Studies of Warning Coloration and Mimicry. IV. A. Mathematical Model of Model-Mimic Frequencies

One of the principal tenets of the Batesian theory of mimicry is that the effectiveness of pseudaposematic coloration is reduced as the incidence of models decreases in relation to that of the mimics. This is to be expected since learning efficiency of predators will be reduced if models become scarce. Field studies (Carpenter 1932, 1949; Ford 1936) have shown that mimicry is, indeed, less effective where the frequency of models is low. Unfortunately, the following generalization is often assumed: if the model outnumbers the mimic, the pseudaposematic coloration is effective, whereas if the mimic outnumbers the model, the system breaks down. Sometimes the principle is stated qualitatively: the mimic must be rare with respect to the model. However, as pointed out by Sheppard (1959) and. Brower (1960), there is nothing unique about the 50-50% ratio. In studies by Brower using caged starlings as predators and meal worms as artificial models and mimics, it was found that the effectiveness of mimicry in a population containing 60% mimics was of the same order of magnitude as that of one containing only 30%. The effectiveness of aposematic and pseudaposematic coloration depends upon a number of variables in addition to model frequency. One is the ability of the predator to remember the aposematic signal. This in turn depends upon the inherent ability of the predator to learn, the nature of the warning signal, and the intensity of the stimulus that the predator receives upon sampling the model. Sheppard (1958) has called attention to the last factor: "An animal sufficiently obnoxious to make predators avoid its colour pattern, even when having encountered only one, could give protection to a mimetic form as common as, or commoner than, the model." In addition, the effectiveness of pseudaposematic coloration depends on the degree of similarity of the model and mimic. Other variables such as hunger, although influencing behavior strongly, may be considered relatively unimportant since their effects would be averaged out over a large sample. It would be advantageous if a mathematical model could be used to describe the situation obtaining between a model and a mimic. Although it should be possible to construct such a model employing all of the variables mentioned above, a simplified model is less cumbersome to handle, and at our present state of knowledge appears entirely adequate. As more data accumulate, it may become desirable to elaborate the simple model presented here. The following assumptions are made: (1) The effectiveness of the mimic pattern is 100%, i.e., the mimic cannot be distinguished visually from the model. This is patently untrue except in the case of artificial models and mimics analyzed below, but it must be admitted that in many cases of known mimicry this condition is approached closely. (2) The strength of the model's stimulus and the ability of the predator to remember it can be expressed in terms of a single parameter, n. This parameter corresponds to the total number of individuals (including the original model) protected by the predator rejecting the model, remembering the stimulus received from it, and thus avoiding both models and mimics until it forgets and renews sampling.