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We investigate a heterogeneous multiplicative growth model with mean-field redistribution and independent local stochastic resetting, modeling populations with quenched growth-rate heterogeneity and random extinction events. At high redistribution or resetting rates, a delocalized phase emerges with computable growth rates and stationary distributions, where reset-free growth generates power-law tails. For weak resetting, favorable sites can localize, leading to a partially localized phase where a finite population fraction resides on a dynamic subset of sites exhibiting multifractal stationary fractions.
A finite fraction of population can localize on a dynamic, subextensive pool of favorable sites in heterogeneous growth models with stochastic resetting.
We study a model of heterogeneous multiplicative growth with mean-field redistribution and independent local stochastic resetting. The model describes populations or wealth distributed over sites with quenched growth-rate heterogeneity, while resets represent random local extinction or bankruptcy events. At large redistribution or resetting rate, we find a delocalized phase and we compute the growth rate as well as the stationary distribution of the population of a given site. We show that reset-free growth episodes generate power-law stationary tails with site-dependent exponents. For weak resetting and redistribution, favorable sites can undergo long reset-free episodes and localize. This produces a partially localized phase, in which a finite fraction of the total population is carried by a small set of sites whose identity changes on an $O(1)$ time scale within a subextensive pool of $N^{u^\star}$ favorable candidates, with $0<u^\star<1$. We derive the corresponding growth rate from a saddle point analysis, and show that the averaged stationary fractions display multifractal behavior.