Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStructural properties of the seed bank and the two-island diffusion
247
0
0.0
(
0
)
Added by
Maite Isabel Wilke Berenguer
Publication date
2017
fields
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We investigate various aspects of the (biallelic) Wright-Fisher diffusion with seed bank in conjunction with and contrast to the two-island model analysed e.g. in Kermany, Zhou and Hickey, 2008, and Nath and Griffiths, 1993, including moments, stationary distribution and reversibility, for which our main tool is duality. Further, we show that the Wright-Fisher diffusion with seed bank can be reformulated as a one-dimensional stochastic delay differential equation, providing an elegant interpretation of the age structure in the seed bank also forward in time in the spirit of Kaj, Krone and Lascoux, 2001. We also provide a complete boundary classification for this two-dimensional SDE using martingale-based reasoning known as McKeans argument.
rate research
Read More
We investigate the compact interface property in a recently introduced variant of the stochastic heat equation that incorporates dormancy, or equivalently seed banks. There individuals can enter a dormant state during which they are no longer subject to spatial dispersal and genetic drift. This models a state of low metabolic activity as found in microbial species. Mathematically, one obtains a memory effect since mass accumulated by the active population will be retained for all times in the seed bank. This raises the question whether the introduction of a seed bank into the system leads to a qualitatively different behaviour of a possible interface. Here, we aim to show that nevertheless in the stochastic heat equation with seed bank compact interfaces are retained through all times in both the active and dormant population. We use duality and a comparison argument with partial functional differential equations to tackle technical difficulties that emerge due to the lack of the martingale property of our solutions which was crucial in the classical non seed bank case.
We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of on/off moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an on/off-branching Brownian motion. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be switched off for an exponential amount of time after which they get switched on again. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed $sqrt{2}$, is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off-branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by $sqrt{sqrt{5}-1}approx 1.111$ under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.
We investigate the behaviour of the genealogy of a Wright-Fisher population model under the influence of a strong seed-bank effect. More precisely, we consider a simple seed-bank age distribution with two atoms, leading to either classical or long genealogical jumps (the latter modeling the effect of seed-dormancy). We assume that the length of these long jumps scales like a power $N^beta$ of the original population size $N$, thus giving rise to a `strong seed-bank effect. For a certain range of $beta$, we prove that the ancestral process of a sample of $n$ individuals converges under a non-classical time-scaling to Kingmans $n-$coalescent. Further, for a wider range of parameters, we analyze the time to the most recent common ancestor of two individuals analytically and by simulation.
We introduce a new Wright-Fisher type model for seed banks incorporating simultaneous switching, which is motivated by recent work on microbial dormancy. We show that the simultaneous switching mechanism leads to a new jump-diffusion limit for the scaled frequency processes, extending the classical Wright-Fisher and seed bank diffusion limits. We further establish a new dual coalescent structure with multiple activation and deactivation events of lineages. While this seems reminiscent of multiple merger events in general exchangeable coalescents, it actually leads to an entirely new class of coalescent processes with unique qualitative and quantitative behaviour. To illustrate this, we provide a novel kind of condition for coming down from infinity for these coalescents using recent results of Griffiths.
The two-parameter Poisson--Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingmans one-parameter Poisson--Dirichlet distribution and to certain Fleming--Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by formulating a $K$-allele Wright--Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a $K$-dimensional diffusion process as $Ntoinfty$. Moreover, the descending order statistics of the $K$-dimensional diffusion converge in distribution to the two-parameter Poisson--Dirichlet diffusion as $Ktoinfty$. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove textit{a priori} that in the limit there is no loss of mass, i.e., that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
