2021/10/06 by Greven, Andreas, Hollander, Frank den, Oomen, Margriet
#60K35 #FOS: Mathematics #Primary 60J70 #Probability (math.PR) #Secondary 92D25
paper · doi:10.48550/arxiv.2110.02714
We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group ΩN of order N∈ℕ. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The migration kernel has a hierarchical structure: individuals hop between colonies at a rate that depends on the hierarchical distance between the colonies. The seed-bank has a layered structure: when individuals become dormant they acquire a colour that determines the rate at which they become active again. The latter allows us to model seed-banks whose wake-up times have a fat tail. We analyse a system of coupled stochastic differential equations that describes the population in the large-colony-size limit. For fixed N∈ℕ, the system exhibits a dichotomy between coexistence and clustering. We identify the range of parameters controlling the migration and the seed-bank for which clustering prevails. We carry out a multi-scale renormalisation analysis in the hierarchical mean-field limit N→∞. We show that block averages on hierarchical space-time scale k ∈ ℕ perform a diffusion with a renormalised diffusion function that depends on k. In the clustering regime, after an appropriate scaling with k, this diffusion function converges to the Fisher-Wright diffusion function as k→∞, irrespective of the diffusion function controlling the resampling. For several subclasses of parameters we identify the speed at which the scaled renormalised diffusion function converges to the Fisher-Wright diffusion as k→∞. We show that the seed-bank reduces the speed compared to the model without seed-bank.