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THE ROLE OF MODELS IN THEORETICAL BIOLOGY 191

physical dimensional variables such as diffusion and convection velocity are important, and dimensionless physical criteria arise as arguments in the stochastic eąuations. Aging and mortality phenomena have been simulated by a Monte Carlo stochastic model in work of Sacher and Trucco (1961), while Niskanen and associates (1963) have used stochastic methods to study induction of cancer by carcinogenic substances.

F. Numerical Models in Ecology

Kinetic and numerical methods of various types have also been used in a number of studies relating to ecological problems. Watt (1961) described complex mathematical models of insect predation on farm crops, using biometric and Computer methods. His work reveals that many different numerical expressions may fit a certain set of data—the simplest ones are generally to be preferred. Garfinkel and Sack (1964) ulitized a Computer program to simulate an ecological problem, based on competition among six species as described by the classic Lotka-Volterra methods. Certain natural invariants were found to arise in this model: to tal biomass of system, population cycle times, and ratios of population numbers. These are examples of numerically invariant properties pertinent to a certain mathematical model. The naturę of the invariant is determined by choice of governing mathematical eąuations. For example, data on population growth in a colony of Daphnia was fitted to a logistic growth curve by F. E. Smith (1963) and this implies that the particular types of invariants in the logistic curve are relatively invariant properties for the entire class of population studies under discussion.

In marinę ecology numerical predictive models are known to be suf-ficiently accurate to be of practical value. Clemens (1963) described a model of albacore migration based on detailed analysis of catch com-position in given localities and at different times. This makes possible fairly accurate prediction of where schools of these fish will be located in the Pacific in various seasons. Ivlev (1961) and Nikol’skiy (1963) have proposed generał eąuations goveming the growth and population density of commercial fish. They obtained results that are helpful in planning optimum fishing strategy and preventing depletion by excessive fishing in a given area. Cassie (1962) utilized statistical distribution models of plankton growth in the ocean. Since fish feed on these smaller organisms a complete model of the marinę environment may have to include plankton.