Dirac concentrations in a chemostat model of adaptive evolution
- 51 Downloads
This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies.
Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.
KeywordsAdaptive evolution Asymptotic behaviour Chemostat Dirac concentrations Hamilton-Jacobi equations Lotka-Volterra equations Viscosity solutions
2000 MR Subject Classification35B25 35K57 47G20 49L25 92D15
Unable to display preview. Download preview PDF.
- Champagnat, N., Mathematical study of stochastic models of evolution belonging to the ecological theory of adaptive dynamics, PhD thesis, Université de Nanterre-Paris X, 2004.Google Scholar
- Mirrahimi, S., Phénomènes de concentration dans certaines EDPs issues de la biologie, PhD thesis, Universit é Pierre et Marie Curie-Paris VI,2011.Google Scholar
- Mirrahimi, S. and Perthame, B., Asymptotic analysis of a selection model with space, J. Math. Pures Appl., to appear.Google Scholar
- Mirrahimi, S., Perthame, B., Bouin, E. and Millien, P., Population formulation of adaptative mesoevolution: Theory and dynamics, J. F. Rodrigues and F. Chalub (eds.), The Mathematics of Darwin’s Legacy, Mathematics and Biosciences in Interaction, Springer-Verlag, New York, 2011.Google Scholar
- Raoul, G., Etude qualitative et numérique d’équations aux dérivées partielles issues des sciences de la nature, PhD thesis, ENS Cachan, 2009.Google Scholar