Properties Of Theoretical Models Research Articles

Constructive properties of theoretical models

This paper isolates various properties of theoretical models which are conducive to the successful fulfilment of the functions of theory in the progress of scientific research. These properties are summarized in the concept of value which refers to the statistical accuracy of a given model along with its complexity which is defined in terms of the testable axioms of the model. These issues are then discussed in the context of specific examples from the psychological research literature.

Dynamics of Spherical Stellar Systems: Properties of Theoretical Models, and Comparison with Clusters and Elliptical Galaxies.

view Abstract Citations (2) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Dynamics of Spherical Stellar Systems: Properties of Theoretical Models, and Comparison with Clusters and Elliptical Galaxies. Michie, Richard W. Abstract A wide range of spherical models are now available and result from a distribution function which for the inner regions is obtained from the Boltzmann equation with encounters. It is assumed that encounters and orbital mixing were important in the early stages of the formation and development of the star clusters and elliptical galaxies, and, since this time period, the structure of these systems has not significantly changed. The orbital mixing allows an extension of the distribution function to the regions far from the center, where mathematically it satisfies the collisionless Boltzmann equation. The gravitational potential is obtained from Poisson's equation. The stellar orbits are not restricted to an isotropic velocity distribution everywhere the models range from complete isotropy to highly radial orbits. While the distribution function is obtained for stars of different mass, the comparison with observation has assumed all stars to have the average stellar mass. In a log density versus log distance plot, it is found that for certain degrees of velocity space anisotropy there is an inflection in the curve, and this is a result of the orbits becoming elongated as well as the energy cutoff becoming important. All the models have finite mass, are self-supporting and satisfy the virial theorem. They are all obtained from an analysis of the Boltzmann equation with particular emphasis placed at the high energies. In dimensionless variables the models are characterized by a single model parameter C. The model parameter involves three physical parameters which indicate the total mass, total energy, and degree of velocity space anisotropy. A comparison with rich galactic clusters, globular clusters, dwarf galaxies and spherical galaxies has been made. The theoretical models appear to be in agreement with the observed systems. For example, the log density versus log distance inflection observed in systems such as NGC 3379 is exactly as predicted by the theory. There may be a small inflection for the globular cluster 47 Tuc. Both star clusters and spherical galaxies show a fairly wide range in the model parameter. The star clusters may favor a somewhat larger value of C, but the data are probably not sufficient to firmly establish this trend. Publication: The Astronomical Journal Pub Date: June 1962 DOI: 10.1086/108880 Bibcode: 1962AJ.....67R.582M full text sources ADS |

ON THEORETICAL MODELS FOR COMPETITIVE AND PREDATORY BIOLOGICAL SYSTEMS

Differences of opinion are obviously possible on the degree to which admittedly oversimplified theoretical models can explain some of the complex observational phenomena to be found in nature. Criticisms from biologists of the mathematical work of Lotka (1925), Volterra (1926) and subsequent writers on the growth and interaction of biological populations have, however, sometimes been justified and sometimes unjustified, for in spite of inevitable limitations such work constitutes a permanent contribution to the understanding of how populations may behave. A significant constructive survey was made by Gause (1934) when he attempted to bridge the gap between theoretical models and natural biological phenomena by controlled laboratory experiments in animal ecology. While experiments also have limitations as representations of nature, the role of both theory and experiment in the physical sciences might be recalled by any biologists inclined to be sceptical of the value of either. The interrelation of these approaches in biology may be illustrated in the field of epidemiology. Here the vicissitudes of infected populations have been studied in the laboratory as well as in the field; but, as I have emphasized elsewhere (Bartlett, 1956, 1957), the properties of theoretical models indicate, among other things, the extent to which population size may sometimes be crucial in the probable sequence of events, and thus indicate to what extent laboratory observations will have any similarity to larger-scale field observations even if the same model applies to both. An essential point is that recent theoretical formulations explicitly recognize the discrete character of populations and the stochastic or random aspect of changes, as distinct from strictly deterministic formulations. The need for this in ecology, which was already perhaps envisaged by Gause (1934, p. 124), is quite apparent in the experiments by Park with the flour-beetle Tribolium, in which one of two species together in a container survived not every time, but with a definite probability (e.g. 30 % of times), that could be estimated by replication and changed by changing the environment (see, for example, Neyman, Park & Scott, 1956). There is now no mathematical difficulty in the formulation of stochastic models (see, for example, Bartlett, 1955 a, 1956), and such formulations have already been made for typical ecological models by Chin Long Chiang (see Kempthorne et al. 1954). The greater intractability of even the simplest of these is, however, a serious obstacle to progress, especially in animal ecology, where even in the deterministic formulations of Lotka and Volterra many simplifications, such as neglect of age structure or of other heterogeneity, were made. One aim of the ensuing discussion is to indicate the enhanced value of deterministic formulations of population dynamics when properly interpreted within more comprehensive stochastic