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  • Springer  (146,830)
  • 1965-1969  (146,830)
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  • 1
    Electronic Resource
    Electronic Resource
    A multidimensional birth process and its application to some problems in the dynamics of biological populations (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 333-345 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper is a sequel to a paper by the author entitled “Restricted Transition Probabilities and Their Applications to Some Problems in the Dynamics of Biological Populations” (Bull. Math. Biophysics, 1966,28, 315–331). The paper is divided into two parts. In part one some aspects of the maximum size attained by the population during a finite time interval are studied for the case the stochastic process underlying the evolution of the population is a birth process. Two interesting by-products emerge from the study presented in part one; namely a combinatorial method of finding solutions to the Kolmogorov differential equations in special cases, and secondly, a set of criteria for the optimum allocation of genotypes in the host population of a host-pathogen system. The optimum allocation of genotypes in the host population is a problem of practical importance in controlling plant pathogens. In part two the theory of restricted transition probabilities developed in the companion paper is applied in finding the distribution of the time to the appearance of the first mutation for the case of a two dimensional birth process. The distribution of the time to the appearance of the first mutation is of importance in understanding the role mutation plays in the evolution of a population, particularly in the pathogen population of a host-pathogen system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476814
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  • 2
    Electronic Resource
    Electronic Resource
    The motion of the left ventricle. I (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 355-362 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The complex arrangement of the muscle fibers in the ventricular wall and the nonsymmetric contraction and expansion of the ventricle preclude the writing of a differential equation of motion for the ventricle as a whole. We can, however, describe the motion of the ventricle by describing the motion of the dimensional parameters length and diameter; the radius, circumference, cross-sectional area, and volume following naturally from these. The ventricle is assumed to be an ellipsoid of revolution and the dimensional parameters to be periodic functions of time. Each of the parameters is expressed as a Fourier series.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476816
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  • 3
    Electronic Resource
    Electronic Resource
    Comparaison des modeles de neurone de Hodgkin-Huxley et de Nelson (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 347-354 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Le modèle de Nelson peut-être considéré comme une approximation du modèle de Hodgkin-Huxley. Moins précis, il est plus maniable. Le modèle de Nelson peut également être considéré comme une généralisation du modèle de Hodgkin-Huxley. En effet, il introduit des liaisons synaptiques localisées ou diffusantes, et un processus de facilitation. Le mécanisme des liaisons synaptiques ne se traduit pas facilement dans le langage de Hodgkin-Huxley. Par contre, le processus de facilitation s'interprète facilement. Nelson's model can be taken as an approximation of Hodgkin-Huxley's model. Its precision is lesser, but it is more usable. Nelson's model can also be taken as a generalization of Hodgkin-Huxley's one; for it introduces localized or diffusing synaptic connexions and a facilitating process. The mechanism of synaptic connexions cannot be easily translated into Hodgkin-Huxley's language. On the contrary, the facilitating process is easily interpreted.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476815
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  • 4
    Electronic Resource
    Electronic Resource
    The theory of the left ventricular aneurysm (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 363-370 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A spherical model for the human left ventricle with two different types of aneurysms, circular and rectangular-square, is proposed and meaningful relations are derived between the parameters of the aneurysms and ventricle. Such ventricular parameters as stroke volume, end-diastolic volume, and end-systolic volume are given normal human values to compute values for end-systolic radius and percentage shortening of muscle for various sized circular and rectangular-square aneurysms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476817
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  • 5
    Electronic Resource
    Electronic Resource
    A note on crime control (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 375-378 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The Volterra theory of two competing populations is extended to the contemporary social problem of crime control. Domains of stability for the time dependence of the numbers in the criminal and enforcement groups are exposed by a numerical example. Both augmentation and reduction of enforcement can produce a stable system. Average values of the ratio of members in each group show great sensitivity to the control policies adopted by the remaining sector of the total population.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476819
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  • 6
