The mathematics describing the seemingly unpredictable movement of a stumbling drunk -- "drunken walk" math -- has helped scientists better understand the logic of ecological invasions.

The spread of disease or invasive species are examples of ecological invasions. The phenomenon is studied by both ecologists and mathematicians. Ecologists focus on field observations and what they can say about ecological invasions. Mathematicians focus on theory.

Both work together to develop models to predict the spread of diseases and invasive species. But the two groups of scientists continue to disagree on the nature of randomness and how it should be accounted for in their predictive models.

Some researchers have suggested randomness accelerates invasions, while others believe randomness impedes the speed of invasions. Some believe randomness has no effect.

To settle the confusion, Tim Reluga, an associate professor of mathematics and biology at Penn State University, built a ecological invasions model based on drunken walk mathematics. Most invasion math models are based on traveling waves solutions.

His model considers three different types of randomness -- spatial, demographic and temporal.

Take the spread of acorn trees in England and Scotland as an example of an invasion. Spatial randomness is introduced by the presence of squirrels, which can carry acorns a variety of distances. Demographic randomness accounts for the variability of reproduction -- how many acorns a given tree might produce. Temporal randomness describes the varying rates of reproduction across time -- how often a tree generates seeds over the course of a year, decade or century.

Reluga believes his model -- detailed in the journal Theoretical Population Biology -- succeeds at reconciling mathematical and ecological models.

"Species invasions are best interpreted not as waves, but as random walks, and that the discreteness of living organisms is fundamentally important," Reluga argued in his newly published paper.

"I hope this paper makes things clear that different kinds of randomness have different effects on invasions," Reluga said in a news release.

Regula's model suggests spatial and temporal randomness accelerate an invasion, just as previous computer simulations have predicted. His model also suggests demographic randomness and greater population density slow invasions.

"This is the way we should be thinking about the problem of randomness in ecological invasions," Reluga said. "If we think about it in this different frame, all the results make natural sense."