RESEARCH

 

I am interested in the a wide variety of equations that define dynamical (or semi-dynamical) systems, including difference equations, matrix equations, ordinary and partial differential equations, integro-difference equations, and functional delay equations. My work focuses on asymptotic dynamics, i.e. stability analysis, bifurcation analysis, oscillations, periodic solutions (forced or unforced), aperiodic dynamics and chaos. My research is motivated by applications to ecology and population dynamics. Current research topics of interest include:  the modeling and analysis of competitive interactions among structured populations; the dynamics of semelparous populations; population dynamics in periodically fluctuating environments and with periodically fluctuation vital rates; inhibited (arrested) maturation due to density effects; the adaptation and evolution of pheotypic traits (for example, the evolution of competitive coexistence versus exclusion); spatial patterns formed by density dependent dispersal in populations with life cycle stages; and the modeling of animal behavior.

Current collaborators include Tom Vincent and Bob Costantino on applications of evolutionary game theory and Shandelle Henson, Jim Haywood and Joe Galusha on the dynamics of seabird behavior ( the Seabird EcologyTeam ). For many years I collaborated an interdisciplinary team biologists and statisticians (see the Beetle Team ). The scope of that work included: the derivation of mathematical models (deterministic and stochastic), the study of the models (both analytical and numerical with the aid of computers), parameter estimation and model validation using laboratory data, and the design and  implementation of laboratory experiments. The over all goal of this work is to show convincingly the value and explanatory power of nonlinear dynamical models for describing and understanding problems in population dynamics and ecology. This goal is addressed by means of carefully derived, biologically based models; carefully designed laboratory experiments; the thorough statistical analysis of gathered data; and rigorous model verification. The first phases of this work were published (and featured in the News & Views section) of the journal Nature, May 1995. Experiments designed to demonstrate, for the first time, that chaos is possible in biological populations have been completed and were reported in Science, January, 1997. A thorough study of this experiment appears in Ecological Monographs 71, No. 2 (2001), 277-303. Follow up theoretical and laboratory work is explored other, sometimes subtle, nonlinear phenomena (Science 294 (19 Oct 2001), 602-605, and Proceedings of the National Academy of Sciences 101, No. 1 (2003), 408-413). This includes the effect of saddle nodes on the dynamics of populations, demographic stochasticity and habitat size, and latticized phase space. Related studies include resonance effects and multi-attractors in periodically fluctuating habitats (see the 29 November 1997 issue of the New Scientist) and competitive interactions (Journal of Animal Ecology 72 (2003), 703-712).

Chaos in Ecology: Experimental Nonlinear Dynamics by J. M. Cushing, R. F. Costantino, B. Dennis, R. A. Desharnais, S. M. Henson, Academic Press, 2003

Matrix Population Models: Construction, Analysis, and Interpretation (Second Edition) by Hal Caswell, Sinauer Associates Inc., 2001

Self-Organization in Complex Ecosystems by Recard V. Solé and Jordi Bascompte, Princeton University Press, Princeton, New Jersey, 2006

Stability in Model Populations by Laurence D. Mueller and Amitabh Joshi, Monographs in Population Biology 31, Princeton University Press, Princeton, New Jersey, 2000

What's Happening in the Mathematical Sciences 1998-1999 by Barry Cipra, published by the American Mathematical Society (ISBN 0-8218--0766-8)
 
Boom time for beetles by Jonathan Knight, New Scientist, 29 November 1997

Chaotic Bugs Make the Leap from Theory to Experiment by Barry Cipra, SIAM NEWS, July/August 1997

Chaotic beetles by H. C. Godfray and M. P. Hassell, Science 275 (1997)

Chaos in a cup of flour by P. Rohani and D.J.D. Earn, Trends in Ecology and Evolution 12 (1997)

 

J. M. Cushing  / Department of Mathematics  / Program in Applied Mathematics  / University of Arizona / Tucson, AZ 85721-0089
(revised 1 January 2008)
Copyright © 2000