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Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations.

Authors :
PERTHAME, BENOÎT
GAUDUCHON, MATHIAS
Source :
Mathematical Medicine & Biology: A Journal of the IMA. Sep2010, Vol. 27 Issue 3, p195-210. 16p. 9 Graphs.
Publication Year :
2010

Abstract

Deterministic population models for adaptive dynamics are derived mathematically from individual-centred stochastic models in the limit of large populations. However, it is common that numerical simulations of both models fit poorly and give rather different behaviours in terms of evolution speeds and branching patterns. Stochastic simulations involve extinction phenomenon operating through demographic stochasticity, when the number of individual ‘units’ is small. Focusing on the class of integro-differential adaptive models, we include a similar notion in the deterministic formulations, a survival threshold, which allows phenotypical traits in the population to vanish when represented by few ‘individuals’. Based on numerical simulations, we show that the survival threshold changes drastically the solution; (i) the evolution speed is much slower, (ii) the branching patterns are reduced continuously and (iii) these patterns are comparable to those obtained with stochastic simulations. The rescaled models can also be analysed theoretically. One can recover the concentration phenomena on well-separated Dirac masses through the constrained Hamilton–Jacobi equation in the limit of small mutations and large observation times. [ABSTRACT FROM PUBLISHER]

Details

Language :
English
ISSN :
14778599
Volume :
27
Issue :
3
Database :
Academic Search Index
Journal :
Mathematical Medicine & Biology: A Journal of the IMA
Publication Type :
Academic Journal
Accession number :
53442911
Full Text :
https://doi.org/10.1093/imammb/dqp018