RESEARCH INTERESTS

My research involves the derivation and analysis of mathematical models that describe population and evolutionary dynamics. I use methods from dynamical systems theory (such as stability analysis and bifurcation theory) applied to a variety of equations types, including difference equations, matrix equations, ordinary and partial differential equations, integro-difference equations, and functional delay equations. I am particularly interested in structured population dynamics, which is a modeling methodology that accounts for different classifications of individuals within a population or species (age classes, size classes, life cycle stages, etc.). A basic goal is to understand how the characteristics of these classes -- their survival rates, birth rates, competitive interactions, evolutionary adaptation, etc. -- affect a population's dynamics, especially with regard to extinction versus survival and the long term nature of the dynamics (equilibration, periodic oscillation, chaos, etc.). I am also study evolutionary game theoretic versions of population models.

Currently I collaborate with Shandelle Henson and Jim Hayward on the dynamics of seabird populations (see the Seabird EcologyTeam webpage and also the article Nature makes sense). In the past I have had interdisciplinary collaborations with the late Tom Vincent and with Robert Costantino on applications of evolutionary game theory in ecology, and with the Robert Costantino, Robert Desharnais, Brian Dennis, Shandelle Henson and Aaron King (the Beetle Team) on experimental nonlinear dynamics. Some readings about some of these research studies are found in my list of publications and in the following books and articles :

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

Mathematicians show how beetles can share a niche by Patrick Barry, Science News, Vol. 175 #3 (2009),  January 31, p 14.

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 9 February 2022)
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