Bill Sears Colloquia - Spring 2003

• February 10 - Fabio Dercole, Department of Electronics and Information Politecnico di Milano, Italy, "Evolutionary dynamics through bifurcation analysis: Methods and application"
  Abstract: Evolution, one of the most important notions in biology, is recognized to be of primary importance in many fields of science. Evolving systems are composed of basic units identified by inheritable traits, with the possibility of mutation during generation. The interactions between groups of homogeneous units are determined by their dimensions and traits and act as a selection process. Classical modeling approaches to evolutionary phenomena fall in the fields of genetics and game theory. They are static approaches, thus lacking a proper dynamic description. Dynamic frameworks have been proposed to mimic the evolutionary dynamics, but they are not derived from underlying mutation-selection processes. By contrast, the recent theory of Adaptive Dynamics explicitly considers the interplay between the fast dynamics of group dimensions and the slow variation of the traits. Under general conditions, the slow variation of the traits is described by a set of ordinary differential equations. The first part of the talk is an overview on Adaptive Dynamics theory, while some applications to problems of remarkable interest in the fields of biology and economics are presented in the second part.
• March 10 - Richard Casey, CAM, "Optimization of parameters in neural models"
  Abstract: Neural models based on the Hodgkin-Huxley equations tend to have a large number of parameters, many of which are often poorly known. One task frequently faced in modelling is to find parameter regimes that produce certain desired behaviour in the model neuron, typically to match firing properties observed experimentally. The difficulties inherent in this task motivates the development of better computational tools, and in this context I'll talk about algorithms we've been working on to better fit neural models to experimental data. We use automatic differentiation to compute periodic orbits in the model vector field, and combine this with a quasi-Newton algorithm to seek optimal fits between the model and the data. One of the key ideas is to define the distance between periodic orbits computed in the model and a reference orbit based on the data, and minimize this distance function using the quasi-Newton algorithm.
• March 31 - Umut Cetin, CAM, "Very Short Introduction to Mathematical Finance"
  Abstract: In this talk, I'll try to illustrate how stochastic calculus is used to model financial phenomenon. I'll start with the concept of self-finacing portfolio, and then describe the relation between existence of martingale measure for the stock price and no-arbitrage. Arbitrage-free pricing of options and complete markets are to be discussed.
• April 14 - Tim Healey, Department of Theoretical and Applied Mechanics, "Computation of Spatial Equilibria of Nonlinealy Elastic Cosserat Rods" (work by Tim Healey & Prashant Mehta)
  Abstract: Rod theory provides an accurate model for long, thin flexible structures. The classical roots of the finite-deformation theory, going back to Kirchhoff, are tied to structural engineering. In recent years there has been a surge of interest in the analysis of such models, due to their applicability in modeling biological structures. We consider the problem of computing global equilibria of elastic Cosserat rods - the later terminology referring to the fact that the rod may suffer shear and extension, as well as bending and twist. As is the case in the pure analysis of such problems, the main difficulty here stems from the effective treatment of the rotation field. Like many others before, we parametrize the rotations via unit quaternions (Euler parameters), which induces a single explicit constraint. Not surprisingly, this leads to inconsistencies in the prescription of boundary conditions. We present an effective formulation, which properly identifies the constraint as a conservation law, enabling consistent specification of boundary conditions. The heart of our approach is inspired by the well known Liapunov Center Theorem, which has been used in the numerical treatment of periodic solutions of systems of ODE's by S. Doedel. We present several concrete examples, employing the software package AUTO, demonstrating the effectiveness of our formulation.
• April 21 - Neo Martinez, San Francisco State University, "Ecological Networks: Integrated Analysis of the Structure and Nonlinear Dynamics of Large Complex Ecological Networks"
  Abstract: Integrated analyses of network structure and dynamics are recognized as one of the most important frontiers of network science. Like other scientists, ecologists have pursued this frontier from two angles. One angle is the study of the nonlinear dynamics of small modules containing a few species or nodes, e.g., a prey and predator or, more recently, 2 or 3 plant species plus a couple of herbivores and a predator. The other angle is the study of the structure of large complex networks containing tens to hundreds of nodes sometimes integrated with linearized analyses of network dynamics. This talk will describe newly integrated studies of the nonlinear dynamics of large complex ecological networks. Effects of the dynamics on structure as well as effects of structure on dynamics will be discussed. Beyond these general results, the integrated structure-dynamic modeling framework provides a robust and ecologically realistic foundation for more sophisticated analyses of the effects of environmental stochasticity, species invasions and extinctions, and spatial heterogeneity in complex ecosystems.