Spatio-temporal Moran dynamics in continuous media

TL;DR

This work addresses how spatial structure alters natural selection beyond well-mixed models by formulating Moran dynamics in continuous media with two fitness components. It derives PDEs for birth-death and death-birth updates, revealing that wave speeds and front shapes depend separately on fecundity $s$ and viability $q$ and differ from the Fisher–Kolmogorov speed $c_{FK} = 2 \sqrt{D s}$. The study provides analytical expressions for the weak-selection wave speeds $c_{Bd}(\bar{φ})$ and $c_{Db}(\bar{φ})$, showing Db often outpaces Bd and FK across parameter regimes, and identifies a duality when swapping $s$ and $q$. It generalizes to heterogeneous media via the isothermal-medium framework and a gauge transform, linking spatial graphs to continuum analogues and enforcing local current conservation. Together, the results offer a principled way to model somatic evolution in crowded tissues and tumors with spatial structure, motility biases, and environmental heterogeneity.

Abstract

Understanding how natural selection unfolds across space and time is a central problem in evolutionary biology. Classic models such as the Moran process capture stochastic birth-death dynamics in structured populations, while reaction-diffusion equations like the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation describe deterministic wave-like spread. In this work, we bridge these perspectives by deriving partial differential equations for the spatiotemporal limit of Moran dynamics in continuous media. Our model incorporates two distinct fitness components: fecundity (birth rate) and viability (death rate). We demonstrate that the resulting selective wave speeds differ substantially in spatial Moran Birth-death (Bd), Moran Death-birth (Db), and FKPP dynamics. When fecundity drives the dynamics, we observe that the selective waves decelerate for the Bd process, whereas in the Db process the wave propagates with a higher, constant speed. In contrast, when viability drives the process, the Db wave accelerates, while the Bd and FKPP waves maintain comparable constant speeds. We extend the framework to heterogeneous media, represented as weighted lattice graphs in one or two dimensions. We derive a continuous space analog of isothermal graphs and establish that the isothermality condition corresponds to the conservation of a local current.

Figures (6)

• Figure 1: Crowded vs dilute population modeling schemes in spatial evolutionary dynamics.(A) A snapshot of a spatially structured population represented by a packed epithelial-like cell lattice. Each polygon denotes a single cell, colored red or blue to indicate mutant or resident genotypes, respectively. Mutants expand via local replacement events at boundaries between neighboring cells. (B) A dilute limit where individuals (cells) move randomly in continuous space and, independently, reproduce based on fitness differences, as modeled by FKPP reaction-diffusion equations.
• Figure 2: Lattice representations and traveling wave profile in spatial Moran dynamics.(A) Schematic of discrete spatial structures used in modeling Moran dynamics. Top: a 1D cycle (ring) of demes (islands), each containing a fixed number of individuals. Mutant frequency $\phi(x)$ varies across space, from $\phi=1$ (blue) to $\phi=0$ (red), forming a selective gradient. Bottom: a 2D square lattice representation where each node contains a well-mixed subpopulation with local mutant frequency $\phi_{ij}$. The continuous approximation yields a mutant field $\phi(x, y)$ with a spatial wavefront. (B) Illustration of the traveling wave solution $\phi(x)$ in the continuum limit. The wavefront connects two homogeneous states, $\phi=1$ and $\phi=0$, and propagates with velocity $c_F$. The width of the transition region corresponds to the diffusive spread of the advantageous allele.
• Figure 3: Wavefront propagation in Bd, Db, and FK dynamics with equivalent net selection coefficient. Panels show snapshots of the mutant frequency $\phi(x,t)$ over time for three models: birth-death (Bd), death-birth (Db), and Fisher-KPP (FK), on a 1D lattice of $K = 100$ islands with a single initial mutant at the center. (A) For selection parameters $s = 2, q = 0$, corresponding to pure birth advantage. (B) For selection parameters $s = 1.2, q = -0.8$, where the net selective advantage $s - q = 2$ remains constant. Each row compares wavefront speeds $c_{\mathrm{Bd}}, c_{\mathrm{Db}}, c_{\mathrm{FK}}$, showing that Db spreads faster than Bd and FK in both scenarios, and that wavefront curvature/width differs across update rules despite identical net selection coefficient.
• Figure 4: Spatiotemporal dynamics of selective wavefronts in 1D for Bd, Db, and FK models. Each panel shows the spatiotemporal heatmap of mutant frequency $\phi(x,t)$ on a 1D lattice with $K=100$ islands, initiated by a single central mutant. Contours correspond to constant frequency levels. (A) For $s = 2, q = 0$, representing pure birth selection. The wavefront in the Bd model exhibits deceleration (nonlinear slope $c_{\mathrm{Bd}}(\bar{\phi})$), while Db and FK show more consistent wave speeds, with Db propagating fastest. (B) For $s = 0, q = 0.5$, representing pure death selection. Here, Bd wave speed is more stable, while Db displays curvature due to frequency-dependent acceleration. In both cases, the Db dynamics yield the highest wave speed compared to Bd and FK, despite equal net selection $s - q$.
• Figure 5: Frequency-dependent wave speeds in birth-death, death-birth, and Fisher-KPP dynamics. Wave propagation speeds $c(\bar{\phi})$ are plotted against the mean mutant frequency $\bar{\phi}$ for three models: birth-death (Bd), death-birth (Db), and Fisher-Kolmogorov (FK). Left: When selection arises solely through increased birth rate ($s = 0.5, q = 0$), Db yields the fastest wave and exhibits weak acceleration, while Bd displays mild deceleration. Right: When selection arises via decreased death rate ($q = -0.5, s = 0$), the Db wavefront accelerates strongly with frequency, in contrast to the nearly constant speeds of Bd and FK. In both cases, Db propagates faster than Bd and FK, despite equal net selection strength $f = s - q$.