Sustainability and Complex Systems in Columbus, or: from IBMs to aggregate models and back

The MBI building

I spent this week in Columbus, Ohio, to attend a workshop on “Sustainability and Complex Systems”, which was organized by Chris Cosner, Volker Grimm, Alan Hastings, and Otso Ovaskainen at the Mathematical Biosciences Institute (MBI). To quote the organizers:

This workshop aims to engage computational and mathematical modelers, empiricists, and mathematicians in a dialogue about how to best address the problems raised by the pressing need to understand complex ecological interactions at many scales.

The focus though has clearly been more on mathematical modeling than on empiricism, with a lot of discussion on analytical vs. simulation-based approaches and how to translate between them. We had a number of really excellent talks and discussions, which I obviously can’t all mention in detail, so check them out here, but I wanted to highlight two contributions in particular, because those covered topics that were quite new to me, and I suspect they may be to others here as well.

Both of those contributions dealt in some way with bridging the gap between simulation models and analytical / mathematical models of biological systems, so coupled ODEs or PDEs or such alike.

The first talk I want to mention was Yannis Kevrekidis from Princeton University about “Coarse-graining computations for complex systems”. In essence, his point was that in many systems, there is a natural separation of time scales between the scale of fast local interactions (which we can/have to describe by simulation approaches such as IBMs), and some slow macroscopic quantities (which we would like to describe by something like ODEs or PDEs). So, if such a separation exists, it may be possible to simulate the fast interactions (for example local competition between plants), derive from that simulations of the “derivatives” for macroscopic quantities that we are interested in (e.g. the population growth rate), and then use these (approximated) derivatives to propagate the macroscopic quantities in space or time as if it were an ODE or PDE. At the new macroscopic state (e.g. higher population size), we can then reinitialize the simulation and reiterate the procedure and so on (details see Kevrekidis & Samaey, 2009). So, this allows us to use all the mathematical tools that require derivatives although we have only a simulation model, which gives access to a lot of interesting mathematical options and may potentially also save some time, although the latter point was a bit unclear because the computational costs for the approximation are likely substantial as well. Yannis was also presenting some further ideas about how to do similar things in space (they call this the “gap-tooth method”, Gear et al., 2003) and how to automatically find good “macroscopic variables” for describing complex dynamic systems by applying geometric diffusion algorithms on the systems’s state space (Coifman et al., 2005). They have used this stuff for chemical models so far, would be interesting to see whether there are applications in ecology.

Illustration of the idea of using a simulation to derive derivatives for macroscopic states from: Kevrekidis, I. G. & Samaey, G. (2009) Equation-Free Multiscale Computation: Algorithms and Applications Annu. Rev. Phys. Chem., 60, 321-344

The second talk was by Otso Ovaskainen on “models and data: from individuals to populations”. Essentially, Otso presented an alternative to standard moment closure methods (e.g. Bolker & Pacala ’97) for analytically solving spatial point-process models. The idea they have is based on working with spatial cumulants instead of spatial moments. Once they do this, for reasons I did not fully catch, it seems to be possible to do a perturbation expansion similar to Ovaskainen & Cornell, 2006 for which they can prove that the solution for the cumulant equations is asymptotically exact (asymptotic = large interaction length). The idea seemed very interesting, although I suspect that technical challenges for implementing this will be comparable to that of moment closure methods, a lot of the steps were very similar as well. Otso seemed to think that the method has a number of advantages though, in particular that there is no need to define a somewhat arbitrary closure. They are working on this idea at the moment, so I guess there will be some publication in the future.

References Yannis KevrekidisKevrekidis, I. G. & Samaey, G. (2009) Equation-Free Multiscale Computation: Algorithms and Applications Annu. Rev. Phys. Chem., 60, 321-344

Gear, C.; Li, J. & Kevrekidis, I. G. (2003) The gap-tooth method in particle simulations Physics Letters A, 316, 190-195

Coifman, R. R.; Lafon, S.; Lee, A. B.; Maggioni, M.; Nadler, B.; Warner, F. & Zucker, S. W. (2005) Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps PNAS, 102, 7426-7431

References Otso Ovaskainen

Ovaskainen, O. & Cornell, S. J. (2006) Space and stochasticity in population dynamics Proc. Natl. Acad. Sci. USA, 103, 12781-12786

Bolker, B. & Pacala, S. W. (1997) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems Theor. Popul. Biol., 52, 179-197