Invasive species are a big deal today. One need only do a simple Google search and see all the exotic species that are hitching a ride on container cargo to find a niche on a new continent. The U.S. Environmental Protection Agency (EPA) has a web site devoted to invasive species; the U.S. National Oceanographic and Atmospheric Agency (NOAA) also has a web site on this topic.

There are a lot of discussions in the scientific literature as well addressing topics from ecology to biological diversity. Interestingly, there is a long history of contributions in the mathematical literature to this topic as well. One such example is the work of Mark Lewis, captured in part in his invited talk at the Mathematical Congress of the Americas in Guanajuato, Mexico in July 2013.

Mathematicians construct and analyze models of biological invasions, asking questions like “Can the invader establish itself (and under what conditions)?” and “will the invading population spread (and if so, how fast)?”

Lewis, in his talk “The Mathematics Behind Biological Invasion Processes,” looked at the second of these questions, focusing on the spread of populations. Such models must take into account the growth rate of the population under various conditions as well as the diffusion of the population.

Populations may compete with other species or cooperate. Lewis gave two examples. The first example was the invasion of the grey squirrel into the U.K., a country where the red squirrel had been prevalent prior to the introduction of the grey squirrel in the 19th century. Interacting species can compete for similar resources. Grey squirrels are larger and more aggressive than their cousins. Will their population eventually replace the red squirrel? Lewis discusses various mathematical models and the conclusion.

A second example is West Nile Virus, introduced into the U.S. in the late 1990s. The spread of the virus depends on hosts (birds and mosquitoes in this case) and has spread rapidly since its introduction.

A notable feature of mathematics is that seemingly disparate phenomena can have very similar mathematical models. The mathematics lends itself to analysis that can be applied generally to many different situations. Lewis traces some of the early history of such models, going back to the work of R.A. Fisher, through to modern dynamical systems.

One can learn about the mathematics behind biological invasions by listening to the recording of the talk online and looking for “Mark A. Lewis – The mathematics behind biological invasion processes.”