The females of the white grub beetle, Dasylepida ishigakiensis, release two enantiomers of 2-butanol, (R)-2-butanol and (S)-2-butanol. The ratio describing the relative proportions of these two enantiomers (R/S ratio) has not yet been investigated. (R)-2-Butanol has been shown to attract males in laboratory and field experiments, whereas (S)-2-butanol tends to inhibit them. To determine the R/S ratio of the 2-butanol emitted by virgin females, we collected 2-butanol from young (53 days old), mature (63 days old) and old females (73 days old) using water, extracted with an SPME fibre and subsequently injected into GC-MS. The major component of the 2-butanol emitted by the young females was (R)-2-butanol, but as the females aged, the component ratio favoured (S)-2-butanol. Young females released an 80:20 mixture of (R)- and (S)-2-butanol, whereas old females released a 45:55 mixture. The EAG response of male antennae to a 50:50 ratio (racemic mixture) showed a similar dose-response curve to that of (R)-2-butanol. The male orientation responses to (R)-2-butanol decreased when the relative proportion of (S)-2-butanol increased. An inhibitory and/or masking effect of (S)-2-butanol on male orientation behaviour was also observed in the flight tunnel assay. These results suggest that males are more strongly attracted to young females than to old females. We also discuss the possibility of using 2-butanol isomers as a control or monitoring agent for this insect.

Protandric oysters generate a relatively uniform reproductive potential over a wide range of environmental conditions that impose variations in growth rate and life span. Sex-at-length keys applied to survey data show that the female fraction routinely fell between 0.4 and 0.5, regardless of location in the salinity gradient. About 70% of population biomass is female over the same salinity range. Due to the necessary local modulation of the rate of male-to-female conversion to limit the influence of varying growth and life span over the salinity gradient, the number of males always exceeds by a small amount the number of females; thus improving the likelihood of a female having neighbouring males, a necessity for an immobile broadcast spawner. However, oysters at the extremes of the estuarine gradient all yielded populations with divergent sex-ratios. One consequence of reduced generation time brought about by increased mortality from disease should be selection favouring the switch from male to female at smaller size, if disease mortality is strongly female-biased. The site with the longest record of high mortality manifests such an increase. Sites up coastal rivers, putative refuges from disease, harbour animals with the slowest male-to-female conversion rates. Arguably these animals are most similar to the ancestral oyster pre-disease. Marketed animals range from 62% to 69% female. The principal influence of the fishery, and of oyster disease, would seem to be a reduction in lifetime egg production. Dermo disease may have reduced lifetime fecundity of females by nearly a factor of four.

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.

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.

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter.

As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk(t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk(t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {Gi(t)}, together with a probability generating function, h(s) = Σk∞= 0pksk. An ith generation particle lives a random length of time, determined by Gi(t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution Gi + 1 (t).

Let Zi(t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Zi(t) and it is shown that for some constants Ni, γ, a, Zi (t)/(Nitγ-1eαt) converges in mean square to a proper random variable.