Correcting mean-field approximations for spatially-dependent advection-diffusion-reaction processes

On the microscale, migration, proliferation and death are crucial in the development, homeostasis and repair of an organism; on the macroscale, such effects are important in the sustainability of a population in its environment. Dependent on the relative rates of migration, proliferation and death, spatial heterogeneity may arise within an initially uniform field; this leads to the formation of spatial correlations and can have a negative impact upon population growth. Usually, such effects are neglected in modeling studies and simple phenomenological descriptions, such as the logistic model, are used to model population growth. In this work we outline some methods for analyzing exclusion processes which include agent proliferation, death and motility in two and three spatial dimensions with spatially homogeneous initial conditions. The mean-field description for these types of processes is of logistic form; we show that, under certain parameter conditions, such systems may display large deviations from the mean field, and suggest computationally tractable methods to correct the logistic-type description

Models of collective cell motion for cell populations with different aspect ratio: diffusion, proliferation & travelling waves

Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this work we provide a new link between individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and population-level partial differential equation models. We work in an exclusion process framework, considering aligned, elongated cells that may occupy more than one lattice site, in order to represent populations of agents with different sizes. Three different idealisations of the individual-level mechanism are proposed, and these are connected to three different partial differential equations, each with a different diffusion coefficient; one linear, one nonlinear and degenerate and one nonlinear and nondegenerate. We test the ability of these three models to predict the population-level response of a cell spreading problem for both proliferative and nonproliferative cases. We also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two-dimensional spreading and invasion. Our results show that each model can accurately predict density data for nonproliferative systems, but that only one does so for proliferative systems. Hence great care must be taken to predict density data with varying cell shape

Models of collective cell spreading with variable cell aspect ration: a motivation for degenerate diffusion models

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multiscale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (PME). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the PME to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models

Low-power radio galaxy environments in the Subaru/XMM-Newton Deep Field at z~0.5

We present multi-object spectroscopy of galaxies in the immediate (Mpc-scale) environments of four low-power (L_1.4 GHz < 10^25 W/Hz) radio galaxies at z~0.5, selected from the Subaru/XMM-Newton Deep Field. We use the spectra to calculate velocity dispersions and central redshifts of the groups the radio galaxies inhabit, and combined with XMM-Newton (0.3-10 keV) X-ray observations investigate the L_X--sigma_v and T_X--sigma_v scaling relationships. All the radio galaxies reside in moderately rich groups -- intermediate environments between poor groups and rich clusters, with remarkably similar X-ray properties. We concentrate our discussion on our best statistical example that we interpret as a low-power (FRI) source triggered within a sub-group, which in turn is interacting with a nearby group of galaxies, containing the bulk of the X-ray emission for the system -- a basic scenario which can be compared to more powerful radio sources at both high (z>4) and low (z<0.1) redshifts. This suggests that galaxy-galaxy interactions triggered by group mergers may play an important role in the life-cycle of radio galaxies at all epochs and luminosities.Comment: 12 pages, 7 figures, accepted for publication in MNRAS. High resolution version available upon reques

The cosmic evolution of radio-AGN feedback to z=1

This paper presents the first measurement of the radio luminosity function of 'jet-mode' (radiatively-inefficient) radio-AGN out to z=1, in order to investigate the cosmic evolution of radio-AGN feedback. Eight radio source samples are combined to produce a catalogue of 211 radio-loud AGN with 0.5<z<1.0, which are spectroscopically classified into jet-mode and radiative-mode (radiatively-efficient) AGN classes. Comparing with large samples of local radio-AGN from the Sloan Digital Sky Survey, the cosmic evolution of the radio luminosity function of each radio-AGN class is independently derived. Radiative-mode radio-AGN show an order of magnitude increase in space density out to z~1 at all luminosities, consistent with these AGN being fuelled by cold gas. In contrast, the space density of jet-mode radio-AGN decreases with increasing redshift at low radio luminosities (L_1.4 < 1e24 W/Hz) but increases at higher radio luminosities. Simple models are developed to explain the observed evolution. In the best-fitting models, the characteristic space density of jet-mode AGN declines with redshift in accordance with the declining space density of massive quiescent galaxies, which fuel them via cooling of gas in their hot haloes. A time delay of 1.5-2 Gyr may be present between the quenching of star formation and the onset of jet-mode radio-AGN activity. The behaviour at higher radio luminosities can be explained either by an increasing characteristic luminosity of jet-mode radio-AGN activity with redshift (roughly as (1+z) cubed) or if the jet-mode radio-AGN population also includes some contribution of cold-gas-fuelled sources seen at a time when their accretion rate was low. Higher redshifts measurements would distinguish between these possibilities.Comment: Accepted for publication in MNRA

Managing urban socio-technical change? Comparing energy technology controversies in three European contexts

A {\em local graph partitioning algorithm} finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph $G$, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number $n$ of vertices in $G$. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the {\em volume-biased evolving set process}, which is a Markov chain on sets of vertices. We prove that for any set of vertices $A$ that has conductance at most $\phi$, for at least half of the starting vertices in $A$ our algorithm will output (with probability at least half), a set of conductance $O(\phi^{1/2} \log^{1/2} n)$. We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is $O(\phi^{-1/2} polylog(n))$. In comparison, the best previous local partitioning algorithm, due to Andersen, Chung, and Lang, has the same approximation guarantee, but a larger ratio of $O(\phi^{-1} polylog(n))$ between the complexity and output volume. Using our local partitioning algorithm as a subroutine, we construct a fast algorithm for finding balanced cuts. Given a fixed value of $\phi$, the resulting algorithm has complexity $O((m+n\phi^{-1/2}) polylog(n))$ and returns a cut with conductance $O(\phi^{1/2} \log^{1/2} n)$ and volume at least $v_{\phi}/2$, where $v_{\phi}$ is the largest volume of any set with conductance at most $\phi$.Comment: 20 pages, no figure

Incorporating spatial correlations into multispecies mean-field models

In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques