When asked to give an invited lecture at the first ever Mathematical Congress of the Americas, I jumped at the chance. This would be an opportunity to meet new colleagues from the Americas and to share my interest in mathematical ecology. I found the meeting, in the beautiful town of Guanajuato, to be well organized and friendly. It was structured so as to allow lots of mixing and time to discuss research over refreshments or lunch.
My particular talk focused on “The Mathematics Behind Biological Invasions,” a subject near and dear to my heart. I enjoy talking about it for three reasons: it has a rich and beautiful history, going back to the work of Fisher, Kolmogorov, Petrovski, Piscounov and others in the 1930’s; the mathematics is challenging and the biological implications are significant; and, finally, it is an area that is changing and growing quickly with much recent research.
The major scientific question addressed in my talk was: how quickly will an introduced population spread spatially? Here the underlying equations are parabolic partial differential equations or related integral formulations. The simplest models are scalar, describing growth and dispersal of a single species, while the more complex models have multiple components, describing competition, predation, disease dynamics, or related processes. Through the combined effects of growth and dispersal, locally introduced populations grow and spread, giving rise to an invasive wave of population density. Thus the key quantity of interest is the so-called spreading speed, the rate at which the invasive wave sweeps across the landscape.
Ideally one would like to have a formula for this speed, based on model parameters, that could be calculated without having to numerically simulate the equations on the computer. It turns out that such a formula can be derived in some situations and not in others. My talk focused on when it was possible to derive a formula. One useful method for deriving a spreading speed formula is based on linearization of the spreading population about the leading edge of the invasive wave, and then associating the spreading speed of the nonlinear model with that of the linearized model. If this method works, the spreading speed is said to be linearly determined.
It turns out that the conditions for a linearly determined spreading speed, while well understood for scalar models, is challenging to analyze for multicomponent models of the sort that include interactions between species. In some cases, such as competition, the results have been worked out, but in many other cases remains an open question.
I was gratified that the talk generated discussion and questions, and I hope that the subsequent follow up will result in new collaborations with colleagues in the Americas who are interested in similar questions.
University of Alberta