Alan J. McKane

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 amplification in epidemics

The role of stochasticity and its interplay with nonlinearity are central current issues in studies of the complex population patterns observed in nature, including the pronounced oscillations of wildlife and infectious diseases. The dynamics of childhood diseases have provided influential case studies to develop and test mathematical models with practical application to epidemiology, but are also of general relevance to the central question of whether simple nonlinear systems can explain and predict the complex temporal and spatial patterns observed in nature outside laboratory conditions. Here, we present a stochastic theory for the major dynamical transitions in epidemics from regular to irregular cycles, which relies on the discrete nature of disease transmission and low spatial coupling. The full spectrum of stochastic fluctuations is derived analytically to show how the amplification of noise varies across these transitions. The changes in noise amplification and coherence appear robust to seasonal forcing, questioning the role of seasonality and its interplay with deterministic components of epidemiological models. Childhood diseases are shown to fall into regions of parameter space of high noise amplification. This type of ‘endogenous’ stochastic resonance may be relevant to population oscillations in nonlinear ecological systems in general.

Stochastic models of evolution in genetics, ecology and linguistics

We give an overview of stochastic models of evolution that have found applications in genetics, ecology and linguistics for an audience of non- specialists, especially statistical physicists. In particular, we focus mostly on neutral models in which no intrinsic advantage is ascribed to a particular type of the variable unit, for example a gene, appearing in the theory. In many cases these models are exactly solvable and furthermore go some way to describing observed features of genetic, ecological and linguistic systems.

Stochastic formulation of ecological models and their applications

The increasing use of computer simulation by theoretical ecologists started a move away from models formulated at the population level towards individual-based models. However, many of the models studied at the individual level are not analysed mathematically and remain defined in terms of a computer algorithm. This is not surprising, given that they are intrinsically stochastic and require tools and techniques for their study that may be unfamiliar to ecologists. Here, we argue that the construction of ecological models at the individual level and their subsequent analysis is, in many cases, straightforward and leads to important insights. We discuss recent work that highlights the importance of stochastic effects for parameter ranges and systems where it was previously thought that such effects would be negligible.

Modeling language change: An evaluation of Trudgill's theory of the emergence of New Zealand English

Trudgill (2004) proposed that the emergence of New Zealand English, and of isolated new dialects generally, is purely deterministic. It can be explained solely in terms of the frequency of occurrence of particular variants and the frequency of interactions between different speakers in the society. Trudgills theory is closely related to usage-based models of language, in which frequency plays a role in the representation of linguistic knowledge and in language change. Trudgills theory also corresponds to a neutral evolution model of language change. We use a mathematical model based on Crofts usage-based evolutionary framework for language change (Baxter, Blythe, Croft, & McKane, 2006), and investigate whether Trudgills theory is a plausible model of the emergence of new dialects. The results of our modeling indicate that determinism cannot be a sufficient mechanism for the emergence of a new dialect. Our approach illustrates the utility of mathematical modeling of theories and of empirical data for the study of language change.

LOCALIZATION AS AN ALTERNATIVE TO GOLDSTONE'S THEOREM

Abstract We study the field theory approach to Anderson localization paying special attention to the way in which localized states avoid Goldstones theorem. We provide a formal derivation of Wegners non-linear σ model for the localization transition and discuss those properties which are unusual, but necessary, if it is to give a correct description of physics near the mobility edge.

Functional determinants by contour integration methods

Abstract We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible, the general idea is first illustrated on the simplest case: a second order differential operator with Dirichlet boundary conditions. The method is applicable to more general situations, and we discuss the way in which the formalism has to be developed to cover these cases. In particular, we also show that simple and elegant formulae exist for the physically important case of determinants where zero modes exist, but have been excluded.

SCALE INVARIANCE, INTERFACES AND NON-EQUILIBRIUM DYNAMICS

Statistical Physics of Growth Processes J. Krug. Shapes and Dynamics of Laplacian Growth V. Hakim. Growth Patterns: From Stable Curved Fronts to Fractal Structures (A Summary) Y. Couder. Non-equilibrium Dynamics of Fluctuating Lines M. Kardar, D. Ertas. An Introduction to the Statistical Mechanics of Driven Diffusive Systems R.K.P. Zia, T. Blum. Cellular Automata Modeling of Hydrodynamics and Reactiondiffusion Processes: Basic Theory B. Chopard. Non-equilibrium Ordering Dynamics and Pattern Formation M. Siegert. Populational Dynamics and Coupled Map Lattices I.M. Janosi. Computer Simulation of Molecular Beam Epitaxy D.E. Wolf. Instabilities in MBE Growth: The SLUG Model A.W. Hunt, et al. Generic Scale Invariance and Self-organized Criticality G. Grinstein. Minimal SOC: Intermittency in Growth and Evolution K. Sneppen. A Self-organized Critical Pinball Machine H. Flyvbjerg. Experiments on Bidimensional Models of Sand: Study of the Dynamics J. Rajchenbach, et al. Index.

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.

Functional determinants for general Sturm-Liouville problems

Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm– Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses the properties of generalized zeta-functions. The general form of the final results is the same for the entire range of problems considered.