The geometric control of boundary-catalytic branching processes

TL;DR

This paper develops a spectral approach to control the growth of boundary-catalytic branching processes by balancing proliferation with absorption, identifying critical parameters for population regulation in complex environments.

Contribution

It introduces a Steklov spectral problem to determine the phase diagram and critical rates for controlling population growth in boundary-catalytic processes.

Findings

Principal eigenvalue determines critical growth-extinction threshold.

Existence of a critical catalytic rate beyond which control is impossible.

Framework applicable to physics, chemistry, and life sciences.

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 the geometric control of the population growth by compensating the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on absorbing regions of the boundary. 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 a bounded domain. In other words, we establish a powerful tool for calculating the growth-regulating 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, chemistry, and life sciences.