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 investigate the growth of clusters within the forest fire model of Ráth and Tóth [22]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Rényi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear.We give a precise description of the process Ct of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that Ct is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to 1 on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process Ct.

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.

We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous gene through such a population.A construction of an appropriate state space for x → w t (x) can be found in Véber and Wakolbinger (2013). Using the identificationthis state space is in one-to-one correspondence with the space M λ of measures on R×{a, A} with 'spatial marginal' Lebesgue measure, which we endow with the topology of vague convergence. By a slight abuse of notation, we also denote the state space of the process (w t ) t∈R by M λ .Definition 2.1 (One-dimensional SΛFV with selection (SΛFVS)) Fix R ∈ (0, ∞) and υ ∈ (0, 1] and let µ be a finite measure on (0, R]. Further, let Π be a Poisson point process on R × R × (0, ∞) with intensity measure dx ⊗ dt ⊗ µ(dr).(2.1)The one-dimensional spatial Λ-Fleming-Viot process with selection (SΛFVS) driven by (2.1) is the M λ -valued process (w t ) t∈R with dynamics given as follows.If (x, t, r) ∈ Π, a reproduction event occurs at time t within the closed interval [x − r, x + r]. With probability 1 − s the event (x, t, r) is neutral, in which case:

We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between our model and a system of branching-coalescing particles. Using this duality, we give a clear and concise explanation for the condensation phenomenon, in which unusually fit vertices may obtain abnormally high degree: it arises from an explosionextinction dichotomy within the branching part of the dual.We show further that, in our model, the condensation is extensive. As the graph grows, unusually fit vertices become, each only for a limited time, neighbouring to a non-vanishing proportion of the current graph.