    Electronic Resource
    Electronic Resource
    Interactions in enzymatic reactions (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 379-390 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The paper deals with interactions of substances via an enzymatic reaction (Bull. Math. Biophysics,25, 141–154, 1963). The substances are the activators, inhibitors and/or substrates of the reaction. Due to the bimolecularity of the processes in the reaction, the quantitative relation between the steady state amount of complexes and the amounts of the substances assumes a typical form. In multiple enzymatic reactions this form is more complicated, though basically similar. Because the substances may influence the steady state amounts of the complexes in opposite directions, the compensation and blocking effects are the properties of enzymatic reactions. The substances with the same direction of influence may potentiate each other. In the enzymatic reaction here considered, the potentiation is always non-negative.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476820
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  • 7
    Electronic Resource
    Electronic Resource
    Decay of cell synchronization: Solutions of the cell-growth equation (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 391-409 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Growth-rate functions in analytic form have been obtained for cell cultures in which the doubling times follow the Gaussian and Poisson distributions. The growth-rate functions are calculated by using Laplace transforms to solve an integral equation previously presented. Oscillatory solutions result if a substantial fraction of the cells in a culture are synchronized to divide at some particular time. The synchrony and, hence, the oscillatory character of the growth-rate function eventually disappear because of the non-zero variance of the doubling-time distribution. If their variances are sufficiently small, the Gaussian and Poisson doubling-time distributions lead to growth-rate functions that become identical in the limit of large time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476821
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  • 8
    Electronic Resource
    Electronic Resource
    On the derivation of a mean growth equation for cell cultures (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 411-416 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract IfN(t) is the expected number of cells in a culture at timet, $$\dot N(t)$$ the corresponding time derivative, andf(t−τ)dt the probability that a cell of aget−τ at timet will divide in the succeeding time intervaldt, then according to Hirsch and Engelberg (this issue) there obtains the integral equation $$\dot N(t) = 2\int_{ - \infty }^t {f(t - \tau )\dot N(\tau )d\tau }$$ for describing the dynamics of the cell population. It is the purpose of this note to give two alternative derivations of this equation, one based on the age density equation of Von Foerster, and the other based on a generalized form of the Harris-Bellman equation describing the first moment of an age dependent, branching process. In addition, a probability model is posed from which the Von Foerster equation and, hence, the Hirsch-Engelberg equation readily follows.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476822
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  • 9
    Electronic Resource
    Electronic Resource
    A mathematical theory of enamel solubility and the onset of dental caries: I. The kinetics of dissolution of powdered enamel in acid buffer (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 417-432 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model of the dissolution kinetics of powdered enamel is developed based on the kinetic rate termq, $$q = K'H - k'C \cdot P_1$$ , whereH=[H +],C=[Ca ++] andP 1=[HPO 4 = ]. The differential equations describing the rate of mineral dissolution (and the linearly related rate of appearance of calcium and phosphate in solution) have been derived and solved for three basic cases: (1) when thepH of the solution and surface area of the enamel are considered constant, (2) when thepH is assumed constant, but the reduction in surface area during dissolution is considered, and (3) when the rise ofpH resulting from the buffering effect of the dissolved enamel is considered along with the change in surface area. Analytical solutions have been obtained for cases (1) and (2), while a numerical solution has been found for case (3). Data from a study on enamel dissolution are presented that agree with the theory of case (3), and it is noted that apH rise as large as 0.5 can occur, as has been shown elsewhere in the literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476823
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  • 10
    Electronic Resource
    Electronic Resource
    Reward and punishment in learning with possible reference to punishment for crime (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 477-481 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract On the basis of Landahl's theory of two-choice learning it is shown that application of punishment for wrong responses, without giving award for correct ones, does not lead to complete learning, no matter how many trials are used. If initially a “wrong response” was learned, then an attempt to inhibit it by punishment alone will in a class of cases lead only to a 50% suppression of that wrong response. Possible connection with the problem of effectiveness of punishment as a deterrent for crime is mentioned.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476827
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