Geometric control of boundary-catalytic branching processes

TL;DR

The paper models boundary-catalytic branching processes as diffusion with branching on a catalytic boundary and absorption elsewhere, linking long-time population growth to a generalized Steklov spectral problem. It derives how to choose the absorption rate to balance branching and achieve steady-state growth or extinction, and establishes a critical catalytic rate above which compensation is impossible, with the threshold depending on geometry. The authors provide analytical results in simple domains (interval, annulus, exterior of a ball), asymptotic formulas for small reactive patches, and numerical FEM validation, framing a spectral-geometry toolkit for controlling diffusion-reaction systems. The work offers a rigorous, geometry-aware approach to optimizing absorption regions to regulate population growth in physical and biological contexts, and highlights open spectral and fluctuation-related questions in the critical regime.

Abstract

Boundary-catalytic branching processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of geometric control of the population growth by compensating the proliferation of particles due to branching events by their absorptions in the bulk or on boundary absorbing regions. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction. In other words, we establish a powerful tool for calculating the optimal absorption rate that equilibrates the opposite effects of branching and absorption events and thus results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling and control of various boundary-catalytic branching processes, with applications in physics and life sciences.

Figures (3)

• Figure 1: (a) Schematic view of a confining domain $\Omega$, whose boundary $\partial\Omega$ is split into three disjoint sets: catalytic region $\Gamma_{\mathrm{c}}$, absorbing region $\Gamma_{\mathrm{a}}$, and reflecting region $\Gamma_{\mathrm{r}}$, each of them may be composed of a finite number of disjoint subsets (e.g., $\Gamma_{\mathrm{a}} = \Gamma_{\mathrm{a}}^1 \cup \Gamma_{\mathrm{a}}^2 \cup \Gamma_{\mathrm{a}}^3$ is shown). As the boundary $\partial\Omega$ does not need to be connected, both $\Gamma_{\mathrm{c}}$ and $\Gamma_{\mathrm{a}}$ can be either patches on the outer boundary (as shown for $\Gamma_{\mathrm{a}}$), or surfaces of impenetrable solids in the bulk (as shown for $\Gamma_{\mathrm{c}}$), or their combinations. (b) A random realization of the BCB process on the interval $\Omega = (0,L)$ with $\Gamma_{\mathrm{c}} = \{0\}$ and $\Gamma_{\mathrm{a}} = \{1\}$, $\kappa_{\mathrm{a}} = 1$, $\kappa_{\mathrm{c}} = 1$, $D = 1$, and $x = 0.5$ (here $\Gamma_{\mathrm{r}} = \emptyset$). Two branching events are indicated by half-circles, one absorption event is indicated by a cross.
• Figure 2: (a,b) Two configurations composed of one catalytic circle of radius $R = 0.1$ at the center, and 9 randomly located absorbing circles of radius $R_{\mathrm{a}}$ (with $R_{\mathrm{a}} = 0.05$(a) and $R_{\mathrm{a}} = 0.1$(b)), enclosed by a reflecting circle of radius $L = 1$. Colors ranging from dark blue to dark red represent variations of the limiting population size $N(\infty|\bm{x})$ for the special case $q_{\mathrm{a}} = \infty$ (perfect sinks) and $q_{\mathrm{c}} = q_{\mathrm{c}}^{\rm crit}$. (c) The optimal absorption rate $\hat{q}_{\mathrm{a}}(q_{\mathrm{c}},0)$ as a function of the catalytic rate $q_{\mathrm{c}}$ for two considered configurations (shown by solid line for $R_{\mathrm{a}} = 0.05$ and by dashed line for $R_{\mathrm{a}} = 0.1$). Dotted lines show the low-rate approximation (\ref{['eq:qa_approx']}), whereas dash-dotted lines present the asymptotic relation (\ref{['eq:qa_asympt']}), with $Q \approx 15.78$ for $R_a = 0.05$ and $Q \approx 19.21$ for $R_a = 0.1$. Vertical lines indicate the critical values $q_{\mathrm{c}}^{\rm crit} \approx 6.61$ and $q_{\mathrm{c}}^{\rm crit} \approx 7.62$ for two configurations. These numerical results were obtained by a finite-element method Chaigneau24, with the maximal meshsize $h_{\rm max} = 0.01$ (see \ref{['sec:FEM']} for details).
• Figure S3: The principal eigenvalue $\mu_0^{(p,q_{\mathrm{a}})}$ from Eq. (\ref{['eq:mu0_shell']}) as a function of $p$ for the spherical shell with $R = 0.2$ and $L = 1$, $D = 1$, and three values of $q_{\mathrm{a}}$ as indicated in the legend. Big red circle indicates the point $\mu_0^{(0,\infty)}$ that corresponds to $\mu_0$ determining the critical catalytic rate $q_{\mathrm{c}}^{\rm crit}$, see Eqs. (\ref{['eq:Steklov']}). Two vertical dash-dotted lines indicate the values $p_{q_{\mathrm{a}}}$, at which $\mu_0^{(p,q_{\mathrm{a}})}$ diverges; note that $\mu_0^{(p,q_{\mathrm{a}})}$ is not shown for $p < p_{q_{\mathrm{a}}}$.