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Spatiotemporal Moran dynamics in continuous media

This paper bridges stochastic Moran processes and deterministic reaction-diffusion models by deriving partial differential equations for continuous media, revealing how distinct fitness components (fecundity vs. viability) and update rules (birth-death vs. death-birth) fundamentally alter selective wave speeds and establishing a continuous analog of isothermal graphs through local current conservation.

Original authors: Melika Gorgi, Kamran Kaveh, Navid Aliakbarian, Mohammad Reza Ejtehadi

Published 2026-07-23

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CC BY 4.0

Original authors: Melika Gorgi, Kamran Kaveh, Navid Aliakbarian, Mohammad Reza Ejtehadi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a bustling city where every neighborhood has a strict rule: exactly 100 people live there, no more, no less. If someone moves out, someone else must move in immediately to keep the count steady. This is how many living things, like the cells in your skin or bacteria in a crowded petri dish, actually live. They don't just float around freely in a big, empty ocean; they are packed tight against their neighbors, jostling for space. In the world of science, this is called "spatial evolution." Scientists have long tried to predict how a new, advantageous trait (like a super-fast bacteria or a cancer cell that grows too much) spreads through these crowded cities.

For decades, the go-to tool for this was a mathematical model called the Fisher-Kolmogorov equation. Think of it like predicting how a rumor spreads in a school where everyone can talk to anyone else at any time. It assumes that if you have a "good" trait, you just reproduce faster, and your family line expands smoothly like a wave rolling across a calm lake. But this model ignores the fact that in a crowded city, you can't just pop into existence; you have to wait for a spot to open up, and that spot only opens if someone else leaves. This paper asks a simple but tricky question: Does the way we model "leaving" (death) versus "arriving" (birth) change how fast that wave of new traits actually moves?

The authors of this paper, Melika Gorgi and her team, decided to bridge the gap between the "crowded city" reality and the "smooth wave" math. They created a new set of equations to describe what happens when evolution happens in a tightly packed, continuous space. They looked at two specific ways a population can update: the "Birth-Death" (BD) method, where a lucky individual is chosen to have a baby first, and that baby pushes out a neighbor; and the "Death-Birth" (DB) method, where a random individual dies first, creating a vacancy that a neighbor then rushes to fill.

Here is the big surprise they found: The old "smooth wave" model is missing a crucial detail. The speed at which a new trait spreads doesn't just depend on how much "better" the new trait is overall. It depends entirely on how that advantage works. If the advantage comes from having more babies (fecundity), the two methods behave very differently. In the "Birth-Death" scenario, the wave of new traits actually slows down as it spreads, like a runner getting tired. But in the "Death-Birth" scenario, the wave zooms ahead at a fast, constant speed, beating the old model's predictions.

On the flip side, if the advantage comes from living longer (viability), the roles reverse. The "Death-Birth" wave starts to speed up as it goes, while the "Birth-Death" wave stays steady. The paper shows that you cannot just look at the "net score" of how much better a mutant is; you have to know if that score comes from being a super-parent or a super-survivor. The authors ran computer simulations to prove this, showing that the old models often get the speed wrong because they treat birth and death as independent events, whereas in a crowded tissue, they are tightly linked like a dance.

They also took this idea a step further to look at uneven landscapes, like a city with some wide streets and some narrow alleys. They defined a special kind of "isothermal" environment where the flow of people (or cells) is perfectly balanced, similar to how temperature balances in a room. They found that in these balanced environments, the rules of evolution stay simple and predictable, but if the environment is messy or biased, the wave can get distorted or speed up in weird ways.

In short, this paper tells us that to understand how life spreads in crowded places—whether it's a tumor growing in a body or a new species taking over an island—we can't just use the old, simple formulas. We have to pay attention to the specific rules of the game: who dies first, who gets to reproduce, and how the crowd reacts. The authors suggest that ignoring these details leads to the wrong predictions about how fast evolution happens, and they provide a new, more accurate map for navigating these crowded biological cities.

Technical Summary: Spatiotemporal Moran Dynamics in Continuous Media

Problem Statement
The paper addresses a central gap in evolutionary biology: bridging the gap between discrete, stochastic spatial models (specifically the Moran process on graphs) and deterministic continuum models (reaction-diffusion equations like Fisher-Kolmogorov-Petrovsky-Piskunov, or FKPP). While classical FKPP models effectively describe the wavelike spread of advantageous alleles in dilute, well-mixed populations, they often fail to capture the dynamics of "crowded" environments, such as epithelial tissues or biofilms, where homeostatic constraints enforce constant population sizes and couple birth, death, and migration events. Furthermore, standard reaction-diffusion frameworks typically collapse fitness into a single selection coefficient, failing to distinguish between the physiological mechanisms of selection: fecundity (birth rate) versus viability (death rate). This distinction is critical in contexts like antimicrobial resistance, where drugs may suppress proliferation or increase mortality via different pathways.

Methodology
The authors derive partial differential equations (PDEs) representing the continuum limit of spatial Moran dynamics in large island lattices (1D cycles and 2D square lattices) under the "weak selection" regime.

Model Formulation: They utilize a two-fitness component model where mutant birth rate $r = 1+s$ and death rate $d = 1+q$ , with $s$ and $q$ representing birth and death selection coefficients, respectively.
Update Rules: The study explicitly contrasts two update rules:
- Death-Birth (DB): An individual is selected for death based on death rate, followed by a neighbor selected for reproduction based on birth rate to fill the vacancy.
- Birth-Death (BD): An individual is selected for reproduction based on birth rate, and the offspring replaces a neighbor selected based on death rate.
Derivation: Starting from stochastic master equations for average mutant frequency, the authors apply a macroscopic scaling limit ( $N, K \to \infty$ ) with rescaled time and space steps. This yields continuum PDEs that include nonlinear diffusion terms and global mean-field coupling terms arising from the constant population constraint.
Heterogeneity: The framework is extended to heterogeneous media by introducing spatially varying motility ( $\mu(x)$ ) and drift/bias fields ( $\alpha(x)$ ), mapping discrete evolutionary graphs to continuous media.

Key Contributions and Results

Derivation of Distinct PDEs: The authors derive specific PDEs for BD and DB dynamics (Eqs. 4 and 6). Unlike the standard FKPP equation, these models contain terms dependent on the mean mutant frequency ( $\bar{\phi}$ ) and, in the BD case, a KPZ-type nonlinearity $(\nabla \phi)^2$ .
Decoupling of Fitness Components: A primary finding is that the net selection coefficient ( $f = s - q$ ) is insufficient to characterize wave propagation in spatial settings. The wave speed and front shape depend on the specific values of $s$ and $q$ individually, not just their difference.
Divergent Wave Dynamics:
- Fecundity-driven selection ( $s \neq 0, q = 0$ ): DB waves propagate with a higher, constant speed. In contrast, BD waves exhibit deceleration as the mutant frequency increases.
- Viability-driven selection ( $s = 0, q \neq 0$ ): DB waves accelerate, while BD and FKPP waves maintain comparable constant speeds.
- Comparison to FKPP: In both scenarios, DB dynamics consistently yield the fastest selective wavefronts compared to BD and FKPP, even when the net selection strength is identical.
Isothermal Media and Gauge Transformations: The authors define "isothermal media" as the continuous analog of discrete isothermal graphs (where node "temperature" is uniform). They demonstrate that the isothermality condition corresponds to the conservation of a local current ( $\nabla \cdot (\alpha + \nabla \mu) = 0$ ). They further show that isothermal media are connected via gauge transformations in the functional space of migration and drift fields.

Significance and Claims
The paper claims to establish a new formalism for studying spatial evolutionary processes in compact, homeostatically regulated systems where standard FKPP approximations are inadequate. By deriving PDEs that explicitly separate birth and death selection mechanisms, the work reveals that the update rule (BD vs. DB) fundamentally alters the spatiotemporal dynamics of selection, leading to accelerating or decelerating waves that cannot be predicted by net fitness alone.

The authors emphasize that the frequency-dependent wave speeds observed are not modeling artifacts but direct consequences of the constant-population constraint, which couples global birth and death events. This provides a more biologically realistic framework for modeling somatic evolution in tissues, microbial colonies, and solid tumors. Additionally, the introduction of "isothermal media" and the associated conservation laws offers a new geometric perspective for analyzing selection in heterogeneous landscapes. The paper concludes that these findings necessitate moving beyond one-parameter fitness approximations in spatial evolutionary models, particularly when distinct physiological mechanisms drive selection.

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