No matter whether you are an ecologist, an evolutionary biologist, a zoologist, botanist, microbiologist, geneticist or whatever, something that most of us in the field of biological science share in common, is a certain fascination and interest for the life and travels of the naturalists of the Age of Revolution (besides, of course, the obvious fascination and curiosity for nature itself!). I am quite sure that most of the readers of this blog have spent some time reading about the great travels of Alexander von Humboldt, Charles Darwin, Alfred Wallace, Joseph Hooker, Thomas Huxley, among so many others. As a Brazilian ecologist, I have always been captivated for this kind of literature, as well. It is always captivating to read about the impressions that these eminent scientists had about our landscapes, fauna, flora, and especially how they viewed the manners and the customs of our Latin American societies, back then.
But, among this great cast of naturalist, I (and many other Brazilian scientists, no doubt) have always been most thrilled by the work of Johannes Friedrich (“Fritz”) Müller (1821 – 1897). Fritz Müller is considered, the first “(naturalized) Brazilian naturalist,” although he was born in Germany (he emigrated to Brazil is his early 30s).
Fritz Müller was a prolific writer; he published over 70 papers, and helped to support and enrich the theory of evolution by natural selection, in the Darwinian era. Darwin came to know Müller’s work, after reading his book Für Darwin, a comprehensive book, where Müller compiled empirical evidence and ideas supporting Darwin’s theory. Impressed by Müller’s argumentation and the richness of details in his writing, Darwin titled him the “Prince of observers”. Although they never met, they kept an extensive exchange of correspondence throughout their lives, after the publication of Für Darwin.
Fritz Müller is also responsible for some ideas that are key to modern evolutionary biology and ecology nowadays. Sadly, his legacy seems to be little known, even among scientists. Some science historians believe that he is not as well known as other early naturalists because most of his work was written in languages other than English (mostly in German).
But one of his most remarkable works is a paper written in 1879, about two mimic butterfly species. This paper not only introduced a new adaptive mechanism for mimicry, which is known as Müllerian Mimicry (i.e., when two or more unpalatable species that share one or more common predators, have come to mimic each other’s warning signal, in order to avoid predation) but also, introduced the first mathematical model of eco-evolutionary dynamics, at least in a Darwinian framework (I would say that Fibonacci’s  and Malthus’  population growth models, are perhaps, the best examples of early ecological models in pre-Darwinian time). His model, also know as “Müller’s number-dependent model of mimicry”, is consider the first mathematical derivation of the fitness gains of adopting a specific trait.
The main aspects of his model are (see Ruxton et al. 2008, especially chapter 9, for further details):
i) The system is formed by at least three species: a predator and two prey species (but the model can be extended for more species);
ii) Prey species are mimic, i.e., present the same color pattern;
iii) Predator learning requires a certain number of unpalatable individuals with a specific color pattern;
iv) As the costs of predator learning will be shared by all prey species, it will always lead to positive and mutualistic advantages to them;
v) The relative fitness gains due to mimicry to each species is given by the inverse square of the relative abundances of the two prey species. Thus, the advantages will be unequal between mimic species (but still positive for both mimic species).
Despite some controversies that were raised in the past 134 years, and the lack of support for some of his predictions (See Sherratt 2008 for a great review and an evaluation of his hypotheses), Müller’s simplistic number-dependent model continues to resonate. In fact, just in the past semester, over 100 papers mentioning “Müllerian mimicry” were published (from Google Scholar; 11 papers in the Web of Knowledge database ). His model, and ideas, has helped ecologists and evolutionary biologists to interpret the benefits of Müllerian mimicry and has laid some of the foundations for co-evolutionary studies.
Beatty, C.D., K. Beirinckx & T.N. Sherratt. 2004. The evolution of Müllerian mimicry in multispecies communities. Nature 431: 63-66. doi:10.1038/nature02818
Mallet, J. 2001. Mimicry: An interface between psychology and evolution. PNAS 98: 8928–8930. doi: 10.1073/pnas.171326298
Mallet, J. & N.H. Barton. 1989. Strong natural selection in a warning-color hybrid zone. Evolution: 421-431.
Pinheiro C.E.G. 2003. Does Müllerian mimicry work in nature? Experiments with butterflies and birds (Tyrannidae). Biotropica 35:356–364. doi: 10.1111/j.1744-7429.2003.tb00589.x
Ruxton, G.D., T.N. Sherratt & M.P. Speed. 2004. Avoiding attack: the evolutionary ecology of crypsis, warning signals, and mimicry. Oxford: Oxford University Press.
Sherratt, T.N. 2008. The evolution of Müllerian mimicry. Naturwissenschaften 95: 681-695. doi: 10.1007/s00114-008-0403-y
West, D.A. 2003. Fritz Müller: A naturalist in Brazil. Ed. Pocahontas Press, Virginia.
Note: As most of you must know, a few months ago, there was a heated debate, on whether or not biologists need math (see for instance, here and here. It started off with an interview given by E. O. Wilson). I won’t get into the discussion, but as this post talks about math and biology, I would like to leave a passage from Darwin’s autobiography:
“During the three years which I spent at Cambridge my time was wasted, as far as the academical studies were concerned, as completely as at Edinburgh and at school. I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.”