The Department of Mathematics at the University of Manchester will host the UK Easter Probability Meeting, a week-long workshop on probability and its applications, with three minicourses, 12 invited talks and opportunities for contributed talks from early-career researchers.
For over a century, mathematical models have played a fundamental role in theoretical population genetics. Indeed, genetics provided some of the earliest applications of the Ito calculus. In turn, population genetics provides a wealth of mathematical challenges.
In these lectures, we shall focus on the models which arise when we try to model the interplay between the forces of evolution (mutation, selection, random genetic drift etc) acting on a population. We shall begin with some classical models for so-called panmictic populations, before turning to the complications that arise when we try to incorporate spatial structure.The challenge is to provide consistent forwards in time models for the way in which the frequencies of different genetic types evolve in the population, and backwards in time models for the ways in which individuals sampled from the population are related to one another.
Based on joint works with Rongfeng Sun and Nikos Zygouras, we show that these regularized solutions undergo a phase transition as the noise strength is varied on a logarithmic scale, with an explicit critical point. In the sub-critical regime, the regularized solutions of both KPZ and SHE exhibit so-called Edwards-Wilkinson fluctuations, i.e. they converge to the solution of the *additive* Stochastic Heat Equation (after centering and rescaling), with a non-trivial constant on the noise. We finally discuss the critical regime, where many questions are open.
This is joint work with Marie Ernst and Yvik Swan.
Based on a joint work with Daniel Contreras and Sebastien Martineau.
Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question in the case in which the growth rate is sublinear and the mass is conserved at fragmentation events. This assumption on the growth ensures that microscopic particles remain microscopic.
In this talk, we present a recent work of the speaker, in which we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. Moreover, we drop the hypothesis of conservation of mass when a fragmentation occurs. With the Feynman-Kac approach, we establish necessary and sufficient conditions on the coefficients of the equation that ensure the so-called Malthusian behaviour with exponential speed of convergence to an asymptotic profile. Furthermore, we provide an explicit expression of the latter.
We discuss a spatial analogue of this result in the presence of selection in random environment by investigating the scaling limits of Spatial Lambda-Fleming-Viot models. We consider two regimes. In the first one, the environment fluctuates in time and space. In the second one, only spatial fluctuations are present. The subpopulation of rare individuals follows the superBrownian motion in random environment and rough superBrownian motion, respectively.
The long time behaviour of the limiting process differs significantly for both cases. While both the standard superBrownian motion and the superBrownian motion in random environment in full space suffers from local weak extinction in dimension $d=1,2$, the rough superBrownian motion persists, even on a torus.
This provides weak circumstantial evidence for Wright's claim that the variation in spatial conditions contributes positively to genetic variety in the populations.
(M) Minicourse (D) Discussion (T) Talk
The meeting will take place in the Department of Mathematics of the University of Manchester.
The room has stepless access.
Registration is opening now. Please use the following link to register:
There will be a conference dinner on Wednesday at no extra charge. The dinner is sponsored by the Applied Probability Trust.
There is an opportunity for early-career researchers and research students to present a short talk (there will be six 20 minute contributed talks) or a poster. Space for the short talks is limited, and participants will be selected based on their proposed abstracts. The deadline for abstract submission is Friday 17 January, and decisions will be communicated by Friday 24 January. Participants selected to present a short talk will receive financial support for their attendance and have the conference fee waived.
We have awarded all our funding for supporting UK research students. Currently there is no funding available.
The organising committee is:
If you have any questions, please email the committee.
This meeting is funded by EPSRC and LMS.