arXiv subjects

Alan J. McKane

Publications and source records attributed to Alan J. McKane.

At least 55 records · Page 3Linked to original sources

arXiv2011

Stability of growing vesicles

We investigate the stability of growing vesicles using the formalism of nonequilibrium thermodynamics. The vesicles are growing due to the accretion of lipids to the bilayer which forms the vesicle membrane. The thermodynamic description is based on the hydrodynamics of a water{/}lipid mixture together with a model of the vesicle as a discontinuous system in the sense of linear nonequilibrium thermodynamics. This formulation allows the forces and fluxes relevant to the dynamic stability of the vesicle to be identified. The method is used to analyze the stability of a spherical vesicle against arbitrary axisymmetric perturbations. It is found that there are generically two critical radii at which changes of stability occur. In the case where the perturbation takes the form of a single zonal harmonic, only one of these radii is physical and is given by the ratio $2 L_p / L_γ$, where $L_p$ is the hydraulic conductivity and $L_γ$ is the Onsager coefficient related to changes in membrane area due to lipid accretion. The stability of such perturbations is related to the value of $l$ corresponding to the particular zonal harmonic: those with lower $l$ are more unstable than those with higher $l$. Possible extensions of the current work and the need for experimental input are discussed.

cond-mat.soft↗

arXiv2011

A unified framework for Schelling's model of segregation

Schelling's model of segregation is one of the first and most influential models in the field of social simulation. There are many variations of the model which have been proposed and simulated over the last forty years, though the present state of the literature on the subject is somewhat fragmented and lacking comprehensive analytical treatments. In this article a unified mathematical framework for Schelling's model and its many variants is developed. This methodology is useful in two regards: firstly, it provides a tool with which to understand the differences observed between models; secondly, phenomena which appear in several model variations may be understood in more depth through analytic studies of simpler versions.

physics.soc-ph↗

arXiv2010

Dynamical description of vesicle growth and shape change

We systematize and extend the description of vesicle growth and shape change using linear nonequilibrium thermodynamics. By restricting the study to shape changes from spheres to axisymmetric ellipsoids, we are able to give a consistent formulation which includes the lateral tension of the vesicle membrane. This allows us to generalize and correct a previous calculation. Our present calculations suggest that, for small growing vesicles, a prolate ellipsoidal shape should be favored over oblate ellipsoids, whereas for large growing vesicles oblates should be favored over prolates. The validity of this prediction is examined in the light of the various assumptions made in its derivation.

cond-mat.soft↗

arXiv2010

Evolutionary dynamics, intrinsic noise and cycles of co-operation

We use analytical techniques based on an expansion in the inverse system size to study the stochastic evolutionary dynamics of finite populations of players interacting in a repeated prisoner's dilemma game. We show that a mechanism of amplification of demographic noise can give rise to coherent oscillations in parameter regimes where deterministic descriptions converge to fixed points with complex eigenvalues. These quasi-cycles between co-operation and defection have previously been observed in computer simulations; here we provide a systematic and comprehensive analytical characterization of their properties. We are able to predict their power spectra as a function of the mutation rate and other model parameters, and to compare the relative magnitude of the cycles induced by different types of underlying microscopic dynamics. We also extend our analysis to the iterated prisoner's dilemma game with a win-stay lose-shift strategy, appropriate in situations where players are subject to errors of the trembling-hand type.

q-bio.PE↗

arXiv2010

Diffusion in a crowded environment

We analyze a pair of diffusion equations which are derived in the infinite system--size limit from a microscopic, individual--based, stochastic model. Deviations from the conventional Fickian picture are found which ultimately relate to the depletion of resources on which the particles rely. The macroscopic equations are studied both analytically and numerically, and are shown to yield anomalous diffusion which does not follow a power law with time, as is frequently assumed when fitting data for such phenomena. These anomalies are here understood within a consistent dynamical picture which applies to a wide range of physical and biological systems, underlining the need for clearly defined mechanisms which are systematically analyzed to give definite predictions.

