Diffusion of two molecular species in a crowded environment: theory and experiments.

Diffusion of a two component fluid is studied in the framework of differential equations, but where these equations are systematically derived from a well-defined microscopic model. The model has a finite carrying capacity imposed upon it at the mesoscopic level and this is shown to lead to nonlinear cross diffusion terms that modify the conventional Fickean picture. After reviewing the derivation of the model, the experiments carried out to test the model are described. It is found that it can adequately explain the dynamics of two dense ink drops simultaneously evolving in a container filled with water. The experiment shows that molecular crowding results in the formation of a dynamical barrier that prevents the mixing of the drops. This phenomenon is successfully captured by the model. This suggests that the proposed model can be justifiably viewed as a generalization of standard diffusion to a multispecies setting, where crowding and steric interferences are taken into account.

Phase lag in epidemics on a network of cities.

We study the synchronization and phase lag of fluctuations in the number of infected individuals in a network of cities between which individuals commute. The frequency and amplitude of these oscillations is known to be very well captured by the van Kampen system-size expansion, and we use this approximation to compute the complex coherence function that describes their correlation. We find that, if the infection rate differs from city to city and the coupling between them is not too strong, these oscillations are synchronized with a well-defined phase lag between cities. The analytic description of the effect is shown to be in good agreement with the results of stochastic simulations for realistic population sizes.

Stochastic oscillations in models of epidemics on a network of cities.

We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of disease spread on a network of n cities. In the model a fraction f(jk) of individuals from city k commute to city j, where they may infect, or be infected by, others. Starting from a continuous-time Markov description of the model the deterministic equations, which are valid in the limit when the population of each city is infinite, are recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple: A unique nontrivial fixed point always exists and has the feature that the fraction of susceptible, infected, and recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have an analogously simple dynamics: All oscillations have a single frequency, equal to that found in the one-city case. We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation of the deterministic equations at the fixed point.

Fixation and consensus times on a network: a unified approach.

We investigate a set of stochastic models of biodiversity, population genetics, language evolution, and opinion dynamics on a network within a common framework. Each node has a state 0

Singular solutions of the diffusion equation of population genetics.

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0,1) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0,1) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.

Exact solution of the multi-allelic diffusion model.

We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus, this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.

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.

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.

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.

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.

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.

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.

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.

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.