# Mathematical Psychology and Psychophysiology

By Stephen Grossberg | Go to book overview

SIAM-AMS Proceedings Volume 13 1981

Sociobiological Variations on a Mendelian Theme

M. FRANK NORMAN

1. Introduction. My articles in this volume illustrate two aspects of my work a psychologically oriented mathematician. The present article shows elementary mathematics in the service of psychology. The other article [1] shows elementary psychology in the service of mathematics. Both papers bear on simple genetic models: deterministic models in this paper and stochastic models in the other one.

In psychology one frequently encounters facile evolutionary explanations of contemporary behaviors and their neural substrates. It is often easy to convince oneself that a certain characteristic of a currently predominant genotype permitted individuals of that genotype to have more offspring than conspecifics, thus ensuring the success of the genotype. The (modest) interest in such exercises is partially predicated on the validity of the transition between individual reproductive success ("fitness") and long-term success of the genotype. This paper shows that such transitions are not always valid.

2. Fitness and survival. According to E. O. Wilson, "Hamilton's theorem on altruism consists merely of a more general restatement of the basic axiom that genotypes increase in frequency if their relative fitness is greater" [2, pp. 415-416, italics added]. W. D. Hamilton's theory of the evolution of altruism will be considered in §4. The present section relates to the italicized proposition, which is an unusually explicit statement of a dominant theme of the literature of Evolutionary Biology.

Consider the following example. Suppose that there are three interbreeding varieties of zebras in a certain region. Call these varieties a, b, and c. They differ in probability of survival from conception to reproductive age, perhaps because of different diets. The common term for this survival probability is viability. Assuming that all varieties are equally fertile, viability is proportional to, and can thus be identified with, the expected number of offspring of a newly

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1980 Mathematics Subject Classification. Primary 92A10, 92A20, 92A25.

-187-

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