Revisiting the 1879 model for Evolutionary Mimicry by Fritz Müller: New mathematical approaches

Abstract

In this paper we present some novel mathematical approaches to describe an extraordinary phenomenon widely cited in the biological literature as “Müllerian Mimicry”. Mimicry in general is an evolutionary phenomenon which was observed and registered in the scientific literature a long time ago by Henry W. Bates in a form named “Batesian” today. Müllerian Mimicry is a subtle phenomenon from an evolutionary point of view and happens when two different species, both of them toxic as well, and under pressure from the same predator, develop a similar strong visual signal in such a way that “teaching casualties” become a shared onus. The tasting of any individual, no matter from which prey population, will turn the hapless predator into a “learned” one which will treat both prey populations with due respect afterwards. This type of Mimicry among butterflies having birds as their main predators was first described by Fritz Müller, a German-Brazilian naturalist in a paper published in 1879 which emphasized the learning dynamics of predators. Besides the formulation of the main principles for studying such phenomenon, Müller argued his ideas with one of the first mathematical model of evolutionary theory. In this paper we present some of Müller's ideas in a framework general enough to be represented by mathematical models. As an example we formulate a conceptual simple discrete-time model and discuss a number of simulations which exemplify many observable aspects of Müllerian Mimicry. In our discussion we will always keep in mind the real examples represented by butterflies as prey populations and birds as their predators which in fact were the case originally studied by Müller and still is one of the more important instances where Mimicry phenomena occur in nature. However, we try to expound a sufficiently general scheme that could support many mathematically specific models for predator learning dynamics and signaling communication