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Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations
Benoît Perthame1, Mathias Gauduchon
1Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, F-75005 Paris, France. benoit.perthame@upmc.fr
Mathematical Medicine and Biology : a Journal of the IMA
|September 8, 2009
Summary
Introducing a survival threshold into deterministic adaptive dynamics models significantly slows evolution and reduces branching patterns. This modification makes model predictions more comparable to stochastic simulations, improving accuracy in evolutionary biology.
Area of Science:
- Evolutionary Biology
- Mathematical Biology
- Population Dynamics
Background:
- Deterministic models of adaptive dynamics are derived from stochastic models but often show discrepancies in simulations.
- Stochastic simulations exhibit extinction phenomena due to demographic stochasticity in small populations.
- Existing models lack a mechanism to account for population size effects on trait evolution.
Purpose of the Study:
- To incorporate a survival threshold into deterministic adaptive dynamics models.
- To investigate the impact of this threshold on evolutionary speed and branching patterns.
- To improve the concordance between deterministic and stochastic models in evolutionary simulations.
Main Methods:
- Focusing on integro-differential adaptive models.
- Implementing a survival threshold in deterministic formulations.
- Conducting numerical simulations to compare model behaviors.
- Theoretical analysis of rescaled models using constrained Hamilton-Jacobi equations.
Main Results:
- The survival threshold drastically slows down evolutionary speed.
- Branching patterns are continuously reduced by the threshold.
- Model predictions with the survival threshold align better with stochastic simulation results.
- Theoretical analysis reveals concentration phenomena in the limit of small mutations.
Conclusions:
- A survival threshold is crucial for accurately modeling adaptive dynamics, especially concerning population size effects.
- This modification bridges the gap between deterministic and stochastic modeling approaches.
- The enhanced deterministic models offer a more realistic representation of evolutionary processes.
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