Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

Hyperbolic predators vs. parabolic prey

Pages: 369 – 400



Rinaldo M. Colombo (Unità INdAM, Università di Brescia, Italy)

Elena Rossi (Dipartimento di Matematica ed Applicazioni, Università di Milano-Bicocca, Milano, Italy)


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.


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

2010 Mathematics Subject Classification

35L65, 35M30, 92D25

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Article revised April 9, 2014