Abstract
The presence of a predator can force the mediation of a coexistence state in three-species ordinary differential equation model, where two competing species are preyed on by a common predator. To understand how the addition of diffusion and prey-taxis affects predator-mediated coexistence in such an ecological system, we consider a general two-competing-prey and one-predator model with double prey-taxes under Neumann boundary conditions. We first show that there is a unique global classical solution to this model with ratio-dependent and nonratio-dependent predator functional responses. Then, we demonstrate the emergence of the so-called stationary patterns. Finally, in detail, we give some sufficient conditions for the existence, nonexistence, and stability of nonconstant positive steady states and time-periodic positive solutions. Surprisingly, we find that the combination of a repulsive prey-taxis and an attractive prey-taxis can also induce the emergence of pattern formations. The theoretical results imply that double prey-taxes play an extremely important part in biological control.
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