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.

Figure 1

Monte Carlo simulation of a collection of $N=2\times {10}^3$ individuals in a two-dimensional box of half-size $L=1$ with reflecting boundary conditions. Each particle undergoes a regular Brownian motion with diffusion coefficient $D={10}^{-2}$. At exponentially distributed times, with rate $\lambda =1$, the walkers disappear and give rise to a random number $k$ of identical and independent descendants (a Galton-Watson birth-death mechanism), with probability $q_k$, each behaving as the parent particle. For (a) and (b) (blue squares), we have chosen $q_1=1$, which means that particles can only diffuse, whereas for (c) and (d) (red circles), we have chosen $q_0=q_2=\frac{1}{2}$, which means that particles can branch (giving rise to two descendants) or be absorbed with equal probability (the so-called binary branching Brownian motion [23]). In either case, the average number of descendants per reproduction event is equal to one. (a) and (c) illustrate the initial configurations at time $t=0$ for the two simulations, respectively: in both cases, the $N$ particles are uniformly distributed in space over the box. The behavior of the two systems at time $t=100$ is displayed in (b) and (d), respectively. In the system with purely diffusive Brownian motions, at a later time the particle positions are just shuffled with respect to the initial configuration because of the random motions of the walkers. Fluctuations are Poissonian and do not sensibly affect the particle distribution, which stays uniform over the box. The evolution of the system with branching Brownian motions is considerably different: at a later time, the walker positions display a strong patchiness, with individuals clustered together. In this latter case, fluctuations are non-Poissonian and deeply affect the behavior of the particle distribution.

Figure 2

Example of realization of a branching Brownian motion in one dimension. A single walker starts to diffuse from position $x_0$ at time $t_0=0$ (solid blue line). At a later time, a branching event occurs, and a new independent Brownian motion starts to diffuse (dashed red line). At the observation time $t$, one of the two walkers is found in the region $V_j$, whereas the other has been absorbed at an earlier time and does not contribute to the counting process.

Figure 3

The pair correlation function $g_t(x_i,x_j=0)$ for $N=100$ particles in a one-dimensional box of half-side $L$ with reflecting boundaries. The branching probabilities are $q_0=0.45$ and $q_2=0.55$, the diffusion coefficient is $D=0.01$, and the reproduction-disappearance rate is $\lambda =1$. The mixing rate is then ${\lambda }_D\simeq 0.0162$, and the fundamental eigenvalue is ${\alpha }_0=0.1$ (supercritical regime). The pair correlation function is displayed at times $t=1$ (blue circles), $t=5$ (red squares), $t=20$ (green diamonds), and $t=30$ (black dots). Solid lines correspond to numerical integration, symbols to Monte Carlo simulations with ${10}^5$ realizations. The dashed line represents the asymptotic limit ${\nu }_2/N({\nu }_1-1)=0.11$.

Figure 4

The pair correlation function $g_t(x_i,x_j=0)$ for $N=100$ particles in a one-dimensional box of half-side $L$ with reflecting boundaries. The branching probabilities are $q_0=0.55$ and $q_2=0.45$, the diffusion coefficient is $D=0.01$, and the reproduction-disappearance rate is $\lambda =1$. The mixing rate is then ${\lambda }_D\simeq 0.0162$, and the fundamental eigenvalue is ${\alpha }_0=-0.1$ (subcritical regime). The pair correlation function is displayed at times $t=1$ (blue circles), $t=5$ (red squares), $t=10$ (green diamonds), and $t=20$ (black dots). Solid lines correspond to numerical integration, symbols to Monte Carlo simulations with ${10}^5$ realizations.

Figure 5

The pair correlation function $g_t(x_i,x_j=0)$ for $N=100$ particles in a one-dimensional box of half-side $L$ with reflecting boundaries. The branching probabilities are $q_0=0.5$ and $q_2=0.5$, the diffusion coefficient is $D=0.01$, and the reproduction-disappearance rate is $\lambda =1$. The mixing rate is then ${\lambda }_D\simeq 0.0162$, and the fundamental eigenvalue is ${\alpha }_0=0$ (critical regime). The pair correlation function is displayed at times $t=1$ (blue circles), $t=10$ (red squares), $t=50$ (green diamonds), and $t=100$ (black dots). Solid lines correspond to numerical integration, symbols to Monte Carlo simulations with ${10}^5$ realizations.