In this paper, we consider spatial predator-prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of nonlinear functional responses, linear and nonlinear predator death terms, linear and nonlinear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey. We identify combinations of the above nonlinearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, nonlinear predator death terms, and nonlinear prey-taxis sensitivity.

Spatially moving predators are often considered for biological control of invasive species. The question arises as to whether introduced predators are able to stop an advancing pest or foreign population. In recent studies of reaction-diffusion models, it has been shown that the prey invasion can only be stopped if the prey dynamics observes an Allee effect.In this paper, we include prey-taxis into the model. Prey-taxis describe the active movement of predators to regions of high prey density. This effect leads to the observation that predators are drawn away from the leading edge of a prey invasion where its density is low. This leads to counterintuitive result that prey-taxis can actually reduce the likelihood of effective biocontrol.