Tim Rogers

Stochastic pattern formation and spontaneous polarisation: The linear noise approximation and beyond

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

Cavity approach to the spectral density of sparse symmetric random matrices.

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, showing excellent agreement.

Modularity and stability in ecological communities

Networks composed of distinct, densely connected subsystems are called modular. In ecology, it has been posited that a modular organization of species interactions would benefit the dynamical stability of communities, even though evidence supporting this hypothesis is mixed. Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method. Results show that modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks.

Demographic noise can reverse the direction of deterministic selection

Significance Demographic stochasticity—the population-level randomness that emerges when the timing of birth, death, and interaction events is unpredictable—can profoundly alter the dynamics of a system. We find that phenotypes that pay a cost to their birth rate to modify the environment by increasing the global carrying capacity can be stochastically selected for, where they would otherwise be deterministically disfavored. Our results hold for a general class of mathematical models but we use a model of public good production for illustration. In this case, demographic stochasticity is exploited by populations of cooperators to turn selection in their favor; it therefore operates as a mechanism that supports the evolution of public good production. Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviors 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 favoring 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 the deterministic analysis as well as by 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 analog to r−K theory, by which small populations can evolve to higher densities in the absence of disturbance.

Maximum-entropy moment-closure for stochastic systems on networks

Moment-closure methods are popular tools for simplifying the mathematical analysis of stochastic models defined on networks, in which high dimensional joint distributions are approximated (often by some heuristic argument) as functions of lower dimensional distributions. Whilst undoubtedly useful, several such methods suffer from issues of non-uniqueness and inconsistency. These problems are solved by an approach based on the maximization of entropy, which is motivated, derived and implemented in this paper. A series of numerical experiments are also presented, detailing the application of the method to the susceptible–infected–recovered model of epidemics, as well as cautionary examples showing the sensitivity of moment-closure techniques in general.

Demographic noise can lead to the spontaneous formation of species

When a collection of phenotypically diverse organisms compete with each other for limited resources, the population can evolve into tightly localised clusters. Past studies have neglected the effects of demographic noise and studied the population on a macroscopic scale, where cluster formation is found to depend on the shape of the curve describing the decline of competition strength with phenotypic distance. Here we show how including the effects of demographic noise leads to a radically different conclusion. Two situations are identified: a weak-noise regime in which the population exhibits patterns of fluctuation around the macroscopic description, and a strong-noise regime where clusters appear spontaneously even in the case that all organisms have equal fitness.

A unified framework for Schelling's model of segregation

Schellings 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 paper a unified mathematical framework for Schellings 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.

Cavity approach to the spectral density of non-Hermitian sparse matrices

The spectral densities of ensembles of non-Hermitian sparse random matrices are analyzed using the cavity method. We present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated. Within this approach, the generalized Girkos law is recovered easily. We compare our results with direct diagonalisation for a number of random matrix ensembles, finding excellent agreement.

Are there species smaller than 1 mm

The rapid advance in genetic sequencing technologies has provided an unprecedented amount of data on the biodiversity of meiofauna. It was hoped that these data would allow the identification and counting of species, distinguished as tight clusters of similar genomes. Surprisingly, this appears not to be the case. Here, we begin a theoretical discussion of this phenomenon, drawing on an individual-based ecological model to inform our arguments. The determining factor in the emergence (or not) of distinguishable genetic clusters in the model is the product of population size with mutation rate—a measure of the adaptability of the population as a whole. This result suggests that indeed one should not expect to observe clearly distinguishable species groupings in data gathered from ultrasequencing of meiofauna.

Stochastic dynamics on slow manifolds

The theory of slow manifolds is an important tool in the study of deterministic dynamical systems, giving a practical method by which to reduce the number of relevant degrees of freedom in a model, thereby often resulting in a considerable simplification. In this paper we demonstrate how the same basic methodology may also be applied to stochastic dynamical systems, by examining the behaviour of trajectories conditioned on the event that they do not depart the slow manifold. We apply the method to two models: one from ecology and one from epidemiology, achieving a reduction in model dimension and illustrating the high quality of the analytical approximations.