Biology – Quantitative Biology – Populations and Evolution
Scientific paper
2011-07-08
Biology
Quantitative Biology
Populations and Evolution
Scientific paper
We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. Following an inverse numerical analysis approach, we obtain a family of partial differential equations (PDE) for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population and weak selection limit. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection, and the particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how good is the convective approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modeled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.
Chalub Fabio A. C. C.
Souza Max O.
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