Theoretical Ecology: Bifurcations in dynamical systems

This first instance of a PhD course on Theoretical Ecology will focus on qualitative changes in dynamics of biological systems that occur when conditions change. Examples of such qualitative changes are when a system changes from being in a constant and stable equilibrium state to exhibiting fluctuations, that is, cyclic dynamics. As another example, even in popular media we may nowadays encounter the term ‘tipping point.’ For example, in the fifth report of the IPCC (the Intergovernmental Panel on Climate Change) a tipping point is defined as an irreversible change in the climate system. These qualitative changes in dynamics occur in many different systems and at many different levels of organization. They are generic phenomena in mathematical models when the value of a particular model parameter changes. These qualitative transitions in dynamics are called bifurcations. The course aims at giving students an understanding about possible phenomena in dynamic models and transitions between different dynamical outcomes, as well as practical training how to apply the theory themselves. The basics are going to be taught while focusing on simple ordinary differential equation (ODE) models. We will then add to it and also discuss the connections to more complex models, such as individual-based models, spatial models or physiologically structured models.