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  • Springer  (62,733)
  • 2010-2014
  • 1980-1984  (42,164)
  • 1960-1964  (20,569)
  • 1925-1929
  • 1980  (42,164)
  • 1964  (20,569)
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  • 2010-2014
  • 1980-1984  (42,164)
  • 1960-1964  (20,569)
  • 1925-1929
Year
  • 1
    Electronic Resource
    Electronic Resource
    Tissue blood flow and the sympathomimetic amines (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 1-7 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In the arteries, blood flow and blood pressure are pulsatile in nature (Roston, 1962a; Roston 1962b). The patterns of blood movement and mural distension in the arteries are important because they may be associated with life-threatening degenerative changes in the arterial walls. As the vascular channels narrow, the pulsation decreases. At the level of the capillaries, almost no pulsation exists (Best and Taylor, 1961). The tissues are affected by the direct flow in the capillaries and not by the pulsation in the arteries. Thus, such quantities as pulse pressure, systolic pressure, and diastolic pressure which characterize blood movement in the arteries are not important as far as the tissues are concerned. Rather, the average pressure and flow in the capillaries are the quantities significant for tissue blood flow. The present study analyzes the local blood circulation in a typical tissue. Logical extension of this analysis results in insights into the physiological behavior of the circulation which integrate a considerable body of experimental data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476615
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  • 2
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    Complementary variational principles for diffusion problems with Michaelis-Menten kinetics (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 131-135 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of complementary variational principles is used to obtain maximum and minimum principles for diffusion problems with Michaelis-Menten kinetics. In an illustrative calculation we obtain an extremely accurate variational solution in good agreement with the numerical solution of McElwain (1978).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462371
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  • 3
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    Structural order and partial disorder in biological systems (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 137-141 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462372
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  • 4
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    Primitivity of the product of two Leslie matrices (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 181-189 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Necessary and sufficient conditions for primitivity of a product of two Leslie matrices are given. Such a product could be used in modeling the growth of a population governed alternately by two different sets of fertility and survival parameters.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02464637
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  • 5
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    Frequency analysis of time-varying elastance model of the left ventricle (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 173-180 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Zadeh's transfer function method for linear time-variable systems is used to apply frequency-domain analysis to a periodically time-varying elastance model of the left ventricle. Left ventricular pressure computed from the system function of the time-varying elastance and the phasors of aortic flow shows a typical waveform of the measured ventricular pressure.
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    URL: http://dx.doi.org/10.1007/BF02464636
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  • 6
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    The 9th Annual Northeast Biomedical Engineering Conference (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 901-901 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461068
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  • 7
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    On the possibilities of error-detecting codes in nucleic acid molecules (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 25-29 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Error-detecting codes have been known to mathematicians and to electrical engineers for over ten years. In general, such codes utilize an additional orparity bit for purposes of detecting errors by the addition of all positive binary bits or “1’s” occurring in any code word. However, since the process of addition is required for such code detection, it is not surprising that these codes have not been applied to the nucleic acid molecule. In 1962, P. I. Hershberg (Trans. I.R.E., CS-10, 280–4, 1962) outlined a categorical constraint which permitted the realization of a class of error-detecting codes which did not require parity bits. This class of codes is applied to the nucleic acid molecule in the present paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476617
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  • 8
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    On some topological properties of the systems of compartments (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 31-38 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Compartment systems are often used as models for tracer and drug kinetics. The structure of a compartment system is here analyzed by means of theory of graphs methods. In particular the precursor-successor relationship between any two compartments is classified according to the structure of the graph of the system and to the values of the elements of the matrix associated with it.
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    URL: http://dx.doi.org/10.1007/BF02476618
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  • 9
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    The equations of growth (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 39-43 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An application of a bifurcation theorem shows the existence of periodic solutions of a system of differential equation used to describe competition between two species. It is then shown that the results are more general than those previously established.
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    URL: http://dx.doi.org/10.1007/BF02476619
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  • 10
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    An analysis of the 2o and 10o field color-matching functions (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 9-24 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The 2o and 10o field color-matching functions are independent: one specification is not a linear transformation of the other, even after correcting for macular pigment effects. Therefore, the “true” color-matching functions which directly describe the linear responses of the eye must be different for the two field sizes. This means that a given stimulus will, in general, have a different chromaticity depending upon the field size, regardless of the choice of any one colorimetric co-ordinate system for all field sizes. However, in spite of these chromaticity differences, a large uniform field usually appears nearly uniform. Such color uniformity implies that even though chromatic differences occur as a function of retinal position or field size, these differences are small. If this is the case, then the underling “true” color-matching functions determining the observed color-matching functions must be nearly, but not quite, identical. These differences vanish as identity between the sets of color-matching functions is approached. This property suggests a method of calculating the “true” color-matching functions. The “true” color-matching functions must approximate those obtained by minimizing the chromaticity differences between two independent sets of data. This can be done by assuming that the coefficients of transformation should be adjusted so as to produce as nearly identical chromaticities for spectrum stimuli as possible. In this paper, it is also assumed that the “true” color-matching functions have no negative values, as if they were based on actual absorption spectra. This article describes the calculation of the “true” 2o and 10o field color-matching functions satisfying these two conditions. For both field sizes, the maxima of the three functions are near 435, 540, and 585 mμ, after correcting for the filtering effects of the ocular media and macular pigment.
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    URL: http://dx.doi.org/10.1007/BF02476616
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  • 11
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    A note on the elementary α-perceptron (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 45-47 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this note the principal convergence theorem (F. Rosenblatt,Principles of Neurodynamics, Spartan Books, Washington D.C., 1962, 111–116) is proved by a new method.
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    URL: http://dx.doi.org/10.1007/BF02476620
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  • 12
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    On the adsorption of ions at charged sites in an electric field: I (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 49-55 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Considering only nearest neighbor interactions, an expression is obtained for the grand partition function for the adsorption of two kinds of monovalent positive ions at a long chain of one type of monovalent negative fixed sites in an electric field. Expressions are obtained for the fractions of sites which are occupied by each kind of ion as well as of those which are unoccupied as a function of the potential of the electric field.
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    URL: http://dx.doi.org/10.1007/BF02476621
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  • 13
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    Two remarks on the mathematical biology of automobile driving (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 57-61 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In connection with a series of previous papers by this author (Bulletin of Mathematical Biophysics,21, 299–308, 375–385;22, 257–262, 263–267;23, 19–29;24, 319–325) results obtained by A. Crawford (Economics 5, 417–428) on the effects of irrelevant lights on reaction times toward a given light stimulus are discussed. The conclusions from a previous paper of this author (Bulletin of Mathematical Biophysics,23, 19–29) are elaborated.
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    URL: http://dx.doi.org/10.1007/BF02476622
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  • 14
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    Analysis of a simple avoidance situation. II. A model (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 77-81 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model has been constructed to describe experimental data recorded in a study of a simple avoidance situation. The theoretical description makes use of the concept of the effective number of shocks. The model explains the existence of oscillations encountered in previous experiments.
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    URL: http://dx.doi.org/10.1007/BF02476624
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  • 15
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    Studies in stochastic learning theory: II. Interpretation of response probabilities (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 63-75 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Response probabilities are interpreted from two points of view. One corresponds to fluctuations in physical parameters suggestive of a neurological basis, and the other corresponds to fluctuations in stimulus sample constitution. The two interpretations are shown to be equivalent under rather general conditions, giving the same type of relation between response and training states. This relation is different from that obtained via the response strength concept used in Part I. As a step toward evaluating the difference in predicted behavior for these different response-training relations, a general functional-difference equation is derived that describes the moments of the corresponding stochastic process in experimenter-subject controlled experiments. As an immediate application, it is used to obtain the continuity condition for the solution of the functional equation treated in Part I, and to justify the differentiability conditions assumed in establishing asymptotic properties of the solution as a function of the reinforcement parameter.
