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

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**Résumé:**

**Abstract
:** Deterministic population models for adaptive dynamics are derived
mathematically from individual centered stochastic models in the limit of large
populations. However numerical simulations of both models do not fit and give
rather different behaviors 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. We
include a similar notion in the deterministic models, 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 is reduced continuously; (iii) they are
comparable to the stochastic simulations. The rescaled models can also be
analyzed 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.

**Mots Clés:** *Evolution; Adaptative Dynamics; Hamilton-Jacobi
equation; Asymptotic analysis; Demographic stochasticity; Individual based
model; Evolutionary branching *

**Date:** 2008-09-19