Global Dynamics of a Threshold Control Discrete Population Model with Allee Effects

Abstract

This paper investigates a threshold control discrete population model with Allee effects, characterized by density-dependent growth functions separated at a critical population threshold. The model captures diverse ecological scenarios through simple switching mechanisms while maintaining biological realism. We overcome analytical challenges in piecewise systems by developing a complete classification of five distinct dynamics, revealing how critical transitions emerge at specific parameter boundaries. One key theoretical contribution identifies the precise conditions generating persistent oscillations, a counterintuitive result demonstrating how discontinuous switching can sustain periodic behavior despite monotonic growth functions. These findings provide actionable conservation strategies, including extinction prevention protocols and sustainable harvesting policies. The framework offers both theoretical advances in piecewise dynamical systems and practical tools for ecological management, with potential applications in species conservation and ecosystem restoration.