When long-term coexistence fails: Finite time extinction in a nonlinear model

In this note, we revisit the predator–prey model with a square root functional response and prey refuge, originally analyzed by Chen and Wang (2018). While the original study asserted the global stability of the coexistence equilibrium under specific conditions, we provide an alternative mathematical proof showing that the prey population can go extinct in finite time. We refine the analytical framework, present an exact extinction criterion, and illustrate our findings with a concrete example: fixed parameter values and initial conditions that satisfy the finite-time extinction condition. Both the analytical and numerical results confirm that extinction occurs in finite time, challenging prior long-term coexistence claims. Our findings underscore the importance of incorporating behavioral dynamics, such as prey aggregation in refuge areas, into ecological models