Abstract
AbstractWe 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. On/off branching Brownian motion shows qualitatively different behaviour to classical branching Brownian motion and is an interesting object for study in itself. 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 $$\sqrt{2}$$
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 $$\sqrt{\sqrt{5}-1}\approx 1.111$$
5
-
1
≈
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.
Funder
Deutsche Forschungsgemeinschaft
Berlin Mathematical School
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Modeling and Simulation,Statistics and Probability
Reference36 articles.
1. Athreya, K.B., Ney, P.E.: Branching Processes. Springer-Verlag, New York-Heidelberg (1972) (Die Grundlehren der mathematischen Wissenschaften, Band 196)
2. Athreya, S.: Probability and Semilinear Partial Differential Equations. Ph.D. dissertation, Univ. Washington (1998)
3. Athreya, S., Tribe, R.: Uniqueness for a class of one-dimensional stochastic PDEs using moment duality. Ann. Probab. 28(4), 1711–1734 (2000)
4. Barlow, M.T., Yor, M.: (semi-) martingale inequalities and local times. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 55(3), 237–254 (1981)
5. Blath, J., Buzzoni, E., González Casanova, A., Wilke-Berenguer, M.: Structural properties of the seed bank and the two island diffusion. J. Math. Biol. 79(1):369–392 (2019)
Cited by
7 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献