Stahl67& bmp

190 WALTER R. STAHL

sition of an allele in a genome. No physical constants other than time are normally encountered in such eąuations and the similarity invariants may take the form of percentages, ratios of rates, ratios of population, or gene numbers.

Numerous reports are available concerning this form of biological simulation. Levins (1965), Edwards (1960), and Mandel (1963) have reviewed the entire problem of genetic modeling. Many different models have been used sińce the pioneering work on genetic analysis of Sewall and Wright in the 1930’s. Fraser (1962) and Crosby (1960) use com-puters to implement complex genetic situations in which phenotypical and genotypical selection are interrelated. Monte Carlo computation methods are employed by Barker (1958) to simulate complex selection phenomena in fruit flies; results from the model study agreed with actual laboratory findings. Models based on probability of birth and death for an organism with a given genome, are considered in Karlin (1962) and Karlin and McGregor (1962).

E. Stochastic Models

Stochastic models are numerical formulations in which central em-phasis is placed on the probabilities of changes in certain parameters with time. Similarity invariants for such models may be numerical invariants or relational ones pertaining to the form of the eąuations. The use of stochastic models in medicine and biology has been reviewed in a symposium edited by Gurland (1964), which contains examples of applications to genetics, epidemics, population migration, psychological gamę theory, cancer induction, and other problems. A volume by Bartlett (1960) deals in detail with stochastic models in ecology and epidemiology. A vo!ume on the mathematical theory of epidemics and devoted largely to stochastic models is available by Bailey (1957); Muench (1959) has written a book called “Catalytic Models in Epidemiology,” which deals mainly with statistical curve-fitting problems for various types of numerical representations of epidemics.

A great many specific reports on biological stochastic models could be cited. Neyman and Scott (1959) and Bamett (1962) have simulated the struggle for survival of two species of flour beetles by a probabilistic approach and notę the possibility of highly nonlinear results of smali random fluctuations in such situations. Sheppard (1962) used stochastic theory for an entirely different prupose. He models distribution of mate-rials in the blood by a “random walk” techniąue applied to individual particles in the bloodstream. In this particular instance a number of