Bulletin of Mathematical Biology

Beckmann, Martin J.

doi: 10.1007/BF02477881pmid: N/A

Abstract Spatial equilibrium distributions of population are derived from the spatial distribution of net rates of reproduction, and from a relationship between migratory flow and gradients of population density and of locational “attractiveness.” Conditions are discussed for which population approaches a uniform spatial density. Under certain conditions a particularly simple statement of the equilibrium conditions is possible in terms of the “potential of population,” a concept introduced by demographers (J. Q. Stewart,Geographical Review,37, 46–85, 1947) to measure the proximity of a point to people.

Rashevsky, N.

doi: 10.1007/BF02477882pmid: N/A

Abstract The imitation effects in a social group depend both on the size of the group and on the distribution of a certain psychobiological quantity ϕ which measures the tendency of an individual towards a given behavior. The distribution function of ϕ determines the ratio μ of the individuals in the society who adopt a given behavior. When the size of the social group is not too large, the actual distribution of ϕ will deviate from the most probable one, and therefore communities of the same size and having the same parameters may have different values of μ. Approximate equations are developed which give the probability of a given μ for a group of a given size. Possible effects of interactions of communities of different sizes are briefly discussed. A generalization of the theory of imitative behavior to any number of mutually exclusive behaviors is given, and its possible sociological implications are discussed.

Kerner, Edward H.

doi: 10.1007/bf02477883pmid: N/A

The system of differential equations proposed by V. Volterra, desoribing the variation in time of the populationsNr of interacting species in a biological association, admits a Liouville's theorem (when logNr are used as variables) and a universal integral of “motion”. Gibbs' microcanonical and canonical ensembles can then provide a thermodynamic description of the association in the large. The “temperature” measures in one number common to all species the mean-square deviations of theNr from their average values. There are several equipartition theorems, susceptible of direct experimental test, a theorem on the flow of “heat” (the conserved quantity in an isolated association) between two weakly coupled associations at different temperatures, a Dulong-Petit law for the heat capacity, and an analog of the second law of thermodynamics expressing the tendency of an association to decline into an equilibrium state of maximal entropy. The analog of the Maxwell-Boltzmann law is a distribution of intrinsic abundance for each species which has been successfully used by ecologists for interpreting experimental data. A true thermodynamics develops upon introducing the idea of work done on an association through a variation of the variables (such as physical temperature) defining the physical and chemical environment. An ergodic theorem is suggested by the agreement of ensemble and time averages in the one case where the latter may be found explicitly.

Greene, Peter H.

doi: 10.1007/BF02477884pmid: N/A

Abstract Interpretations of the mechanisms of perception of contours or of Mach bands have stressed either the role of various spatial derivatives of light intensity at the retina or the importance of various forms of inhibitory effects between neighboring retinal elements. Evidence is presented here in support of the latter type of interpretation. It is considered that the brightness contrast and perceived contours arise from neural elements, each of which is stimulated in proportion to the intensity of photo-receptor excitation at a point of the retina and inhibited in proportion to the mean intensity in some neighborhood of that point. The role of the spatial derivatives is best seen as a particular manifestation of the inhibitory mechanism. Predictions based upon this hypothesis appear to be consistent with experimentally observed evidence.

Landahl, H. D.

doi: 10.1007/BF02477885pmid: N/A

Abstract The effects of area and intensity on the critical flicker frequency, threshold, and reaction time are considered in terms of neural net theory. An attempt is made to develop a mechanism which can account for the phenomena associated with the empirically observed laws of Ricco, Granit, Talbot, and Ferry-Porter as well as observations on reaction time and threshold. A simple model gives results which are substantially in agreement with observation except for a few apparent discrepancies. Experimental procedures are suggested which can determine whether these are apparent or real.