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  • Springer  (83,398)
  • 1995-1999
  • 1980-1984
  • 1975-1979  (83,398)
  • 1940-1944
  • 1977  (43,877)
  • 1975  (39,521)
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  • 1995-1999
  • 1980-1984
  • 1975-1979  (83,398)
  • 1940-1944
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  • 1
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    Bulletin of mathematical biology 37 (1975), S. 37-49 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The chromosomal theory of inbreeding based on a gametic interaction system lead us to define a depression coefficientD. Comparison of random, sib and half-sib matings (with inbreeding coefficientF=0, 1/4 and 1/8) shows thatD depends on the structure of the starting population and on values of the model parameters. This result accounts for responses of lines whose depression does not depend directly on the inbreeding coefficient and which theories of inbreeding based on increasing homozygosity fail to explain.
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  • 2
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    Bulletin of mathematical biology 37 (1975), S. 59-69 
    ISSN: 1522-9602
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    Notes: Abstract An idealization of chemical combination is formulated as a model of computability, and it is shown that this model has universal computational power just in case assembly has at least two-dimensional space in which to occur. It is also shown that this model, under reinterpretation, corresponds to a cellular automaton in which growth occurs by differentiation only (i.e., the state into which any cell is born is thereadfter fixed). Hence this latter model of growth is also computationally universal.
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  • 3
    ISSN: 1522-9602
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    Notes: Abstract Kinetics of biological light emission processes do not mean what they seem to mean, because measured light intensity is not proportional to reactant concentration but to reaction rate. Therefore, the differential equation for light decay is usually different from that of concentration decay, so that mass action interpretations cannot be applied directly to light intensity decay. An observed second order light decay for Chlorella at 6.5°C, implies Elovich solid state reaction kinetics, which agrees with other evidence for solid state processes in photosynthesis. An observed 1.5 order light decay for Cholorella at 28°C implies second order liquid or solid state reaction kinetics. First ordere light decay implies first order reaction kinetics.
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  • 4
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    Bulletin of mathematical biology 37 (1975), S. 71-78 
    ISSN: 1522-9602
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    Notes: Abstract Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ R θ(x 1,x 2,t) dx 1 dx 2 of a biological species with an area-density distribution function θ=θ(x 1,x 2,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇2θ +fθ −gθ n+1 whereD (〉0),n (〉0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(x 1,x 2, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn〉1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ R θ(x 1,x 2, 0)2 dx 1 dx 2 sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.
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  • 5
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    Bulletin of mathematical biology 37 (1975), S. 127-138 
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    Notes: Abstract The equilibrium probability distribution of the process level is studied for a general class of reversible stochastic reactions. A calculationally convenient approximation for equilibrium probabilities is derived and its accuracy is investigated over a range of values of the equilibrium constant. A method of approximating the equilibrium means and variance is developed and illustrated forQ th-order processes.
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  • 6
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    Bulletin of mathematical biology 37 (1975), S. 565-572 
    ISSN: 1522-9602
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    Notes: Abstract Beside the concept of material inputs and outputs of components of the representation of biological systems given to us by Rosen, the concept of energy is incorporated. The interaction of material and energy is represented by a cartesian product; and separate material and energetical mappings are considered as the new representation of components. These developments generate aMα category, and it is shown thatMα is isomorphic to theM category of previous developments.
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  • 7
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    Bulletin of mathematical biology 37 (1975), S. 555-564 
    ISSN: 1522-9602
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    Notes: Abstract This paper discusses the solution of a generaln-compartment system with time dependent transition probabilities utilizing the technique described by Cardenas and Matis (1975) (hereafter abbreviated (CM)). In addition, the cumulant generating function is derived for a special class of reversiblen-compartment systems where the time-dependent intensity coefficients corresponding to the migration and death rates are some multiple of each other. The immigration rates can be any integrable function of time. The moments are also obtained and the solution to the two-compartment system is presented explicitly. The solution is illustrated with a linear and a periodic function which forms have been widely reported in the literature.
