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We are interested in the invasion phase for stochastic processes withinteractions when a single mutant with positive fitness arrives in a residentpopulation at equilibrium. By a now classic approach, the first stage of theinvasion is well approximated by a branching process. The macroscopic phase,when the mutant population is of the same order of the resident population, isdescribed by the limiting dynamical system. We obtain sharper estimates andcapture the intermediate mesoscopic phase for the invasive population. Itallows us to characterize the hitting times of thresholds, which inherit alarge variance from the first stages. These issues are motivated in particularby quantifying times to reach critical values for cancer population orepidemics.

Sharp approximation and hitting times for stochastic invasion processes