cond-mat.stat-mech↗

arXiv2010

A spatial model of autocatalytic reactions

Biological cells with all of their surface structure and complex interior stripped away are essentially vesicles - membranes composed of lipid bilayers which form closed sacs. Vesicles are thought to be relevant as models of primitive protocells, and they could have provided the ideal environment for pre-biotic reactions to occur. In this paper, we investigate the stochastic dynamics of a set of autocatalytic reactions, within a spatially bounded domain, so as to mimic a primordial cell. The discreteness of the constituents of the autocatalytic reactions gives rise to large sustained oscillations, even when the number of constituents is quite large. These oscillations are spatio-temporal in nature, unlike those found in previous studies, which consisted only of temporal oscillations. We speculate that these oscillations may have a role in seeding membrane instabilities which lead to vesicle division. In this way synchronization could be achieved between protocell growth and the reproduction rate of the constituents (the protogenetic material) in simple protocells.

cond-mat.stat-mech↗

arXiv2009

Limit cycles, complex Floquet multipliers and intrinsic noise

We study the effects of intrinsic noise on chemical reaction systems, which in the deterministic limit approach a limit cycle in an oscillatory manner. Previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude. We show that a similar effect occurs when the stable attractors are limit cycles. We compute the correlation functions and spectral properties of the fluctuations in suitably co-moving Frenet frames for several model systems including driven and coupled Brusselators, and the Willamowski-Roessler system. Analytical results are confirmed convincingly in numerical simulations. The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.

cond-mat.stat-mech↗

arXiv2008

Thermodynamics of vesicle growth and instability

We describe the growth of vesicles, due to the accretion of lipid molecules to their surface, in terms of linear irreversible thermodynamics. Our treatment differs from those previously put forward by consistently including the energy of the membrane in the thermodynamic description. We calculate the critical radius at which the spherical vesicle becomes unstable to a change of shape in terms of the parameters of the model. The analysis is carried out both for the case when the increase in volume is due to the absorption of water and when a solute is also absorbed through the walls of the vesicle.

cond-mat.stat-mech↗

arXiv2008

Effects of food web construction by evolution or immigration

We present results contrasting food webs constructed using the same model where the source of species was either evolution or immigration from a previously evolved species pool. The overall structure of the webs are remarkably similar, although we find some important differences which mainly relate to the percentage of basal and top species. Food webs assembled from evolved webs also show distinct plateaux in the number of tropic levels as the resources available to system increase, in contrast to evolved webs. By equating the resources available to basal species to area, we are able to examine the species-area curve created by each process separately. They are found to correspond to different regimes of the tri-phasic species-area curve.

q-bio.PE↗

arXiv2008

Quasi-cycles in a spatial predator-prey model

We show that spatial models of simple predator-prey interactions predict that predator and prey numbers oscillate in time and space. These oscillations are not seen in the deterministic versions of the models, but are due to stochastic fluctuations about the time-independent solutions of the deterministic equations which are amplified due to the existence of a resonance. We calculate the power spectra of the fluctuations analytically and show that they agree well with results obtained from stochastic simulations. This work extends the analysis of these quasi-cycles from that previously developed for well-mixed systems to spatial systems, and shows that the ideas and methods used for non-spatial models naturally generalize to the spatial case.

q-bio.PE↗

arXiv2008

How limit cycles and quasi-cycles are related in systems with intrinsic noise

Fluctuations and noise may alter the behavior of dynamical systems considerably. For example, oscillations may be sustained by demographic fluctuations in biological systems where a stable fixed point is found in the absence of noise. We here extend the theoretical analysis of such stochastic effects to models which have a limit cycle for some range of the model parameters. We formulate a description of fluctuations about the periodic orbit which allows the relation between the stochastic oscillations in the fixed point phase and the oscillations in the limit cycle phase to be elucidated. In the case of the limit cycle, a suitable transformation into a co-moving frame allow fluctuations transverse and longitudinal with respect to the limit cycle to be effectively decoupled. While longitudinal fluctuations are of a diffusive nature, those in the transverse direction follow a stochastic path more akin to an Ornstein-Uhlenbeck process. Their power spectrum is computed analytically within a van Kampen expansion in the inverse system size. This is carried out in two different ways, and the subsequent comparison with numerical simulations illustrates the effects that can occur due to diffusion in the longitudinal direction.

