A sharp survival threshold for branching Brownian motion in a self-generated resource environment

Abstract

We study branching Brownian motion in a resource environment created by the particles themselves. Particles reproduce at rate $β$ times the resource nearby, resource is consumed at rate $c$ times the number of particles nearby, and each death returns resource mass $m$. Let $θ=βm/c$, which represents the number of births supported by resource from one corpse, assuming perfect efficiency. We prove that the population dies out almost surely if $θ\leq1$, even with perfect recycling $\left(θ=1\right)$. If $θ>1$ and the interaction kernel has finite range and is bounded below at short distances, the population survives with positive probability from every initial state with at least two particles, or with one particle and some resource. The threshold does not depend on the diffusivity, the death rate or the interaction kernel. We prove extinction for $θ\leq 1$ by showing the total mass in the system to be a supermartingale. The proof of survival rests on two estimates: a large colony grows if it consumes its corpses with little waste, and a dense colony recycles corpses efficiently. Together these imply that a large, dense colony grows with exponentially high probability, and a standard doubling and restart argument then yields survival.