A DIFFUSIVE BEDDINGTON–DEANGELIS PREDATOR–PREY MODEL WITH NONLINEAR PREY-TAXIS, PROPORTIONAL REMOVAL, AND A COMMON FREE BOUNDARY

Abstract

This paper studies a one-dimensional diffusive predator–prey model incorporating the
Beddington–DeAngelis functional response, nonlinear prey-taxis, proportional removal, and a common free
boundary driven by both species. The existence, positivity, global continuation, and boundedness of solutions
are established. Conditions for exponential extinction, finite habitat expansion, and the spreading–vanishing
alternative are derived. A critical habitat length and sufficient conditions for vanishing are also determined,
revealing how proportional removal and joint boundary movement affect the invasion threshold.