Series: Mathematics Colloquium

Date: Thursday, April 17, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Howard Weiss

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: J. M. Cushing, University of Arizona

Title: Nonlinear Dynamics and Ecology

Abstract:

Nonlinear mathematical models have a long history in theoretical
population dynamics and ecology. The famous logistic and
Lotka/Volterra differential equations are the prototypical examples
that appear today in undergraduate textbooks. However, the impact of
nonlinear theory on ecology has been limited primarily to the simplest
of dynamic phenomena, namely, stable equilibria and limit cycles. One
reason for this is the lack of close connections between theoretical
models and data. Unlike in the hard sciences, in ecology there is a
scarcity of models that have been validated and parameterized using
data and that provide quantitatively accurate descriptions and
predictions.  As a result, the tools necessary for a rigorous
investigation of complex nonlinear phenomena are not in general
available. In this talk I will describe some results obtained from a
decade long project in which many nonlinear phenomena have been
studied using mathematical models in conjunction with biological
experiments. This project, carried out by an interdisciplinary team of
mathematicians, statisticians, and biologists, entails model
derivation and validation, mathematical and statistical analyses of
models and data, and laboratory experiments designed to study model
predicted phenomena. The results include complex, subtle, and often
unintuitive nonlinear effects that have been documented for the first
time in the dynamics of a real biological population. The phenomena
about which I will speak include bifurcations, chaos, saddle
equilibrium (and cycles) and their stable manifolds, lattice effects,
periodic forcing and resonance, multiple attractors, phase switching,
and effects related to stochasticity.