Georgia Institute of Technology College of Sciences

Interactions between trophic levels are influenced not only by species abundances, but also by the behavioral, life history, morphological traits of the interacting species as well. Adaptive changes in these traits can be heritable or plastic in nature and both yield phenotypic change that occurs as fast as changes in population abundances. I present how fast-slow systems theory can be used to understand the effects rapid adaptation has on community dynamics in predator-prey systems. This analysis emphasizes that heritable and plastic traits have different effects on community dynamics.

Many infectious agents spread via close contact between infected and susceptible individuals. The nature and structure of interactions among individuals is thus of fundamental importance to the spread of infectious disease. Heterogeneities among host interactions can be modeled with contact networks, and analyzed using tools of percolation theory. Thus far, the field of contact network epidemiology has largely been focused on the impact of network structure on the progression of disease epidemics. In this talk, we introduce network models which incorporate feedback of the disease spread on network structure, and explore how this feedback limits the potential for future outbreaks. This has implications for seasonal diseases such as influenza, and supports the need for more adaptive public health policies in response to disease dynamics.

The study of collective behavior---of animals, mechanical systems, or even abstract oscillators---has fascinated a large number of researchers from observational geologists to pure mathematicians. We consider the collective behavior of herds of cattle. We first consider some results from an agent-based model and then formulate a mathematical model for the daily activities of a cow (eating, lying down, and standing) in terms of a piecewise affine dynamical system. We analyze the properties of this bovine dynamical system representing the single animal and develop an exact integrative form as a discrete-time mapping. We then couple multiple cow "oscillators" together to study synchrony and cooperation in cattle herds, finding that it is possible for cows to synchronize less when the coupling is increased. [This research is in collaboration with Jie Sun, Erik Bollt, and Marian Dawkins.]

Individual chemical reactions between molecules are inherently stochastic, although for a large collection of molecules, the overall system behavior may appear to be deterministic. When deterministic chemical reaction models are sufficient to describe the behavior of interest, they are a compact way to describe chemical reactions. However, in other cases, these mass-action kinetics models are not applicable, such as when the number of molecules of a particular type is small, or when no closed-form expressions exist to describe the dynamic evolution of overall system properties. The former case is common in biological systems, such as intracellular reactions. The latter case may occur in either small or large systems, due to a lack of smoothness in the reaction rates. In both cases, kinetic Monte Carlo simulations are a useful tool to predict the evolution of overall system properties of interest. In this talk, an approach will be presented for generating approximate low-order dynamic models from kinetic Monte Carlo simulations. The low-order model describes the dynamic evolution of several expected properties of the system, and thus is not a stochastic model. The method is demonstrated using a kinetic Monte Carlo simulation of atomic cluster formation on a crystalline surface. The extremely high dimension of the molecular state is reduced using linear and nonlinear principal component analysis, and the state space is discretized using clustering, via a self-organizing map. The transitions between the discrete states are then computed using short simulations of the kinetic Monte Carlo simulations. These transitions may depend on external control inputs―in this application, we use dynamic programming to compute the optimal trajectory of gallium flux to achieve a desired surface structure.