Diffusion rate determines balance between extinction and proliferationin birth-death processes

We here study spatially extended catalyst induced growth processes.This type of process exists in multiple domains of biology, rangingfrom ecology (nutrients and growth), through immunology (antigensand lymphocytes) to molecular biology (signaling molecules initiatingsignaling cascades). Such systems often exhibit an extinction-proliferationtransition, where varying some parameters can lead to either extinctionor survival of the reactants.
   When the stochasticity of the reactions, the presence of discretereactants and their spatial distribution is incorporated into theanalysis, a non-uniform reactant distribution emerges, even when allparameters are uniform in space.
   Using a combination of Monte Carlo simulation and percolation theorybased estimations; the asymptotic behavior of such systems is studied.In all studied cases, it turns out that the overall survival of thereactant population in the long run is based on the size and shapeof the reactant aggregates, their distribution in space and the reactantdiffusion rate. We here show that for a large class of models, thereactant density is maximal at intermediate diffusion rates and lowor zero at either very high or very low diffusion rates. We give multipleexamples of such system and provide a generic explanation for thisbehavior. The set of models presented here provides a new insighton the population dynamics in chemical, biological and ecologicalsystems.