-spatial Neveu process
for the case
< 2, in preparation.
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[Contents] | [Index] |
Collaborator: M. Birkner
Cooperation with: J. Blath, M. Capaldo, A. Etheridge (University of Oxford, UK), A. Depperschmidt (Technische Universität Berlin), J. Geiger (Universität Kaiserslautern), G. Kersting, A. Wakolbinger (Universität Frankfurt am Main), M. Möhle (Universität Tübingen), J. Schweinsberg (University of California, San Diego, USA)
Supported by: DFG: Dutch-German Bilateral Research Group ``Mathematics of random spatial models from physics and biology''
Description:
The general aim is to understand mathematical properties of models for randomly reproducing, possibly spatially extended populations. In particular, we are interested in describing the genealogy and, in scenarios with explicit spatial distribution, the spatial distribution of individuals in an ``old'' population (i.e. in an equilibrium). See, e.g., [7] for an overview over this area.
An important problem for biological applications (e.g., estimation of mutation rates) is to describe the distribution of the genealogy connecting a finite sample from a population. For the ``classical'' finite variance superprocess, it is well known that the genealogy is, up to a time change, depending on the total mass, given by the so-called Kingman coalescent (which was originally introduced in connection with a simpler class of models where the population size is artificially kept fixed), see [8]. In joint work with several co-authors ([3]) we could clarify for which general continuous state branching processes the genealogy is given by a time change of a general coalescent: This is exactly the case for the stable continuous state branching processes.
This project is connected with the studies of the renormalized spatial Neveu branching process by K. Fleischmann and V. Vakhtel: The Neveu process is a special example of a stable continuous branching process, and the methods developed in [3] could in fact be adapted to settle a technical question of Fleischmann and Vakhtel, see [4].
Stochastic models for spatially extended populations with local self-regulation (in particular in two spatial dimensions) which remove the strong independence assumptions inherent in classical branching random walks and at the same time are able to predict a non-trivial long-time behavior, are desirable from the point of view of ecological modeling. In order to do this, one introduces a ``feedback'', rendering individual reproduction super- oder subcritical in dependence on the ``local'' configuration. Up to now, such models have been mostly studied via computer simulation, among the first mathematically rigorous papers are [5], [6]. [5] studied the phenomenon using continuous masses, i.e. in a limit with many particles with small individual masses.
In collaboration with A. Depperschmidt, we are studying a particle-based analog of the model in [5] in discrete time: In a preliminary investigation, we could rigorously show that long-time survival is possible in an appropriate part of the parameter space.
Branching random walks in random environments can be viewed as another step to remove the strong independence assumptions in ``classical'' branching random walks. Non-spatial branching processes in random environment have a rich variety of possible limit behaviors, a similar (and exhaustive) classification in the spatial setting is still missing. Some steps in this direction were taken in [2] where we could give conditions for extinction and for supercritical growth of a single family by analyzing the genealogy of a sampled individual under the annealed measure. Notably, space facilitates survival in this model: There are constellations in which the spatial population grows exponentially even though its non-spatial analog would die out almost surely.
The so-called directed polymer in random environment, a model known from statistical physics, appears as a conditional mean of the (local) population size, given the environment. Methods established in the field of branching processes, namely representations of size-biased laws, could be adapted to yield an improved criterion for weak disorder, see [1].
References:
-spatial Neveu process
for the case
< 2, in preparation.
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[Contents] | [Index] |