Adaptation of autocatalytic fluctuations to diffusive noise

Evolution of a system of diffusing and proliferating mortal reactants is analyzed in the presence of randomly moving catalysts. While the continuum description of the problem predicts reactant extinction as the average growth rate becomes negative, growth rate fluctuations induced by the discrete nature of the agents are shown to allow for an active phase, where reactants proliferate as their spatial configuration adapts to the fluctuations of the catalyst density. The model is explored by employing field theoretical techniques, numerical simulations, and strong coupling analysis. For $d<~2,$ the system is shown to exhibits an active phase at any growth rate, while for $d>\mathrm{}2\mathrm{}$ a kinetic phase transition is predicted. The applicability of this model as a prototype for a host of phenomena that exhibit self-organization is discussed.