Interdisciplinary Science

Per Enflo  ■  Interdisciplinary Science

I think that mathematics can successfully be used in many more areas of science than is done today - to help us better understand the world around us. The three areas below, that I have worked in, represent three different ways to apply mathematics - or mathematical thinking.

My  Human Evolution/Population Genetics work was initially motivated by the challenge to understand an apparent contradiction between fossil data and DNA data concerning the Neandertals. It was not clear that mathematics would play any role in the solution of this problem, but it actually did. Mathematics certainly played a role in the evaluation of different scenarios, that I studied. The second project is a study of how much of Human Evolution can be understood as a consequence of the reproduction strategy of humans. No mathematics is involved.

The Mathematical Biology/Ecology work was initially to study, whether changes in the composition of algae in Lake Erie after the zebra mussel invasion, could be understood as consequences of basic ecological phenomena: Different species competing for the same nutrients, and the predator-prey relation. In the big picture, there are equations for these basic phenomena - systems of equations of Lotka-Volterra type. Thus it was fairly clear what type of mathematical equations one should look for. One of the biggest challenges was to find realistic parameter values in these equations.

In the Acoustics work, the mathematical equations were already there. The challenge was to find as precise and explicit solutions as possible.

Human Evolution/Population Genetics

In the 1990:s there was an apparent contradiction between fossil data and DNA data concerning the Neandertals. Some fossil data indicated that there had been interbreeding between Neandertals and Early Moderns. On the other hand, all Neandertal DNA seemed to have disappeared in today's human population. I showed that the concept of "regions, where the populations do not reproduce themselves" could be used to give a simple explanation of the loss of Neandertal genes, even under a scenario of total and random interbreeding between Neandertals and Early Moderns. This model also predicted that one might find a fossil with modern anatomy and mitochondrial DNA very different from ours. Such a find was done in Australia in the end of the year 2000.

The low genetic variation in today's human population has been attributed to a bottleneck in the past in the human population. The concept of "regions where the populations do not reproduce themselves" gives a possible explanation of this low genetic variation, without assuming any bottleneck.

Background and Annika Jensfelt's

     ■ Article in Svenska Dagbladet 2001-01-14


Presentation at the Yearly Meeting of American Association of Physical Anthropologists, Kansas City, Missouri, March 28 to March 31, 2001

     ■ A simple reason, why Neandertal ancestry can be consistent with current DNA information 

     ■ Abstract of talk extracted from Program [American Journal of Physical Anthropology, 114 p. 62]


Presentation for a Panel discussion on the topic "How unique, in fact, is the animal man? (Hur unik är djuret människa, egentligen?)" in 2008 at Göteborg Book Fair, Gothenburg, Sweden

     ■ Reproduction strategy as a driving force for Human Evolution



    1. Race, Culture and Human Evolution

    From: University of Michigan | By: Rachel Caspari, Milford Wolpoff

    "A second cause of genetic variation over space was recognized by the mathematician Per Enflo. He noted that human populations differed (as they still differ) in what he calls their reproductive stability. Some maintain constant population sizes for long time periods, while others fluctuate in response to diseases, or ecological or climatic changes. People will tend to move from reproductively stable to unstable regions, which can lead to some genes dispersing widely but not necessarily evenly. Selection also can distribute features across broad gradients of variation when the features are not neutral."

    Source: Fathom the source for online learning

    2. No evidence of Neandertal mtDNA contribution to Early Modern Humans

    PLOS Biology 2004

Mathematical Biology/Ecology

We worked out (joint work with A.Spalsbury, R.Heath and myself) numerical solutions of Lotka-Volterra systems with 3 equations - involving zebra mussels, "edible" algae and "inedible" algae and, as a parameter, phosphorus loading of Lake Erie.

The model suggests the following: 1) Loading more phosphsorus into the lake will have a negative effect on the composition of algae. 2) The system will have oscillations, and one can expect more oscillations nearshore than offshore, in the quantity of zebra mussels as well as in the quantity of the different types of algae . 3) If algae had been grazed by a short-lived aggressive species (e.g. Daphnia) rather than by zebra mussels, one would have more oscillations in the quantity of algae than have been observed.

Refugia for edible algae - larger offshore than nearshore - stabilize the system. In fact, the modeling suggests the following general ecological principle: Introducing refugia for prey, will change an oscillating ecological system into an ecological system which is in a stable equilibrium state. So, also outside of the refugium, the oscillations will stop.


Modeling the effects of nutrient concentrations on community production and ecosystem stability: Framework for a Great Lakes Model. (with R.T.Heath, R.Sturtevant and D.Shoup). In Great Lakes Modeling Summit--Focus on Lake Erie. L.Tulen and J.dePinto, Eds., published by Int. Joint Commission, Windsor, ONT., ISBN 1-894280-17-2 (2000), 37-50

Long-term Plankton Community Effects of Zebra Mussels: Mathematical Analysis of a Simple Predator-prey Model With and Without Refugia. (with R.T. Heath). 48th Annual Conference of the  International Association for Great Lakes Research, Abstracts,  Ann Arbor, 2005, Great lakes ecosystem forecasting : improving understanding and prediction. (2005) p 53


The background of the first paper is that perturbations at the vertex of an acoustic sawtooth wave in a slightly dispersive medium (i.e. water) have been observed. By solving a variant of Korteweg-de Vries-Burgers' equation, we showed that such perturbations will happen as a consequence of small bubbles in the medium.

In the second paper we gave an exact solution formula of the sound field from a point source over a large surface with two different materials (the "land-sea" problem). The formula is complex and in the years 1982-86 we worked it out to be easily interpreted near the surface. Before our work, such formulas had only been given for large homogeneous surfaces.

The referee report starts: "The authors have tackled and solved a very difficult problem."



Perturbations at the vertex of an acoustic sawtooth wave in a slightly dispersive medium (with B. Enflo), Report/Institut Mittag-Leffler, 99-0178853-6 ; 1982:1 Stockholm

An exact solution formula for sound wave propagation from a point source over a surface with an impedance discontinuity (with B. Enflo), J. of Acoustical Soc. of America, Vol. 82 (1987) 2123-2134

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