Biomedical subjects

A J McKane

Publications and source records attributed to A J McKane.

8 recordsLinked to original sources

PubMed2006

Utterance selection model of language change.

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of population genetics, but there are significant differences. The continuous time formulation of the model is expressed in terms of a Fokker-Planck equation. This equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight. Whilst the stationary properties of this system have much in common with the single-speaker case, time-dependent properties are richer. In the particular case where linguistic forms can become extinct, we find that the presence of many speakers causes a two-stage relaxation, the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations.

Journal Article↗

PubMed2005

Spatiotemporal dynamics in marginal populations.

Population dynamics across a mortality gradient at an ecological margin are investigated using a novel modeling approach that allows direct comparison of stochastic spatially explicit simulation results with deterministic mean field models. The results show that demographic stochasticity has a large effect at population margins such that density profiles fall off more sharply than predicted by mean field models. Substantial spatial structure emerges at the margin, and spatial correlations (measured parallel to the margin) exhibit a sharp maximum in the tail of the density profile, indicating that spatial substructuring is greatest at an intermediate point across the ecological gradient. Such substructuring may have a substantial impact on Allee effects and evolutionary processes in marginal populations.

Animals↗

PubMed2005

Predator-prey cycles from resonant amplification of demographic stochasticity.

We present the simplest individual level model of predator-prey dynamics and show, via direct calculation, that it exhibits cycling behavior. The deterministic analogue of our model, recovered when the number of individuals is infinitely large, is the Volterra system (with density-dependent prey reproduction) which is well known to fail to predict cycles. This difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite. Our results indicate that additional biological mechanisms, such as predator satiation, may not be necessary to explain observed predator-prey cycles in real (finite) populations.

Animals↗

PubMed2004

Path integrals and fluctuations in irreversible thermodynamics.

We express the set of stochastic differential equations which describe fluctuations in linear irreversible thermodynamics in terms of path integrals. The stochastic terms which are added to the linearized macroscopic equations have a correlation matrix that is singular, which implies that the straightforward formulation of the problem in terms of path integrals fails. We therefore begin by constructing a path-integral representation which is valid whether or not the correlation matrix is singular. We apply this to linearized irreversible thermodynamics, but the technique is designed to be applicable to more general versions of the theory. The approach emphasizes the role of the response and correlation functions as basic elements of the theory, and we calculate these quantities explicitly for the case of density fluctuations in a fluid.

Journal Article↗

PubMed2004

Stochastic models in population biology and their deterministic analogs.

We introduce a class of stochastic population models based on "patch dynamics." The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

Adaptation, Physiological↗

PubMed2001

Fluctuation dissipation theorems and irreversible thermodynamics.

We investigate the statistics of fluctuations in macroscopic systems described by thermodynamics. We begin by reviewing fluctuations in the context of linear irreversible thermodynamics and show that a more direct characterization of the fluctuations is possible, if velocity fluctuations are explicitly included in the second variation of the entropy, delta2S, about the equilibrium state. A similar procedure is then applied to what is the main goal of this paper: elucidating the nature of fluctuations in hyperbolic macroscopic systems, where signals have a finite transmission velocity. We find that, once again, velocity fluctuations have to be explicitly included, which takes us outside of extended irreversible thermodynamics as it is often defined. We find the explicit form of the fluctuation-dissipation theorem in this case, and determine the statistics of the stochastic variables in terms of the quantities appearing in the deterministic dynamics. The fluctuating theory is then reformulated in order to elucidate the relationship between the extended theory and linear irreversible thermodynamics. This has the effect of bringing out the general structure more clearly: the real, frequency-independent transport coefficients of linear irreversible thermodynamics are replaced by their complex, frequency-dependent counterparts in the extended theory.

Journal Article↗

PubMed2001

Connecting the vulcanization transition to percolation.

The vulcanization transition is addressed via a minimal replica-field-theoretic model. The appropriate long-wavelength behavior of the two- and three-point vertex functions is considered diagrammatically, to all orders in perturbation theory, and identified with the corresponding quantities in the Houghton-Reeve-Wallace field-theoretic approach to the percolation critical phenomenon. Hence, it is shown that percolation theory correctly captures the critical phenomenology of the vulcanization transition associated with the liquid and critical states.

Journal Article↗

PubMed2001

The influence of predator--prey population dynamics on the long-term evolution of food web structure.

We develop a set of equations to describe the population dynamics of many interacting species in food webs. Predator-prey interactions are nonlinear, and are based on ratio-dependent functional responses. The equations account for competition for resources between members of the same species, and between members of different species. Predators divide their total hunting/foraging effort between the available prey species according to an evolutionarily stable strategy (ESS). The ESS foraging behaviour does not correspond to the predictions of optimal foraging theory. We use the population dynamics equations in simulations of the Webworld model of evolving ecosystems. New species are added to an existing food web due to speciation events, whilst species become extinct due to coevolution and competition. We study the dynamics of species-diversity in Webworld on a macro-evolutionary time-scale. Coevolutionary interactions are strong enough to cause continuous overturn of species, in contrast to our previous Webworld simulations with simpler population dynamics. Although there are significant fluctuations in species diversity because of speciation and extinction, very large-scale extinction avalanches appear to be absent from the dynamics, and we find no evidence for self-organized criticality.

Animals↗