TY - JOUR

T1 - Steady states in a non-conserving zero-range process with extensive rates as a model for the balance of selection and mutation

AU - Grange, Pascal

N1 - Publisher Copyright:
© 2019 IOP Publishing Ltd.

PY - 2019/8/13

Y1 - 2019/8/13

N2 - We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and removed from the system with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model of particles is mapped to a model of population dynamics: the site label is interpreted as a level of fitness, the site-dependent creation rate is interpreted as a selection function, and the hopping process is interpreted as the introduction of mutants. In the limit of large density, the fraction of the total population occupying each site approaches the limiting distribution in the house-of-cards model of selection-mutation, introduced by Kingman. A single site can be occupied by a macroscopic fraction of the particles if the mutation rate is below a critical value (which matches the critical value worked out in the house-of-cards model). This feature generalises to classes of selection functions that increase sufficiently fast at high fitness. The process can be mapped to a model of evolving networks, inspired by the Bianconi-Barabási model, but involving a large and fixed set of nodes. Each node forms links at a rate biased by its fitness, moreover links are destroyed at a uniform rate, and redirected at a certain rate. If this redirection rate matches the mutation rate, the number of links pointing to nodes of a given fitness level is distributed as the numbers of particles in the non-conserving zero-range process. There is a finite critical redirection rate if the density of quenched fitnesses goes to zero sufficiently fast at high fitness.

AB - We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and removed from the system with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model of particles is mapped to a model of population dynamics: the site label is interpreted as a level of fitness, the site-dependent creation rate is interpreted as a selection function, and the hopping process is interpreted as the introduction of mutants. In the limit of large density, the fraction of the total population occupying each site approaches the limiting distribution in the house-of-cards model of selection-mutation, introduced by Kingman. A single site can be occupied by a macroscopic fraction of the particles if the mutation rate is below a critical value (which matches the critical value worked out in the house-of-cards model). This feature generalises to classes of selection functions that increase sufficiently fast at high fitness. The process can be mapped to a model of evolving networks, inspired by the Bianconi-Barabási model, but involving a large and fixed set of nodes. Each node forms links at a rate biased by its fitness, moreover links are destroyed at a uniform rate, and redirected at a certain rate. If this redirection rate matches the mutation rate, the number of links pointing to nodes of a given fitness level is distributed as the numbers of particles in the non-conserving zero-range process. There is a finite critical redirection rate if the density of quenched fitnesses goes to zero sufficiently fast at high fitness.

KW - condensation

KW - network science

KW - out-of-equilibrium systems

KW - population dynamics

KW - zero-range process

UR - http://www.scopus.com/inward/record.url?scp=85072793418&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/ab3370

DO - 10.1088/1751-8121/ab3370

M3 - Article

AN - SCOPUS:85072793418

SN - 1751-8113

VL - 52

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 36

M1 - 365601

ER -