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    URL: http://dx.doi.org/10.1007/BF02476623
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  • 16
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    An avoidance learning situation. A neural net model (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 83-89 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A simple avoidance situation is considered in terms of a neural net learning model. Data for the control situation can be represented by an expression having three parameters which determine the initial and the steady state activities together with the transient aspects. The introduction of a learning parameter then allows one to calculate satisfactorily the results obtained in the experimental situation in which shock is applied.
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    URL: http://dx.doi.org/10.1007/BF02476625
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  • 17
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    Comparison of two models for a simple avoidance learning situation (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 101-101 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476627
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  • 18
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    Group representation of genetic recombinations (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 91-100 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An algebraic representation of operations of genetic recombinations is illustrated. It is shown that the recombinations between chromosomes in the two-strand model can be represented by groups, in the sense of the theory of groups. Recombinations between chromosomes with inversions and a translocation are considered as well as cases without them. It is found that the groups derived from such cases are Abelianp-groups (p=2) and that the types of the Abelian groups for the various pairs of chromosomes are different from each other. Differences among those recombination groups are illustrated by showing the sets of generators of the various groups, which generate the corresponding recombination groups by multiplication.
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    URL: http://dx.doi.org/10.1007/BF02476626
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  • 19
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    Abstract biological systems as sequential machines (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 103-111 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that a rather close relationship exists between the (ℳ,ℛ)-systems, defined previously as prototypes of abstract biological systems, and the sequential machines which have been studied by various authors. The theory of sequential machines is reformulated in a way suitable for its application to the study of the intertransformability of (ℳ,ℛ)-systems as a result of environmental alteration. The important concept of strong connectedness is most useful in this direction, and is used to derive a number of results on intertransformability. Some suggestions are made for further studies along these lines.
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    URL: http://dx.doi.org/10.1007/BF02476831
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  • 20
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    The arterial circulation of the heart (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 113-120 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Blood flows into the aorta and its branches during left ventricular systole. Most of the arterial walls in the body stretch during systole in accordance with their elastic properties (Roston, 1962a, b). During diastole, the rebound of the distended walls supplies an additional propulsive force pushing the blood forward. Since the metabolic exchange between most of the tissues in the body and their blood vessels is ordinarily the same throughout the cardiac cycle, it makes little difference whether or not the blood flow occurs during systole or diastole. The circulation in the coronary arteries behaves in a quite different way. Because the muscle fibers of the heart contract during systole and relax during diastole, different conditions for blood flow and metabolic exchange exist during the phases of the cardiac cycle. As a result, specification of whether blood flows in the coronary arteries during systole or diastole may be important. Such specification complicates the study of the coronary artery circulation. For example, because of the arterial elasticity, some of the blood which enters the coronary arteries during diastole comes in contact with the muscle fibers during systole. The present work contains a theoretical study of the coronary artery circulation.
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    URL: http://dx.doi.org/10.1007/BF02476832
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  • 21
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    On the adsorption of ions at charged sites in an electric field: II (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 139-146 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A study is made of the adsorption of one kind of monovalent positive ion at a long chain of alternating monovalent negative fixed charged (“lattice”) and uncharged (“interstitial”) sites both of one type in an electric field. Considering only nearest neighbor interactions an expression is obtained for the grand partition function. The fractions of sites of both types which are occupied and unoccupied are determined. It is shown that an equilibrium constant can be defined for the adsorption of ions at oppositely charged sites.
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    URL: http://dx.doi.org/10.1007/BF02476834
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  • 22
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    Color perception using a single cone type with distributed maximum sensitivity (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 121-138 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper proposes a model for color perception which accounts for variations in the dimension of the space of perceived colors. The model assumes that there is only one type of cone with only one shape of response curve, but that different cone's response curves differ by translation. It also assumes that the final discrimination system, learned from originally random connections, maximizes discrimination in the normal visual environment. Learning mechanisms are discussed, and the form which the final discrimination system ought to take is plausibly derived. An algorithm for the tristimulus curves is obtained from this model, and it is shown that a good fit of the empirical data can be obtained.
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    URL: http://dx.doi.org/10.1007/BF02476833
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  • 23
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    Some remarks on Rosen's quantum-mechanical approach to genetics (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 187-191 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It has been suggested by Robert Rosen (Bull. Math. Biophysics,22, 227–255, 1960) that multiple alleles or pseudoalleles correspond to multiple cites of degenerate states of some quantum mechanical observable which acts as a source of primary genetic information. It is pointed out here that if the quantum mechanical states are determined by the different sequences of the purine and pyrimidine bases in the DNA molecule, the expected number of pseudoalleles would be much too large. The expected number is considerably reduced if we assume that a quantum mechanical state determines the coupling between a molecule of transfer RNA and the corresponding amino acid.
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    URL: http://dx.doi.org/10.1007/BF02476837
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  • 24
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    A mathematical model of the human thermal system (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 147-166 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper describes a mathematical model developed to simulate the physical characteristics of the human thermal system in the transient state. Physiological parameters, such as local metabolic heat generation rates, local blood flow rates, and rates of sweating, must be specified as input data. Automatic computation of these parameters will be built into the model at a later date when it is used to study thermal regulation in the human. Finite-difference techniques have been used to solve the heat conduction equation on a Control Data Corporation 1604 computer. Since numerical techniques were used, it was possible to include many more factors in this model than in previous ones. The body was divided into 15 geometric regions, which were the head, the thorax, the abdomen, and the proximal, medial, and distal segments of the arms and legs. Axial gradients in a given segment were neglected. In each segment, the large arteries and veins were approximated by an arterial pool and a venous pool which were distributed radially throughout the segment. Accumulation of heat in the blood of the large arteries and veins, and heat transfer from the large arteries and veins to the surrounding tissue were taken into account. The venous streams were collected together at the heart before flowing into the capillaries of the lungs. Each of the segments was subdivided into 15 radial sections, thereby allowing considerable freedom in the assignment of physical properties such as thermal conductivity and rate of blood flow to the capillaries. The program has been carefully checked for errors, and it is now being used to analyze some problems of current interest.
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    A physicomathematical model of sodium balance (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 193-198 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model is introduced in which the reabsorption of sodium is governed by an enzymatic process. This process is in turn assumed to be influenced by the extracellular volume which depends on the amount of sodium in the body at a given time. The model allows for damped oscillations when the sodium intake lies within range of values and thus can account for observed oscillations.
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  • 26
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    A neurobiophysical model of schizophrenias and of their possible treatment (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 167-185 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A neurobiophysical model is proposed for the explanation of some characteristics of schizophrenic behavior. The normal reactions to exogenous stimuli are mediated through a set of centers, while some endogenous stimuli result in abnormal reactions removed from reality, such as dreamlike states, paranoias, hallucinations, etc. The two sets of centers are cross-inhibited and the usual equations for such cross-inhibited systems are applied. In normal life exogenous stimuli as a rule result preponderantly in pleasant reactions, and the corresponding pathways are therefore reinforced. This results in an inhibition of the abnormal reactions. If the life history of an individual is such that a sufficiently large number of ordinarily experienced stimuli leads to unpleasant reactions and, therefore, the corresponding pathways are inhibited, the endogenously stimulated centers for abnormal reactions prevail and various schizophrenic symptoms occur. The same result may be achieved with a normal life history but through organic changes in the system, which differentially affect various thresholds and excitation parameters. The model thus leads to the conclusion that what appears now to be a large array of contradictory findings in the “organic” versus the “psychological” controversy is actually not a contradiction, but is a result of the dependence of normal and abnormal behaviors on a large number of neurobiophysical parameters. Some general comparisons between the conclusions drawn from the model and some known facts are made. The model also provides a first step toward a neurobiophysical interpretation of the mechanism of psychotherapy.
