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)).

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.

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.

The imago of the peppered moth exists in two sharply distinguished forms, speckled and black. Clarke and Sheppard (1966) have shown that the frequency of the black moth decreases from 99% in Liverpool to 10% in North Wales. If these frequencies are in equilibrium, albeit dynamic equilibrium, then such a gradient is called a cline. Haldane (1948) briefly describes several clines, and then defines a model that may represent the life-cycle of certain species. The assumptions defining his model are as follows.

In 1958 Kimura introduced his Maximum Principle in the Genetical Theory of Natural Selection, which has since been described in several books (Ewens (1969), Crow and Kimura (1970), Kojima (1970)). I shall use Ewens' notation and treatment which, although it uses the discrete-generation model, is formally equivalent to Kimura's continuous-time treatment.

There is now a large literature on linkage disequilibrium between pairs of loci, both for selection in infinite populations and for neutral genes in finite populations, but there have been few studies with more loci. Bennett (1954) showed how the frequencies of chromosomes with any number of neutral genes would change in an infinite population, and the author (unpublished) has extended Bennett's results to find expected changes in chromosome frequencies with up to six loci in finite populations. For two linked neutral genes in finite populations the expected disequilibrium is zero, but the variance of the disequilibrium or the correlation of gene frequencies in segregating populations has been found. This has been done by Monte Carlo simulation (Hill and Robertson (1968)), but an approximation can be obtained by diffusion methods (Ohta and Kimura (1969)) and the asymptotic values using inbreeding theory (Sved (1971)). In this note we discuss the case of disequilibrium between three neutral loci and show how it relates to that between two loci.

The paper discussed the value of the ‘Optimum Model” in describing the inter-relationship between metric characters and natural fitness. This model represents one extreme of a whole spectrum of models. It can be used to predict changes in the mean of the character under natural selection, but the basic premises underlying such use are not well understood.

A classical question in population genetics is ‘In a species in which there is obligate outcrossing is there an increased amount of heterozygosity?’. This somewhat vague question can have two different meanings: (1) Is the number of alleles maintained at a locus greater? and (2) If there are two alleles at a particular locus is the number of heterozygotes formed greater? It is the last question which is discussed here.

Let {XN(t)} be a sequence of continuous time Markov population processes on an n-dimensional integer lattice, such that XN has initial state Nx(0) and has a finite number of possible transitions J from any state X: let the transition X → X + J have rate NgJ(N–1X), and let gJ(x) and x (0) be fixed as N varies. The rate of convergence of √N(N–1XN(t) — ζ (t)) to a Gaussian diffusion is investigated, where ζ(t) is the deterministic approximation to N–1XN(t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic.

We consider a finite controlled Markov chain, the description of which depends on an unknown parameter a, and investigate the following control policy. To each a an optimal stationary control is associated. a is estimated recurrently from the trajectory by the minimum contrast method, and the optimal stationary control corresponding to the estimate is used. We present asymptotic properties of the estimate and of the criterion function. They follow from the law of large numbers and from the central limit theorem for controlled Markov chains derived with the aid of martingales.

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

In this paper we consider two optimization problems and two game problems. In each problem, a particle is hidden on the real line (sometimes randomly, and sometimes by an antagonistic hider), and a seeker, starting at the origin, wishes to find the particle with minimal expected cost. We consider a fairly wide class of cost functions depending upon the position of the particle and the time used to discover it. For the games we obtain the values and (∊-) optimal strategies. For the optimization problems we obtain qualitative features of (∊-) optimal searches.

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, flats, and networks in Euclidean spaces, pattern recognition, random coverage and packing, random search, stereology, and probabilistic aspects of integral geometry.

The following properties of entropies, as measures of expected information, seem natural. The amount of information expected from an experiment does not change if we add outcomes of zero probability (expansibility). The expected information is symmetric in the (probabilities of the) outcomes. The information expected from a combination of two experiments is less than or equal to the sum of the informations expected from the single experiments (subadditivity); equality holds here if the two experiments are independent (additivity).

In this paper it is shown that linear combinations of the Shannon and Hartley entropies and only these have the above properties. The Shannon and the Hartley entropies are also individually characterized.

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by mn spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and mn grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.

Kennedy (1972) showed that the standard single-server queueing model is continuous. These results are extended to the standard multi-server model here. Even when there is only one server, an additional condition is needed for the queue length process.