The Latent Roots of Some Genetic Drift ModelsAuthor(s): C. CanningsSource: Advances in Applied Probability, Vol. 6, No. 2 (Jun., 1974), pp. 232-233Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426271 .

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232 3RD CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

Age distribution in some size-dependent populations

W. A. O'N. WAUGH, University of Toronto

It is well known that the population size Z, in many branching processes settles down to steady exponential growth as t -+ oo. We have, approximately

Zt e, e(t-T)

where 2 (> 0) is a growth rate sometimes called the Malthusian parameter, and T is a random variable. The author has recently developed a classification of Markovian birth (and death) processes with size-dependent rates, and shown that in one of these classes, the population size grows like D(t - T), where (D is a continuous function. This class generalizes the Markovian branching processes in the sense that when the size-effect vanishes the branching process is recovered

(see Waugh (1970), (1972a, b)). Similar results can be proved for some age-de- pendent birth processes (Waugh (1974)).

For processes that ultimately settle down to steady exponential growth R. A. Fisher (1930) gave a heuristic argument which leads to the expected proportion A(x) of age < x when t is large. In the present paper this argument is generalized to obtain the proportion of the population in a given age-range in populations of the class described above. Since the growth rate is changing with increasing time in the size-dependent case, the expected proportion is no longer independent of the time, even for large time, but simple expressions are nevertheless obtainable for it.

References FISHER, R. A. (1930) The Gen3tical Theory of Natural Selection. Oxford University Press.

Chapter II. WAUGH, W. A. O'N. (1970) Transformation of a birth process into a Poisson process. J. R.

Statist. Soc. B 32, 418-431. WAUGH, W. A. O'N. (1972a) Uses of the sojourn time series for the Markovian birth process.

Proc. 6th Berkeley Symposium 3, 501-514. WAUGH, W. A. O'N. (1972b) Taboo extinction, sojourn times and asymptotic growth for

the Markovian birth and death process. J. Appl. Prob. 9, 486-506. WAUGH, W. A. O'N. (1974) Asymptotic growth of a class of size-and-age dependent birth

processes. J. Appl. Prob. 11, 000-000.

II E. Biological Processes

The latent roots of some genetic drift models

C. CANNINGS, University of Sheffield

Moran and Watterson (1959) introduced the conditioned branching process as a tool for the study of genetic drift. Its use has since been extended to many complex models (e.g., Karlin (1968)). In particular the classical models of Wright

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Sheffield, 13-17 August 1973 233

(1931) and Moran (1958) have been generalized by Karlin and McGregor (1965) and Chia and Watterson (1969).

An alternative and simpler approach through the use of exchangeable random variables is demonstrated, which at the same time permits greater generality. Non-independent mutation, and interaction between birth events and death events can be introduced. The form of the latent roots obtained is extremely simple, e.g., for a generalization of Wright's model suppose the number of off- spring of the N (constant) individuals at each time point is given by ?1, ,2, , ",

**, 1N with an appropriate distribution function so that the ?'s are exchangeable, then

are the latent ro k=l,2,sN, are the latent roots.

References

CHIA, A. B. AND WATTERSON, G. A. (1969) Demographic effects on the rate of genetic evolu- tion. I. Constant size populations with two genotypes. J. Appl. Prob. 6, 231-249.

KARLIN, S. (1968) Equilibrium behavior of population genetics models with non-random mating. II. Pedigrees, homozygosity and stochastic models. J. Appl. Prob. 5, 487-566.

KARLIN, S. AND MCGREGOR, J. (1965) Direct product branching processes and related in- duced Markoff chains. I. Calculation of rates of approach to homozygosity. Bernoulli, Bayes, Laplace Anniversary Volume. Springer-Verlag, Berlin. 11-145.

MORAN, P. A. P. (1958) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60-71. MORAN, P. A. P. AND WATTERSON, G. A. (1959) The genetic effects of family structure in

natural populations. Austral. J. Biol. Sciences 12, 1-15. WRIGHT, S. W. (1931) Evolution in Mendelian populations. Genetics 16, 97-159.

Percolation processes and tumour growth

D. MOLLISON*, King's College Research Centre, Cambridge

Although there exist environments such as seashores and railway embank- ments which are essentially one-dimensional, the principal justification for work on one-dimensional population processes (such as the spread of an advantageous gene or of epidemics [2], [5]-[8] is as a step towards the analysis of the more difficult two-dimensional cases. For continuous deterministic models the exten- sion to two dimensions does not appear to involve qualitative differences [4], [8]; but for the more realistic stochastic case problems arise in two dimensions which are not met in one. As the example described here shows, the continuous case can no longer be expected to provide an upper bound for propagation as in one dimension [8].

* Now at Heriot-Watt University, Edinburgh.

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