Hermann J. Eberl

Fire blight is a bacterial plant disease that affects apple and pear trees. We present a mathematical model for its spread in an orchard during bloom. This is a PDE-ODE coupled system, consisting of two semilinear PDEs for the pathogen, coupled to a system of three ODEs for the stationary hosts. Exploratory numerical simulations suggest the existence of travelling waves, which we subsequently prove, under some conditions on parameters, using the method of upper and lower bounds and Schauder’s fixed point theorem. Our results are likely not optimal in the sense that our constraints on parameters, which can be interpreted biologically, are sufficient for the existence of travelling waves, but probably not necessary. Possible implications for fire blight biology and management are discussed.

We present and discuss the Streeter–Phelps equations, which were the first river quality model. If the parameters are constants, then the model in its linear formulation can be solved explicitly. This reveals, however, that depending on parameters and initial data, the model might predict negative oxygen concentrations, which marks a breakdown of the model. To address this shortcoming, we introduce a nonlinear modification which, in the case of constant parameters, we can study in the phase plane. In real-world applications, parameters are never constant and are usually known not exactly, but instead with some uncertainty. We show how we can use the solutions for the constant parameter case to obtain estimates for the unknown solutions from estimates of the model parameters, using differential inequalities.

We derive a one-dimensional macroscopic model for biofilm formation in a porous medium reactor to investigate the role of longitudinal diffusion of substrate and suspended bacteria on reactor performance. By comparing an existing base model—one without longitudinal diffusion, which was the point of departure for our work, to the new model—we noticed significant changes in system dynamics. Our results suggest that neglecting it can lead to underestimation of quenching length and biofilm accumulation downstream, even in the advection-dominated regime. The effects of attachment and detachment of suspended bacteria on biofilm formation and substrate degradation were also examined. In the one-dimensional model, it was found that attachment has a stronger influence on substrate depletion, which becomes more pronounced as diffusion in the pore space increases.

Bioventing is an established technique extensively employed in the remediation of soil contaminated with petroleum hydrocarbons. In this study, the objective was to develop an improved foundational bioventing model that characterizes gas flow in vadose zones where aqueous and non-aqueous phase liquid (NAPL) are present and immobile, accounting for interphase mass transfer and first order biodegradation kinetics. By incorporating a correlation for the biodegradation rate constant, which is a function of soil properties including initial population of petroleum degrader microorganisms in soil, sand content, clay content, water content, and soil organic matter content, this model offers the ability to integrate a specific biodegradation rate constant tailored to the soil properties for each site. The governing equations were solved using the finite volume method in OpenFOAM employing the “porousMultiphaseFoam v2107” (PMF) toolbox. The equation describing gas flow in unsaturated soil was solved using a mixed pressure-saturation method, where calculated values were employed to solve the component transport equations. Calibration was done against a set of experimental data for a meso-scale reactor considering contaminant volatilization rate as the pre-calibration parameter and the mass transfer coefficient between aqueous and NAPL phase as the main calibration parameter. The calibrated model then was validated by simulating a large-scale reactor. The modelling results showed an error of 2.9% for calibrated case and 4.7% error for validation case which present the fitness to the experimental data, proving that the enhanced bioventing model holds the potential to improve predictions of bioventing and facilitate the development of efficient strategies to remediate soil contaminated with petroleum hydrocarbons.

The Anaerobic Digestion Model 1 is the quasi-industry standard for modelling anaerobic digestion, and it has seen several new implementations in recent years. It is assumed that these implementations would give the same results; however, a thorough comparison of these implementations has never been reported. This paper considers four different implementations of ADM1: one in Julia, one in Java, and two in Python. The Julia code is a de novo implementation of the ODE formulation of ADM1 that is reported here for the first time. The existing Java and Python codes implement the more common DAE formulation. Therefore, this paper also examines how DAE implementations compare to ODE implementations in terms of computational speed as well as solutions returned. As expected, the ODE and DAE forms both return comparable solutions. However, contrary to popular belief, the Julia ODE implementation is faster than the DAE implementations, namely by one to three orders of magnitude of compute time, depending on the simulation scenario and the reference implementation used for comparison.

The human gut microbiota rely on complex carbohydrates for energy and growth, particularly dietary fiber and host-produced mucins. These complex carbohydrates must first be hydrolysed by certain microbial groups to enable cross-feeding by the gut microbial community. We consider a mathematical model of the enzymatic hydrolysis of complex carbohydrates into monomers by a microbial species. The resulting monomers are subsequently digested by the microbial species for growth. We first consider the microbial species in a single compartment continuous stirred-tank reactor where dietary fiber is the only available substrate. A two compartment configuration in which a side compartment connected by diffusion is also studied. The side compartment is taken to be the mucus layer of the human colon, providing refuge from washout and an additional source of complex carbohydrate in the form of mucins. The two models are studied using stability analysis, numerical exploration, and sensitivity analysis. The delay in substrate availability due to hydrolysis results in bistability and the unconditional asymptotic stability of the trivial equilibrium. The addition of the mucus compartment allows the microbial species to survive under conditions that would otherwise result in washout in a comparable single compartment reactor. This would suggest that depending on the features of the gut microbiota being studied, extracellular hydrolysis and a representation of the mucus layer should be included in mathematical and lab reactor models of the human gut microbiota.

The interactions between marine bacteria and particulate matter play a pivotal role in the biogeochemical cycles of carbon and associated inorganic elements in the oceans. Eutrophic plumes typically form around nutrient-releasing particles and host intense bacterial activities. However, the potential of bacteria to reshape the nutrient plumes remains largely unexplored. We present a high-resolution numerical analysis for the impacts of nutrient uptake by free-living bacteria on the pattern of dissolution around slow-moving particles. At the single-particle level, the nutrient field is parameterized by the Péclet and Damköhler numbers (0 < Pe < 1000, 0 < Da < 10) that quantify the relative contribution of advection, diffusion and uptake to nutrient transport. In spite of reducing the extent of the nutrient plume in the wake of the particle, bacterial uptake enhances the rates of particle dissolution and nutrient depletion. These effects are amplified when the uptake timescale is shorter than the plume lifetime (Pe/Da < 100, Da > 0.0001), while otherwise they are suppressed by advection or diffusion. Our analysis suggests that the quenching of eutrophic plumes is significant for individual phytoplankton cells, as well as marine aggregates with sizes ranging from 0.1 mm to 10 mm and sinking velocities up to 40 m per day. This microscale process has a large potential impact on microbial growth dynamics and nutrient cycling in marine ecosystems.

We propose a new mathematical framework for the addition of stochastic attachment to biofilm models, via the use of random ordinary differential equations. We focus our approach on a spatially explicit model of cellulolytic biofilm growth and formation that comprises a PDE-ODE coupled system to describe the biomass and carbon respectively. The model equations are discretized in space using a standard finite volume method. We introduce discrete attachment events into the discretized model via an impulse function with a standard stochastic process as input. We solve our model with an implicit ODE solver. We provide basic simulations to investigate the qualitative features of our model. We then perform a grid refinement study to investigate the spatial convergence of our model. We investigate model behaviour while varying key attachment parameters. Lastly, we use our attachment model to provide evidence for a stable travelling wave solution to the original PDE-ODE coupled system.