Ecological interactions and spatial dynamics in microbial aggregates: A novel modelling framework

We present a mathematical model based on a system of partial differential equations (PDEs) with cross-diffusion and reaction terms to describe ecological interactions between multiple bacterial species and substrates within microaggregates, where bacteria proliferate in response to substrate availability and undergo passive dispersal driven by population pressure gradients. The ecological interactions include interspecific competition for shared substrates, and commensalism, whereby one species benefits from the metabolic by-products of another. The main motivation comes from individual-based models (IBMs) of microbial aggregates, where simulations reveal that substrate-limited conditions can give rise to rich spatial patterns. Our numerical experiments demonstrate that our PDE-based model captures the key qualitative features of three verification scenarios that have previously been investigated with IBMs. Moreover, we formally derive a competition system from an on-lattice biased random walk, and establish local well-posedness for a parameter-symmetric subcase of it. We then formally analyse the travelling wave behaviour of this case in one spatial dimension and compare the minimal travelling wave speed with the wave speed measured in the simulations.

💡 Research Summary

This paper introduces a continuum‐scale mathematical framework for describing the spatial and temporal dynamics of multi‑species microbial aggregates interacting through shared substrates. Building on observations from individual‑based models (IBMs) that reveal rich pattern formation under substrate‑limited conditions, the authors derive a system of coupled partial differential equations (PDEs) that incorporates (i) pressure‑driven cross‑diffusion, (ii) Monod‑type growth kinetics, (iii) density‑dependent mortality, and (iv) substrate production/consumption via conversion factors.

The general model (equation 2.1) treats the density of each bacterial species u_i(x,t) and the concentration of each substrate c_j(x,t) in a two‑dimensional domain Ω. Bacterial movement is modeled as the sum of a small random diffusion term (coefficient d_i) and a dominant advective flux a_i u_i ∇f(ρ), where ρ=∑_i u_i is the total biomass and f(ρ)=ρ is a linear pressure function. This term captures the “shoving” effect observed in densely packed aggregates, where cells are displaced by crowding pressure rather than by Brownian motion.

Growth follows Monod kinetics r_i c_j/(K_{ij}+c_j) scaled by a yield coefficient Y_{ij}. A mortality term b_i ρ accounts for crowding‑induced death. Substrate dynamics obey standard Fickian diffusion (coefficient D_j) together with consumption (δ_{ij}=1) and production (σ_{ij}>0) terms, allowing the model to represent both competition for a common resource and commensal cross‑feeding. Boundary conditions can be Neumann (flux) or Dirichlet (fixed concentration) depending on experimental set‑up.

Three specific ecological scenarios are extracted from the general system:

1. Pure competition – n species compete for a single substrate. The resulting reduced system (2.3) reproduces the radial sectoring and columnar stratification observed in IBM simulations of three‑species aggregates.

2. Pure commensalism – three species form a linear metabolic chain: species 1 consumes an external substrate and produces a metabolite that fuels species 2, which in turn produces a second metabolite for species 3. The PDE model predicts layered radial structures matching IBM outcomes.

3. Mixed interaction (nitrifying bacteria) – a combination of competition and commensalism, illustrating the flexibility of the framework.

Mathematically, the authors prove local‑in‑time well‑posedness for a symmetric parameter case (all a_i, d_i, r_i, K_i, Y_i, b_i equal) using Amann’s theory for quasilinear parabolic equations. They then perform a formal travelling‑wave analysis in one spatial dimension, reducing the system to a single equation for total biomass w(x,t). Linearisation yields a KPP‑type minimal wave speed c_* = 2√{a r c_0/(K + c_0)}. Numerical simulations of the full PDE system confirm that the observed front speed closely matches this analytical prediction, providing a rare quantitative bridge between IBM simulations and continuum theory.

Numerically, an implicit‑explicit finite‑difference scheme is employed, with diffusion treated implicitly for stability and advection handled explicitly. Simulations reproduce:

• Radial segregation of three competing species, with each sector maintaining access to the limiting substrate.
• In a two‑species competition where one strain has a higher maximal growth rate (growth strategist) and the other a higher yield (yield strategist), the yield strategist dominates under low‑substrate conditions, echoing IBM findings.
• Layered stratification in the commensal chain, where each species occupies a distinct radial band corresponding to its preferred metabolite.

The paper discusses the advantages of pressure‑driven cross‑diffusion over traditional density‑dependent diffusion models, emphasizing its mechanistic link to cell crowding. Limitations include the restriction of rigorous analysis to the symmetric case and the current focus on two‑dimensional domains. Future work is suggested on extending the theory to non‑symmetric parameters, three‑dimensional geometries, and coupling with fluid flow to better represent bioreactor environments.

In summary, this work delivers a comprehensive, analytically tractable, and computationally efficient PDE framework that captures the essential spatial ecology of multi‑species microbial aggregates. By validating against IBM results and providing explicit travelling‑wave speed formulas, it offers a valuable tool for researchers in microbial engineering, environmental microbiology, and theoretical ecology seeking to predict and control pattern formation and community composition in spatially structured microbial systems.