Central limit theorems describing isolation by distance under varying population size

TL;DR

The authors develop a forwards-in-time central limit theorem for a spatial Λ-Fleming-Viot model with fluctuating population size, deriving an SPDE-driven description of genetic-fluctuation dynamics and a Wright-Malécot formula for identity by descent under variable density. The approach combines a LLN for population size (reaction-diffusion limit), a CLT for fluctuations (martingale problem and SPDE), and a careful derivation of a two-lineage coalescent structure through the weighted propagators $G_{s,t}$ and $P_{s,t}$. The results show lineages experience drift toward population-mass centers and coalesce at rate inversely related to local size, providing a realistic, tractable analytical framework for isolation by distance with demographic fluctuations. This work bridges analytical SLFV models and applied genetic-pattern inference by incorporating carrying-capacity-driven growth and frequency of reproductive events, with simulations validating the Wright-Malécot predictions in landscapes featuring barriers to gene flow.

Abstract

We derive a central limit theorem for a spatial $Λ$-Fleming-Viot model with fluctuating population size. At each reproduction, a proportion of the population dies and is replaced by a not necessarily equal mass of new individuals. The mass depends on the local population size and a function thereof. Additionally, as new individuals have a single parental type, with growing population size, events become more frequent and of smaller impact, modelling the successful reproduction of a higher number of individuals. From the central limit theorem we derive a Wright-Malécot formula quantifying the asymptotic probability of identity by descent and thus isolation by distance. The formula reflects that ancestral lineages are attracted by centres of population mass and coalesce with a rate inversely proportional to the population size. Notably, we obtain this information despite the varying population size rendering the dual process intractable.

Figures (2)

• Figure 1: Comparison between the population size of the process run for a time of $125$ on the left and the analytical solution on the right, both corresponding to \ref{['eq:growthfunctionexample']}. As the radius of events is positive, in events which fall right next to the edge of the valley, the growth function is positive leading to folds on the rim of the population density valley. The analytical solution depicts the solution to a discretized version of \ref{['eq:reactiondiffusion']}.
• Figure 2: Comparison of the probabilities of identity (on a logarithmic scale) from three reference positions $45, 60, 75$ relative to all other locations. For example, red captures the probability of identity sampling one individual from $45$ and one from the population at the corresponding position on the x-axis. The continuous line shows the prediction $\Theta$ from the analytical calculation based on a calculation of the transition probabilities of ancestral lineages. The dotted lines illustrate the average probability of identity from the simulations of the process, with the lighter regions representing the $50$-th and $90$-th percentiles after $2000$ simulations.

Theorems & Definitions (59)

• Definition 2.2
• Proposition 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Proposition 3.1
• Theorem 3.2
• Remark 3.3
• Conjecture 3.4
• Theorem 3.4