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, finding excellent agreement.

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.

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 noiseinduced 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.

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. O ver the past century, mathematical biology has provided a framework with which to begin to understand the complexities of evolution. Historically, development has focused on deterministic models (1). However, when it comes to questions of invasion and migration in ecological systems, it is widely acknowledged that stochastic effects may be paramount, because the incoming number of individuals is typically small. The importance of demographic (intrinsic) noise has long been argued for in population genetics; it is the driver of genetic drift and can undermine the effect of selection in small populations (2, 3). This concept has also found favor in game theoretic models of evolution that seek to understand how apparently altruistic traits can invade and establish in populations (4). However, the past decade has seen an increase in the awareness of some of the more exotic and counterintuitive aspects of demographic noise: It has the capacity to induce cycling of species (5), pattern formation (6, 7), speciation (8), and spontaneous organization in systems that do not display such behavior deterministically.Here we explore the impact of demographic noise on the direction of selection in interactions between multiple phenotypes or species. Historically, a key obstacle to progress in this area has been the analytical intractability of multidimensional stochastic models. This is particularly apparent when trying to investigate problems related to invasion, where systems are typically far from equilibrium. A promising avenue of ...