## The Hopf-Hopf bifurcation and chaos in ecological systems

This post arises from the fact that there seems to be some constructive interference between various directions I am pursuing at the moment. The first has to do with the course on dynamical systems I just finished giving. This course was intended not only to provide students in Mainz with an extended introduction to the subject but also to broaden my own knowledge. I wrote lecture notes for this in German and having gone to the effort of producing this resource I thought I should translate the notes into English so as to make them more widely available. The English version can be found here. Both versions are on the course web page. The second thing is that I will be organizing a seminar on bifurcation theory next semester and I want it to achieve a wide coverage, even at the risk that its waters, being broad, may be shallow (this is paraphrase of a quote I vaguely remember from Nietzsche). The connections between these two things are that I treated simple bifurcation theory and a little chaos in the course and that going further into the landscape of bifurcations necessarily means that at some point chaos rears its ugly head. The third thing is the fact that it has been suggested that the MAPK cascade, a dynamical system I am very interested in from the point of view of my own research, may exhibit chaotic behaviour, as described in a paper of Zumsande and Gross (J. Theor. Biol. 265, 481). This paper attracted my attention when it appeared on arXiv but it is only now that I understood some of the underlying ideas and, in particular, that the Hopf-Hopf bifurcation plays a central role. This in turn led me to a paper by Stiefs et. al. on chaos in ecological systems (Math. Biosci. Eng. 6, 855). They consider models with predator-prey interaction and a disease of the predators.

A Hopf-Hopf (or double Hopf) bifurcation arises at a stationary point where the linearization has two pairs of non-vanishing purely imaginary eigenvalues. Of course it is necessary to have a system of at least dimension four in order for this to occur. The subset of parameter space where it occurs has codimension two and lies at the intersection of two hypersurfaces on which there are Hopf bifurcations. For this system there is an approximate normal form. In other words the system is topologically equivalent to a system given by simple explicit formulae plus higher order error terms. The dynamics of the model system ignoring error terms can be analysed in detail. For simple bifurcations a system in approximate normal form is topologically equivalent to the model system. For the Hopf-Hopf bifurcation (and for the simpler fold-Hopf bifurcation with one zero and one pair of non-zero purely imaginary eigenvalues) this is no longer the case and the perturbation leads to more complicated dynamics. For instance, a heteroclinic orbit in the model system can break as a result of the perturbation. A lot of information on these things can be found in the book of Kuznetsov. In the paper on ecological systems mentioned above a Hopf-Hopf bifurcation is found using computer calculations and this is described as ‘clear evidence for the existence of chaotic parameter regions’. My understanding of chaos is still too weak to be able to appreciate the precise meaning of this statement.

Using computer calculations Zumsande and Gross find fold-Hopf bifurcations in the MAPK cascade (without explicit feedback) indicating the presence of complex dynamics. If chaos occurs in the ecological system and the MAPK cascade what biological meaning could this have? Ecosystems can often be thought of as spatially localized communities with their own dynamics which are coupled to each other. If the dynamics of the individual communities is of a simple oscillatory type then they may become synchronized and this could lead to global extinctions. If the local dynamics are chaotic this cannot happen so easily and even if a fluctuation which is too big leads to extinctions in one local community, these can be avoided in neighbouring communities, giving the ecosystem a greater global stability. One point of view of chaos in the MAPK cascade is that it is an undesirable effect which might interfere with the signalling function. It might be an undesirable side effect of other desirable features of the system. In reality MAPK cascades are usually embedded in various feedback loops and these might suppress the complex  behaviour in the free cascade. Zumsande and Gross investigated this possibility with the conclusion that the feedback loops tend to make things worse rather than better.