Stahl67( bmp

192 WALTER R. STAHL

A rather different approach to ecological modeling has been chosen by Meier and associates (1964). These authors apply gamę theory, somewhat as used in psychological studies, and heuristic programming techniąues to create a gamę theory model of animals in a given ernńron-ment. Humań players have also been used to simulate the behavior for an animal group when presented with certain choices pertaining to obtaining food and water, predation, etc. A model of this type involves some numerical features, but the fact that it is roughly isomorphic with a psychological gamę model reveals that relational or cybernetic simi-larity criteria are as important as the numerical invariants.

G. Interpretations of Numerical Kinetic Models

Numerical and kinetic models imply a considerable variety of numerical and relational invariant properties for the simulated system. The similarity criteria for such models are not as obvious as the dimensionless numbers of the physical analog theory, but have been discussed by authors such as Berman and Schoenfeld (1956). Choice of a particular kind of invariant for a class of models is ąuite arbitrary and based on the naturę of the eąuations, prior experience with similar systems, and actual comparison of numerical data from different systems which appear to be governed by the same type of eąuations and are therefore models of each other.

Two slightly different uses of the expression numerical model are im-plicit in the above. One may ref er to an actual set of eąuations as “model eąuations” for a particular physical phenomena. In this case numerical data from a Computer or hand solution of the eąuations is compared with experimental findings. Altematively, a given eąuation may be held to govern a class of phenomena and generate appropriate invariants for comparisons of members of this class, which are then models of each other in respect to the governing eąuations. For example, the ratio of numbers of predators to prey is an invariant suitable for analyzing actual experimental findings from certain types of ecological studies for which a mathematical model or theory has been formulated.

In the case of physical analogs and numerical kinetic models the in-variants are usually numerical in naturę, but may also be relational or topological and pertain to the form of governing eąuations. Physical and numerical modeling methods have in common the reąuirement that an algorithm must be stated for obtaining numerical invariance criteria that relate the modę to the prototype.