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  • 8
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    Bulletin of mathematical biology 37 (1975), S. 573-588 
    ISSN: 1522-9602
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    Notes: Abstract The relations (inflow) = (dose)/(area under indicator curve), and (volume of distribution) = (throughflow) × (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem, is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution.
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  • 9
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    Bulletin of mathematical biology 37 (1975), S. 219-219 
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    Bulletin of mathematical biology 37 (1975), S. 291-299 
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    Notes: Abstract Perturbation methods are applied to a differential equation predator-prey model to find the approximate amplitudes and period of limit cycle solutions. In the model the feeding rate per unit predator per unit prey decreases as the prey become scare. The rigorous applicability of the perturbation technique depends on the assumptions that the limit cycle amplitude is relatively small and that near the equilibrium point the growth rate of each species is most sensitive to changes in the density of the other species. The second assumption is usually roughly satisfied in practice and examples are considered which suggest that the first assumption can be greatly relaxed.
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    Bulletin of mathematical biology 37 (1975), S. 367-387 
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    Notes: Abstract Signal Detection Theory can be used to provide a mathematical model describing the choice of a predator trying to distinguish between a model and a Batesian mimic. The mathematical model yields a number of a deductions, in particular that it may or may not assist the mimic population if mimics more closely resemble their models. The assumptions underlying the analysis are discussed in some detail.
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    Bulletin of mathematical biology 37 (1975), S. 419-425 
    ISSN: 1522-9602
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    Notes: Abstract A new type of physical transition, denotedS→S *, has been detected in irradiated organic molecules (λ=546 nm) through their interaction with specific biological macromolecules. In a specific enzyme-substrate interaction, a clear enhancement of the reaction rate is observed, when the substrate is irradiated with sharply well defined times. These “efficient irradiation times” are always of the 5k sec type (k=1, 2, 3, …). They have been consistently revealed in a great number of specific biological interactions. The present note demonstrates an important property, i.e. that forevery irradiation time aS→S * transition is induced in organic molecules. It is shown that for any irradiation times different from the 5k sec type (k=1, 2, 3, …) states of theS * type may occur, but the biological macromolecules may “detect” only theS * states induced by irradiations of the 5k sec type.
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    Bulletin of mathematical biology 37 (1975), S. 459-470 
    ISSN: 1522-9602
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    Notes: Abstract A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. It envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. It has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematocrit. The wall layer thickness is found to decrease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the case of blood may be attributed to the deformability of the erythrocytes.
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    Bulletin of mathematical biology 37 (1975), S. 489-504 
    ISSN: 1522-9602
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    Notes: Abstract Three dimensional laminar, viscid flow is developed for Newtonian fluids which provides absolute values for axial, radial and tangential velocity fields everywhere if the dimensions of the vessel are known and two simultaneous axial velocities e.g. on and off the central axis in the same plane, and the central axis axial velocity gradient are measured. In addition, normal and shear stresses are determinable. The equation set satisfies geometric and other known flow limiting conditions such as no slip at surfaces etc. and are amenable for inclusion in general, dynamic flow expressions. Alternatively they may be used alone for certain problems involving gradients and secondary flows. A range of illustrations are shown for a distorting vessel with elliptic cross-section and small axial taper (analogous to the pulmonary trunk during ejection).
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    Bulletin of mathematical biology 37 (1975), S. 521-553 
    ISSN: 1522-9602
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    Notes: Abstract A regulated left ventricular dynamics model is presented which involves interaction of the dynamics of the left ventricular and circulatory systems and their regulation by the central nervous system. On-line human parametric simulation (parameter estimation) and consequential prognostic implications (based on parametric values) are demonstrated. Model responses to simulated physiologic stresses help delineate tolerances of subjects. In order to have an estimate of the reliability of the model, the sensitivity of the model's responses to changes in the values of its intrinsic parameters is assessed. Also determined is the extent to which errors in measuring the pressure affect the calculated values of the model's simulation parameters and subsequently influence the values of other diagnostically useful variables (such as contractility, oxygen consumption rate, heart rate), when the model is used to determine the limiting physiological stress sustainable by the subject. A comparison of the model's composition with those of other similar cardio-circulatory models is included.
