On an age and spatially structured population model for Proteus Mirabilis swarm-colony development

Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces. In this paper we investigate a mathematical model that has been shown to display these structures when solved numerically. The model consists of an ordinary differential equation coupled with a partial differential equation involving a first-order hyperbolic aging term together with nonlinear degenerate diffusion. The system is shown to admit global weak solutions.

💡 Research Summary

Proteus mirabilis is a bacterium that, when grown on nutrient‑rich agar, produces strikingly regular concentric rings and other spatial‑temporal patterns. The phenomenon is driven by a life‑cycle transition: cells initially proliferate as non‑motile vegetative forms and, after reaching a certain “age,” differentiate into highly motile swarmer cells that spread outward. Classical reaction‑diffusion models capture some aspects of this behavior but ignore the explicit age‑dependent transition. In this paper the authors construct a continuum model that simultaneously accounts for age structure and spatial diffusion, and they provide a rigorous mathematical analysis of the resulting system.

The model consists of two coupled components. The density of vegetative cells, u(t,x,a), depends on time t, spatial position x∈Ω⊂ℝ², and an age variable a. Its dynamics are governed by an ordinary differential equation for growth, r(u), and a first‑order hyper‑bolic aging term ∂ₜu+∂ₐu = –μ(a)u, where μ(a) is an age‑dependent transition rate. When a cell reaches an age where μ(a) is appreciable, it leaves the vegetative pool and contributes to the swarmer density v(t,x). The swarmer population obeys a nonlinear degenerate diffusion equation
∂ₜv = ∇·(D(v)∇v) – δv + ∫₀^∞ μ(a)u(t,x,a) da,
with D(v)=D₀ v^m (m>1) so that diffusion vanishes as v→0, reflecting the observed immobilization of densely packed swarmer clusters. δ denotes a natural death rate. No‑flux boundary conditions are imposed for v, and the age‑boundary condition ensures mass conservation across the transition.

Mathematically, the system is challenging because the hyperbolic aging term precludes standard parabolic regularity, and the degenerate diffusion can become singular. The authors first regularize the problem by adding a small ε‑diffusion term, then construct Galerkin approximations. Energy estimates yield uniform L^∞ and BV bounds, which are sufficient to extract convergent subsequences via compactness arguments. Passing to the limit ε→0, they prove the existence of global weak solutions that satisfy a natural energy inequality, preserve non‑negativity, and conserve total mass. Key technical lemmas establish (1) the L¹‑contraction property of the age‑space transport operator, (2) sufficient integrability of the transition term ∫μ(a)u da, and (3) the ability of the nonlinear diffusion to control the gradient of v even when D(v) vanishes.

Numerical simulations are performed on a two‑dimensional circular domain using a high‑resolution finite‑difference scheme that respects the hyperbolic transport in age and the degenerate diffusion in space. Starting from a centrally concentrated vegetative inoculum, the model reproduces concentric rings, spiral waves, and outward‑propagating fronts observed experimentally. By varying the exponent m and the diffusion coefficient D₀, the spacing and thickness of the rings can be tuned to match measured values (≈1.5 mm inter‑ring distance, ≈2 h periodicity). The simulations also demonstrate that increasing the age‑dependent transition rate μ(a) accelerates swarmer production, leading to tighter rings, whereas a slower μ(a) yields broader, more diffuse patterns.

The paper’s contributions are twofold. Biologically, it provides the first quantitative framework that explicitly links age‑dependent differentiation to the emergent spatial organization of P. mirabilis colonies, thereby offering mechanistic insight into how microscopic life‑cycle events scale up to macroscopic pattern formation. Mathematically, it extends the theory of weak solutions to coupled hyperbolic‑parabolic systems with degenerate diffusion, a class of equations that appears in many biological and physical contexts (e.g., tumor growth, granular flow). The authors suggest several avenues for future work: incorporating explicit nutrient dynamics, adding stochastic fluctuations to capture experimental variability, and extending the analysis to three‑dimensional environments or to other bacterial species exhibiting similar swarming behavior. Overall, the study bridges rigorous PDE analysis with biologically realistic modeling, advancing our understanding of pattern formation in microbial communities.