Looking forwards and backwards: dynamics and genealogies of locally regulated populations

TL;DR

The paper develops a comprehensive, mechanistic framework for spatial populations where birth, death, and juvenile establishment depend on location and local density via smoothing kernels. Through three scaling regimes, the authors derive (i) interacting superprocess limits for stochastic density evolution when demographic feedback remains significant, (ii) nonlocal PDE limits that capture density-dependent growth in a deterministic setting, and (iii) classical PDE limits under simultaneous, fine-scale scaling that yields local diffusion-reaction equations such as Fisher–KPP, Allen–Cahn, and PME with logistic growth. A key innovation is explicitly modeling a juvenile dispersal step and retaining genealogical information via a lookdown construction, which yields explicit ancestral lineages and reveals how lineage motion can differ even when equilibrium densities coincide. The work also analyzes traveling-wave scenarios, showing how lineage dynamics vary across Fisher–KPP, Allen–Cahn, and PME fronts, including stationary ancestral distributions behind fronts and potential shifts in coalescent regimes. Overall, the results provide a rigorous bridge between microscopic, density-dependent population models and macroscopic PDE descriptions, while emphasizing the informational value of lineage dynamics for identifiability and inference in spatial ecology. The methodology and lookdown approach offer a versatile toolkit for studying genealogies in high-density, spatially structured populations with nonlocal interactions.

Abstract

We introduce a broad class of spatial models to describe how spatially heterogeneous populations live, die, and reproduce. Individuals are represented by points of a point measure, whose birth and death rates can depend both on spatial position and local population density, defined via the convolution of the point measure with a nonnegative kernel. We pass to three different scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE. The classical PDE is obtained both by first scaling time and population size to pass to the nonlocal PDE, and then scaling the kernel that determines local population density; and also (when the limit is a reaction-diffusion equation) by simultaneously scaling the kernel width, timescale and population size in our individual based model. A novelty of our model is that we explicitly model a juvenile phase: offspring are thrown off in a Gaussian distribution around the location of the parent, and reach (instant) maturity with a probability that can depend on the population density at the location at which they land. Although we only record mature individuals, a trace of this two-step description remains in our population models, resulting in novel limits governed by a nonlinear diffusion. Using a lookdown representation, we retain information about genealogies and, in the case of deterministic limiting models, use this to deduce the backwards in time motion of the ancestral lineage of a sampled individual. We observe that knowing the history of the population density is not enough to determine the motion of ancestral lineages in our model. We also investigate the behaviour of lineages for three different deterministic models of a population expanding its range as a travelling wave: the Fisher-KPP equation, the Allen-Cahn equation, and a porous medium equation with logistic growth.

Figures (3)

• Figure 1: Snapshots of two simulations, with small $\alpha=\theta/N$ (left) and large $\alpha = \theta/N$ (right). Simulations are run with a Fisher-KPP-like parameterization: birth and establishment are constant, while death increases linearly with density, at slope $1/\theta$. Left: $\alpha=0.1$. Right: $\alpha=10$. Other parameters were the same: dispersal ($q_\theta$) and interactions (here, only $\rho_F$) are Gaussian with standard deviation 1, and the equilibrium density ($N$) is 10 individuals per unit area. The remaining parameters are constant: $r \equiv \gamma \equiv 1$.
• Figure 2: Simulated populations under a porous medium equation with logistic growth (\ref{['PME']}) in $d=1$, $\theta/N$ small on the top; large on the bottom. Values of $\theta$ in top and bottom figures are 1 and 100, respectively, and both have $N$ set so that the density is roughly 100 individuals per unit of habitat (as displayed on the vertical axis). See text for details of the simulations.
• Figure 3: Left: A snapshot of individual locations in a two-dimensional simulation in which the constant density is unstable and a stable, periodic pattern forms. Right: Population density in an expanding wave in a one-dimensional simulation forming a periodic pattern; each panel shows the wavefront in three periods of time; within each period of time the wavefront at earlier times is shown in blue and later times in pink. In both cases, $\gamma(m) = 3/(1 + m)$, $\mu \equiv 0.3$, and $r \equiv 1$; dispersal is Gaussian with $\sigma=0.2$ and density is measured with $\rho_\gamma(x) = p_9(x)$, i.e., using a Gaussian kernel with standard deviation 3.

Theorems & Definitions (66)

• Remark 2.1
• Remark 2.3
• Definition 2.4: Martingale Problem Characterisation
• Definition 2.5: Dispersal generator
• Remark 2.6
• Remark 2.7
• Lemma 2.9
• Theorem 2.10
• Corollary 2.11
• Definition 2.12: Weak solutions