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    URL: http://dx.doi.org/10.1007/BF02476836
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  • 27
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    Fluctuation analysis in small chemically reacting systems (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 147-160 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A theory of noise fluctuations is developed which is applicable to systems of any size in which unimolecular or bimolecular reactions are occurring. The main difference between small and large reacting systems is that in the former the probability of finding a particle in a particular state does not obey a Gaussian distribution, but satisfies a distribution which reflects the mechanism of the chemical reaction. This difference is reflected in the main result of the theory: an autocorrelation function that is expressible as a sum of exponentials, the amplitudes of which are explicit functions of the moments of the distribution. Thus, by using small systems, the autocorrelation function,in principle, allows the elucidation of reaction mechanisms. Numerical simulations indicate that for reacting systems having ten or fewer particles, the deviation of the autocorrelation function from a single exponential should be easily detectable, and that estimates of the first four moments of the distribution should be possible. Accurate inference of the distribution, however, will require further mathematical and experimental advances.
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  • 28
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    Concept of energy in biological systems and the effects of irradiations of low energies on enzyme-substrate systems (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 161-172 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The recent mathematical formalization of the concepts of matter and extrinsical energy, which are used for the relational representation of biological systems, is employed in the analysis of the important experimental discoveries of Comorosanet al. related to low energy electromagnetic irradiations on enzyme substrates. By means of the present analysis one of the properties inherent to the experimental phenomena is more precisely exposed, and theoretical developments corresponding to “energetical evolutions” in a biological system (Leguizamón, 1976) may now have an experimental basis. Important limitations are introduced for the validity of the commutativity and associativity of cartesian product of sets, when they represent matter and its linked extrinsical energy. In connection with this last aspect, new important knowledge is obtained for the relational mathematical representation of biological systems.
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  • 29
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    Traveling waves in a simple population model involving growth and death (1980)
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    Bulletin of mathematical biology 42 (1980), S. 397-429 
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    Notes: Abstract The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.
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    Intracellular mechanisms of adaptation and self-regulation in self-organizing networks: The role of chemical transducers (1980)
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    Bulletin of mathematical biology 42 (1980), S. 365-396 
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    Notes: Abstract This paper describes mechanisms of intracellular and intercellular adaptation that are due to spatial or temporal factors. The spatial mechanisms support self-regulating pattern formation that is capable of directing self-organization in a large class of systems, including examples of directed intercellular growth, transmitter production, and intracellular conductance changes. A balance between intracellular flows and counterflows causes adaptation. This balance can be shifted by environmental inputs. The decrease in Ca2+-modulated outward K+ conductance in certain molluscan nerve cells is a likely example. Examples wherein Ca2+ acts as a second messenger that shunts receptor sensitivity can also be discussed from this perspective. The systems differ in basic ways from recent diffusion models. Chemical transducers driven by membrane-bound intracellular signals can establish long-range intercellular interactions that compensate for variable intercellular distances and are invariant under developmental size changes; diffusional signals do not. The intracellular adaptational mechanisms are formally analogous to intercellular mechanisms that include cellular properties which are omitted in recent reaction-diffusion models of pattern formation. The cellular models use these properties to compute size-invariant properties despite wide variations in their intercellular signals. Mechanisms of temporal adaptation can be derived from the simplest laws of chemical transduction by using a correspondence principle. These mechanisms lead to such properties of intercellular signals as transient overshoot, antagonistic rebound, and an inverted U in sensitivity as intracellular signals or adaptation levels shift. Such effects are implicated in studies of behavioral, reinforcement, motor control, and cognitive coding.
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    A model for early death caused by radiation pneumonitis and pulmonary fibrosis after inhaling insoluble radioactive particles (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 447-459 
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    Notes: Abstract Large radiation doses to the lung can cause early death from cardiopulmonary insufficiency resulting from radiation pneumonitis and pulmonary fibrosis. A model for early death following inhalation of insoluble radioactive particles is propose. The model is based on three assumptions: (1) early death results from damage to a cluster of cells from a large number of cell clusters at risk, (2) the dose that causes early death depends on how the radiation is delivered in time and (3) the cell clusters at risk to damage are equally sensitive ro radiation. Results from asymptotic theory of extreme values, along with biophysical considerations, suggest that the cumultive distribution function for the absorbed radiation dose to the production of pulmonary injury sufficient to cause early death is best estimated by the third asymptotic distribution without a threshold. This distribution function is identical to the Weibull cumulative distribution function. Data for Beagle dogs after inhaling relatively insoluble forms of alpha- or beta-gamma-emitting particles are shown to support the Weibull model.
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    Models of human lung airways and their application to inhaled particle deposition (1980)
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    Bulletin of mathematical biology 42 (1980), S. 461-480 
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    Notes: Abstract Models of the human respiratory tract were developed based on detailed morphometric measurements of a silicone rubber cast of the human tracheobronchial airways. Emphasis was placed on the “Typical Path Lung Model” which used one typical pathway to represent a portion of the lung, such as a lobe, or to represent the whole lung. The models contain geometrical parameters, including airway segment diameters, lengths, branching angles and angles of inclination to gravity, which are needed for estimating inhaled particle deposition. Aerosol depositions for various breathing patterns and particle sizes were calculated using these lung models and the modified Findeisen-Landahl computational scheme. The results agree reasonably well with recent experimental data. Regional deposition, including lobar deposition fractions, are also calculated and compared with results based on the ICRP lung deposition model.
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    On the completely symmetric compartmental system (1980)
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    Bulletin of mathematical biology 42 (1980), S. 481-488 
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    Notes: Abstract The completely symmetrical system is defined as having identical transfer coefficients between pairs of compartments and the same loss coefficient for each compartment. The eigenvalues and eigenvector are explicitly found along with the inverses of the system matrix and the matrix of eigenvectors. Many properties, special instances of more general theorems, can be seen at once from the explicit analytic solution of the initial value, washout and washin problems. The system serves as a known case for testing estimation procedures, algorithms for solutions of linear systems, eigenvalue-eigenvector and inversion routines and is of considerable tutorial value.
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    Natural transformations of organismic structures (1980)
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    Bulletin of mathematical biology 42 (1980), S. 431-446 
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    Notes: Abstract The mathematical structures underlying the theories of organismic sets, (M, R)-systems and molecular sets are shown to be transformed naturally within the theory of categories and functors. Their natural transformations allow the comparison of distinct entities, as well as the modelling of dynamics in “organismic” structures.
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    The dipole flip-flop theory for the excitation and conduction of excitable tissues—I. Dipole's behavior and gating currents (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 489-505 
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    Notes: Abstract To explain the sodium conductance change using Wei's dipole model (Wei, 1969), we may expect that during depolarization the dipole's population difference, ΔN, is first reduced and then returns more slowly to its resting value. This paper shows that the experimental results of gating currents support this idea. Such time course of ΔN, however, is not a usual relaxation process. To account for the unusual behavior of ΔN, we propose two additional assumptions: (1) there exists a special coupling system (probably the intramolecular vibrations) whose coupling strength with the dipoles is much stronger than with the thermal bath (intermolecular vibrations), and (2) there also exist “traps” for the dipole's excitation energy so that this energy is transformed into other energy forms at a rate increasing with the increase of depolarization. Experiments suggest that the traps are proteins located at the inner membrane surface.
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    Some generalized conjugacy theorems and the concepts of fitness and survival in logistic growth models (1980)
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    Bulletin of mathematical biology 42 (1980), S. 507-528 
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    Notes: Abstract Current research into the dynamics of iterative ecological and biological models has lead to a number of theorems concerning the existence of various types of iterative dynamical behavior. In particular, much study has been done on the dynamical behavior of the “simplest dynamical system”f b(x)=bx(1−x), which is just the canonical discrete form of logistic growth equations found in ecology, sociobiology, and population biology. In this paper, we make use of some of the techniques and concepts of topological dynamics to construct a number of generalized conjugacy theorems. These theorems are then used to demonstrate that the mappingf b has a number of conjugacy classes in which the dynamics of the iterates is equivalent to within a change of variables. The concepts of fitness and survival in logistic equations are then shown to be independent, if we follow certain intuitive definitions for these concepts. This conclusion follows from a comparison of the conjugacy classes of the functionf b and the extinction sets off b.