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    Bulletin of mathematical biology 37 (1975), S. 659-673 
    ISSN: 1522-9602
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    Notes: Abstract Then-stage harvesting strategy of Elizarov and Svirezhev is examined. As a result, some important new features appear. A discussion is presented on whether or not one should harvest a species at one time stage or wait until a later time. The paper is concerned with contributions which are primarily mathematical formulations and results for continuous, as well as discrete time, logistic growth of a single species being harvested. Age class structure is ignored.
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    Bulletin of mathematical biology 39 (1977), S. 663-678 
    ISSN: 1522-9602
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    Notes: Abstract A number of apparently different lines of inquiry into fundamental biological processes point to the central role played by the notion of observation in the theory of biological systems. Not only do we use the results of our own observations to obtain the system descriptions which are the starting-points for an understanding of biological processes, but it is a basic postulate of physics that the interactions between biological systems themselves can be regarded as observations. On this basis, it is clear that we cannot properly understand biological interactions unless the observables we employ for system description are the same as those involved in the interactions we are describing. To do this requires a general theory of observables and system description, establishing the relationship between different modes of description. A sketch of such a theory is developed in the present paper, using only two postulates: (a) that all interactions are determined by the values of observables of a system evaluated on specific states, and (b) that real-valued observables suffice. As an application, a specific test is proposed whereby it can be determined whether the observables employed to describe interacting systems are sufficient to specify the observables involved in the interaction itself.
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    Bulletin of mathematical biology 39 (1977), S. 679-691 
    ISSN: 1522-9602
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    Notes: Abstract Differential equations are derived whose solution gives the cross-sectional shape of a flexible tube as a function of the transmural pressure. These equations are solved digitally to produce a series of closed curves, each curve representing the shape of a cross section for a particular set of conditions. These are then applied to the case of systemic arteries, pulmonary arteries, and large veins. The results predict that systemic arteries must always be circular, even when the internal and external pressures are equal. In veins, a small positive internal pressure causes them to become circular, regardless of their initial state, with negligible stretching. Further increases in internal pressure cause the area of the cross section to increase due only to stretching, the shape remaining essentially circular. With pulmonary arteries, known to be noncircular, changes in the cross-sectional area result from a combination of stretching and changes of shape.
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    Bulletin of mathematical biology 39 (1977), S. 693-704 
    ISSN: 1522-9602
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    Notes: Abstract Stochastic models of human reproduction are beginning to play significant roles in the evaluation of family planning programs. A class of stochastic processes called absorbing, agedependent, semi-Markov processes frequently arises in the construction of such models. The paper begins with a discussion of some technicalities regarding absorbing, age-dependent, semi-Markov processes. Then, an algorithm due to Littman, which makes possible the computerization of this class of stochastic processes, is presented. Briefly, Littman’s algorithm provides an efficient method for numerically solving systems of renewal type integral equations, provided the system does not contain a large number of equations. After setting down a concrete model for a large clinical trial of intrauterine devices conducted in Taiwan, the paper concludes with a discussion of a method for validating the model based on the data collected in the clinical trial.
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    Bulletin of mathematical biology 39 (1977), S. 743-743 
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    Bulletin of mathematical biology 39 (1977), S. 705-719 
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    Notes: Abstract Some theoretical requirements for the valid use of hemolytic plaque inhibition as a method for studying cellular selection and recognition in an immune response are reviewed. Aside from providing a rational basis for the use of this technique, the theory resolves a growing body of apparently conflicting results on the affinity regulation of IgM secreting cells and can also be used to deduce structural (in particular, morphological) features of the IgM molecule. The diverse predictions of the theory are examined in light of experimental results and find support in a wide data base. Additional specific experiments are proposed which can be used in conjunction with the theory to help clarify the relation between cellular proliferation and maturation.
