arXiv subjects

Alan J. McKane

Publications and source records attributed to Alan J. McKane.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic quasi-cycles as a simple explanation for the time evolution of the Cape Rodney-Okakari Point Marine ecological reserve

The dataset collected at the Cape Rodney-Okakari Point Marine (CR-OPM) reserve on the North Island of New Zealand is rather unique. It describes the cyclic time evolution of a rocky intertidal community, with the relative abundances of the various coastal species that have been meticulously monitored for more than 20 years. Past theoretical studies, anchored on a deterministic description, required invoking ad hoc mechanisms to reproduce the observed dynamical paths. Following a maximum likelihood approach to interpolate individual stochastic trajectories, we here propose quasi-cycles as an alternative and simpler mechanism to explain the oscillations observed in the population numbers of the ecosystem. From a general standpoint, we also show that it is possible to return conclusive evidence on the existence of stochastic quasi-cycles, without resorting to global fitting strategies which necessitate handling a large collection of independent replicas of the dynamics, a possibility that is often precluded in real life applications.

q-bio.PE↗

arXiv2018

Exploiting fast-variables to understand population dynamics and evolution

We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are caused by individual events while the dynamics are described in terms of the time-evolution of a probability density function. In general, the application of the diffusion approximation still leaves a description that is quite complex. However, in many biological applications one or more of the processes happen slowly relative to the system's other processes, and the dynamics can be approximated as occurring within a slow low-dimensional subspace. We review these time-scale separation arguments and analyse the more simple stochastic dynamics that result in a number of cases. We stress that it is important to retain the demographic noise derived in this way, and emphasise this point by showing that it can alter the direction of selection compared to the prediction made from an analysis of the corresponding deterministic model.

q-bio.PE↗

arXiv2018

Reduction of a metapopulation genetic model to an effective one island model

We explore a model of metapopulation genetics which is based on a more ecologically motivated approach than is frequently used in population genetics. The size of the population is regulated by competition between individuals, rather than by artificially imposing a fixed population size. The increased complexity of the model is managed by employing techniques often used in the physical sciences, namely exploiting time-scale separation to eliminate fast variables and then constructing an effective model from the slow modes. Remarkably, an initial model with 2$\mathcal{D}$ variables, where $\mathcal{D}$ is the number of islands in the metapopulation, can be reduced to a model with a single variable. We analyze this effective model and show that the predictions for the probability of fixation of the alleles and the mean time to fixation agree well with those found from numerical simulations of the original model.

q-bio.PE↗

arXiv2017

A mapping of the stochastic Lotka-Volterra model to models of population genetics and game theory

The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection and the Moran model with frequency-dependent selection (equivalently, a game-theoretic formulation of the Moran model). We demonstrate how these mappings can be used to calculate extinction probabilities and the times until a species' extinction in the SLVC model.

q-bio.PE↗

arXiv2016

Stochastic epidemic dynamics on extremely heterogeneous networks

Networks of contacts capable of spreading infectious diseases are often observed to be highly heterogeneous, with the majority of individuals having fewer contacts than the mean, and a significant minority having relatively very many contacts. We derive a two-dimensional diffusion model for the full temporal behavior of the stochastic susceptible-infectious-recovered (SIR) model on such a network, by making use of a time-scale separation in the deterministic limit of the dynamics. This low-dimensional process is an accurate approximation to the full model in the limit of large populations, even for cases when the time-scale separation is not too pronounced, provided the maximum degree is not of the order of the population size.

physics.soc-ph↗

arXiv2016

Suppressing escape events in maps of the unit interval with demographic noise

We explore the properties of discrete-time stochastic processes with a bounded state space, whose deterministic limit is given by a map of the unit interval. We find that, in the mesoscopic description of the system, the large jumps between successive iterates of the process can result in probability leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating these escape events, and we compare the results with those obtained for the metastable behavior of the microscopic model, where escape events are not possible. We find that, rather than truncating the noise distribution, censoring this distribution to prevent escape events leads to results which are more consistent with the microscopic model. The addition of higher moments to the noise distribution does not increase the accuracy of the final results, and it can be replaced by the simpler Gaussian noise.

