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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