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    Bulletin of mathematical biology 39 (1977), S. 721-741 
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    Notes: Abstract A theory of the cell volume is presented with emphasis on the swelling effect of high concentrations of KCl and other chloride salts. In this theory a particular cell volume represents a state of balance between the tendency of the cell water to build deeper layers of polarized water and the restraining forces exerted by the salt linkages and H-bonds. Taking into account also the different structure-breaking effects of different salts, theoretical curves can be constructed which describe the complex multiple peak-plateau of swelling curve observed in frog muscle in response to increasing concentrations of different chloride salts.
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    Bulletin of mathematical biology 37 (1975), S. 97-100 
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    Bulletin of mathematical biology 37 (1975), S. 1-9 
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    Notes: Abstract The study of systems exhibiting a band-pass function is completed for systems whose parameters are time-dependent. In the case of periodic parametric excitations, it is demonstrated that some systems can get into “resonance” for a particular frequency. By studying this problem, a new and probably fruitful approach of some rhythmic behaviours can be made.
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    Bulletin of mathematical biology 37 (1975), S. 19-35 
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    Notes: Abstract Analytical techniques are developed which permit objective control of asiist device driving systems. In addition to being objective, the techniques described in this paper are optimal in the sense of minimizing a performance index which consists of a term involving left ventricular power and a term involving deviations of aorta hemodynamic parameters from normal values. Comparisons are included of off-line computations and measurements on dogs with experimentally induced myocardial infarctions undergoing intraaortic balloon pumping.
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    Bulletin of mathematical biology 37 (1975), S. 427-458 
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    Notes: Abstract A mathematical model simulating a cell growing in a culture medium is obtained. Using this model, various behavioral patterns of the cell are obtained under different types of disturbances, in particular when (i) a Mg2+ deficiency experiment and, (ii) a split-dose ionizing radiation experiment are carried out, (iii) when disturbances on the rate constants of the biochemical reactions taking place in the nucleus of the cell are applied, and (iv) when the cell's interior components are perturbed. The cell model results obtained agree well with experimental results for the Mg2+ and split dose experiments, and explain the mechanism of the split dose radiation experiment without the need to introduce additional axioms (e.g. healing processes) into the dynamics of the cell. Conditions are obtained which cause the cell to behave in a rapidly growing ‘tumor-like’ mode; it is shown that once the cell moves into this ‘tumor-like’ mode, its behavior is irreversible, i.e. if a disturbance of opposite type is then applied to the ‘tumor’ cell, the cell will not revert back to its original normal behavior.
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    Bulletin of mathematical biology 37 (1975), S. 85-90 
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    Notes: Abstract Solution of the equation that describes simulatenous liquid flow and diffusion in a spherical model of the vitreous body of the eye shows that a small dissolved specie can move both anteriorly and posteriorly from a source behind the lens even though there is a slow liquid movement almost entirely in the posterior direction. This results explains why tracer studies using large particles (dyes or colloids) show only a posterior flow, whereas studies using sodium ion show anterior movement as well.
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    Bulletin of mathematical biology 37 (1975), S. 101-107 
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    Bulletin of mathematical biology 37 (1975), S. 111-111 
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    Bulletin of mathematical biology 37 (1975), S. 221-221 
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    Bulletin of mathematical biology 37 (1975), S. 255-268 
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    Notes: Abstract The equations relating hybridized RNA to free RNA, in the case of simple hybridization, or to ratio of labelled and unlabelled RNA in competitive hybridization, are derived. Analysis of the equations shows how hybridization data may be used to infer properties of the distribution of components in an RNA mixture, or the relation between two distributions in competitive hybridization. A critical examination of the assumptions underlying the equations indicates that some of then may be violated in certain cases, or have no current support, evidential or theoretical. The consequences of such qualifications for the interpretation of hybridization data are indicated.
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    Bulletin of mathematical biology 37 (1975), S. 589-636 
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    Notes: Abstract The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.
