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Random mapping based on a genetic model

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  • Falk, H.
  • Ventevogel, W.J.

Abstract

We consider the discrete-time dynamics of a one-locus, two-allele (two-gene) model of a genetic population experiencing natural selection in a random environment. The dynamics is described by a stochastic recursion relation. We construct hierarchies of upper and lower bounds for the nth generation average gene frequency 〈Xn〉. The bounds provide very sharp estimates of 〈Xn〉 (n = 1,2,…) and may be interpreted in the spirit of mean-field and spin-cluster approximations. We also find a phase boundary in the space of viability parameters and environment probabilities. On one side of the boundary one of the two genes eventually becomes extinct (1 - 〈Xn〉å 0 for n å ∞); whereas, on the other side of the boundary both genes survive. For the special case where the environment is restricted to be a two-valued random variable, the average gene frequency 〈Xn〉 is, in principle, derivable from a partition function for an n-spin, Ising model.

Suggested Citation

  • Falk, H. & Ventevogel, W.J., 1979. "Random mapping based on a genetic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 95(2), pages 191-207.
    • Handle: RePEc:eee:phsmap:v:95:y:1979:i:2:p:191-207
    DOI: 10.1016/0378-4371(79)90051-7
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