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    Use of transient and steady-state measurements of the unidirectional flux ratio for the determination of the free energy change of chemical reactions and active transport systems (1980)
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    Bulletin of mathematical biology 42 (1980), S. 529-537 
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    Notes: Abstract For chemical reactions not at equilibrium but proceeding in the forward direction in the steady state, a result found by a method first introduced by H. G. Britton (1963, 1965) is generalized to prove that if $${{\vec J} \mathord{\left/ {\vphantom {{\vec J} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}} \right. \kern-\nulldelimiterspace} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}$$ is the unidirectional flux ratio, $${{\vec J} \mathord{\left/ {\vphantom {{\vec J} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}} \right. \kern-\nulldelimiterspace} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}$$ exp (−ΔG/RT). The conditions under which the equality or inequality applies are discussed. If the unidirectional fluxes are not in the steady state, the unidirectional flux ratio is time invariant in certain specific situations. One such important case is for chemical reaction systems with an ordered sequence of reactions. For systems with more than one pathway, $${{\vec J} \mathord{\left/ {\vphantom {{\vec J} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}} \right. \kern-\nulldelimiterspace} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}$$ is not constant except for special cases. These results also apply to diffusional and active transport systems.
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    On Varma's prey-predator problem (1980)
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    Bulletin of mathematical biology 42 (1980), S. 599-600 
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    On the evolution of prebiotic enzyme coupled macromolecular systems (1980)
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    Bulletin of mathematical biology 42 (1980), S. 539-549 
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    The geometry of chaos: Dynamics of a nonlinear second-order difference equation (1980)
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    Bulletin of mathematical biology 42 (1980), S. 551-597 
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    Notes: Abstract The nonlinear second-order difference equationx n+1=axn(1-xn−1), where 0≦x nX≦1 anda ≧1, is examined from varying points of view, analytical, numerical and geometrical. An analytic expression is obtained for an invariant attracting curveC ∞ (a) in phase space, which becomes the central object of study. This basic curve, which replaces the simple parabolic shape typical of many analogous first-order models, may have a complicated geometrical structure. As the parametera increases,C ∞(a) undergoes transformations characterized by the dynamical descriptions: stable node→stable focus→stable limit cycle →chaotic attractor. Although the limited characterization ofchaos by the appearance of nonperiodic solutions and solutions of arbitrarily large period is relied upon, this appears to be only a simplified approximation of the real behavior of solutions. Trajectories (x n, xn+1),n=0,1,…, are calculated using the related nonlinear planar mapT a(x,y)=(y,ay(1−x)), and regions of persistence and escape are described for characteristic values ofa. The study of persistence, of even more fundamental interest than the associated problems of periodicity and stability, receives special attention. We introduce a geometrical model, similar in many respects to that for the well-known analoguex n+1=axn(1−x n), but having several new and important features. It appears that as the parametera increases in the chaotic regime there are infinitely many intermittent bursts of increase in the probability that any initial point (x 0, x1) will persist in the unit square under successive iterations of the mappingT a, an unexpected property that should be of interest for applications. A discussion of the applicability of these results to population dynamics theory is given, and it is suggested that such equations might find useful application to problems in developmental biology as well.
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    A mathematical model of the functional state of the oxygen transport system (1980)
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    Bulletin of mathematical biology 42 (1980), S. 627-645 
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    Notes: Abstract Based on the principle of minimum power, a mathematical model of the functional state of the oxygen transport system is presented. The optimization model minimizes the power expenditure of the heart, bone marrow, lung and other tissues. The model is used to determine the functional parameters of the oxygen transport system in man under both normal and varying barometric pressures. Theoretical results are compared with experimental data.
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    Theory of cation-phospholipid-induced shape changes in a lipid bilayer couple (1980)
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    Bulletin of mathematical biology 42 (1980), S. 601-625 
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    Notes: Abstract A quantitative model of ion binding and molecular interactions in the lipid bilayer membrane is proposed and found to be useful in examining the factors underlying such membrane characteristics as shape, sidedness, stability and vesicle size at various cation concentrations. The lipid membrane behaves as a bilayer couple whose preferential radius of curvature depends on the expansion or contraction of one monolayer relative to the other. It is proposed that molecular packing may be altered by electrostatic repulsion of adjacent like-charged phospholipid headgroups, or by bringing two headgroups closer together by divalent cation crossbridging. The surface concentrations of each type of cation-phospholipid complex can be described by simple binding equilibria and the Gouy-Chapman-Stern formulation for the surface potential in a diffuse double layer. The asymmetric distribution of acidic phospholipids in most biological membranes can account for the differential effects of identical ionic environments on either side of the bilayer. The fraction of vesicle material which tends to have a right-side-out orientation may be approximated by a normal distribution about the mean curvature. The theory generates vesicle sidedness distributions that, when fitted to experimental results from human erythrocyte membranes, provide an alternative method of estimating intrinsic cationphospholipid dissociation constants and other molecular parameters of the bilayer. The results also corroborate earlier suggestions that the Gouy-Chapman theory tends to overestimate free counter-ion concentrations at the surface under large surface potentials.
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    Photoelectric properties of the “yellow strips” inVespa orientalis; a mathematical model (1980)
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    Bulletin of mathematical biology 42 (1980), S. 681-689 
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    Notes: Abstract The “yellow strips” on the cuticle of the Oriental Hornet (Vespa orientalis, Hymenoptera, Vespinae), present photoelectric properties. A mathematical model for the relative changes in resistance as a photoconductive process conforms to the general model for a semiconductor with traps.
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    Damped oscillating processes in biological and biochemical systems (1980)
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    Bulletin of mathematical biology 42 (1980), S. 701-718 
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    Notes: Abstract Damped nonlinear oscillations in biological and biochemical systems are investigated by the extended Krylov-Bogoliubov-Mitropolskii (KBM) method. A review on the extension made by Popov to the KBM method is given and also further improvements are presented. Applications are made to models of oscillating chemical reactions (Lefever and Nicolis, 1971), FitzHugh (1961) equations, and population dynamics (Gatto and Rinaldi, 1977). Comparison to damped oscillating physical and engineering systems is made.
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    The problem of time unit in Leslie’s population model (1980)
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    Bulletin of mathematical biology 42 (1980), S. 719-728 
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    Notes: Abstract The conditions that will allow the lumping together of several age classes in the Leslie model are investigated. We show that if the lumping is to be valid for all population distributions, then the parameters of the model must be periodic. Lumping is valid when the population is in equilibrium, but equilibrium should be tested before the model is lumped.
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    Applied catastrophe theory in the social and biological sciences (1980)
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    Bulletin of mathematical biology 42 (1980), S. 647-679 
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    Notes: Abstract Catastrophe theory is a mathematical theory which, allied with a new and controversial methodology, has claimed wide application, particularly in the biological and the social sciences. These claims have recently been heatedly opposed. This article describes the debate and assesses the merits of the different arguments advanced.
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    Effects of red cell permeability on transcapillary tracer transport: The case of negligible back diffusion (1980)
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    Bulletin of mathematical biology 42 (1980), S. 765-795 
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    Notes: Abstract Estimates of capillary tracer permeability calculated using multiple indicator data depend upon the particular model adopted to describe blood tissue exchange. The model proposed by Crone (1963) is appropriate when some of the injected tracer diffuses into the tissue but does not return appreciably to the bloodstream before data collection is terminated. Under these conditions extraction of tracer by the tissue depends on a single dimensionless parameter, αcap, defined as the ratio of capillary permeability surface area to water flow. The effects of finite red cell tracer permeability on the Crone model estimate of capillary permeability are examined in the present study. The results indicate that even when back diffusion from the extravascular space is negligible, significant errors in the Crone model estimate can be expected when capillary permeability is relatively high and the ratio of red cell to capillary permeability is less than unity. However, when an aliquot of blood is equilibrated with tracer prior to injection and the dimensionless capillary permeability is relatively low (i.e. αcap ≦ 0.25 for a haematocrit≦50%), the whole blood Crone model estimate of αcap will be within 10% of the actual value, irrespective of red cell permeability. Red cell-plasma exchange for commonly used tracer-organ combinations should not significantly affect Crone estimates of capillary permeability under normal physiological conditions, but may be important in low flow situations.