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    Bulletin of mathematical biology 37 (1975), S. 109-109 
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    Bulletin of mathematical biology 37 (1975), S. i 
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    Bulletin of mathematical biology 37 (1975), S. 139-146 
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    Notes: Abstract Krylov-Bogoliubov-Mitropolsky perturbation method was used to study the effect of nonlinearity in the Volterra-gause-Witt (VGW) model for a two species prey-predator system. The first order corrections to both the frequency of oscillation and the amplitude of the linearized system were computed. It was found that the basic qualitative features of the nonlinearity are exhibited by the first order result. We have also discussed the Lotka-Volterra problem which is a special case of VGW model.
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    Bulletin of mathematical biology 37 (1975), S. 147-160 
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    Notes: Abstract This paper presents a mathematical analysis of a tumor model first proposed by Skipper and Zubrod. The tumor model is comprised of three compartments, a proliferative compartment, a nonproliferative but viable compartment, and a dead compartment. By the suitable selection of functions describing loss of cells from the proliferative and nonproliferative compartments, the model is capable of describing tumor behavior during periods of growth and drug treatment. The loss functions during treatment are related to pharmacokinetic functions and may be chosen according to known drug properties. Tumor properties may be simulated by the appropriate choice of cell cycle parameters. It therefore seems feasible to simulate tumor behavior for scheduled treatment with chemotherapeutic agents. Another important result of this analysis is the derivation of a fraction labelled mitosis function which incorporates the nonproliferative compartments.
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    Bulletin of mathematical biology 37 (1975), S. 161-180 
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    Notes: Abstract Techniques of modelling and simulation are discussed as they relate to bioengineering systems. The advantages and disadvantages of different analytical engineering methods utilized to gather information concerning the behavior of complex physiological and neuromuscular control mechanisms are explained. An Inners Criterion is developed to determine if the roots of a model lie within a certain “biologically realistic region” ΓB, in the complex plane which contains the roots of linearized models for a large variety of neuromuscular systems. Several algorithmic methods based on the Jury Inners Test are described which specify whether the model roots lie within the desired region, thereby providing an indication as to the validity of the proposed model. This technique can help to eliminate tedious simulation on an unrealistic model with roots lying far outside this region. An exemplary model for control of vergence eye movements is presented and shown to satisfy the ΓB criterion; several counter-examples are also discussed. The Inners approach can be adapted to other classes of bioengineering systems by specifying the region based on models that are contained in the class of interest.
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    Bulletin of mathematical biology 37 (1975), S. 215-218 
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    Bulletin of mathematical biology 37 (1975), S. 220-220 
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    Bulletin of mathematical biology 37 (1975), S. 223-254 
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    Notes: Abstract In this paper I consider how information is required to specify various systems. It is shown that the transitive information of any physical system, is distributed among three distinct components. One of these, the selective component, is required to specify the elemental parts of the system. Another, the connective component, is required to specify the macrostructure of the system; that is the way the parts are put together. And a third, the conformative component, is required to specify the intrinsic complexion or microstructure of the system. An interesting method for analyzing branched systems which takes account of connective ambiguity is described in some detail. The relationship between information and entropy, known as the Clausius-Shannon Identity, is then discussed with reference to selected thermodynamic models: and that aspect of information which is often overlooked, namely the distinction, between transitive and intransitive information is highlighted. The applications (or perhaps more correctly, the limitations of applying this treatment) to problems of biological interest are also indicated.
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    Bulletin of mathematical biology 37 (1975), S. 269-275 
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    Notes: Abstract This paper discusses two compartment models with interaction allowed between the compartments. The total number of particles in the system at any time is discussed along with the number to the found in each separate compartment. An interesting result is that the number of particles in each of the two compartments areindependent random variables. Some asymptotic results are also given. The paper is a continuation of some earlier work by the author.
