Spatial Populations with Seed-Bank: Renormalisation on the Hierarchical Group

We consider a system of interacting diffusions labeled by a geographic space that is given by the <italic>hierarchical group</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript upper N"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Omega _N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">N\in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Individuals live in colonies and are subject to resampling and migration as long as they are <italic>active</italic>. Each colony has a seed-bank into which individuals can retreat to become <italic>dormant</italic>, suspending their resampling and migration until they become active again. The migration kernel has a <italic>hierarchical structure</italic>: individuals hop between colonies at a rate that depends on the hierarchical distance between the colonies. The seed-bank has a <italic>layered structure</italic>: 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">N\in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the system exhibits a dichotomy between <italic>coexistence</italic> (= locally multi-type equilibria) and <italic>clustering</italic> (= locally mono-type equilibria). To identify the <italic>clustering regime</italic>, i.e., the range of parameters controlling the migration and the seed-bank for which clustering prevails, we apply a necessary and sufficient criterion derived in (Electron. J. Probab. 27 (2022), Paper No. 18, 88, DOI 10.1214/21-ejp728. MR4377129) that is valid for any geographic space given by a countable Abelian group endowed with the discrete topology. The diffusion function controlling the resampling does not play a role in this criterion. We carry out a <italic>multi-scale renormalisation analysis</italic> in the <italic>hierarchical mean-field limit</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we show that block averages on hierarchical space-time scale <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">k \in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> perform a diffusion with a renormalised diffusion function that depends on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the clustering regime, after an appropriate scaling with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this diffusion function converges to the Fisher-Wright diffusion function as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">k\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, irrespective of the diffusion function controlling the resampling. Thus, the system exhibits <italic>full universality</italic> on large space-time scales in terms of the scaling limit. For several subclasses of parameters we identify the speed at which the scaled renormalised diffusion function converges to the Fisher-Wright diffusion as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">k\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This speed in turn determines the speed at which <italic>mono-type clusters grow in space and time</italic>. We show that the seed-bank reduces the speed compared to the model without seed-bank. This reduction is the result of a delicate interplay between migration and seed-bank, which in the limit as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be worked out explicitly.