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    A geometric model for the myocardium: Biventricular wall stresses in normal and hypertrophied states (1980)
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    Bulletin of mathematical biology 42 (1980), S. 807-828 
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    Notes: Abstract Assuming truncated ellipsoidal geometry for the right and left ventricles, a model is developed for the myocardium enabling biventricular mechanical behavior to be studied. Employing pressure-volume data taken from normal dog hearts and from hearts in which the pulmonary artery has been banded over periods of 2–40 weeks, it is shown that: (a) right ventricular wall stresses are higher than left ventricular stresses; (b) right ventricular wall stress increases initially to a maximum after 3–4 weeks followed by a decline to normal and even subnormal levels, attaining a minimum value at 32–33 weeks; (c) left ventricular stresses behave in a similar manner, attaining their maximum and minimum levels after 7–8 weeks and 32–33 weeks respectively. These results suggest that surgical or medical therapy in patients with hypertrophied ventricles might be more appropriate during the period of wall stress reduction.
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    A mathematical model of the oscillations in the aphakic eye (1980)
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    Bulletin of mathematical biology 42 (1980), S. 837-845 
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    Notes: Abstract In this paper we describe a mathematical model of the oscillations of the diaphragm which limits the vitreous body from the anterior segment of the human eye after the lens has been removed in a cataract operation. We study the motion of this diaphragm driven by movements of the eye. Firstly, a mathematical statement of the problem is given and then we solve the problem exactly for a given class of eye movements. From the analysis we deduce that significant oscillations of the membrane are driven by saccades and that it is the angular acceleration of the eye which causes these types of oscillations. A numerical example is given.
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    Pursuit-evasion wave trains in prey-predator systems with diffusionally coupled delays (1980)
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    Bulletin of mathematical biology 42 (1980), S. 871-887 
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    Notes: Abstract The Lotka-Volterra system of prey-predator equations is considered with a special type of continuous time delay. In the case of equal diffusion coefficients Hopf’s bifurcation technique is used to show the existence of travelling wave train solutions for the prey-predator system.
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    Mathematical model of the activation of prothrombin by factor X a and factor V t (1980)
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    Bulletin of mathematical biology 42 (1980), S. 861-870 
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    Notes: Abstract A mathematical model of prothrombin activation is being proposed which includes the feedback mechanism of thrombin and the alteration of factor V by thrombin. This model is in good agreement with experimental data for the dependence of the rate of thrombin formation on the concentrations of factors V and X a . In particular, it correctly predicts the existence and location of a maximum in both of these cases.
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    A three-dimensional model of the upper tracheobronchial tree (1980)
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    Bulletin of mathematical biology 42 (1980), S. 847-859 
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    Notes: Abstract A new model of the upper tracheobronchial tree is proposed to account for the three-dimensional nature of the airway system. In addition to the tube length, the tube diameter, and the branching angle, the model includes information on the orientation angle of each tube relative to its parent tube. The orientation angle, defined as the angle between two successive bifurcations, is useful for calculating the gravitational inclination of each tube. The information on orientation angle is further used to construct a binary coding system for identifying individual tubes in the airway tree. The proposed model is asymmetrical, but the same principles can be readily used to construct a symmetrical one.
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    On control and optimal control in biodynamic systems (1980)
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    Bulletin of mathematical biology 42 (1980), S. 889-897 
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    Notes: Abstract In any control system for which the number of independent controls is smaller than the number of degrees of freedom to be controlled, our choice of control in any state is restricted to a submanifold of smaller dimension than the tangent space. This simple fact has a number of important consequences for questions of biological import; we consider its implications for adaptation, for senescent phenomena and for the determination of tertiary structures of polypeptides through control of certain average properties. We also formulate the Pontryagin Maximum Principle of Optimal control theory in such a way as to inquire whether specific biodynamic systems can be regarded as optimal with respect to rate of accumulation of particular quantities of the system. We find that if this is possible, the quantity in question must play the role of a clock.
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    Mathematical models in the health sciences: A computer-aided approach (1980)
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    Bulletin of mathematical biology 42 (1980), S. 899-900 
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    Journal of applied probability (1964)
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    Bulletin of mathematical biology 26 (1964), S. 199-199 
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    Training program in computer use for biomedical scientists (1964)
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    Bulletin of mathematical biology 26 (1964), S. 201-201 
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    1964 International Congress for Logic, Methodology, and Philosophy of Science (1964)
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    Bulletin of mathematical biology 26 (1964), S. 203-203 
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    Theoretical evaluation of aerosol deposition in anatomical models of mammalian lung airways (1980)
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    Bulletin of mathematical biology 42 (1980), S. 1-15 
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    Notes: Abstract A computational model to predict deposition of a wide variety of particulate pollutants in several species of mammals is presented. The model incorporates breathing pattern and detailed anatomical models of the respiratory tract based on extensive morphometric measurements of individual airways. The predicted deposition from this general model is in close agreement with observed deposition of monodisperse aerosols in rats. Particle size and density and respiratory breathing patterns are the critical parameters affecting regional deposition.
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    Theory of antigen-antibody induced particulate aggreagation—I. General assumptions and basic theory (1980)
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    Bulletin of mathematical biology 42 (1980), S. 17-36 
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    Notes: Abstract A theory of antigen-antibody induced particulate aggregation is developed by investigating the stability of model systems of particles. Conditions for the formation of large aggregates are derived by imposing the requirement that at equilibrium a statistically significant number of redundant bonds would occur in a reduced monomer-dimer model system. A relationship is obtained which predicts the fractional agglutination in the reduced dimer system as a function of the antigen, antibody and particulate concentrations: $$\frac{g}{{2f c_0 (1 - g)^{2^ - } }} = \frac{{s_1 }}{r} + \frac{{s_1 s_2 }}{{2!r^2 }} + ... + \frac{{s_1 s_2 ...s_j }}{{j!r^j }},$$ wherec 0 is the initial concentration of monomer,f is a proximity factor,g is the fractional agglutination,s i is the average rate of formation of theith bond from an (i−1)th bound dimer, andr is the average rate of dissociation of a single antibody-antigen bond.
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    Theory of antigen-antibody induced particulate aggregation—II theoretical analysis and comparison with experimental results (1980)
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    Bulletin of mathematical biology 42 (1980), S. 37-56 
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    Notes: Abstract The roles of the concentrations of the three interacting constituents in the aggregation process (antibodies, antigens and particulates) are analyzed in detail. It is shown that the basic equation derived in Part I is consistent over a broad range of conditions with experimental findings previously reported.
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    Hypothalamic regulation of pituitary secretion of luteinizing hormone—II feedback control of gonadotropin secretion (1980)
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    Bulletin of mathematical biology 42 (1980), S. 57-78 
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    Notes: Abstract A general mathematical model describing the biochemical interactions of the hormones luteinizing hormone releasing hormone (LHRH), luteinizing hormone (LH) and testosterone (T) in the male is presented. The model structure consists of a negative feedback system of three ordinary differential equations, in which the qualitative behavior is either a stable constant equilibrium solution or oscillatory solutions. A specific realization of the model is used to describe the experimental observations of pulsatile hormone release, its experimental suppression, the onset of puberty, the effects of castration, and several other qualitative and quantitative results. This model is presented as a first step in understanding the physicochemical interactions of the hypothalamic-pituitary-gonadal axis.
