Scaling limits for a population model with growth, division and cross-diffusion

TL;DR

The paper develops a size- and space-structured stochastic particle model for growing and dividing populations, derives a mean-field limit to a nonlocal aggregation-diffusion-growth-fragmentation equation, and rigorously establishes a localisation limit in the absence of growth/division via entropy methods. It introduces a dimensionless formulation and a macroscopic scaling showing how short-range interactions converge to a local kernel, while providing a rigorous proof only in the growth-free case. Complementary numerical experiments compare microscopic, mesoscopic, and macroscopic models across three regimes, demonstrating strong cross-scale agreement that improves with increasing particle number and that remains robust even when localisation is not extreme. The results illuminate how nonlocal interactions, diffusive smoothing, and size-structured reproduction interplay across scales and offer practical guidance for choosing modelling scales in biological aggregation contexts.

Abstract

Originally motivated by the morphogenesis of bacterial microcolonies, the aim of this article is to explore models through different scales for a spatial population of interacting, growing and dividing particles. We start from a microscopic stochastic model, write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. Under smoothness and symmetry assumptions for the interaction kernel, we then obtain entropy estimates, which provide us with a localization limit at the macroscopic level. Finally, we perform a thorough numerical study in order to compare the three modeling scales.

Figures (9)

• Figure 1: Schematic representation of our main models and results
• Figure 2: Numerical simulations without particle fragmentation and with constant growth, at times $t=4$ (panel A) and $t=60$ (panel B). In each panel, the first column shows the spatial density $u^{spatial}(x,y,t)$ plotted as function of the space variables $x$ (abscissa) and $y$ (ordinates). The second column shows the radial density $u^{radial}(\lambda,r,t)$ as function of the radial variable $\lambda$ (abscissa) and size variable $r$ (ordinates). In each column, the first line shows the solution of the microscopic model for $N = 100$ particles, the second line for $N_0=2000$ particles, the third line the solution of the mesoscopic model and the fourth line the solution of the macroscopic model. The figure on the top right represents the size density $u^{size}(r,t)$ as function of the particle size variable $r$, while the bottom right panel shows the radial distribution $u^{radial}(\lambda, t)$ as function of the radial variable $\lambda$. Blue curves are the macroscopic quantities, orange curves the mesoscopic quantities, and the dotted lines the micro quantities (yellow for $N_0=100$, green for $N_0 = 500$, light blue for $N_0=2000$ and red for $N_0 = 4000$).
• Figure 3: $L^1$ relative errors between the three quantities of interest as function of time without particle fragmentation and with constant growth: $E_{tot}$ (left figure), $E_{spatial}$ (middle figure) and $E_{size}$ (right figure). Dotted lines are the relative errors between the meso and micro models for $N_0=100$ (blue curves), $N_0=500$ (yellow curves), $N_0=2000$ (yellow curves) and $N_0=4000$ (purple curves). Black continuous lines are the relative errors between the meso and macro models.
• Figure 4: Numerical simulations with particle fragmentation and no growth, at times $t=4$ (panel A) and $t=60$ (panel B). In each panel, the first column shows the spatial density $u^{spatial}(x,y,t)$ plotted as function of the space variables $x$ (abscissa) and $y$ (ordinates). The second column shows the radial density $u^{radial}(\lambda,r,t)$ as function of the radial variable $\lambda$ (abscissa) and size variable $r$ (ordinates). In each column, the first line shows the solution of the microscopic model for $N = 100$ particles, the second line for $N_0=2000$ particles, the third line the solution of the mesoscopic model and the fourth line the solution of the macroscopic model. The figure on the top right represents the size density $u^{size}(r,t)$ as function of the particle size variable $r$, while the bottom right panel shows the radial distribution $u^{radial}(\lambda, t)$ as function of the radial variable $\lambda$. Blue curves are the macroscopic quantities, orange curves the mesoscopic quantities, and the dotted lines the micro quantities (yellow for $N_0=100$, green for $N_0 = 500$, light blue for $N_0=2000$ and red for $N_0 = 4000$).
• Figure 5: $L^1$ relative errors between the three quantities of interest as function of time with particle fragmentation and without growth: $E_{tot}$ (left figure), $E_{spatial}$ (middle figure) and $E_{size}$ (right figure). Dotted lines are the relative errors between the meso and micro models for $N_0=100$ (blue curves), $N_0=500$ (yellow curves), $N_0=2000$ (yellow curves) and $N_0=4000$ (purple curves). Black continuous lines are the relative errors between the meso and macro models.

Theorems & Definitions (12)

• Remark 1
• Theorem 1
• proof : Proof of Theorem \ref{['prop: existunicit']}
• Theorem 2
• Remark 2
• proof
• Lemma 1
• Remark 3
• Theorem 3
• Proposition 1