Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis
The work deals with the question of equation-based conditions, which define the existence of solutions for the distributed predator-prey model with the taxis term included into the PDE related to the predators' dynamics. The title of the article may be slightly ambiguous: ``prey-taxis'' does not mean a taxis of preys under any stimulus but the taxis of predators induced by the gradient of preys distribution, i.e., the reaction-diffusion equation for predator density \(u\) contains an additional term \(-\chi\nabla(q(u)\nabla v)\), where \(v\) is the prey density, \(q(u)\) is some function of the predator density, and \(\chi\) is the taxis coefficient. The last quantity is the main control parameter, which determines the existence of a unique global classical solution to the system. The proven theorem determines its boundaries as an explicit expression, which depends on the space dimensionality, the diffusion coefficient for preys, and their maximal density.
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