≥ 0. Within this domain, the existence and local stability of the boundary and positive equilibria are analyzed, and sufficient conditions for the possible configurations of positive equilibria are derived from the discriminant of the associated cubic equation. The positivity of the uncontrolled and controlled maps is also examined through analytical one-step conditions and direct numerical feasibility checks. By employing center manifold reduction and normal form theory, analytical conditions for Flip and Neimark-Sacker bifurcations at the positive equilibrium are obtained. Two stabilization approaches are then investigated. The state-feedback scheme yields an explicit triangular stability region in the gain-parameter plane, whereas the weighted hybrid scheme is equivalent to replacing the Euler step size h by an effective step size h_eff = ωh. Numerical experiments, including convergence tests, bifurcation diagrams, maximum Lyapunov exponent (MLE) spectra, time series, and phase portraits, support the theoretical analysis and illustrate the transition among stable, periodic, quasi-periodic, and chaotic dynamics of the Euler-generated map.

Keywords

discrete predator-prey model / Holling type-IV / Leslie-Gower / bifurcation analysis / chaos control

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