## A reaction-diffusion system modeling predator-prey with prey-taxis.(English)Zbl 1156.35404

Summary: We are concerned with a system of nonlinear partial differential equations modeling the Lotka-Volterra interactions of predators and preys in the presence of prey-taxis and spatial diffusion. The spatial and temporal variations of the predator’s velocity are determined by the prey gradient. We prove the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. The linearized stability around equilibrium is also studied. A finite volume scheme is build and numerical simulation show interesting phenomena of pattern formation.

### MSC:

 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)
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### References:

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