Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Effects of an advection term in nonlocal Lotka–Volterra equations

Pages: 1181 – 1188

(Fast Communication)

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a16

Authors

Rebecca H. Chisholm (School of Biotechnology and Biomolecular Sciences, University of New South Wales, Sydney, Australia)

Tommaso Lorenzi (ENS Cachan and CNRS, Centre de Mathématiques et de Leurs Applications, Cachan, France; and INRIA-Paris-Rocquencourt, MAMBA Team, Domaine de Voluceau, Le Chesnay, France)

Alexander Lorz (Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 6, Sorbonne Universités, Paris, France; and INRIA-Paris-Rocquencourt, MAMBA Team, Domaine de Voluceau, Le Chesnay, France)

Abstract

Nonlocal Lotka–Volterra equations have the property that solutions concentrate as Dirac masses in the limit of small diffusion. In this paper, we show how the presence of an advection term changes the location of the concentration points in the limit of small diffusion and slow drift. The mathematical interest lies in the formalism of constrained Hamilton–Jacobi equations. Our motivations come from previous models of evolutionary dynamics in phenotype-structured populations [R.H. Chisholm, T. Lorenzi, A. Lorz, et al., Cancer Res., 75, 930–939, 2015], where the diffusion operator models the effects of heritable variations in gene expression, while the advection term models the effect of stress-induced adaptation.

Keywords

nonlocal Lotka–Volterra equations, Dirac masses, phenotype-structured populations, stress-induced adaptation

2010 Mathematics Subject Classification

35R09, 45M05, 92D15, 92D25

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