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    Bulletin of mathematical biology 37 (1975), S. 277-289 
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    Notes: Abstract General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ∂c i /∂t =D i ∇2 c i +Q i (c) subject to boundary conditions of practical interest, where the enumerator indexi runsl ton, c i =c i (x,t) denotes the concentration or density of theith participating molecular or biological species,D i is the diffusivity constant for theith species, andQ i (c), an algebraic function of then-tuplec=(c 1,...,c n ), expresses the local rate of production of theith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals ofc which decrease monotonically with time can often be found, as exemplified here for Volterra and Verhulst-Volterran-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reaction-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, $$\mathop {\lim }\limits_{t \to \infty } c_i (x, t) = \hat c_i (x)$$ , if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl ofQ(c) inc-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of “weak diffusion” with the reaction rate terms dominant in the governing equations.
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    Bulletin of mathematical biology 37 (1975), S. 323-365 
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    Notes: Abstract A model nonlinear network involving chemical reactions and diffusion is studied. The time evolution and bounds on the steady state solutions are analyzed. Spatially ordered solutions of the equations of the dissipative structure type are found by bifurcation theory. These solutions are calculated analytically and their qualitative properties are discussed.
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    Bulletin of mathematical biology 37 (1975), S. 637-657 
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    Notes: Abstract Models based on molecular mechanisms are presented for pattern formation in developing organisms. It is assumed that there exists a diffusion governed gradient in the morphogenetic field. It is shown that cellular differentiation and the subsequent pattern formation result from the interaction of the diffusing morphogen with the genetic regulatory mechanism of cells. In a second stage it is shown that starting from a homogeneous distribution of morphogen, polarity can be generated spontaneously in the morphogenetic field giving rise to the establishment of a gradient. The stability of these gradients is demonstrated. The onset of a morphogenetic gradient and pattern formation are combined in a single coherent model. Size invariance and its biological implications are discussed.
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    Bulletin of mathematical biology 37 (1975), S. 11-17 
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    Notes: Abstract A simple population model consisting of one adult and two larval stages with cannibalism or competition among the larval stages is presented. The solutions are found to be either periodic or of a steady state nature depending on the ratios of fertility and cannibalism among the larvae. Two similar cannibalism pressure functions are compared and the conditions that lead to steady or periodic solutions, or to extinction, are examined.
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    Bulletin of mathematical biology 37 (1975), S. 301-321 
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    Notes: Abstract Thyroid hormone synthesis shows two important characteristics at each iodine organification step: iodine fixation to thyroglobulin tyrosyl residues in the vicinity of the thyroid cell microvilli, and the storage of thyroglobulin in the follicular lumen. In order to study the influence of kinetic parameters (chemical reactions, diffusion coefficient) and, of the structure (follicular radius and reactional space width) we have determined the impulsional response of a diffusion sphere exchanging matter with a compartment where the chemical reactions are taking place. This study gives the starting point of a mathematical model of hormonal secretion by a thyroid follicle. Moreover it suggests a simple way of estimating the thyroglobulin diffusion coefficient by the mean transit time determination. Finally, we discuss respectively the validity limits of a compartmental description of the model, and of a continuous description by an infinite sum of exponentials of a system where chemical reactions interfere with a storage process by diffusion.
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    Bulletin of mathematical biology 37 (1975), S. 389-405 
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    Notes: Abstract In order to represent the biological evolution of a predator-prey ecology it is necessary to add to the equations of population dynamics terms corresponding to spontaneous mutation. Using a Volterra-Lotka ecology as an example, a model is developed for this. It is based on the assumption of two levels of description; a local one containing mutation probabilities, and the other the macroscopic average equations for the whole system. Diffusion processes link the two. The “evolutionary state” of a species is interpreted as an average effectiveness in terms of a genetic parameter space and it is shown that as a result of random mutations the ecosystem drifts irreversibly through this space.
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    Bulletin of mathematical biology 37 (1975), S. 407-417 
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    Notes: Abstract A number of recent experiments have revealed the existence of mutants with different free run periods in their circadian rhythms. Parameter variations in mathematical models can be used to simulate such changes. In addition, phase response curves (PRC) are derived and the effect of parameter variation in their shape is studied. It is shown that changes in global parameters can also distort their shape. Therefore one cannot conclude that genetic experiments provide evidence in favor of “chronon” models since “kinetic” models can also simulate their outcome.
