A predator-prey system in which the prey's attack-vulnerability is not a constant but a saturating decreasing function of predator density, representing an inducible defence such as camouflage, burrowing, evasion or counterattack, paid for by a reduction in prey growth. Consumption follows the Beddington-DeAngelis response, so mutual interference among predators enters the denominator alongside handling time. The temporal model is shown to be well posed with positive, uniformly bounded solutions inside an explicit absorbing region; its interior equilibria are the positive roots of a quartic and exist once the conversion efficiency exceeds the death rate times the sum of the half-saturation and handling parameters. Analysis of the resulting bifurcations shows that defence is stabilizing and raises prey abundance, that predator abundance increases with moderate interference only when defence is weak and otherwise declines monotonically, and that varying the handling rate opens a bistable region in which the initial population sizes decide which state is reached. Extending to a bounded two-dimensional domain with diffusion and no-flux boundaries, stronger defence shrinks the Turing domain and lowers the critical prey diffusivity, so patterns become less likely, producing cold spots inside the Hopf-stable Turing region and labyrinthine patterns in the Turing-Hopf region. Adding repulsive predator-taxis for the prey and attractive prey-taxis for the predator shrinks the pattern-forming region further, making directed movement a stabilizing rather than destabilizing ingredient.
A mathematical-ecology research paper (31 pages, 30 figures) that builds and analyses a predator-prey model in which prey mount an inducible defence, and then follows that model from ordinary differe…