Ancestral lineages in spatial population models with local regulation

@article{Birkner2019AncestralLI,
  title={Ancestral lineages in spatial population models with local regulation},
  author={Matthias C. F. Birkner and Nina Gantert},
  journal={arXiv: Probability},
  year={2019}
}
We give a short overview on our work on ancestral lineages in spatial population models with local regulation. We explain how an ancestral lineage can be interpreted as a random walk in a dynamic random environment. Defining regeneration times allows to prove central limit theorems for such walks. We also consider several ancestral lineages in the same population and show for one prototypical example that in one dimension the corresponding system of coalescing walks converges to the Brownian… 
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References

SHOWING 1-10 OF 52 REFERENCES
Random walks in dynamic random environments and ancestry under local population regulation
We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of…
Random walks on oriented percolation and in recurrent environments
We present a central limit theorem for random walks on oriented percolation clusters. These random walks are representations of ancestral lineages in a population model with local competition. We…
Ergodic behavior of locally regulated branching populations
For a class of processes modeling the evolution of a spatially structured population with migration and a logistic local regulation of the reproduction dynamics we show convergence towards an upper…
Neutral evolution in spatially continuous populations.
We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum.…
Survival and complete convergence for a spatial branching system with local regulation.
We study a discrete time spatial branching system on $\mathbb{Z}^d$ with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show…
Coalescing directed random walks on the backbone of a 1+1-dimensional oriented percolation cluster converge to the Brownian web
We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of…
Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems
  • Bolker, Pacala
  • Environmental Science
    Theoretical population biology
  • 1997
TLDR
This work presents a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments of ensembles of distributions of organisms, and finds that the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model.
The Stepping Stone Model, II: Genealogies and the Infinite Sites Model, Submitted
TLDR
It is shown that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent, and mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site, are investigated.
Coexistence in locally regulated competing populations and survival of branching annihilating random walk
We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic…
Multi-type spatial branching models for local self-regulation I: Construction and an exponential duality
We consider a spatial multi-type branching model in which individuals migrate in geographic space according to random walks and reproduce according to a state-dependent branching mechanism which can…
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