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    Bulletin of mathematical biology 37 (1975), S. 51-57 
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    Notes: Abstract A methematical description of a coupling between a chemical reaction, the diffusion through a cellular membrane and a fluid flow is presented. This coupling may occur at the membrane levels of the cells bathed by fluid flows (e.g. endothelial cells). By such a coupling, the fluid flow and the diffusion can act as drivers of some intracellular endergonic reactions.
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    Bulletin of mathematical biology 37 (1975), S. 91-95 
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    Notes: Abstract The classical epidemic equations of Kermack and McKendrick are cast into dimensionless form. This allows discussion of the assumptions underlying the standard approximate solutions.
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    Bulletin of mathematical biology 37 (1975), S. 471-488 
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    Notes: Abstract A study is made of blood flow by assuming that the blood constitutes a suspension of cells in plasma instead of a simple homogeneous fluid. A macroscopic theory governing the motion of plasma in a plasma-cell system is derived from the local volume averaging method for a system without mass transfer between the phases, and its characteristic length is much larger than the size of the cells. The equations governing the motion of the local averaged fluid quantities include one additional term in the equation of motion and two additional terms in the energy equation. These terms represent, respectively, the force exerted upon the fluid by the particles, and the rate of heat transfer and work done upon the fluid by the particles. The theory is applied to obtain the effective viscosity as the explicit function of the volume concentration of the cells by assuming that the cells behave like rigid spherical particles with slip-collision, and the plasma is an incompressible Newtonian fluid. Comparison with existing experimental results shows a good agreement. The theory is also used to obtain the effects of cell distribution upon the overall effective viscosity in a circular tube. The quantitative result shows that there is a decrease in overall effective viscosity as the concentration of cells increases toward the center of the tube, and the overall effective viscosity is smaller than the flow with evenly distributed cells.
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    Bulletin of mathematical biology 37 (1975), S. 505-519 
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    Notes: Abstract The distribution of the two-compartment, reversible system with time-dependent transitions is proposed and verified. Inasmuch as the required probabilities cannot, in general, be expressed in closed form, a method of approximating these probabilities is described. An example with specific inverse functions of time is presented.
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    Bulletin of mathematical biology 37 (1975), S. 113-126 
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    Notes: Abstract The airway system of the lung from the mouth to the pulmonary membrane is modelled by matching a cylindrical model of a pathway through the respiratory region of the lung onto a one-dimensional trumpet model for the conducting airways. The concentration of O2 in gas expired from this model airway system is investigated following an inspiration of air at two different flow rates (10 litres/min and 85 litres/min). In each case, expiration occurs at the same constant flow rate as that during the previous inspiration. The inspirations, which are studied in an earlier paper, are each of 2 sec duration and begin at a lung volume of 2300 ml and a lung oxygen tension of 98 mm Hg. The equations are solved numerically and plots of expired O2 concentration against time and against expired volume are shown. It is found that at 85 litres/min, gas mixing in the lung is complete after about 0.7 sec of expiration whereas at 10 litres/min, about 2.6 sec of expiration is required for complete equilibration. It is suggested that the experimental alveolar plateau slope is not in general caused by a slow approach to equilibrium of gas concentrations; except at very low flow rates in the early part of the concentration/time plateau.
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    Bulletin of mathematical biology 37 (1975), S. 181-192 
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    Notes: Abstract On the basis of an abstract, simple bistable reaction system (‘homogeneous Eccles-Jordan trigger’) used as anRS flip-flop, an abstract homogeneousastable flip-flop is devised. It can be run also as amonostable flip-flop and as aT flip-flop. The qualitative behavior of the three systems can be understood, in the limiting case, with the aid of Poincaré's notion of bifurcation of steady states. The reaction system is proposed as a paradigm for a specific class of ‘decomposable’ chemical and dynamical systems (so-called DC-type dynamical automata). Two possible biological applications are mentioned.
