Renormalisation of hierarchically interacting Cannings processes
Greven A, den Hollander F, Kliem S, Klimovsky A (2014)
Publication Type: Journal article, Original article
Publication year: 2014
Journal
Publisher: Instituto nacional de matemática pura e aplicada
Book Volume: 11
Pages Range: 43-140
Journal Issue: 1
URI: http://alea.impa.br/english/index_v11.htm
Abstract
In order to analyse universal patterns in the large space-time behaviour
of interacting multi-type stochastic populations on countable geographic spaces, a
key approach has been to carry out a renormalisation analysis in the hierarchi-
cal mean-field limit. This has provided considerable insight into the structure
of interacting systems of finite-dimensional diffusions, such as Fisher-Wright or
Feller diffusions, and their infinite-dimensional analogues, such as Fleming-Viot or
Dawson-Watanabe superdiffusions.
The present paper brings a new class of interacting jump processes into focus. We
start from a single-colony C Λ -process, which arises as the continuum-mass limit of
a Λ-Cannings individual-based population model, where Λ is a finite non-negative
measure that describes the offspring mechanism, i.e., how individuals in a single
colony are replaced via resampling. The key feature of the Λ-Cannings individual-
based population model is that the offspring of a single individual can be a positive
fraction of the total population. After that we introduce a system of hierarchi-
cally interacting C Λ -processes, where the interaction comes from migration and reshuffling-resampling on all hierarchical space-time scales simultaneously. More
precisely, individuals live in colonies labelled by the hierarchical group Ω N of or-
der N , and are subject to migration based on a sequence of migration coefficients
c = (c k ) k∈N 0 and to reshuffling-resampling based on a sequence of resampling mea-
sures Λ = (Λ k ) k∈N 0 , both acting in k-macro-colonies, for all k ∈ N 0 . The reshuffling
is linked to the resampling: before resampling in a macro-colony takes place all in-
dividuals in that macro-colony are relocated uniformly, i.e., resampling is done in
c,Λ
a locally “panmictic” manner. We refer to this system as the C N -process. The
Λ
dual process of the C -process is the Λ-coalescent, whereas the dual process of the
c,Λ
C N -process is a spatial coalescent with multi-scale non-local coalescence.
For the above system, we carry out a full renormalisation analysis in the hierarchical
mean-field limit N → ∞. Our main result is that, in the limit as N → ∞, on
c,Λ
each hierarchical scale k ∈ N 0 , the k-macro-colony averages of the C N -process at
k
the macroscopic time scale N (= the volume of the k-macrocolony) converge to a
random process that is a superposition of a C Λ k -process and a Fleming-Viot process,
the latter with a volatility d k and with a drift of strength c k towards the limiting
(k + 1)-macro-colony average. It turns out that d k is a function of c l and Λ l for all
0 ≤ l < k. Thus, it is through the volatility that the renormalisation manifests itself.
We investigate how d k scales as k → ∞, which requires an analysis of compositions
of certain Möbius-transformations, and leads to four different regimes.
We discuss the implications of the scaling of d k for the behaviour on large space-
c,Λ
time scales of the C N -process. We compare the outcome with what is known from
the renormalisation analysis of hierarchically interacting Fleming-Viot diffusions,
pointing out several new features. In particular, we obtain a new classification
for when the process exhibits clustering (= develops spatially expanding mono-
type regions), respectively, exhibits local coexistence (= allows for different types
to live next to each other with positive probability). Here, the simple dichotomy
of recurrent versus transient
migration for hierarchically interacting Fleming-Viot
P
(1/c
diffusions, namely,
k ) = ∞ versus < ∞, is replaced by a dichotomy
k∈N 0
that expresses a trade-off between migration and reshuffling-resampling, namely,
P
P k
l=0 Λ l ([0, 1]) = ∞ versus < ∞. Thus, while recurrent migrations
k∈N 0 (1/c k )
still only give rise to clustering, there now are transient migrations
P that do the same
when the non-local resampling is strong enough, namely,
l∈N 0 Λ l ([0, 1]) = ∞.
Moreover, in the clustering regime we find a richer scenario for the cluster formation
than for Fleming-Viot diffusions. In the local-coexistence regime, on the other hand,
we find that the types initially present only survive with a positive probability, not
with probability one as for Fleming-Viot diffusions. Finally, we show that for
finite N the same dichotomy between clustering and local coexistence holds as for
N → ∞, even though we lack proper control on the cluster formation, respectively,
on the distribution of the types that survive.
Authors with CRIS profile
Involved external institutions
Netherlands (NL)
Germany (DE)How to cite
APA:
Greven, A., den Hollander, F., Kliem, S., & Klimovsky, A. (2014). Renormalisation of hierarchically interacting Cannings processes. ALEA : Latin American Journal of Probability and Mathematical Statistics, 11(1), 43-140.
MLA:
Greven, Andreas, et al. "Renormalisation of hierarchically interacting Cannings processes." ALEA : Latin American Journal of Probability and Mathematical Statistics 11.1 (2014): 43-140.
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