Abstract
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. 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}, 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.
Highlights
Introduction and main results1.1 MotivationOne of the most fundamental models in spatial population genetics and ecology, describing the spread of a beneficial allele subject to directional selection, was introduced by Fisher in [14]
A very interesting feature of the F-KPP Equation and a main reason for the amenability of its analysis is given by the fact that the solution to (1.1) is dual to branching Brownian motion (BBM), as was shown by McKean [29]
Another major tool needed for deriving the uniqueness result is the powerful duality technique, i.e. we prove a moment duality with an “on/off branching coalescing Brownian motion” which as in [3] we define slightly informally as follows
Summary
One of the most fundamental models in spatial population genetics and ecology, describing the spread of a beneficial allele subject to directional selection, was introduced by Fisher in [14]. Corresponding discrete-space population genetic models have recently been studied in [15] and non-spatial models, where dormancy and resuscitation are modeled in the form of classical migration between an active and an inactive state, have been derived and investigated in [6] and [7] (these papers provide biological background and motivation). Where c, c ≥ 0 are the switching rates between active and dormant states, s ≥ 0 is the selection parameter, ν ≥ 0 the reproduction parameter and m1, m2 ≥ 0 are the mutation parameters One may view this as a continuous stepping stone model (cf [33]) with seed bank. We expect new technical problems, since the second component v(t, x) comes without the Laplacian, so that all initial roughness of v0 will be retained for all times, preventing jointly continuous solutions
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