Surviving the Attack of the Clones

Abstract

We consider a population dynamics model in which each diffusing particle that hits a catalytic surface can split into two independent copies (clones). The particles of such a growing-in-size population search in parallel for a hidden partially reactive target to trigger a reaction event (e.g., a viral attack). We investigate the statistics of the fastest first-reaction time (FRT) among all the particles. We establish a nonlinear integral equation for the survival probability and then analyze the associated probability density of the FRT and its moments. Lower and upper bounds on the mean FRT are then deduced in terms of the system parameters (target reactivity, catalytic rate, diffusivity, etc.). Because autocatalytic replication can rapidly increase the number of searchers, it can substantially accelerate the diffusive search. We solve the nonlinear equations numerically in a basic geometric setting and reveal advantages and limitations on the autocatalytic search.