P. Holgate

Population survival and life history phenomena.

Abstract The theory of stochastic branching processes is used to investigate the effect of different features of the life history of an individual, on the chance that an isolated population of such individuals will die out as a result of random fluctuations.

A prey-predator model with switching effect

A system involving one predator and two prey species is studied, in which the predator feeds more intensively on the more abundant species. For the model studied, more general than that of previous writers, the three species equilibrium can be stable or unstable, depending on the birth and predation rates.

Canonical multiplication in the genetic algebra for linked loci

Abstract It is shown that in the absence of mutation the canonical multiplication table takes an extremely simple form. This can be exploited to cut down drastically the amount of computation involved in evaluating a finite pedigree. It also facilitates the calculation of some numerical invariants of the algebra.

Jordan algebras arising in population genetics

The non-associative algebras arising in genetics ( 1 ), are rather isolated from other branches of non-associative algebra ( 6 ). However, in a paper ( 5 ), in which he studied these algebras in terms of their transformation algebras, Schafer proved that the gametic and zygotic algebras for a single diploid locus are Jordan algebras.

The lognormal characteristic function

A number of different ways are examined of representing the characteristic function φ(t) of the lognormal distribution, which cannot be expanded in a Taylor series based on the moments. In §2 the use of a finite Taylor series is examined. A method of summing the divergent formal expansion is discussed in §3. In §4 the fact that φ(t) is a boundary analytic function is exploited. Asymptotic approximation of the integral defining φ(t) is studied in §5. Each approach produces some interesting information about the distribution.

Selfing in genetic algebras

SummaryThe effect of self fertilization on the distribution of genetic types in a population can be represented algebraically by a linear transformation. In this paper the relationship of the transformation to the genetic algebra governing the population is investigated. In particular, the problems of multiple alleles, polyploidy and linked loci are studied.

The interpretation of derivations in genetic algebras

Abstract Meanings are assigned to a linear transform of an element in a genetic algebra representing a probability distribution over the possible genetic types, and to products of elements where only one of the factors is such a probability element. These lead on to a characterization of a derivation on a genetic algebra in terms of the equality of two genetically meaningful expressions.

Contributions to the Mathematics of Animal Trapping

The paper is concerned with probability distributions arising in live trapping studies on mammal populations, with emphasis on the effects of heterogeneity between animals. The distribution of observed length of residence of an animal in a defined study area is obtained for cases where: (i) the chance of capture on a given occasion varies between animals, having a beta distribution, while the probability of emigration during a given period is the same for all, and (ii) the probability of capture is constant but the emigration probability has a beta distribution. In the former case the distribution is a new one, whose probability generating function is essentially an Appell function of the fourth kind. Distributions of the number of times each animal would be captured in a fixed period are also discussed, and in a simple case the joint distribution of number of captures and length of residence is derived. Problems of estimation are considered, and the theory is illustrated by fitting data from recent work on voles and field mice.

Varieties of stochastic model: a comparative study of the Gompertz effect.

A comparative study is made of various models for the Gompertz phenomenon, which is a form of growth rate limitation in population dynamics. Deterministic, Markov birth-death, diffusion and stochastic differential equation models are studied, with a view to assessing their advantages and limitations.

Fitting a Straight Line to Data from a Truncated Population

It is usually assumed in linear regression theory that for a given value of the regressor variable t, the dependent variable y is distributed normally with expectation a linear function of t, say y N(a + bt, c-) However, situations arise in practice where the distribution of y must necessarily be truncated at a value independent of t. The following example which illustrates this arose in connection with studies of the wearing away of the tooth crowns of Red Deer, carried out by Dr. B. Mitchell of the Nature Conservancys Speyside Research Station, Aviemore, Scotland. The following simplified model of the wearing process is assumed. Suppose that for all the deer in a given herd, the crowns finish growing at the same age, and that thereafter the rate of wear is the same for all animals, and constant in time. The randomness in the model is due to the fact that crown weights at maturity are distributed normally about their mean with variance o-. When a crown wears away completely, the deer is no longer able to eat, and dies. The exact age at which the crowns mature need not be known, provided all those in the sample are above that age. Consider an individual whose tooth crown weighed ym at maturity, (age t..). At age t (>t7n) it would weigh