A logistic equation with nonlocal interactions

Luis Caffarelli; Serena Dipierro; Enrico Valdinoci

doi:10.3934/krm.2017006

Article Contents

2017, Volume 10, Issue 1: 141-170. Doi: 10.3934/krm.2017006

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A logistic equation with nonlocal interactions

1.
The University of Texas at Austin, Department of Mathematics and Institute for Computational Engineering and Sciences, 2515 Speedway, Austin, TX 78751, USA
2.
School of Mathematics and Statistics, University of Melbourne, 813 Swanston St, Parkville VIC 3010, Australia
3.
School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
4.
Weierstraß-Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 5/7,10117 Berlin, Germany
5.
CNR, Istituto di Matematica Applicata e Tecnologie Informatiche, via Ferrata 1,27100 Pavia, Italy
6.
Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50,20133 Milan, Italy

*Corresponding author: Luis Caffarelli
*Corresponding author: Luis Caffarelli

Received: January 31, 2016

Revised: April 30, 2016

Published: September 2017

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Abstract

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms.

More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case.

As ambient space, we can consider:
  $ \bullet $bounded domains,
  $ \bullet $periodic environments,
  $ \bullet $transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one.

In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

Keywords:

Mathematics Subject Classification: Primary: 35Q92, 46N60, 35R11; Secondary: 60G22.

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