arXiv subjects

Hermann J. Eberl

Publications and source records attributed to Hermann J. Eberl.

4 recordsLinked to original sources

arXiv2026

A Combined ODE Model of Carbohydrate Fermentation and Colorectal Cancer

We formulate and analyze a system of non-linear ordinary differential equations that describe key metabolic and immunological interactions between butyrate produced by fiber-fermenting gut microbiota, colorectal cancer cells and host cell populations. The model is studied both independently and in conjunction with a pre-existing carbohydrate fermentation model. The parameter space is explored through sensitivity analyses. Simulation experiments are conducted to illustrate the emergence of varying dynamical behaviour driven by butyrate availability. Our model predicts that butyrate production is driven by fiber consumption and further supported by probiotics in the case of microbial dysbiosis. It also suggests that butyrate may help in suppressing tumour growth. We also show that by adding noise with sufficiently high intensity, cancer elimination occurs almost surely in infinite time and that this threshold level of noise intensity decreases with increasing butyrate concentrations.

q-bio.TO↗

arXiv2024

A PDE-ODE Coupled Spatio-Temporal Mathematical Model for Fire Blight During Bloom

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.

math.AP↗

arXiv2022

Travelling waves in a PDE-ODE coupled system with nonlinear diffusion

We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the TWs.

math.AP↗

arXiv2015

Simulating biofilm deformation and detachment with the immersed boundary method

We apply the immersed boundary (or IB) method to simulate deformation and detachment of a periodic array of wall-bounded biofilm colonies in response to a linear shear flow. The biofilm material is represented as a network of Hookean springs that are placed along the edges of a triangulation of the biofilm region. The interfacial shear stress, lift and drag forces acting on the biofilm colony are computed by using fluid stress jump method developed by Williams, Fauci and Gaver [Disc. Contin. Dyn. Sys. B 11(2):519-540, 2009], with a modified version of their exclusion filter. Our detachment criterion is based on the novel concept of an averaged equivalent continuum stress tensor defined at each IB point in the biofilm which is then used to determine a corresponding von Mises yield stress; wherever this yield stress exceeds a given critical threshold the connections to that node are severed, thereby signalling the onset of a detachment event. In order to capture the deformation and detachment behaviour of a biofilm colony at different stages of growth, we consider a family of four biofilm shapes with varying aspect ratio. Our numerical simulations focus on the behaviour of weak biofilms (with relatively low yield stress threshold) and investigate features of the fluid-structure interaction such as locations of maximum shear and increased drag. The most important conclusion of this work is that the commonly employed detachment strategy in biofilm models based only on interfacial shear stress can lead to incorrect or inaccurate results when applied to the study of shear induced detachment of weak biofilms. Our detachment strategy based on equivalent continuum stresses provides a unified and consistent IB framework that handles both sloughing and erosion modes of biofilm detachment, and is consistent with strategies employed in many other continuum based biofilm models.

physics.flu-dyn↗