Hyperbolic predators vs. parabolic prey

We present a nonlinear predator–prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for prey. The drift term in the predators’ equation is a nonlocal function of the prey density, so that the movement of the predators can be directed towards regions with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions, and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions.

Keywords

nonlocal conservation laws, predatory-prey systems, mixed hyperbolic-parabolic problems

2010 Mathematics Subject Classification

35L65, 35M30, 92D25

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Published 3 December 2014

Article revised April 9, 2014