Central limit theorems describing isolation by distance under various forms of power-law dispersal

TL;DR

This paper develops a comprehensive forward-in-time analysis of isolation-by-distance under power-law dispersal within a spatial $\\Lambda$-Fleming-Viot framework. By introducing two radii with heavy-tail distributions, the authors prove a law of large numbers to Lebesgue measure and a central limit theorem that yields a mild form of a stochastic partial differential equation with a fractional diffusion operator $\\mathcal{D}_\\alpha$ and a coalescence-driven noise structure $\\mathcal{Q}$. They derive generalized Wright–Malécot formulae across one- and two-tail regimes, identify a dimension-dependent phase transition in the coalescence regime, and elucidate how the asymmetry between parent-search and replacement radii shapes dispersal and coalescence. The approach combines semimartingale decompositions with Walsh martingale-measure theory to connect forward spatial processes to backward genealogies, offering insights for parameter inference of long-range dispersal in populations. The results extend prior work on SLFV by handling dual radii and heavy-tailed event sizes, with potential applications to ecological and evolutionary inference of dispersal patterns from genetic data.

Abstract

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $Λ$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and Véber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.

Figures (3)

• Figure 1: Example of the effect of a single reproduction event on the frequency of one allele in a spatial $\Lambda$-Fleming-Viot model with two alleles ($0$ and $1$). The characteristics of the reproduction event are $(t, x, u, r_1, r_2)$ with $u = 0.4$, and we plot $z \mapsto \rho_{t^-}(z, \lbrace 1 \rbrace)$ on the top row and $z \mapsto \rho_t(z, \lbrace 1 \rbrace)$ on the bottom row. On the left, the parent-search radius $r_1$ is larger than the replacement radius $r_2$ and the parent is selected at a point with high proportion of allele 1, even though this allele is relatively rare in the replacement area. We then plot the most likely outcome: that $k_0 = 1$, and a fraction $u$ of the population in the ball of radius $r_2$ is replaced by new offspring carrying allele 1. On the right, the parent-search radius $r_1$ is small and the replacement radius $r_2$ is relatively large. The parent is then selected near the centre of the event, where allele 1 is less abundant. We again plot the most likely outcome: that $k_0 = 0$, and a fraction $u$ of the population in a large area is replaced by new offspring carrying allele 0. In both cases, the average distance between parent and offspring is large, but the spatial correlations induced on allele frequencies are very different. Note also that, when $r_1 > r_2$, the region occupied by allele 1 may become disconnected.
• Figure 2: Two realisations of the spatial $\Lambda$-Fleming-Viot model with two alleles (0 and 1) in two dimensions. Initially, the population inside the white circle is monomorphic of type 0, while the population outside the circle is monomorphic of type 1. The plots show the proportion of the type 1 allele after some time. The parameters of the SLFV are chosen according to the one-tail regime, one where $\mathop{\mathrm{b}}\nolimits = 2$ (on the left) and one where $\mathop{\mathrm{b}}\nolimits = 0.5$ (on the right). In both simulations, the parameter $\mathop{\mathrm{a}}\nolimits$ is chosen so that $\alpha = 1.3$, and $\mathop{\mathrm{c}}\nolimits = 1$.
• Figure 3: Graph of the function $r \mapsto F_{\mathop{\mathrm{d}}\nolimits,\alpha, \beta}(r)$ defined in \ref{['eq:functionofwmf']} for several choices of $\alpha$ and $\beta$. The function is computed after setting $\zeta = 1$, $\sigma^2= 1$, $\mu = 0.2$, and $\gamma$ is chosen in order to normalize the functions at $r = 3$. We do not specify $u_0, \mathop{\mathrm{a}}\nolimits, \mathop{\mathrm{b}}\nolimits, \mathop{\mathrm{c}}\nolimits$ and as such the functions are only defined up to multiplicative constants. We thus normalize by our choice of $\gamma$; the aim of the figure is to show how alpha and beta affect the way in which the probability of identity decreases as a function of the distance between the samples. On the left, $\mathop{\mathrm{d}}\nolimits = 2$, and four curves are plotted: the blue curve corresponds to long range dispersal ($\alpha = 1.5 < 2$) and long range coalescence ($\beta =1.5 < d$), the orange dashed curve corresponds to long range dispersal ($\alpha < 2$) with short range coalescence ($\beta = 2.2 \geq \mathop{\mathrm{d}}\nolimits$), the green and red curves correspond to short range dispersal ($\alpha = 2$) and short range coalescence ($\beta \geq \mathop{\mathrm{d}}\nolimits$). The last two curves are superposed because of the normalization. On the right plot, four curves with the same parameters are plotted in dimension $\mathop{\mathrm{d}}\nolimits = 3$: the blue and orange curves correspond to long range dispersal ($\alpha = 1.5 < 2$) and long range coalescence ($\beta < \mathop{\mathrm{d}}\nolimits$), the green curve corresponds to short range dispersal ($\alpha = 2$) and long range coalescence ($\beta = 2.2 < \mathop{\mathrm{d}}\nolimits$), and the red curve corresponds to short range dispersal ($\alpha = 2$) and short range coalescence ($\beta = 3 \geq \mathop{\mathrm{d}}\nolimits$). It can be noted that having either long range dispersal or long range coalescence significantly slows down the decrease of the probability of identity with geographic distance, and that this effect is more pronounced for small values of $\alpha$ and $\beta$.

Theorems & Definitions (49)

• Definition 2.1
• Theorem 2.2
• Remark 3.1
• Remark 3.2
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Remark 3.5
• Theorem 3.6
• Lemma 4.1