The delayed reaction-diffusion Filippov system with threshold control: Modeling and analysis of plateau pika-vegetation dynamics

Abstract

Plateau pika (Ochotona curzoniae) plays dual ecological roles: enhancing biodiversity at moderate population densities, but driving severe grassland degradation during population outbreaks. To capture these phenomena, we develop delayed reaction-diffusion Filippov systems incorporating Holling-III functional response, maturation time delays, prey-taxis behavior, and threshold-based control strategies. Two types of threshold controls are considered: a single-threshold model, where intervention is triggered only when vegetation density drops below a threshold $T$; and a dual-threshold model, which introduces an additional pika density threshold $HT$ to enable preventive control. We establish the well-posedness of solutions, analyze equilibrium stability and Hopf bifurcations, and characterize sliding mode dynamics via Filippov’s convex method, thereby identifying pseudoequilibria and their stability conditions. Numerical simulations demonstrate that the dual-threshold strategy substantially outperforms the single-threshold approach by preventing extreme pika aggregation, maintaining more uniform vegetation distribution, and reducing grassland degradation. This dual-threshold mechanism highlights a paradigm shift from reactive to proactive ecological management. Our findings provide theoretical support for sustainable grassland management on the Tibetan Plateau.