Despite their recognized importance, the genetic changes that a population may experience while it is invading new territory have received little attention from modelers. To advance understanding of the genetics of invading populations, two types of mathematical models will be developed and analyzed. The first is a system of integrodifference equations for the spatial moments of mutant and wild type population densities. By analyzing the stability of traveling waves for the equations, the researchers will determine how the probability of so-called "mutation surfing"--in which mutants take over the leading edge of an advancing wave of population density--depends on biologically relevant model parameters. The second type of model is a system reaction-diffusion equations for population density and trait mean and/or variance of a quantitative trait. The existence and stability of traveling wave solutions to such systems will be studied in order to model the evolution of a quantitative trait during an invasion. Rigorous analysis of both types of systems will be complemented both by numerical solution of the equations and stochastic population simulations as appropriate. Finally, in a separate but related part of the project, statistical tests useful to biologists who wish to infer the age and point of origin of an allele based on mutation surfing or the presence of diversifying selection in a subdivided population will be developed. The project will provide training opportunities for a postdoctoral fellow and two undergraduate students.
It is widely acknowledged that nonnative species are causing heavy damage to native ecospecies as well as economic losses (over US $300 billion per year in six large countries alone, according to one study). Effective environmental management requires assessing the risk of degradation of native species as well as the economic risk (e.g. to crops) posed by a newly introduced species or strain. All three parts of the proposed project will contribute to our ability to measure and predict changes in ecosystems, including anthropogenic changes such as the spread of nonnative species or of pesticide-resistant or engineered genotypes. This study of adaptive evolution during invasions will contribute to improvements in prediction of the rates of invasion by nonnative species. These results will allow improved estimates of the benefits and costs of proposed control strategies. The study of mutation surfing will also be useful to biologists whose primary goal is to infer the migration histories of populations--for example, the spread of humans from Africa to Europe, Asia and the Americas. In particular, it will indicate the degree of uncertainty involved in making certain types of inferences the migration histories of populations.
