A prey-predator system with Allee effect and anti-predator behavior: a continuous and discrete-time study

Abstract This study investigates the dynamics of a prey-predator system that incorporates two ecologically significant mechanisms of prey population: the Allee effect and anti-predator behavior. The model is formulated as a system of nonlinear ODEs and further extended to its corresponding discrete-time version. The mathematical analysis begins with establishing the positivity and boundedness of solutions, ensuring ecological viability. Conditions for the existence and stability of ecologically relevant equilibrium points are derived, and a detailed bifurcation analysis is conducted. It is observed that the system exhibits rich dynamics, including saddle-node, transcritical, Hopf, and Bogdanov-Takens bifurcations that are analyzed using underlying dynamical system approaches and illustrated through numerical simulations to reveal transitions in system behavior. To further explore the dynamics, a discrete-time version of the model is constructed using the forward Euler method. The stability and bifurcation structures (transcritical and Neimark-Sacker bifurcations) of the discrete system are examined. The center manifold theory is utilized to establish the existence of these bifurcations in the discrete framework. To validate these theoretical predictions, numerical simulations are performed, showing the emergence of invariant closed curves and bifurcation transitions as key parameters are varied. This work provides a valuable insight into the influence of ecological thresholds and behavioral responses on predator-prey interactions across both continuous and discrete-time domains.