TY - JOUR

T1 - The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations

AU - Chalub, Fábio Augusto da Costa Carvalho

AU - Souza, Max O.

N1 - FACCC was partially supported by CMA/FCT/UNL, financiamento base 2011 ISFL-1-297 and projects PTDC/FIS/101248/2008, PTDC/FIS/70973/2006 from FCT/MCTES/Portugal. FACCC also acknowledges the hospitality of CRM/Barcelona where part of this work was performed and discussions with J. J. Velazquez (Madrid). MOS was partially supported by CNPq grants #s 309616/2009-3 and 451313/2011-9, and FAPERJ grant # 110.174/2009. We thank the careful reading and comments of three anonymous referees.

PY - 2014/4

Y1 - 2014/4

N2 - 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. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. 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 accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.

AB - 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. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. 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 accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.

KW - Diffusion approximations

KW - Wright-Fisher process

KW - Replicator equation

KW - Continuous limits

KW - Continuous limits

KW - Diffusion approximations

KW - Replicator equation

KW - Wright-Fisher process

U2 - 10.1007/s00285-013-0657-7

DO - 10.1007/s00285-013-0657-7

M3 - Article

C2 - 23503810

VL - 68

SP - 1089

EP - 1133

JO - Journal Of Mathematical Biology

JF - Journal Of Mathematical Biology

SN - 0303-6812

IS - 5

ER -