T H E many attempts which have been made to simulate curves of limited growth by mathematical formulations either aim at accounting for their form through certain fundamental postulates about the growth process, or else are severely practical in scope and inspired by the desire to obtain any relatively simple equation which contains the essence of the numerical data. In the empirical approach the magnitudes of the constants in the fitted equations may be used to assess the importance in growth of experimentally controllable factors, but the constants themselves are not regarded as having any absolute significance for the theory of growth. Obviously the usefulness of any empirical equation is enhanced if its constants yield easily information of direct biological interest. The three best-known 'growth functions' are as follows:

A Flexible Growth Function for Empirical Use

The diffusion of an innovation traditionally has been defined as the process by which that innovation is “communicated through certain channels over time among the members of a social system” (Rogers, 1983, p. 5). As such, the diffusion process consists of four key elements: innovation, communication channels, time, and the social system.

New Product Diffusion Models in Marketing: A Review and Directions for Research:

Regional differences in species number have puzzled naturalists ince the early 1800's, and explanations account for a large part of modern ecological research. Two venerable observations form the cornerstone of our knowledge on the subject: The number of species within a taxonomic group tends to increase with decreasing latitude (see Fischer 1960; Pianka 1966); and the number of species within a taxonomic group tends to increase with increasing area (see Preston 1960, 1962; Williams 1964; MacArthur and Wilson 1967; Simberloff 1972). Despite early research on the latter trend (the species-area relationship), ecologists have studied it intensely only in the last 50 yr. The relationship was originally envisioned as an empirical tool and used in three principle ways: (1) to determine optimal sample size and sample number, (2) to determine the minimum area of a \"community,\" and (3) to predict the number of species in areas larger than those sampled. All three uses are discussed by Kilburn (1966). More recently interest in the species-area relationship has focused on mechanistic explanations, its precise mathematical descriptions, and interpretations of parameters derived from these mathematical descriptions. Williams (1964) and Preston (1960, 1962) have proposed that the exponential and power function models (\"exponential model\" throughout his paper also refers to the species/log area transformation, and \"power function\" also refers to the log species/log area transformation) of the species-area relationship result from the way in which individuals are distributed among species. Williams' (1964) exponential model, which emphasizes habitat heterogeneity, was considered important by many plant ecologists but is now largely ignored. Preston's (1960, 1962) power function model was based on the assumption of a dynamic equilibrium of species exchanges between islands in an archipelago. This assumption led to the equation of the power function model with the idea of a dynamic equilibrium as expounded by MacArthur and Wilson (1963, 1967), such that an adequate fit of this model to observed species numbers has been viewed as support of the equilibrium hypothesis (Grant 1970; Diamond 1973; Simpson 1974). The interplay of the equilibrium hypothesis and the power function model of the species-area relationship has led to interpretation of the slope and intercept of the power function model exclusively in the context of the equilibrium hypothesis. In particular, specific values of the slope of the power function are often construed to * Order of authorship determined by the toss of a coin. t Present address: Department of Biology, University of South Florida, Tampa, Florida 33620. Am. Nat. 1979. Vol. 113, pp. 791-833. c) 1979 by The University of Chicago. 0003-0147/79/1306-0002$03.26

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The statistics and biology of the species-area relationship

A tool changer is disclosed for automatically removing a working tool after having been used from the spindle of a machine tool such as a vertical milling machine, transferring the used working tool to a rack which stores a plurality of working tools, picking up from the rack another working tool for the next machining operation, transferring such new working tool to the spindle, and operatively mounting such new working tool on the spindle.

Temperature and Organism Size—A Biological Law for Ectotherms?

Fish lengths have been measured in many different ways The differences arise from choosing different reference points near the anterior end and near the posterior end of the fish, and from using different methods of making the measurement Methods of making the measurement include using calipers; using a tape held along the curve of the body; laying the fish on a measuring board with the front end pressed against an upright piece; laying the fish on a board with a movable cross hair above it, attached to an indicator running along a scale In theory, any combination of reference points and methods might be used, but practice is considerably more restricted Some of the commoner combinations have special names; these are provided in the chapter, together with the reference points used In addition to these methodological differences, length varies with the condition of the fish—for example, whether it is alive, recently killed, after rigor mortis has set in, or at different intervals of time after preservation in formalin or alcohol

11 – Growth Rates and Models

T _ HE work reviewed in this paper is aimed at establishing connections between two fundamental aspects of living organisms, their metabolism and growth. What we call growth of even a simple organism is a tremendously complex phenomenon from the biochemical, physiological, cytological, and morphological viewpoints. There are, however, certain aspects that are amenable to quantitative analysis, and such an approach appears to lead to some insight into the connections between metabolism and growth, and to some answer to the seemingly trivial, but in fact hardly explored question, \"Why does an organism grow at all, and why, after a certain time, does its growth come to a stop?\

Ludwig von Bertalanffy

Papers

Quantitative Laws in Metabolism and Growth