Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results
Alexandre Nolasco de Carvalho Stefanie Sonner
We construct exponential pullback attractors for time continuous asymptotically compact evolution processes in Banach spaces and derive estimates on the fractal dimension of the attractors. We also discuss the corresponding results for autonomous processes.
keywords: Exponential attractors non-autonomous dynamical systems fractal dimension. pullback attractors
Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications
Alexandre Nolasco de Carvalho Stefanie Sonner
This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.
keywords: non-autonomous damped wave equation. non-autonomous dynamical systems Exponential attractors non-autonomous Chafee-Infante equation pullback and forwards attractors
On a coupled SDE-PDE system modeling acid-mediated tumor invasion
Sandesh Athni Hiremath Christina Surulescu Anna Zhigun Stefanie Sonner

We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.

keywords: Multiscale model intra-and extracellular proton dynamics tumor acidity stochastic differential equations partial differential equations pH-taxis
A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor
Peter E. Kloeden Stefanie Sonner Christina Surulescu
A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed. The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion equation for the extracellular proton concentration on the macroscale. In a more general context the existence and uniqueness of solutions for local and nonlocal SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model, both in its local version and in the case with nonlocal path dependence. Numerical simulations are performed to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.
keywords: partial differential equations nonlocal sample dependence intra- and extracellular proton dynamics tumor acidity stochastic differential equations mean-field model. Multiscale model
Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects
Blessing O. Emerenini Stefanie Sonner Hermann J. Eberl

We analyze a mathematical model of quorum sensing induced biofilm dispersal. It is formulated as a system of non-linear, density-dependent, diffusion-reaction equations. The governing equation for the sessile biomass comprises two non-linear diffusion effects, a degeneracy as in the porous medium equation and fast diffusion. This equation is coupled with three semi-linear diffusion-reaction equations for the concentrations of growth limiting nutrients, autoinducers, and dispersed cells. We prove the existence and uniqueness of bounded non-negative solutions of this system and study the behavior of the model in numerical simulations, where we focus on hollowing effects in established biofilms.

keywords: Quorum sensing biofilm cell dispersal density dependent diffusion existence uniqueness numerical simulation
Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces
Tomás Caraballo Stefanie Sonner

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.

keywords: Random dynamical system exponential attractor random pullback attractor fractal dimension ε-entropy
Stochastic models in biology and the invariance problem
Jacky Cresson Bénédicte Puig Stefanie Sonner
Invariance is a crucial property for many mathematical models of biological or biomedical systems, meaning that the solutions necessarily take values in a given range. This property reflects physical or biological constraints of the system and is independent of the model under consideration. While most classical deterministic models respect invariance, many recent stochastic extensions violate this fundamental property. Based on an invariance criterion for systems of stochastic differential equations we discuss several stochastic models exhibiting this behavior and propose classes of modified, admissible models as possible resolutions. Numerical simulations are presented to illustrate the model behavior.
keywords: Itô's and Stratonovich's interpretation. Invariance criteria stochastic differential equations biological applications model validation

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