The sex of a zygote is often decided by an XX/XY sex-determination mechanism. One important property of this system is the approximate numerical equality of the two sexes (at least at fertilization) due to mendelian segregation. Does the obvious evolutionary stability of the XX/XY sex-determination system pose a difficult problem to population genetics? We study the fate of new mutations occurring in populations with an XX/XY system, where the mutations are such that they change the sex-determination system. If such mutations occur frequently and the conditions for their initial increase are weak, then the evolutionary stability of the XX/XY system is hard to understand.

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

The paper develops a formula for inbreeding probabilities in the multi-locus case, this formula being analogous to that given by Malecot for single-locus chromosomes. A detailed example provides a comparison between the present approach and that by Malecot and it is found that the two methods produce slightly different results. This slight difference is then shown to be due to certain paths in pedigrees which seem to have been overlooked by Malecot. An indication of calculations necessary for higher-locus cases concludes the paper.

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 < m 2 < ··· < m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

The model of a population which grows logistically, except for random disasters that reduce the population by a given amount at the instants of a Poisson process, which was examined by Hanson and Tuckwell (1978), is generalised in two ways:

In the incompatibility system operating in Lythrum salicaria, the three incompatibility types Long, Mid and Short will be equally frequent in a population of infinite size. But in small populations the complete system cannot be maintained indefinitely because of random loss of incompatibility types. A simulation model for such populations shows that this may be an important factor for populations consisting of less than 20 plants. When loss actually occurs, it will in most cases involve the Short type and in some cases the Mid type, but only rarely the Long type. This result may explain why the Long type appears to be the most common one in nature and Short the most infrequent one.

The treatment of selfing in genetic algebras [3] can be extended by the definition of an analogous operator on the copular algebra which corresponds to brother–sister mating, one on the duplicate of the copular algebra corresponding to mating between double first cousins, etc.

Suppose, to take a simple example, we have a set of sibships. Each individual (sib) has a certain measured character x. We wish to estimate the interclass (or sib–sib) correlation for this character. Our model is as follows. In sibship s the observed values x are supposed drawn from a distribution with (true) mean m s and variance v s . The sibships are drawn from a population of sibships, so that the m s have a mean M and variance B, and the v s have mean A. (We also suppose that the number n s of sibs in sibship s is independent of m s and v s .) The interclass correlation is then easily seen to be

ρ = B/(A + B).

The obvious way to proceed is to estimate A and B first, and then find ρ. Take a sample of k sibships. In sibship s the observed mean is and the deviance (sum of squares of deviations) δs . Let n = σ n s = the total number of individuals in the sample, and δ = σ δs = the total deviance within sibships. Proceed iteratively as follows:

Consider a single gene locus for which the associated character is not sex-linked, and suppose that the character is exhibited as one of alleles. Under the Mendelian model of inheritance, if genotype frequencies are constant, the distribution of zygotes is stable from the second generation on, regardless of the value of n. (This is the celebrated Hardy–Weinberg law—see Hardy (1908).) In 1924 Bernstein raised the question of whether the Mendelian model is, in fact, a necessary consequence of the assumption that the zygote distribution is stable from the second generation on; when n = 2 or 3 Bernstein gave an affirmative solution, using elementary but tedious manipulation of probabilities. Lyubich (1971) used methods of convex analysis to obtain a partial solution of Bernstein's problem for general n. Holgate (1975) introduced algebraic methods in order to present a clearer picture of the underlying structures, and to relate Bernstein's problem to the algebras studied by Etherington (1939), Schafer (1949) and others. The purpose of this paper is to expound recent work of the author and others, building on Holgate's and Lyubich's ideas. The subject-matter of the paper is contained in a forthcoming joint paper by the author and A. E. Stratton.

The probability distribution of the heterogenic (non-identical by descent) fraction of the genome in a finite monoecious random mating population has been derived. It was assumed that in any generation both the lengths of heterogenic and homogenic segments are exponentially distributed. An explicit expression is given for the expected number of ‘external junctions’ (sites that mark the end of a heterogenic segment) per unit map length in any generation. The latter necessitates the introduction of two higher-order identity relations between three genes, and their recurrence relations. Theoretical results were compared with the outcome of a series of simulation runs (showing a very good fit), as well as with the results predicted by Fisher's ‘theory of junctions’. In contrast to Fisher's approach, which only applies when the average heterogeneity is relatively small, the present model applies to any generation.