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    Dipole theory of interactions of nerve signals (1980)
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    Bulletin of mathematical biology 42 (1980), S. 79-94 
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    Notes: Abstract Based upon the transition rate equation of dipoles in the membrane, we deal with two important aspects of interaction of nerve signals: (1) conditions for nerve excitation and (2) frequency spectrum analysis of nerve impulse. Interrelations between signal amplitudes and frequencies are formulated in detail. There are several important conclusions which can be drawn from our calculations. First, toexcite the nerve, low frequencies are generally more effective than high frequencies. Second, tosedate the nerve (i.e. to suppress undesired activities), high frequencies would suit better. Third, harmonics produced through interactions of nerve signals are not necessarily weaker than the fundamental frequencies. The great significance of our theory is that it indicates in principle the feasibility to alter or rewrite the information contents of a nerve message in our body by applying stimulations of appropriate strengths and frequencies. Thus, the theory provides a physical basis and hence some understanding for a new branch of medicine—neuro therapy such as Nogier's auriculotherapy, Lamy's phonophoresis, Voll's electroacupuncture and the fast rising TENS (transcutaneous electro-neuro stimulation).
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    On a new symmetry in biological systems (1980)
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    Bulletin of mathematical biology 42 (1980), S. 107-117 
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    Notes: Abstract A new physical property, called resonance of the B-type is hypothetically attached to the λ =546 nm irradiated crystalline (small) molecules. In this respect an up or down configuration is assumed for those states obtained through irradiation times that are multiples of 5 sec. With these assumptions, the cellular receptors that may detect these states appear to belong to three classes: the up, down and alternatively mixed up-down. Using the classic formalism of eigenvectors and eigenvalues, a simple spectroscopic type of formula is derived, through which all the possible states of the above characteristic may be obtained.
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    A model of hierarchical ecosystems with migration (1980)
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    Bulletin of mathematical biology 42 (1980), S. 119-130 
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    Notes: Abstract A model of ecosystems with migration is proposed from the viewpoint of flow. This model explains the following two points: (1) How the density-dependent terms in population dynamics arise as a consequence of migration. (2) How the ecosystem exhibits a hierarchy in energy per unit biomass.
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    Announcements (1980)
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    Bulletin of mathematical biology 42 (1980), S. 143-145 
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    On the fisher and the cubic-polynomial equations for the propagation of species properties (1980)
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    Bulletin of mathematical biology 42 (1980), S. 95-106 
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    Notes: Abstract For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.
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    Cation binding to membranes: Competition between mono-, di- and trivalent cations (1980)
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    Bulletin of mathematical biology 42 (1980), S. 191-220 
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    Notes: Abstract The binding of mono-, di- and trivalent cations to negatively charged surfaces is studied within the framework of a modified Gouy-Chapman equation. For any given combination of ions of the above valences, the existence and uniqueness of the solution for the surface potential is shown. The treatment provides the surface potential and charge density. For a system containing only monovalent and divalent ions, analytical solutions are given. When trivalent ions are also present, a procedure based on numerical integration is described. The distance dependence of the electrostatic potential for planar surfaces is given. The calculations provide the amount of cations tightly bound and the amount trapped in the double layer region. The competition between cations for binding to surfaces is elucidated.
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    Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells. II. Complete system (1980)
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    Bulletin of mathematical biology 42 (1980), S. 221-238 
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    Notes: Abstract This paper deals with a model describing the behavior of barium-treatedApalysia neurons. The model is represented by a dynamical system, so-called “complete system”, defined in R4 and depending on a small parameter. The study of this system under zero membrane current conditions was performed with the use of the qualitative theory of singular perturbations. We show that this system has a stable periodic solution of the discontinuous type when the small parameter tends to 0+. A reduced system defined in R3, associated to the complete system was also studied: it corresponds to a constant activation of the inward current. We demonstrate that the corresponding hypothetical cell remains silent under zero current conditions.
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    Chemical modification of enzymes: Critical evaluation of the graphical correlation between residual enzyme activity and number of groups modified (1980)
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    Bulletin of mathematical biology 42 (1980), S. 239-255 
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    Notes: Abstract In the study of chemical modification of enzymes and other biologically active proteins, plots of fractional residual activity as a function of number of groups modified per enzyme molecule are often used to establish a correlation between the chemical modification and enzyme inactivation reactions and to determine the stoichiometry of the modification reaction. This paper presents a critical examination of the underlying theoretical framework of such graphs. Whereas these plots are usually presented as linear functions, it is shown here that the general equation describing the relationship between inactivation and modification contains an exponential term; therefore, in the general case, the plot is actually a curve. It is suggested that caution be exercised in the interpretation of such plots and that equations such as those derived in the text be used to fit theoretical curves to the data, in order to maximize the information gained from chemical modification experiments.
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    The nonexistence of “hermaphoroditic” tracer systems (1980)
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    Bulletin of mathematical biology 42 (1980), S. 257-265 
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    Notes: Abstract This communiction argues that so-called “hermaphroditic” tracer systems, which are neither open nor closed, do not exist physically. The argument is based on the assumption that any observable (possibly nonhomogeneous) macroscopic compartment can be approximated by a compartmentC with a finite number of entry points for the tracer, each associated with an abstract subcompartment ofC. It is shown that the “hermaphroditic” property requires that the mean waiting time be infinite in at least one of the subcompartments, or in a subcompartment elsewhere in the system. A subcompartment with infinite mean waiting time must have some sort of memory, of infinite duration, which knows how long a given particle has been retained, however long that is, and thereby determines its probability of departure. Assuming, as seems likely, that no physical basis exists for such an infinite memory, it follows that “hermaphroditic” systems do not exist.
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    Heat generation in tissues (1980)
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    Bulletin of mathematical biology 42 (1980), S. 273-274 
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    Temperature distribution in muscle (1980)
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    Bulletin of mathematical biology 42 (1980), S. 275-275 
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    Book reviews (1980)
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    Bulletin of mathematical biology 42 (1980), S. 277-281 
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    A note on the structure of system state spaces and its implications on the existence of non-repeatable experiments (1980)
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    Bulletin of mathematical biology 42 (1980), S. 267-272 
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    Notes: Abstract The implication of state space structure on the existence of a repeatable experimentE designed to determine if a states∈L has propertyP or notP is investigated. It is shown that if a state spaceL is connected, then no experimentE is repeatable. This formalism is used to demonstrate that if a propertyP has an associated set of points inL which is dense with dense complement inL, then there exists no repeatable experimentE which can be used to test whethers has propertyP or notP. Other consequences of this formalization are discussed.
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    Some results about nonlinear chemical systems reprosented by trees and cycles (1980)
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    Bulletin of mathematical biology 42 (1980), S. 282-282 
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    Effects of stenosis on non-newtonian flow of the blood in an artery (1980)
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    Bulletin of mathematical biology 42 (1980), S. 283-294 
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    Notes: Abstract It has been shown that the resistance of flow and the wall shear increase with the size of the stenosis but these increases are comparatively small due to non-Newtonian behaviour of the blood indicating the usefulness of its rheological character in the functioning of the diseased arterial circulation.
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    On the Hamiltonian content of biodynamics (1980)
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    Bulletin of mathematical biology 42 (1980), S. 327-337 
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    Notes: Abstract The modern theory of generalized Hamiltonian systems is used to construct a unified canonical description of the linear Lagrangian biodynamics introduced by Kerner.
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    Asymptotic method for peristaltic transport (1980)
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    Bulletin of mathematical biology 42 (1980), S. 305-325 
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    Notes: Abstract The purpose of this paper is to justify an asymptotic method developed for the study of peristaltic transport in a tube of arbitrary cross section. Within the framework of long wave approximation, the three-dimensional nonlinear Navier-Stokes equations are reduced to a sequence of two-dimensional linear boundary value problems of Laplace and biharmonic operators. It is shown that, if a Reynolds number is less than some constant, the solution of the approximate equations is indeed an asymptotic approximation to the exact solution of the problem as the ratio of the maximum radius of the tube to the wave length of the peristaltic motion of the wall tends to zero, and the error estimates are expressed inL 2 norms. Furthermore, under the same condition the exact solution is shown to be unique and stable under arbitrary perturbation of spatially periodic disturbance. Application of the stability condition to peristaltic transport in a tube of circular cross section is given.
