Chapter 8 Bifurcations in dynamical systems

The focus in this part of the course is 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 parameter in such a model changes. These qualitative transitions in dynamics are called bifurcations.

The types of bifurcations we will encounter in this course are:

• Limit points, also called saddle-node bifurcation points
  At a limit or saddle-node bifurcation point a stable steady state (equilibrium point) and an unstable steady state merge and disappear. At one side of the parameter values at which the limit point occurs there are therefore both a stable and an unstable steady state, at the other side of the parameter value these two steady states have disappeared. The stable steady state is called a node, the unstable steady state is called a saddle, hence the name saddle-node bifurcation point. Saddle-node bifurcation or limit points are the most prominent type of tipping point because a small change in a parameter, which would cause its value to cross the threshold value at which the bifurcation point occurs, leads to a collapse of the system state.

• Branching points, also called transcritical bifurcation points
  Branching or transcritical bifurcation points separate two ranges of conditions with on the one side conditions that allow for persistence of a species or population and on the other side extinction. For example in the context of epidemics of infectious diseases, an important branching point occurs at the threshold above which a disease can cause an epidemic. As we know from the Covid-19 epidemic this happens at the threshold value where \(R_0=1\), that is where every infected individual infects exactly one other individual. If \(R_0<1\) the epidemic dies out, while for \(R_0>1\) the epidemic will grow. The point where \(R_0=1\) corresponds to a branching or transcritical bifurcation point.

• Hopf bifurcation points
  Many students will have previously encountered the fascinating phenomenon of cycles in population abundance of predators and their prey. The most classic example of such population cycles are the fluctuations in the densities of the Canadian Lynx and its prey the Snowshoe Hare in Canada, as recorded by the trappers hunting for the pelts of these species in the 19th century. This example is even regularly taught in Dutch high schools. Cycles are however a common dynamic phenomenon that occurs in many different biological systems at different scales of organization, for example, in the breakdown of glucose in yeast cells (Bier et al. 1996), in axons of neurons (Hindmarsh and Rose 1984), neural networks (Borisyuk and Kirillov 1992) and laboratory systems of prey and predators (Fussmann et al. 2000). Cyclic dynamics in a particular system, however, do not just occur independently or at random. In general cycles originate when a stable, but overdamped equilibrium looses its stability. An overdamped equilibrium is characterized by the fact that a system exhibits dissipating fluctuations before settling in the equilibrium, like a pendulum coming to rest after swinging back and forth. When conditions change, that is when a parameter in the model changes, the overdamped fluctuations toward the equilibrium may become undamped. The point at which this transition occurs between an overdamped approach to the equilibrium and undamped fluctuations is referred to as a Hopf bifurcation point.