For family data where some of the children may have a certain inherited disorder, the number of affected children has usually been assumed to be binomially distributed given the total number of children in the family. In this assumption is included the assumption that the distribution of the total number of children does not depend on the probability that a child is affected. For many disorders this assumption is unlikely to hold because the birth of an affected child may lead either to some sort of family limitation or to some sort of overcompensation. In such cases models other than the binomial distribution have to be used.

Many medical genetics studies are at least partially sequential, and in considering problems of ascertainment it seems desirable to consider such schemes, in addition to classical ascertainment models. Cannings and Thompson (1978) indicated the correction to the likelihood required when sampling a pedigree sequentially from some predetermined initial event, and Thompson and Cannings (1979) have extended this result and compared sequential sampling with more classical ascertainment models.

Fischer (1930) stated his fundamental theorem of natural selection (ftns) as ‘the rate of increase in fitness of any organism at any time is equal to its genetic (additive) variance in fitness at that time’. The theorem related to an idealised population and he thought that it held ‘the supreme position among the biological sciences’. His proof of the theorem is, however, not understood. This has given rise to questions concerning not only the validity of the proof but also concerning the importance of the theorem.

For N = 1, 2, ···, let {(X N (k), Y N (k)), k = 0, 1, ···} be a homogeneous Markov chain in ℝm x ℝn . Suppose that, asymptotically as N → ∞, the ‘infinitesimal’ covariances and means of X N ([·/∊N ]) are a ij (x, y) and b i (x, y), and those of Y N ([·/δN ]) are 0 and c l (x, y). Assume limN→∞ δN = limN→∞ ∊N /δN = 0 and the zero solution of ý = c(x, y) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i) X N ([·/∊N ]) converges weakly to a diffusion process with coefficients a ij (x, 0) and b i (x, 0), and (ii) Y N ([t/∊N ]) → 0 in probability for every t > 0. (The case limN→∞ δN = δ∞ > 0 = limN→∞ ∊N is also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups.

The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.

The ray solution of Felsenstein's n-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t 1, t 2) if Pr {x(t) > i} is increasing (decreasing) with t on (t 1, t 2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.

We study clustering (‘polymerisation’) processes under the assumptions that new bonds may or may not form within an existing cluster. We show that both models exhibit criticality (‘gelation’) at the same point, and investigate statistics of polymer size and structure beyond criticality in the first case. Gelation may or may not be accompanied by critical effects at unit level; we examine a prototype of each case: the cosh model and the Poisson model.

We consider a polymerisation process stratified in that space is divided into regions, between which migration occurs, but with bonding occurring only within a region. In the case of a process whose specification is spatially homogeneous, criticality (gelation) is then easily detectable as the point at which statistical equidistribution over regions becomes unstable. Stratification does import a new feature, however, in that the equipartition solution can become metastable below criticality; local fluctuations of density can induce ‘gelational collapse’ at a density below the critical value. We derive also detailed results for the inhomogeneous case, both below and above criticality. Statistics of polymer size and structure are also easily determined in the stratified case, although one can locate the critical point without recourse to these. Finally, one can to a large extent treat the case in which inter- and intrapolymer reaction rates differ, and show that such difference affects the onset of metastability rather than of instability.

The solution for the multitype version of the polymerisation process treated in earlier papers is examined, and found to present several new features, because of the necessarily complex integration path, and because of the greater number and variety of potential singularities. Cases of ‘Poisson interaction’ are particularly amenable to treatment, however, and we examine two particular cases of interest in Sections 5 and 6. One is a version of the Ising model, which shows phase transition and its character very simply. The other is a version of the farmer–trader model considered previously, which now manifests a number of features, including a discontinuous transition between agrarian and mercantile economies. A stratified Poisson-interaction model is examined in Section 7. The roles of the various singularities are clarified, and a behaviour analogous to that of the single-type case confirmed.