We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial Λ-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right. * etheridg@stats.ox.ac.uk, supported in part by EPSRC Grant EP/I01361X/1 †

We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles' masses decay. In standard BBM, we may define the front displacement at time t as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from Θ(1) to o(1). We show that one can find arbitrarily large times t for which this occurs at a distance Θ(t 1/3 ) behind the front displacement for standard BBM.

We ask the question "when will natural selection on a gene in a spatially structured population cause a detectable trace in the patterns of genetic variation observed in the contemporary population?". We focus on the situation in which 'neighbourhood size', that is the effective local population density, is small. The genealogy relating individuals in a sample from the population is embedded in a spatial version of the ancestral selection graph and through applying a diffusive scaling to this object we show that whereas in dimensions at least three, selection is barely impeded by the spatial structure, in the most relevant dimension, d = 2, selection must be stronger (by a factor of log(1/µ) where µ is the neutral mutation rate) if we are to have a chance of detecting it. The case d = 1 was handled in Etheridge et al. (2015).The mathematical interest is that although the system of branching and coalescing lineages that forms the ancestral selection graph converges to a branching Brownian motion, this reflects a delicate balance of a branching rate that grows to infinity and the instant annullation of almost all branches through coalescence caused by the strong local competition in the population.

Motivated by the study of branching particle systems with selection, we establish global existence for the solution (u, µ) of the free boundary problemx u + u for t > 0 and x > µ t , u(x, t) = 1 for t > 0 and x ≤ µ t ,when the initial condition v : R → [0, 1] is non-increasing with v(x) → 0 as x → ∞ and v(x) → 1 as x → −∞. We construct the solution as the limit of a sequence (u n ) n≥1 , where each u n is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi et al. [5] show that this global solution can be identified with the hydrodynamic limit of the so-called N -BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by killing the leftmost particle at each branching event.• If v (1) ≤ v (2) are two valid initial conditions and (u (i) , µ (i) ) is the solution with initial condition v (i) , then u (1) ≤ u (2) and µ (1) ≤ µ (2) .We say that (u, µ) is a classical solution to (FBP) above if (u, µ) satisfies the equation (FBP), and u(·, t) → v(·) in L 1 loc as t ց 0. Remark 1. We will show that u(t, x) → v(x) at all points of continuity of v as t ց 0 (since v is non-increasing, it is differentiable almost everywhere).Remark 3. As discussed below, the condition that v is non-increasing can be relaxed to some extent.Our motivation for studying the problem (FBP) stems from its connection with the so-called N -BBM, a variant of branching Brownian motion in R in which the number of active particles is kept constant (and equal to N ) by removing the leftmost particle each time a particle branches. More details are given in Section 2 below, but in a nutshell, De Masi et al. [5] show that as N → ∞, under appropriate conditions on the initial configuration of particles, the N -BBM has a hydrodynamic limit whose cumulative distribution can be identified with the solution of (FBP), provided such a solution exists.The overall idea behind the proof is to construct u as the limit of a sequence of functions u n , where, for each n, u n satisfies an n-dependent non-linear equation, but where all the u n have the same initial condition. More precisely, let v : R → [0, 1] be a measurable function and, for n ≥ 2, let (u n (x, t), x ∈ R, t ≥ 0) be the solution to(1.1)For each n ≥ 2, this is a version of the celebrated Fisher-KPP equation about which much is known (see e.g. [12,1,15,17,10,16]). In particular,• u n exists and is unique,• u n (x, t) ∈ (0, 1) for x ∈ R and t > 0 (unless v ≡ 0 or v ≡ 1).Since the comparison principle applies, we see furthermore that for every x ∈ R, t > 0 fixed, the sequence n → u n (x, t) is increasing. Therefore, the following pointwise limit is well defined:with u(x, t) ∈ (0, 1] for t > 0 (unless v ≡ 0). Indeed, in most of the cases we are interested in, there are regions where u(x, t) = 1. We have the following results on u: Theorem 1.2. Let v : R → [0, 1] be a measurable function. The function u(x, t) as defined by (1. 1) and (1.2) satisfies the following properties:• u is continuous ...

We study the evolution of gene frequencies in a population living in R d , modelled by the spatial Λ-Fleming-Viot process with natural selection ([BEV10], [EVY14]). We suppose that the population is divided into two genetic types, a and A, and consider the proportion of the population which is of type a at each spatial location. If we let both the selection intensity and the fraction of individuals replaced during reproduction events tend to zero, the process can be rescaled so as to converge to the solution to a reaction-diffusion equation (typically the Fisher-KPP equation, [EVY14]). We show that the rescaled fluctuations converge in distribution to the solution to a linear stochastic partial differential equation. Depending on whether offspring dispersal is only local or if large scale extinction-recolonization events are allowed to take place, the limiting equation is either the stochastic heat equation with a linear drift term driven by space-time white noise or the corresponding fractional heat equation driven by a coloured noise which is white in time. If individuals are diploid (i.e. either AA, Aa or aa) and if natural selection favours heterozygous (Aa) individuals, a stable intermediate gene frequency is maintained in the population. We give estimates for the asymptotic effect of random fluctuations around the equilibrium frequency on the local average fitness in the population. In particular, we find that the size of this effect -known as the drift load -depends crucially on the dimension d of the space in which the population evolves, and is reduced relative to the case without spatial structure.AMS 2010 subject classifications. Primary: 60G57 60F05 60J25 92D10. Secondary: 60G15.2. Update q as in (6).Note that if we let w = |B(x, r)| −1 B(x,r) q t − (z) dz, then at a neutral reproduction event, P (k = a) = w and at a selective event, P (k = a) = w − F (w).Remark. The existence of a unique Ξ-valued process following these dynamics under condition (4) is proved in [EVY14, Theorem 1.2] in the special case F (w) = w(1 − w) (in the neutral case s = 0, this was done in [BEV10]). In our general case, the condition on w − F (w) allows us to define a dual process and hence prove existence and uniqueness in the same way as in [EVY14].We shall consider two different distributions µ for the radii of events, i) the fixed radius case : µ(dr) = δ R (dr) for some R > 0, ii) the stable radius case : µ(dr) = 1 r≥1 r d+α+1 dr for a fixed α ∈ (0, 2 ∧ d).