An approximation is found to the distribution of the maximum number of infectives present at any time during the course of a closed epidemic. The technique used is applicable to a commonly occurring type of random walk problem where there is a curved absorbing boundary which is far from the mean path except over a narrow range.

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

When two closely-related species have a partially-overlapping geographical range, it sometimes happens that populations of the two species are very similar in the non-overlapping parts of the range where only one species is present, but that they diverge and are easily distinguishable in the area of overlap where both species occur together. This phenomenon is known as character displacement. One of the best known examples occurs in two species of Rock Nuthatches (Sitta neumayer and S. tephronota) in their area of overlap in Iran (Vaurie (1951)). A plausible explanation of this phenomenon is that it is due to competition between the two species. The purpose of this paper is to construct a model for character displacement based on this idea. A full account of this work will be published elsewhere (Bulmer (1973)).

A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the “bang-bang” type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.

A single realization {Zn, 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained.

Wright (1931) introduced a simple model for a bisexual, diploid population of fixed size and with non-overlapping generations. With N1 males and N2 females, the rate of drift was λ0 = 1 – (N/8N1N2) where N = N1 + N2.

The Hardy-Weinberg law is generally regarded as one of the most important results of population genetics. It was originally proved for the case of populations with distinct generations (Hardy (1908), Weinberg (1908)); a general proof for populations with overlapping generations has apparently not been given before. The case of a single autosomal locus with an arbitrary number of alleles is considered here. Births and deaths are assumed to occur in continuous time. The weak ergodicity property of the birth rate and age structure of such a population, first derived by Norton (1928), is used to establish the fact that allele frequencies tend to constant limits in the absence of mutation, migration, selection and genetic drift.

This paper develops a comparison method for critical branching processes. The method is applied to prove the exponential limit law for the multi-type age-dependent process under second moment conditions.

An abstract gambler's problem is formulated in a continuous-time setting and analogues are proved for some of the discrete-time results of Dubins and Savage in their book How to Gamble if You Must. Applications are made to problems of controlling a Brownian motion process.

A single server faces a renewal stream of low and another one of high priority customers, one of them being a Poisson stream. For the preemptive resume and head-of-the line disciplines the transient behaviour of the queue is studied. Various limiting distributions are also obtained.

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.

Information is available from a diallel cross consisting of all possible crosses among n parental lines, excluding the parental lines themselves and assuming that reciprocal crosses are identical. The lines are assumed to be a random sample from an infinite population of such lines. The general combining abilities of the lines may be more important than the specific combining abilities of the individual crosses. If so, we may decide to rank the lines on the basis of their estimated general combining abilities and make crosses for further testing, or for commercial production, using only lines with high estimated general combining abilities. Existing statistical selection theory can be used to predict the expected genetic superiority of these crosses. For n = 6, 10, and 20, the best five crosses would be expected to result from crossing the lines as follows, where the line number is its rank order on estimated general combining ability: 1 × 2, 1 × 3, 1 × 4, 2 × 3, 1 × 5. Crosses 2×3 and 1 × 5 may not necessarily be expected to be in the stated order, depending on the value of n. In a particular trial, the estimated general combining abilities may suggest that other crosses should be made, but the practical importance of this is doubtful. Note that if only three crosses are to be selected, 1 × 4 is expected to be better than 2 × 3, i.e., the best three crosses are not obtained by crossing the best three lines. This method of selection based on general combining abilities will be referred to as line selection.