cond-mat.stat-mech↗

arXiv2016

Demographic noise can reverse the direction of deterministic selection

Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviours will be driven to extinction by mutant cheats that absorb common benefits but do not themselves contribute. Here we show that when demographic stochasticity is accounted for, selection can in fact act in the reverse direction to that predicted deterministically, instead favouring cooperative behaviors that appreciably increase the carrying capacity of the population. Populations that exist in larger numbers experience a selective advantage by being more stochastically robust to invasions than smaller populations, and this advantage can persist even in the presence of reproductive costs. We investigate this general effect in the specific context of public goods production and find conditions for stochastic selection reversal leading to the success of public good producers. This insight, developed here analytically, is missed by both the deterministic analysis as well as standard game theoretic models that enforce a fixed population size. The effect is found to be amplified by space; in this scenario we find that selection reversal occurs within biologically reasonable parameter regimes for microbial populations. Beyond the public good problem, we formulate a general mathematical framework for models that may exhibit stochastic selection reversal. In this context, we describe a stochastic analogue to r-K theory, by which small populations can evolve to higher densities in the absence of disturbance.

q-bio.PE↗

arXiv2016

Simplification and analysis of a model of social interaction in voting

A recently proposed model of social interaction in voting is investigated by simplifying it down into a version that is more analytically tractable and which allows a mathematical analysis to be performed. This analysis clarifies the interplay of the different elements present in the system --- social influence, heterogeneity and noise --- and leads to a better understanding of its properties. The origin of a regime of bistability is identified. The insight gained in this way gives further intuition into the behaviour of the original model.

physics.soc-ph↗

arXiv2016

Staged Models for Interdisciplinary Research

Modellers of complex biological or social systems are often faced with an invidious choice: to use simple models with few mechanisms that can be fully analysed, or to construct complicated models that include all the features which are thought relevant. The former ensures rigour, the latter relevance. We discuss a method that combines these two approaches, beginning with a complex model and then modelling the complicated model with simpler models. The resulting "chain" of models ensures some rigour and relevance. We illustrate this process on a complex model of voting intentions, constructing a reduced model which agrees well with the predictions of the full model. Experiments with variations of the simpler model yield additional insights which are hidden by the complexity of the full model. This approach facilitated collaboration between social scientists and physicists -- the complex model was specified based on the social science literature, and the simpler model constrained to agree (in core aspects) with the complicated model.

physics.soc-ph↗

arXiv2016

The role of demographic stochasticity in a speciation model with sexual reproduction

Recent theoretical studies have shown that demographic stochasticity can greatly increase the tendency of asexually reproducing phenotypically diverse organisms to spontaneously evolve into localised clusters, suggesting a simple mechanism for sympatric speciation. Here we study the role of demographic stochasticity in a model of competing organisms subject to assortative mating. We find that in models with sexual reproduction, noise can also lead to the formation of phenotypic clusters in parameter ranges where deterministic models would lead to a homogeneous distribution. In some cases, noise can have a sizeable effect, rendering the deterministic modelling insufficient to understand the phenotypic distribution.

q-bio.PE↗

arXiv2016

Intrinsic noise in systems with switching environments

We study individual-based dynamics in finite populations, subject to randomly switching environmental conditions. These are inspired by models in which genes transition between on and off states, regulating underlying protein dynamics. Similarly switches between environmental states are relevant in bacterial populations and in models of epidemic spread. Existing piecewise-deterministic Markov process (PDMP) approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching. Here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear-noise approximation. Specifically we derive the stationary distributions of a number of model systems, in good agreement with simulations. This improves existing approaches which are limited to the regimes of fast and slow switching.

cond-mat.stat-mech↗

arXiv2015

Onset of collective motion in locusts is captured by a minimal model

We present a minimal model to describe the onset of collective motion seen when a population of locusts are placed in an annular arena. At low densities motion is disordered, while at high densities locusts march in a common direction, which may reverse during the experiment. The data is well-captured by an individual-based model, in which demographic noise leads to the observed density-dependent effects. By fitting the model parameters to equation-free coefficients, we give a quantitative comparison, showing time series, stationary distributions and the mean switching times between states.

