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December 2016 A piecewise deterministic model for a prey-predator community
Manon Costa
Ann. Appl. Probab. 26(6): 3491-3530 (December 2016). DOI: 10.1214/16-AAP1182


We are interested in prey–predator communities where the predator population evolves much faster than the prey’s (e.g., insect-tree communities). We introduce a piecewise deterministic model for these prey–predator communities that arises as a limit of a microscopic model when the number of predators goes to infinity. We prove that the process has a unique invariant probability measure and that it is exponentially ergodic. Further on, we rescale the predator dynamics in order to model predators of smaller size. This slow–fast system converges to a community process in which the prey dynamics is averaged on the predator equilibria. This averaged process admits an invariant probability measure which can be computed explicitly. We use numerical simulations to study the convergence of the invariant probability measures of the rescaled processes.


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Manon Costa. "A piecewise deterministic model for a prey-predator community." Ann. Appl. Probab. 26 (6) 3491 - 3530, December 2016.


Received: 1 March 2015; Revised: 1 October 2015; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1358.92077
MathSciNet: MR3582809
Digital Object Identifier: 10.1214/16-AAP1182

Primary: 60J25
Secondary: 60J75 , 92D25

Keywords: averaging techniques , ergodicity , Invariant measures , irreducibility , Piecewise deterministic Markov processes , Prey–predator communities , slow–fast systems

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.26 • No. 6 • December 2016
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