Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field

Competition in a system of Brownian particles: Encouraging achievers

We introduce and analytically and numerically study a simple model of interagent competition, where underachievement is strongly discouraged. We consider N≫1 particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of N→∞, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the "swarm" of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a nonstationary "halo" around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-N effects at a distance of order sqrt[N] from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin-the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as sqrt[t]. Finally, we extend the inter-agent competition from n=2 to an arbitrary number n of competing Brownian particles (n≪N). Our analytical predictions are supported by Monte Carlo simulations.

On the maximal displacement of near-critical branching random walks

We consider a branching random walk on $$\mathbb {Z}$$ Z started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring $$1+\theta /n$$ 1 + θ / n . For $$t\ge 0$$ t ≥ 0 , we study $$M_{nt}$$ M nt , the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that $$M_{nt}/\sqrt{n}$$ M nt / n converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of $$M_{nt}$$ M nt . We also confirm that when $$\theta >0$$ θ > 0 , the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky (Ann Probab 23(4):1748–1754, 1995). The rightmost position over all generations, $$M:=\sup _t M_{nt}$$ M : = sup t M nt , is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when $$\theta <0$$ θ < 0 .

Volume explored by a branching random walk on general graphs

Branching processes are used to model diverse social and physical scenarios, from extinction of family names to nuclear fission. However, for a better description of natural phenomena, such as viral epidemics in cellular tissues, animal populations and social networks, a spatial embedding-the branching random walk (BRW)-is required. Despite its wide range of applications, the properties of the volume explored by the BRW so far remained elusive, with exact results limited to one dimension. Here we present analytical results, supported by numerical simulations, on the scaling of the volume explored by a BRW in the critical regime, the onset of epidemics, in general environments. Our results characterise the spreading dynamics on regular lattices and general graphs, such as fractals, random trees and scale-free networks, revealing the direct relation between the graphs' dimensionality and the rate of propagation of the viral process. Furthermore, we use the BRW to determine the spectral properties of real social and metabolic networks, where we observe that a lack of information of the network structure can lead to differences in the observed behaviour of the spreading process. Our results provide observables of broad interest for the characterisation of real world lattices, tissues, and networks.

A limit theorem for patch sizes in a selectively-neutral migration model

Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. LetN(t) be the (random) size of a block intersecting a fixed colony at timet.ThenE[N(t)] grows like √tin one dimension,t/logtin two, andtin three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove thatN(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution ofN(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rateu> 0 is imposed, the ‘block structure' has an equilibrium distribution. IfN(u) is the size of a block intersecting a fixed colony at equilibrium, then asu→ 0N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systemsu≈ 10–6is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.

A limit theorem for patch sizes in a selectively-neutral migration model

Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like √t in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rate u &gt; 0 is imposed, the ‘block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u → 0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u ≈ 10–6 is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.

Residence times of branching diffusion processes

The residence time of a branching Brownian process is the amount of time that the mother particle and all its descendants spend inside a domain. Using the Feynman-Kac formalism, we derive the residence-time equation as well as the equations for its moments for a branching diffusion process with an arbitrary number of descendants. This general approach is illustrated with simple examples in free space and in confined geometries where explicit formulas for the moments are obtained within the long time limit. In particular, we study in detail the influence of the branching mechanism on those moments. The present approach can also be applied to investigate other additive functionals of branching Brownian process.

Neutron fluctuations: The importance of being delayed

The neutron population in a nuclear reactor is subject to fluctuations in time and in space due to the competition of diffusion by scattering, births by fission events, and deaths by absorptions. As such, fission chains provide a prototype model for the study of spatial clustering phenomena. In order for the reactor to be operated in stationary conditions at the critical point, the population of prompt neutrons instantaneously emitted at fission must be in equilibrium with the much smaller population of delayed neutrons, emitted after a Poissonian time by nuclear decay of the fissioned nuclei. In this work, we will show that the delayed neutrons, although representing a tiny fraction of the total number of neutrons in the reactor, actually have a key impact on the fluctuations, and their contribution is very effective in quenching the spatial clustering.

Spatial extent of branching Brownian motion

We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost (X(max)≥0) and leftmost (X(min)≤0) visited sites up to time t. At each time step the existing particles in the system either diffuse (with diffusion constant D), die (with rate a), or split into two particles (with rate b). We focus on the regime b≤a where these two extreme values X(max) and X(min) are strongly correlated. We show that at large time t, the joint probability distribution function (PDF) of the two extreme points becomes stationary P(X,Y,t→∞)→p(X,Y). Our exact results for p(X,Y) demonstrate that the correlation between X(max) and X(min) is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF p(ζ) of the (dimensionless) span ζ=(X(max)-X(min))/√[D/b], which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior p(ζ)∼1/2(1+Δ)ζ for small spans, with Δ=(a/b-1). In the critical case (Δ=0) this distribution has a nontrivial power law tail p(ζ)∼8π√[3]/ζ(3) for large spans. On the other hand, in the subcritical case (Δ>0), we show that the span distribution decays exponentially as p(ζ)∼(A(2)/2)ζexp(-√[Δ]ζ) for large spans, where A is a nontrivial function of Δ, which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between X(max) and X(min). Finally we verify our results via direct Monte Carlo simulations.

Clustering of branching Brownian motions in confined geometries

We study the evolution of a collection of individuals subject to Brownian diffusion, reproduction, and disappearance. In particular, we focus on the case where the individuals are initially prepared at equilibrium within a confined geometry. Such systems are widespread in physics and biology and apply for instance to the study of neutron populations in nuclear reactors and the dynamics of bacterial colonies, only to name a few. The fluctuations affecting the number of individuals in space and time may lead to a strong patchiness, with particles clustered together. We show that the analysis of this peculiar behavior can be rather easily carried out by resorting to a backward formalism based on the Green's function, which allows the key physical observables, namely, the particle concentration and the pair correlation function, to be explicitly derived.

Spatial extent of an outbreak in animal epidemics

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, and therefore, fluctuations around their average are important: then, it is commonly assumed that the susceptible-infected-recovered mechanism can be described by a stochastic birth-death process of Galton-Watson type. The displacements of the infected individuals can be modeled by resorting to brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, like in the case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and we compare them with Monte Carlo simulations.

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.