Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Λ-Wright--Fisher processes provide an important modeling framework within mathematical population genetics. We present a variety

of parameter-dependent long-term behaviors for a broad class of such
processes and explain how to discriminate the different boundary
behaviors by explicit criteria. In particular, we describe situations
in which both boundary points are asymptotically inaccessible – an
apparently new phenomenon in this context. This has interesting
biological implications, because it leads to a class of stochastic
population models in which selection alone can maintain genetic
variation. If at least one boundary point is asymptotically accessible,
we derive decay rates for the probability that the boundary is not
essentially accessed. To prove this result, we establish and employ
Siegmund duality. The dual process can be sandwiched at the boundary in
between two transformed Lévy processes. This allows us to relate the
boundary behavior of the dual to fluctuation properties of the Lévy
processes and it sheds new light on previously established
accessibility conditions. 
This is joint work with Fernando Cordero and Grégoire Véchambre.