Differential equations

Differential equations are the mathematical descriptions of physical and biological systems that continuously change in time. These equations are defined by relationships between the current state of the system and the rate of its change. Our goal is to predict the future behaviour of the system, using analytic, qualitative or numerical investigations of the solutions.

Mathematical epidemiology

Mathematical models can predict how infectious diseases progress to show the likely outcome of an epidemic. By disease modelling we can provide important information to public health by designing, evaluating and comparing different strategies to control the outbreak. The basic mathematical framework uses systems of differential equations called compartmental models.

Nonlinear dynamics

Nonlinear dynamics describe the time evolution of systems where the output is not proportional to the input. To understand complex systems we need to take into account all the interactions of variables and complicated feedbacks. A major goal is to describe the geometric structure of attractors, that encapsulate the most important information about the long term dynamics.

Time delays

Time delays arise in various fields of engineering, physics and biology. Dynamical systems including delays can be written as delay differential equations (DDEs). Their right hand side is a functional, and the corresponding phase space is infinite dimensional. Nonlinear DDEs can show very interesting dynamics and lead to fascinating and sophisticated mathematics.

Modelling of cell biology processes

Cell proliferation, death and motility are key events in many important cell biological processes, such as embryonic development, tissue regeneration or the progression of cancer. Mathematical models help to understand the collective behaviour and the spatio-temporal dynamics of various cell populations, which may lead to, for example, more efficient treatments of cancer.

Bifurcation theory

Bifurcations represent situations when the behaviour of a system suddenly changes as a parameter is crossing a critical value. They are associated to loss of stability, the appearance or destruction of equilibria, sudden emergence of periodic or more complex behaviours. The birth of chaos from simple dynamics can be understood through a cascade of subsequent bifurcations.