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    A one tube flow problem arising in physiology (1980)
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    Bulletin of mathematical biology 42 (1980), S. 295-304 
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    Notes: Abstract A mathematical analysis, including existence and uniqueness, is given for some boundary value problems which model the flow of a fluid-solute mixture in a tube which is placed in an interstitium. The model permits an interchange of fluid and solute across the tube walls.
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    Topographic organization of nerve fields (1980)
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    Bulletin of mathematical biology 42 (1980), S. 339-364 
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    Notes: Abstract The vertebrate nervous system has topographic interconnections in many parts, known for example as retinotopy, somatotopy, etc. It is plausible that modifiable synapses play an important role in forming and refining these connections together with the sensory experiences. To elucidate the mechanism of topographic organization, we propose a simple model consisting of two nerve fields connected by modifiable excitatory synapses. The model also includes modifiable inhibitory synapses. The behavior of the model is described by a set of simultaneous non-linear integro-differential equations. By analyzing the equations, we obtain the equilibrium solution of topographic connections. It is also proved that a part of the presynaptic field which is frequently stimulated comes to be mapped on a large area of the postsynaptic field so that it has a good resolution.
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    Steady-state analysis of a two-compartment barrier-limited capillary-tissue model with michaelis-menten saturation kinetics (1980)
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    Bulletin of mathematical biology 42 (1980), S. 691-700 
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    Notes: Abstract The steady-state solution of the equations governing substrate exchange between vascular and extravascular compartments separated by a membrane with finite, symmetrical substrate permeability is presented. Substrate removal from the extravascular compartment by Michaelis-Menten saturation type kinetics with negligible diffusion in the axial and instantaneous diffusion in the transverse directions in both compartments are assumed. It is shown that the solution degenerates into known expressions for special linearized and asymptotic cases. The method of solution is also applied to an extension of the original model incorporating autoregulatory feedback effects upon the process responsible for substrate removal.
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    Time lags and global stability in two-species competition (1980)
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    Bulletin of mathematical biology 42 (1980), S. 729-737 
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    Notes: Abstract Global asymptotic stability and equilibrium coexistence is established in two species Lotka-Volterra-type competition when there are time delays in interspecific interaction terms and the intraspecies competition is stronger than the interspecies competition.
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    A note on the dominance hierarchy index (1980)
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    Bulletin of mathematical biology 42 (1980), S. 739-746 
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    Notes: Abstract The realized (observed) value of Landau’s dominance hierarchy index is examined. Under a model of constant pairwise dominance probabilities, the observed index is shown to be a strongly consistent estimator of the underlying (true) index. However, a large number of encounters between animals is shown to be required in order to reduce bias and variance to practical levels except when the pairwise dominance probabilities are near one.
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    Food webs and niche spaces (1980)
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    Bulletin of mathematical biology 42 (1980), S. 747-748 
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    Cation binding to membranes; competition between mono-, di-and trivalent cations (1980)
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    Bulletin of mathematical biology 42 (1980), S. 749-749 
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    On the theory of interfacial double-layer potential differences and periaxonal accumulation of potassium in squid giant axons (1980)
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    Bulletin of mathematical biology 42 (1980), S. 751-763 
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    Notes: Abstract The steady state potassium conductance as a function of measured membrane potential difference (p.d.) ϕ of the squid giant axon is corrected for the effect of accumulation of potassium in the periaxonal space. This correction is made on the assumption that several mathematical models of the axon are valid. These are (i) the McIlroy (1975), McIlroy-Hahn (1978) model of membrane conductanceg i(i=K, Na) which is a detailed model of passive transport of ions across the axonal membrane with the aid of mobile, negatively-charged carriers, (ii) the Adelmanet al. (1973) compartmental model of the periaxonal and external bathing-solution spaces, (iii) the enzymatic theory of nervous conduction due to McIlroy (1970 a, b, c), (iv) the Wien dissociative effect of the axolemmic electric field on the weak membrane buffer proposed by Bass and Moore (1968) as a trigger mechanism in nervous excitation and (v) the model (McIlroy, 1979) of the interfacial double-layer p.d.s. which are assumed to exist at the membrane’s surfaces because of the presence of a fixed surface charge. From the correctedg k (ϕ) curves the values of the double-layer p.d.s. of model (v) are deduced and these are shown to lead to a consistent, physically reasonable solution for the distance (approx. 6.8Å) between the fixed surface charges and for the dissociation constants of these sites in their interactions with the ions of the extra-membrane electrolytes. Assuming that the selectivity coefficint of the potassium conducting system for the squid giant axon is approx. 52 it is deduced that the potassium permeability,P ks , of the periaxonal barrier ≈1.37(±0.5)×10−4 cm sec−1 and the thickness of the periaxonal space ≈451±159Å.
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    Effects of peripheral layer viscosity on blood flow through the artery with mild stenosis (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 797-805 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The effects of peripheral layer viscosity on physiological characteristics of blood flow through the artery with mild stenosis have been studied. It has been shown that the resistance to flow and the wall shear decrease as the peripheral layer viscosity decreases.
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    On the Higgins model of glycolysis (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 829-836 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A study was made of Higgins’ model of glycolysis incorporating molecular diffusion of intermediates, utilizing an earlier conjecture due to Landau. Conditions for the existence of asymptotically stable spatio-temporal periodic solutions are obtained.
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    URL: http://dx.doi.org/10.1007/BF02461061
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    Jagdbetriebliche Folgerungen aus dem Verhältnis zwischen der Größe der Jagdbezirke und dem Wohnraum einiger jagdbarer Tiere (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 2-8 
    Details
    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary The Shooting Law in vigour within the Federal Republic is based upon the so-called districtsystem; there are 2 kinds of hunting territories, the hunting district of owners and the common hunting district. For each of these 2 kinds minimum sizes are required, 75 resp. 150 ha. The shooting district is the unit of management. It is investigated if the size of the management unit corresponds to the biological circumstances of the kinds of deer at the disposal of the shooting enterprise. Considering only the most important biological factor, the size of the yearly district — i. e. of the area which is inhabited by 1 piece of one kind of deer in the course of 1 year —, it has to be stated that this yearly district covers a multiple of the minimum size of a shooting district. This proportion - one should say disproportion - is recognized by some legal measures (for instance the possibility of rounding up the shooting district, shooting according to shooting plan); but these are insufficient. Especially considering the restriction of movement of free living animals because of the increase of human population, the intensification of agriculture and traffic, other measures (i. e. such of shooting management) ought to be taken.
    Abstract: Résumé La loi cynegétique coutumier de la République Féderale d'Allemagne est basé sur le système des districts. Elle connait deux formes de terrains de chasse: districts de chasse particuliers et districts de chasse communs. L'étendue pour chaque forme est fixé à un minimum de 75 ou 150 has. Chaque district de chasse est une unité d'exploitation. La question est posée si l'étendue d'une unite d'exploitation est approprié aux données biologiques de gibier vivant sur cette domaine. Regardant une des plus importantes données biologiques, l'étendue du district annuel, c'est à dire l'aréal donc a besoin une seule bête d'une espèce pendant une année de chasse, on constate alors que le district annuel est largement au dessus de l'étendue minimum d'un district de chasse. On tache de rendre compte à cette proportion - mieux vaut dire disproportion - par quelques mésures législatives (p. e. possibilité d'arrondir les districts de chasse, abattement suivant un plan conçu) toutes ces mésures restent cependant inefficace. Quand on regarde le rétrécissement pour bêtes fauves de circuler librement, rétrécissement dû à l'augmentation de la population humaine, l'intensité de la culture des terres et densité de la circulation sur routes, il fallait envisager encore d'autres mésures, c'est à dire ceux pour l'exploitation des chasses.
