Diagnostics and Experimental Design in Nonlinear Dynamics

Cornell University, Ithaca NY

Investigators

Abstract

This grant is focused on developing statistical methodology for the parameters of ordinary differential equations and non-linear stochastic models that describe the evolution of physical and biological systems. The proposed methodological development can be broadly divided into two parts: (i) inferential methods and (ii)design of experiments. In inferential methodology, the investigator proposes diagnostic tests to assess mis-specification of the drift processes and the presence of state variables that have not been included in a proposed differential equation model. For the design of experiments, control-theoretic methods are suggested for experimental dynamical systems. Specifically, methods to choose system inputs so as to maximize the Fisher Information concerning the parameters of interest for the observed system are described. While models for rapid evolution in experimental ecologies provide real-world a real-world application, the proposed methods have much wider applicability, for example in chemical engineering, pharmacokinetics and neurodynamics. The research in this grant will provide new, improved methods for making statistical inferences about real-world systems that change over time. As a specific example, we consider models that describe an experimental ecology in which different species of micro-organisms are grown together in a laboratory tank. However, this research has broader applications in areas as diverse as human immune systems, models for drug absorbtion by the body and chemical engineering. (I) A first task is to answer the question "Is the proposed model adequate to describe the data observed from this system and, if not, how should it be improved?" A particular question in experimental ecology is whether the models are missingspecies that are present in the system and which must therefore have evolved during the experiment. This research will provide the first statistical test of recently-advanced theories that evolution can occur at the same time-scales as ecological processes; theories that have profound implications for environmental protection, the management of algal ponds producing biofuels and many other domains. (II) A second focus of this work is to develop tools to improve the design of these experiments so as to obtain maximal information about the system from the data they generate. In ecological experiments, system inputs such as nutrients can be manipulated over time so as to produce the behavior that is most informative about the system's dynamics. Similar experimental design problems exist in neural dynamics, chemical engineering and pharmacokinetics. This grant will support the development of novel mathematical, statistical, and computational methods for addressing these issues.