A Model with Darwinian Dynamics on a Rugged Landscape

Abstract

We discuss the population dynamics with selection and random diffusion, keeping the total population constant, in a fitness landscape associated with Constraint Satisfaction, a paradigm for difficult optimization problems. We obtain a phase diagram in terms of the size of the population and the diffusion rate, with a glass phase inside which the dynamics keeps searching for better configurations, and outside which deleterious ‘mutations’ spoil the performance. The phase diagram is analogous to that of dense active matter in terms of temperature and drive.

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Notes

  1. 1.

    If one allows for many mutations to exist, while still having a single dominant population at almost all times, a somewhat different regime is obtained [27]. Then Eq. (4) no longer holds due to the population ‘cloud’ of deleterious mutations. If these mutants do not reproduce (\(\lambda =0\)), Eq. (4) may be mended by considering an effective \(N_{eff}=N-N_{cloud}\), but for more general deleterious mutations a simple prescription is hard to give. However, this correction is small when mutation rates are low \(\mu \ll 1\) (but not necessarily very low \(\mu N\ll 1\)). More precisely, Eq. (4) holds when the fraction of deleterious mutations is small, \(\frac{\mu \lambda }{\lambda -\lambda _{del}}\ll 1\), where \(\lambda \) is the fitness of the dominant population and \(\lambda _{del}\) is a typical fitness of deleterious mutations.

  2. 2.

    This entails that the width of the fitness distribution in the population at a given time is \(\sigma _{\lambda }^{2}\sim \frac{N}{L^{2}\tau _0}\sim \frac{1}{N^{2}}\) (following arguments as in [33, 34] ). The time-scale for a given spin flip is on average \(\tau _{point}=L\tau _0\). which corresponds also to the time-scale for an individual to shuffle its entire configuration.

  3. 3.

    This phase diagram seem very similar to the one obtained by Neher and Shraiman, where recombination and epistasis play the roles of mutation and selection. See: Neher et al. [39].

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Acknowledgments

We would like to thank JP Bouchaud, and D.A. Kessler for helpful discussions.

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Correspondence to Jorge Kurchan.

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Brotto, T., Bunin, G. & Kurchan, J. A Model with Darwinian Dynamics on a Rugged Landscape. J Stat Phys 166, 1065–1077 (2017). https://doi.org/10.1007/s10955-016-1637-2

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Keywords

  • Population dynamics
  • Glasses
  • Genetic algorithm
  • Glassy landscape