Mathematical models of mimetic systems

Because of the advent of data and the desire to predict and to model mimetic systems, the past two decades have seen a number of attempts to construct mathematical models, often utilizing the simulative power of the computer. Thus Huheey (1964, 1976) has presented models for Batesian and Mullerian mimicry, respectively, based originally on the experimental data of J. Brower (1960) with respect to the feeding habits of starlings on quinine-dipped mealworms, but augmented by data on the reaction of toads and treefrogs to honeybees, Apis mellifera (Huheey, 1980b). Other workers who have made valuable 

The basis of the Batesian model is a simplistic assumption about predator behavior A predator will cease eating a particular noxious model upon experiencing a traumatic encounter with it, and thereafter it will avoid eating that model (and its Batesian mimics) for the next n-1 encounters. A simple mathematical model can then be derived from this assumption to give (Huheey, 1964)  

The main difficulties with this mathematical model, aside from the simplistic picture of predator memory, are that (i) time does not enter into the model as a real variable, and (ii) there is no allowance for alternative prey. The latter should be particularly important since the reluctance of the predator to accept unpalatable prey will be a function of its hunger. Holling (1%5) has presented a mathematical treatment of mimicry which considers type and quantity of alternative food. Dill (1975) has extended these ideas and provided some experimental support (see also Schuler, 1980). 

A number of workers have been intrigued by the value and meaning of n. Most have agreed that long-waiting strategies are inefficient and have opted for 

Arnold (1978) has called attention to the fact that all previous workers had assumed a uniform distribution of models and mimics in space. He was able to show mathematically that if the models clump, there will be reduced predation upon them. Selection will favor values for the predator s mean movements and waiting time (n) such that it will move out of the clump of models and into a region with a higher density of mimics. Individual models should be selected for clumping including traits, such as pheromones, that promote it (Eisner and Kafatos, 1962). 

---