Toby is from Yorkshire in the UK. He was a philosophy and maths undergraduate and got his PhD in maths from Warwick University. Since then he has been an itinerant mathematician, working on “Coupled Cell Systems” at the University of Houston (2003-2005) and adaptive radiation and “Pod Systems” at the University of British Columbia (2005-2007). He is currently using pod systems, bifurcation theory, coupled cell systems and singularity theory to study resilience in coupled social-ecological systems.
Adaptive radiation is the diversification of a monomorphic population into distinct types as it adapts to different ecological niches. It has long been controversial in the context of sympatric (geographically connected) populations, and it is thought by many that geographic isolation (allopatry) is required for diversification. In sympatry, the argument goes, we should expect the population to spread out smoothly across phenotype space, rather than form into clumps. I will present a novel class of dynamical systems — called “Pod Systems” — that can shed light on the problem.
If a monomorphic population is seen as a unimodal density distribution over phentoype space then the question of what to expect when monomorphism becomes suboptimal can be phrased mathematically as: “What happens when a unimodal distribution becomes unstable?” Generally, one would use partial differential equations to study this sort of question, but current theory is not well-equipped to deal with the loss of stability of non-uniform distributions. Pod systems rememdy this, and also bring to bear the powerful conceptual tool of symmetry. In particular, the pod approach presents adaptive radiation as a “symmetrybreaking bifurcation”. I will explain how a monomorphic population has more symmetry than a polymorphic one, and how the loss of symmetry is often associated with the formation of pattern. Analysis using pod systems shows that the the loss of symmetry associated with adaptive radiation is natural, and should be expected in a large range of ecological scenarios.
In addition to talking about pod systems and adaptive dynamics, I will also touch on the subjects of structure in biological systems, and how and why it can and should be exploited. Although much of this talk has a mathematical basis, I aim to make it accessible (and hopefully intetesting) to biologists as well.