One way that organisms cope with environmental heterogeneity is by evolving a repertoire of phenotypes, called a "reaction norm." In a collaborative research project with Dr. Mark Kirkpatrick, we devised a general method for constructing quantitative genetic models for the evolution of reaction norms over continuous environmental gradients, such as temperature and humidity gradients. We extended previous research on the evolution of phenotypic plasticity over discrete environments to continuous environments by converting techniques developed for modeling "infinite-dimensional traits" to reaction norms (infinite-dimensional traits are characters, such a wing shape, that require an infinite number of measurements to be completely described). Our work resulted in a mathematical theory of reaction norm evolution for continuous environments that can fluctuate in space and/or in time. A promising new avenue of research I am pursuing concerns the evolution of delayed phenotypic responses to changing environments. Current models of reaction norm evolution assume, unrealistically, that phenotypic adjustments are made immediately in response to environmental fluctuations. I plan to examine how delayed responses might affect the evolution of reaction norms as well as how the delay itself might evolve. I expect this work to result in a more complete understanding of how the environmental induction of phenotypically plastic traits can evolve. It will also provide a natural framework for incorporating developmental considerations in population genetics.

My work on infinite-dimensional quantitative characters and
reaction norms has led to a stimulating collaboration with
a mathematical statistician, Dr. Jay Beder. The primary goal of our work has been to establish a mathematically rigorous framework for describing the selection and evolutionary of infinite-dimensional traits. Our results to date suggest that intuition concerning adaptation derived from consideration of simpler quantitative traits holds for infinite-dimensional traits as well. This research has unexpectedly led to
the discovery of entirely new results of potentially fundamental importance in mathematical probability theory.

**Selected publications in this research area**

- Gomulkiewicz, R., and M. Kirkpatrick. (1992). Quantitative genetics and the evolution of reaction norms.
*Evolution***46**:390-411. - Via, S., R. Gomulkiewicz, G. de Jong, C. Schlichting, S. Scheiner, and P. van Tienderen. (1995). Adaptive phenotypic plasticity: Consensus and controversy.
*Trends in Ecology and Evolution***10**:212-217. - Gomulkiewicz, R., and J. H. Beder. (1996). The selection gradient of an infinite-dimensional trait.
*SIAM Journal of Applied Mathematics***56**:509-523. - Gomulkiewicz, R., (In press). Game theory, optimization, and quantitative genetics. In L.A. Dugatkin & H.K. Reeve, eds.
*Advances in Game Theory and the Study of Animal Behavior*. Oxford University Press. Oxford. - Beder, J.H., and R. Gomulkiewicz. (In press). Computing the selection gradient and evolutionary response of an infinite-dimensional trait.
*Journal of Mathematical Biology*.