Reaction-diffusion systems from kinetic models for bacterial communities on a leaf surface

TL;DR

This work develops a kinetic-to-macroscopic framework that derives reaction-diffusion systems with nonlinear and cross-diffusion from mesoscopic kinetic equations for interacting cell populations in a host medium. By employing a diffusive scaling and a Hilbert expansion, the authors first obtain linear self-diffusion in the diffusive limit, then introduce turning operators to generate cross-diffusion, all while linking macroscopic coefficients to microscopic interaction rules. They apply the approach to two bacterial strains on a leaf surface, incorporating a dense host layer and nutrient-related dynamics, and perform Turing-instability analyses—both with self-diffusion and cross-diffusion—to predict pattern formation, which is corroborated by numerical simulations. The resulting framework provides a mechanistic, multiscale method to connect microscopic cellular interactions to macroscopic ecological patterns, with potential applications in phyllosphere microbiology and pattern formation in biological systems.

Abstract

Many mathematical models for biological phenomena, such as the spread of diseases, are based on reaction-diffusion equations for densities of interacting cell populations. We present a consistent derivation of reaction-diffusion equations from systems of suitably rescaled kinetic Boltzmann equations for distribution functions of cell populations interacting in a host medium. We show at first that the classical diffusive limit of kinetic equations leads to linear diffusion terms only. Then, we show possible strategies in order to obtain, from the kinetic level, macroscopic systems with nonlinear diffusion and also with cross-diffusion effects. The derivation from a kinetic description has the advantage of relating reaction and diffusion coefficients to the microscopic parameters of the interactions. We present an application of our approach to the study of the evolution of different bacterial populations on a leaf surface. Turing instability properties of the relevant macroscopic systems are investigated by analytical methods and numerical tools, with particular emphasis on pattern formation for varying parameters in two-dimensional space domains.

Figures (4)

• Figure 1: On the left panel: $s_1$ and $s_2$ defined in \ref{['cond2.1c']}-\ref{['cond2.2c']}, for three different choices of the cross-diffusion parameters $\delta_{12}, \delta_{21}$, as functions of $\delta$. In each case, the minimal values $\overline{\delta}$, such that \ref{['cond2.1c']}-\ref{['cond2.2c']} are satisfied for all $\delta>\overline{\delta}$, are indicated by a small square of the same colour. On the right panel, the values of $\overline{\delta}$ are plotted as a function of $\delta_{12}$ and $\delta_{21}$. In the absence of cross-diffusion (small red square), $\overline{\delta}$ is about 2.6. Such a value is kept constant over the red line crossing the origin. The values of the other parameters are $\zeta = 3$, $\beta = 1.5$, $\nu = 1.4$.
• Figure 2: Parameters space $\zeta-\delta$ in which, holding stability condition \ref{['zetabar']}, conditions for Turing instability and formation of patterns are satisfied (region III in each panel), taking functions $c_i$ constantly equal to $1$ (panels (a), (b), (d)) or as in \ref{['Funci1']} (panel (c)) and functions $\lambda_i$ as in \ref{['Funlami1']} (panels (a), (c)), null (panel (b)) or as in \ref{['Funlami2']} (panel (d)). Other parameters are taken as in \ref{['MacPar']}.
• Figure 3: A collection of four different numerical solutions to \ref{['SistMac2.1']}-\ref{['SistMac2.1_bc']}, for different choices of the diagonal and cross-diffusion coefficients, at the final time $T=1000$. In the first column, I, the solutions $n_1$ and $n_2$ for $c_1=c_2=1$ and cross-diffusion terms $\lambda_1,\lambda_2$ as in \ref{['Funlami1']} are shown. In the second column, II, $c_1=c_2=1$ and $\lambda_1 = \lambda_2 = 0$ (i.e. the system reduces to the self-diffusion case \ref{['SistMac2']}). In III, the diagonal diffusion has terms $c_1, c_2$ as in \ref{['Funci1']} and cross-diffusion as in \ref{['Funlami1']}. In the last case, IV, $c_1=c_2=1$ and cross-diffusion functions as in \ref{['Funlami2']}. The yellow dashed segments over the spots refer to Figure \ref{['fig:numerical simulations_profiles']}, where the profiles of the spots along such segments are compared for the 4 cases. The remaining parameters are $\zeta = 3$, $\beta = 1.5$, $\tau = 2$, $\nu = 1.4$, $D_1 = 0.1$, $\delta_{22} = 2.7$.
• Figure 4: Some details of the solutions $n_1$ and $n_2$ are reported in Figure \ref{['fig:numerical simulations']}. The first box refers to the function $n_1$ for the cases I--IV. In the first image, we plot its integral over time, in the second image, the profile of the spots, along the cuts highlighted in Figure \ref{['fig:numerical simulations']} with yellow dashed segments. In the same way, the second box relates to the function $n_2$.

Theorems & Definitions (2)

• Theorem 1
• Lemma 1