Bone remodeling is a lifelong process where mature bone tissue is removed from the skeleton and new bone tissue is formed. These processes also control the reshaping or replacement of bone following injuries like fractures but also micro-damage, which occurs during normal activity. We discuss adequate general classes of two- and three-dimensional population models for the cell types involved in this process and show that two populations are not enough to explain what biologists observe. A three-dimensional model class not only explains the observations but also explains what is called Paget's disease of oscillating or disorganized bone remodeling. The mathematical tools are developed step by step and describe the basics of a 'bifurcation theory for an infinity number of ordinary differential equations with the same coupling structure'. This is joint work with Thilo Gross, Martin Zumsande and Dirk Stiefs.
Dr. Siegmund's research focusses on dynamical systems in a broad sense. A wide range of different model classes is necessary to understand complex phenomena, including models with or without delay, deterministic or random, low-dimensional or generated by partial differential equations, time-invariant or time-varying. Some of the key interests for various classes of dynamical systems are spectral theory, invariant and inertial manifolds, timescale separation, reduction by Hartman-Grobman and normal form results, stability radii, bifurcation theory and attractors, nonlinear dynamical systems and control theory, transient dynamics and coherent structures, systems on graphs, network theory and applications e.g. in systems biology und fluid dynamics.
Host: Igor Mezic