nlin.AO↗

arXiv2008

Predicting the species abundance distribution using a model food web

A large number of models of the species abundance distribution (SAD) have been proposed, many of which are generically similar to the log-normal distribution, from which they are often indistinguishable when describing a given data set. Ecological data sets are necessarily incomplete samples of an ecosystem, subject to statistical noise, and cannot readily be combined to yield a closer approximation to the underlying distribution. In this paper we use empirical data obtained from an ecosystem model to study the predicted SAD in detail, resolving features which can distinguish between models but which are poorly seen in field data. We find that the power-law normal distribution is superior to both the log-normal and logit-normal distributions, and that the data can improve on even this at the high-population cut-off.

q-bio.PE↗

arXiv2005

Functional determinants in the presence of zero modes

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 we illustrate the general ideas using the Laplacian with Dirichlet boundary conditions on the interval. Afterwards, we indicate how more general operators as well as general boundary conditions can be covered.

hep-th↗

arXiv2004

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 generalised zeta-functions. The general form of the final results are the same for the entire range of problems considered.

math-ph↗

arXiv2001

Food Web Structure and the Evolution of Ecological Communities

Simulations of the coevolution of many interacting species are performed using the Webworld model. The model has a realistic set of predator-prey equations that describe the population dynamics of the species for any structure of the food web. The equations account for competition between species for the same resources, and for the diet choice of predators between alternative prey according to an evolutionarily stable strategy. The set of species present undergoes long-term evolution due to speciation and extinction events. We summarize results obtained on the macro-evolutionary dynamics of speciations and extinctions, and on the statistical properties of the food webs that are generated by the model. Simulations begin from small numbers of species and build up to larger webs with relatively constant species number on average. The rate of origination and extinction of species are relatively high, but remain roughly balanced throughout the simulations. When a `parent' species undergoes speciation, the `child' species usually adds to the same trophic level as the parent. The chance of the child species surviving is significantly higher if the parent is on the second or third trophic level than if it is on the first level, most likely due to a wider choice of possible prey for species on higher levels. Addition of a new species sometimes causes extinction of existing species. The parent species has a high probability of extinction because it has strong competition with the new species. Non-parental competitors of the new species also have a significantly higher extinction probability than average, as do prey of the new species. Predators of the new species are less likely than average to become extinct.

nlin.AO↗

arXiv2001

Connecting the vulcanization transition to percolation

The vulcanization transition is addressed via a minimal replica-field-theoretic model. The appropriate long-wave-length 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.

cond-mat.dis-nn↗

arXiv2000

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 non-linear, 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 timescale. 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-organised criticality.

nlin.AO↗

arXiv1998

Modelling coevolution in multispecies communities

We introduce the Webworld model, which links together the ecological modelling of food web structure with the evolutionary modelling of speciation and extinction events. The model describes dynamics of ecological communities on an evolutionary timescale. Species are defined as sets of characteristic features, and these features are used to determine interaction scores between species. A simple rule is used to transfer resources from the external environment through the food web to each of the species, and to determine mean population sizes. A time step in the model represents a speciation event. A new species is added with features similar to those of one of the existing species and a new food web structure is then calculated. The new species may (i) add stably to the web, (ii) become extinct immediately because it is poorly adapted, or (iii) cause one or more other species to become extinct due to competition for resources. We measure various properties of the model webs and compare these with data on real food webs. These properties include the proportions of basal, intermediate and top species, the number of links per species and the number of trophic levels. We also study the evolutionary dynamics of the model ecosystem by following the fluctuations in the total number of species in the web. Extinction avalanches occur when novel organisms arise which are significantly better adapted than existing ones. We discuss these results in relation to the observed extinction events in the fossil record, and to the theory of self-organized criticality.

adap-org↗