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Abstract

Explaining species’ changing ranges, including the speed and ultimate extent of biological invasions, is a central goal in evolutionary ecology. While some species’ range limits have obvious causes (a land species will not colonize an ocean), others do not. Why might a tree species grow on a mountainside only up to an altitude of, say, 1200 meters and not 1300?Kirkpatrick and Barton (1997) proposed a system of partial differential equations (PDE) that modeled the genetics and population dynamics of a population in an extended spatial habitat. (Biologically, the genetics are those of a quantitative trait such as body size, whose value is determined by many genes together with environmental factors.)

Kirkpatrick and Barton used numerical solutions of the system to model a population that cannot invade new habitat because it cannot evolve to thrive in a new environment. Garcia-Ramos and Rodriguez (2002), also numerically, solved the same system to model an invasive population whose rate of advance into new habitat is slowed by the need to evolve adaptively. Despite its high profile in biology, however, almost no rigorous analysis of the Kirkpatrick-Barton system has been carried out.

This talk centers on new theorems and numerical results that shed light on the behavior of solutions of the Kirkpatrick-Barton and related systems. In particular, we prove the existence of certain types of traveling waves (representing biological invasions) and localized stationary solutions (representing “halted” invasions) for the original system. We also highlight biologically significant differences between the predictions of genetic and nongenetic models in spatial ecology, and mathematical challenges that arise in the study of spatial models of quantitative traits.