    Notes: Zusammenfassung Das in der Bundesrepublik Deutschland geltende Jagdgesetz fußt auf dem sogenannten Reviersystem; es kennt zwei Arten von Jagdbezirken, den Eigenjagdbezirk und den Gemeinschaftlichen Jagdbezirk. Für jede dieser zwei Arten bestehen Mindestgrößen, 75 bzw. 150 ha. Der Jagdbezirk ist die Betriebseinheit. Es wird untersucht, ob die Größe der Betriebseinheit den biologischen Gegebenheiten der dem Jagdbetrieb zur Verfügung stehenden Wildarten angemessen ist. Wenn man dabei als eine der wichtigsten biologischen Gegebenheiten die Größe des Jahresbezirkes - also des Areals, das ein Stück einer Wildart im Laufe eines Jagdjahres einnimmt -herausgreift, so stellt man fest, daß dieser Jahresbezirk ein Vielfaches der Mindestgröße eines Jagdbezirkes beträgt. Diesem Verhältnis - besser gesagt Mißverhältnis - wird wohl durch einige gesetzliche Maßnahmen (z. B. Möglichkeit zur Abrundung der Jagdbezirke, Bejagung nach Abschußplan) Rechnung getragen; diese sind jedoch unzureichend. Insbesondere mit Rücksicht auf die Einengung der Bewegungsfreiheit der freilebenden Tiere infolge der Zunahme der Bevölkerung, der Intensivierung der Landeskultur und des Verkehrs, sollten noch andere Maßnahmen (nämlich solche des Jagdbetriebes) Platz greifen.
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    IV. Referate (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 38-40 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02282259
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    Der Marderhund (Nyctereutes procyonoides Gray) in Rumänien (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 51-54 
    Details
    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary The raccoon-dog (Nyctereutes procyonoides Gray) has been signalized for the first time in Roumania in 1951 in a locality in the South of Roumania, near Bucharest. Lately this dog has been identified in several localities in the North and North-East of the country in Moldavia. Recently, the raccoon-dog gains, more and more, ground, especially in the Danube delta, where the largest number has been observed. In Roumania, the penetration of the raccoon-dog was accomplished by migration-way, by the specimens colonised in U.R.S.S. (Ukraina and the Moldavian Republic).
    Abstract: Résumé Le chien raton (Nyctereutes procyonoides Gray) a été signalé pour la première fois en Roumanie en 1951, dans une localité dans le sud de la Roumanie, près de Bucarest. Plus tard il a été identifié dans plusieurs localités dans le nord el le nord-est du pays — en Moldavie. Dernièrement, le chien raton gagne, de plus en plus, du terrain, spécialement dans le delta du Danube, où il a été observé le plus grand nombre de spécimens. En Roumanie, la pénétration du chien raton s'est faite par voie de migration, des exemplair s colonisés en U.R.S.S. (l'Ukrain et la République Moldave).
    Notes: Zusammenfassung Der Marderhund wurde in Rumänien zum ersten Male im Jahre 1951 beobachtet, und zwar im Süden des Landes, in der Umgebung von Bukarest. Später wurde er an mehreren Stellen im Norden und Nordosten des Landes, in der Moldova, festgestellt. In letzter Zeit besetzt der Marderhund immer mehr Gebiete im Donaudelta, wo die größte Zahl beobachtet wurde. Das Eindrigen des Marderhundes nach Rumänien erfolgte durch Zuwanderung aus den in der UdSSR geschaffenen Ansiedlungen (Ukraine und der Moldauischen Republik).
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    Vergleich einiger amerikanischer Wildschadenverhütungsmittel mit deutschen Präparaten (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 62-68 
    Details
    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary Seven wildlife repellents made in USA were tested in a series of preliminary experiments. In these preliminary tests succes was required for admittance to the decisive field test, which is a part of the official recognition procedure of german wildlife repellents. The repulsivity, observed for fenced animals (red-, fallow- and roe-deer, mufflon and rabbits) was compared with nine german official recognized repellents. The results of phytotolerancy using the pelargonium-test and those relating to the technical properties are shortly explained.
    Abstract: Résumé Sept répulsifs de fabrication américaine et utilisés contre les dégâts causés par le gibier ont été soumis à une série d'essais préliminaires. Ce test préliminaire doit avoir été subi avec succès pour que, dans la procédure de reconnaissance officielle des répulsifs pour le gibier mis sur le marché en Allemagne fédérale, ceux-ci soient admis à subir un test décisif sur le terrain. L'action répulsive, observée chez des animaux en enclos (cerfs, daims, chervreuils, mouflons et lapins), a été comparée à celle de neuf produits allemands officiellement reconnus. Les résultats de phytotolérance d'après le test au Pelargonium et ceux relatifs aux propriétés techniques sont brièvement exposés.
    Notes: Zusammenfassung Sieben amerikanische Präparate zur Abwehr von Wildschäden wurden nach der Methode, die bei der Vorprüfung deutscher Wildschadenverhütungsmittel im Rahmen des amtlichen Anerkennungsverfahrens eingeführt ist, getestet. Die im Gehege ermittelte Abwehrwirkung gegen Rotwild, Damwild, Rehwild, Muffelwild und Kaninchen wird mit den Ergebnissen für neun deutsche Präparate verglichen. Weiterhin werden Ergebnisse zur Pflanzenverträglichkeit nach dem Test mit der Pelargonie und einige Ergebnisse der technischen Prüfung vorgelegt.
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    Von Steinadler (Aquila chrysaëtos) in Tirol (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 54-61 
    Details
    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary After the population of golden eagles in Tyrol had been much reduced by shooting at least 20 head in the years 1958/59, it was demanded that further shooting must be subject to exact scientific investigation concerning density of population ecc. During the following years 13 eyries were controlled and the young eagles were ringed. Offhand tests in the course of these investigations informed us also of the young eagles' diet. Finally the act of putting the rings on is described and the first results of distant findings (South of France) are treated. The possibilities of shooting golden eagle are considered and it is recommended to prefer the clearing of eyries in certain cases.
    Abstract: Résumé L'état en grand-aigles dans le pays Tyrol fut pendant les années 1958/1959 fortement réduit par l'abattement d'au moins 20 exemplaires. La demande fut alors posée que des recherches scientifiques exacte sur l'état etc. devraient précéder futures abattements. Pendant les années suivantes on tenait donc sous contrôle 13 aires d'aigles couverts de jeunes pousses et garnissait les aiglons d'un anneau. Recherches sur épreuves au hasard donnaient des éclaircissements sur la nourriture des jeunes. On parle enfin du garnissage avec anneau et des premiers resultats sur trouvailles éloignés (venant du Midi de la France). Pour conclure on se répand sur les possibilités d'une chasse aux grand-aigles mais croit plus efficase une vidange de l'aire.
    Notes: Zusammenfassung Nachdem im Jahre 1958/59 in Tirol der Bestand an Steinadlern durch Abschuß von mindestens 20 Exemplaren stark reduziert worden war, wurde gefordert, daß weiteren Abschüssen genaue wissenschaftliche Untersuchungen über Bestand u. dgl. vorausgehen müssen. In den folgenden Jahren wurden 13 beflogene Horste kontrolliert und die Jungadler beringt. Stichprobenartige Untersuchungen gaben dabei auch über die Ernährung der Jungen Aufschluß. Schließlich wird über die Beringung, berichtet, erste Ergebnisse von Fernfunden (aus Südfrankreich) werden behandelt. Abschließend wird auf die Möglichkeit der Bejagung des Steinadlers eingegangen und eher eine fallweise Aushorstung für günstig angesehen.
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    Zu “Folgen des Winters 1962/63” (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 69-70 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Type of Medium: Electronic Resource
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    Abwurf des Bastgehörns (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 70-71 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Doppelkopfbildung beim Rothirsch (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 71-72 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Laufverletzung bei einem Rehbock (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 72-74 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01964660
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    Ehrenpromotion Dr. Lindner (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 74-77 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01964661
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    Professor Dr. Gösta Notini (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 77-78 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    URL: http://dx.doi.org/10.1007/BF01964662
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    Eugen Fischer 90 Jahre (1964)
    Springer
    European journal of wildlife research 10 (1964), S. 78-78 
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    ISSN: 1439-0574
    Source: Springer Online Journal Archives 1860-2000
    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    URL: http://dx.doi.org/10.1007/BF01964663
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