This paper is about the total number of individuals who are heterozygotic for a specified allele, before it is either lost or fixed. The exact distribution is found for small populations, and two limiting processes are investigated.

This note points out an error made in an earlier paper treating a diffusion equation for the probability distribution of gene frequencies under selection. The main interest lies in the orthogonality properties of certain eigenfunctions and the determination of the corresponding Fourier coefficients. Two particular cases of selection parameter values are considered. Approximate formulas are given for the two smallest eigenvalues for arbitrary values of the selection parameters.

We study a stochastic differential equation model of prey-predator evolution. To keep in line with a known deterministic model we include the social and interaction terms with the drift in our model, and the randomness arises as fluctuations in the ecosystem. The notion of equilibrium population level and special types of fluctuations force us to work with degenerate elliptic operators. We consider the propagation of the population system in a sufficiently large but bounded domain. This enables us to look at not only the population extinction, but also the explosion beyond a certain level. Both extinction and explosion are possible; and, when they are not, we show that the population asymptotically reaches the equilibrium level. We show that the extinction, explosion and saturation probabilities satisfy, as functions of the initial population size, an integral equation arising out of a Dirichlet problem for a non-degenerate elliptic equation; and these probabilities are also smooth solutions of the Dirichlet problems. They are also used to express the solution of another Dirichlet problem for a degenerate elliptic equation.

An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate, and that the population is sufficiently large for the diffusion model to apply. Bounds on some moments are calculated from the density, and some applications are made to the problem of which allele is oldest in a population. A postulate made by Ewens (1972), concerning the distribution of allele numbers in a finite random sample from the neutral diffusion population, is shown to be correct.

The extinction probability q of a supercritical simple branching process is well known to be less than unity. Intuitively, it is apparent that when the offspring mean is close to one, so, usually, will q be. This notion is made rigorous, and simple bounds are given for q in terms of the second and third factorial moments, which are applicable when the offspring mean is close to unity. A comparison is made of various upper bounds for q. The note contains some numerical examples.

The spatial dispersion of a neutral allele is described using the theory of multitype branching processes in which the types represent colonies between which individuals can migrate. Each mutant individual averages less than one offspring, so the mutant population faces certain extinction. Expressions are given for the first two moments of the total number of individuals to visit specified colonies in one, two and three dimensions. Data from Drosophila populations are used to show the improbability of the same neutral allele occurring at widely separated localities.

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

A stochastic model for inter-plant competition in a single-species population is presented, in which the probability that a given plant will be eliminated is defined in terms of the sizes and relative spatial locations of rival plants. The properties of the model are then investigated and illustrations of particular cases given, with particular emphasis on the distribution of the sizes of surviving plants. Empirical comparisons are made between the properties of the model and some experimental results due to Ford (1975b). Some possible extensions to the basic model are suggested.

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 < α < ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

The parity of an individual in a population process is the number of offspring that individual has produced. If Nk(t) is the number of individuals of parity k at time t in a birth-and-death process and we show that, conditional on N(t)→ ∞,

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.

Using an integral equation of Darling and Siegert in conjunction with the backward Kolmogorov equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of a temporally homogeneous Markov process from a set in the phase space. The results, which are similar to those for diffusion processes, are used to find the expectation of the time between impulses of a Stein model neuron.

The motion of a particle is investigated in the presence of a velocity-dependent random force assumed to be proportional to velocity. Two different possibilities are considered, namely, the presence and absence of random driving force. In the absence of random driving force, the velocity and displacement auto-correlation function are calculated. The probability distribution in velocity space is also evaluated. It is found that in the absence of intrinsic damping, the energy of the particle increases without limit. The condition for the energetic stability of the particle in the presence of random driving force is obtained. The Fokker-Planck equation for the probability distribution in velocity space is derived from the stochastic Liouville equation for delta-correlated velocity-dependent random force.

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.

The method is also applied to the total progeny in the critical process.

We consider a p-type general age-dependent branching process (g.a.d.b.p) in which each individual lives a random length of time and at random times during its life and possibly after its death gives birth to offspring. If this process starts with one individual of type i then it turns out that the number of individuals of type i alive at time t forms a one-type g.a.d.b.p. This fact is exploited to establish a NASC for a properly normalized, supercritical p-type g.a.d.b.p. to have a limiting distribution which is not degenerate at zero.