Effects and biological consequences of the predator-mediated apparent competition II: PDE models

In Lou et al. (Lou Y, Tao W, Wang Z-A. Effects and biological consequences of the predator-mediated apparent competition I: ODE models. J. Math. Biol. 91 (2025), 47, 37 pages), the authors investigated the effects and biological consequences of the predator-mediated apparent competition using a temporal (ODE) system consisting of one predator and two prey species (one is native and the other is invasive) with Holling type I and II functional responses. This paper is a sequel to Lou et al. (Lou Y, Tao W, Wang Z-A. Effects and biological consequences of the predator-mediated apparent competition I: ODE models. J. Math. Biol. 91 (2025), 47, 37 pages.), by including spatial movements (diffusion and prey-taxis) into the ODE system and examining the spatial effects on the population dynamics under the predator-mediated apparent competition. We establish the global boundedness of solutions in a two-dimensional bounded domain with Neumann boundary conditions and the global stability of constant steady states in certain parameter regimes, by which we find a threshold dynamics in terms of the predator's death rate. For the parameters outside the global stability regimes, we conduct a linear stability analysis to show that diffusion and/or prey-taxis can induce instability by both steady-state and Hopf bifurcations. We further use numerical simulations to illustrate that various spatial patterns are all possible, including stable spatial aggregation patterns, spatially homogeneous but time-periodic patterns, and spatially inhomogeneous and time-oscillatory patterns. It comes with a surprise that either of diffusion and prey-taxis can induce steady-state or Hopf bifurcations to generate intricate spatial patterns in the one predator-two prey system, which is sharply different from the one predator-one prey system for which neither diffusion nor prey-taxis can induce spatial patterns. These results show that spatial movements play profound roles in the emerging properties for predator-prey systems with multiple prey species. We also find that prey-taxis may play dual roles (stabilization and destabilization) and facilitate the predator-mediated apparent competition to eliminate the native prey species under the moderate initial mass of invasive prey species.