q-bio.PE↗

arXiv2015

Modes of competition and the fitness of evolved populations

Competition between individuals drives the evolution of whole species. Although the fittest individuals survive the longest and produce the most offspring, in some circumstances the resulting species may not be optimally fit. Here, using theoretical analysis and stochastic simulations of a simple model ecology, we show how the mode of competition can profoundly affect the fitness of evolved species. When individuals compete directly with one another, the adaptive dynamics framework provides accurate predictions for the number and distribution of species, which occupy positions of maximal fitness. By contrast, if competition is mediated by the consumption of a common resource then demographic noise leads to the stabilization of species with near minimal fitness.

q-bio.PE↗

arXiv2015

WKB versus generalized van Kampen system-size expansion: the stochastic logistic equation

Stochastic fluctuations are central to the understanding of extinction dynamics. In the context of population models they allow for the description of the transition from the vicinity of a non-trivial fixed point of the deterministic dynamics to a trivial fixed point, where the population has become extinct. To characterize analytically the fluctuations of a given stochastic population model, one can operate within the so-called linear-noise approximation. Here the fluctuations are taken to be Gaussian, and the phenomenon of extinction is so rare as to be negligible, for all practical purposes. When the size of the population becomes small, non-Gaussian fluctuations are instead found. Two analytical schemes are in principle available to determine the nature of the distribution of associated fluctuations: the generalized van Kampen system-size expansion beyond the conventional order of approximation and a WKB-like method. Here we investigate the accuracy of these two different approximation schemes, with reference to a simple stochastic process that converges to the logistic equation in the deterministic limit.

cond-mat.stat-mech↗

arXiv2015

Stationary solutions for metapopulation Moran models with mutation and selection

We construct an individual-based metapopulation model of population genetics featuring migration, mutation, selection and genetic drift. In the case of a single `island', the model reduces to the Moran model. Using the diffusion approximation and timescale separation arguments, an effective one-variable description of the model is developed. The effective description bears similarities to the well-mixed Moran model with effective parameters which depend on the network structure and island sizes, and is amenable to analysis. Predictions from the reduced theory match the results from stochastic simulations across a range of parameters. The nature of the fast-variable elimination technique we adopt is further studied by applying it to a linear system, where it provides a precise description of the slow-dynamics in the limit of large timescale separation.

q-bio.PE↗

arXiv2015

Models of genetic drift as limiting forms of the Lotka-Volterra competition model

The relationship between the Moran model and stochastic Lotka-Volterra competition (SLVC) model is explored via timescale separation arguments. For neutral systems the two are found to be equivalent at long times. For systems with selective pressure, their behavior differs. It is argued that the SLVC is preferable to the Moran model since in the SLVC population size is regulated by competition, rather than arbitrarily fixed as in the Moran model. As a consequence, ambiguities found in the Moran model associated with the introduction of more complex processes, such as selection, are avoided.

q-bio.PE↗

arXiv2015

Impact of commuting on disease persistence in heterogeneous metapopulations

We use a stochastic metapopulation model to study the combined effects of seasonality and spatial heterogeneity on disease persistence. We find a pronounced effect of enhanced persistence associated with strong heterogeneity, intermediate coupling strength and moderate seasonal forcing. Analytic calculations show that this effect is not related with the phase lag between epidemic bursts in different patches, but rather with the linear stability properties of the attractor that describes the steady state of the system in the large population limit.

q-bio.PE↗

arXiv2015

The statistics of fixation times for systems with recruitment

We investigate the statistics of the time taken for a system driven by recruitment to reach fixation. Our model describes a series of experiments where a population is confronted with two identical options, resulting in the system fixating on one of the options. For a specific population size, we show that the time distribution behaves like an inverse Gaussian with an exponential decay. Varying the population size reveals that the timescale of the decay depends on the population size and allows the critical population number, below which fixation occurs, to be estimated from experimental data.

cond-mat.stat-mech↗