Abstract

This study aims to analyze the dynamical behavior of a predator–prey model incorporating a Holling type II functional response and an anti-predator mechanism. The methods employed include dynamical systems analysis, namely equilibrium point determination, stability analysis using the Jacobian matrix, and bifurcation analysis. This analytical approach is supported by numerical simulations to construct phase portraits and illustrate system trajectories around equilibrium points. The results reveal various complex behaviors, particularly changes in stability and the emergence of bifurcation phenomena, which are influenced by variations in key parameters such as the saturation parameter, anti-predator parameter, and predator–prey interaction rate. Specifically, three types of bifurcations are identified: Hopf, transcritical, and saddle-node bifurcations. Hopf bifurcation leads to stable periodic oscillations, transcritical bifurcation results in an exchange of stability between equilibrium points, while saddle-node bifurcation causes the appearance or disappearance of equilibrium points. Numerical simulations further support these findings by illustrating system behavior before, at, and after bifurcation. Overall, the interaction between functional response saturation and anti-predator mechanisms plays a crucial role in determining the stability and dynamics of predator–prey populations.

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