Small populations corrections for selection-mutation models

Pierre-Emmanuel Jabin; Pierre-Emmanuel Jabin

doi:10.3934/nhm.2012.7.805

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 805-836. doi: 10.3934/nhm.2012.7.805

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Small populations corrections for selection-mutation models

Pierre-Emmanuel Jabin ^{1
,}

1.
CSCAMM and Department of Mathematics, University of Maryland, College Park, MD 20742-4015

Received: 01 March 2012 Revised: 01 August 2012
Primary: 35B25, 35K55, 92D15.

We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of [20], this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.
- Adaptive dynamics,
- Hamilton-Jacobi equation with constraints,
- Dirac concentration,
- small populations
Citation: Pierre-Emmanuel Jabin. Small populations corrections for selection-mutation models[J]. Networks and Heterogeneous Media, 2012, 7(4): 805-836. doi: 10.3934/nhm.2012.7.805

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Abstract

We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of [20], this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.

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