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    Bulletin of mathematical biology 37 (1975), S. 193-213 
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    Notes: Abstract The stochastic theory of a nonlinear game is presented which incorporates some of the essential properties of living systems: metabolism, reproduction and mutability. The steady state distribution function as well as the complete time development are given explicitly. The second law of thermodynamics is generalized to a certain class of nonequilibrium systems. An order parameter is introduced as a measure of the system's internal organization. From the point of view of phase transition theory, the model exhibits a transition at the absolute zero of temperature, with critical behaviour showing up in the low temperature region.
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    Bulletin of mathematical biology 37 (1975), S. 675-689 
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    Notes: Abstract A theory for environmental systems is defined on the basis of two elements, termed ‘environmental unity’ and ‘behavior’. Environmental systems are regarded as non-living systems, each one related with only one biological system. We construct a material-energetic environmental diagram, which is introduced in terms of the theory of categories, thereby giving rise to a new categoryE. By means of two biological conditions, and the definition of static property of the biological system (related to its own environment), a set of theorems is obtained, exhibiting mathematical consequences for the represented theory.
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    The Geneva risk and insurance review 4 (1977), S. 3-3 
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    The Geneva risk and insurance review 4 (1977), S. 16-40 
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    The Geneva risk and insurance review 4 (1977), S. 5-15 
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    The Geneva risk and insurance review 4 (1977), S. 41-55 
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    The Geneva risk and insurance review 4 (1977), S. 56-57 
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    The Geneva risk and insurance review 4 (1977), S. 58-63 
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    The Geneva risk and insurance review 4 (1977), S. 64-64 
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    The Geneva risk and insurance review 5 (1977), S. 3-8 
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    The Geneva risk and insurance review 5 (1977), S. 22-34 
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    The Geneva risk and insurance review 5 (1977), S. 9-21 
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    Notes: Zusammenfassung Insgesamt liegen für eine güterliche Theorie des Wirtschaftens im Versicherungsunternehmen bereits heute viele Ansätze vor, insbesondere soweit es den Risikoausgleichsprozeß betrifft. Dagegen stecken die produktionstheoretischen Ansätze für die übrigen Input-Output-Beziehungen noch in den Kinderschuhen. Dies entpricht dem allgemein zu beobachtenden Rückstand in der betriebswirtschaftlichen Erforschung von Dienstleistungsunternehmen.
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    The Geneva risk and insurance review 5 (1977), S. 35-42 
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    The Geneva risk and insurance review 6 (1977), S. 3-16 
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    The Geneva risk and insurance review 6 (1977), S. 17-35 
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    The Geneva risk and insurance review 6 (1977), S. 36-52 
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    The Geneva risk and insurance review 6 (1977), S. 76-89 
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    The Geneva risk and insurance review 6 (1977), S. 53-75 
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    Notes: Concluding remarks In order to learn from the relations between business cycle and single branches of insurance, we should not only analyse the kind of their cyclical dependences but also regard the quantitative effects. But already, the submitted investigation allows the conclusion, that the oscillations of the claims not always are random but to a large degree cyclical. p ]Whether it will be possible to use leading business indicators for the forecast of premium income can not yet be proofed for sure. Nevertheless, the present, study suggests the assumption that further investigations in this direction may lead to success.
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    Bulletin of mathematical biology 39 (1977), S. 339-347 
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    Notes: Abstract As is well-known from the classical applications in the electrical and mechanical sciences, energy is a suitable Liapunov function: thus, by analogy, all energy functions proposed in ecology are potential Liapunov functions. In this paper, a generalized Lotka-Volterra model is considered and the stability properties of its non-trivial equilibrium are studied by means of an energy function first proposed by Volterra in the context of conservative ecosystems. The advantage of this Liapunov function with respect to the one that can be induced through linearization is also illustrated.
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    Bulletin of mathematical biology 39 (1977), S. 359-372 
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    Notes: Abstract A commonly accepted mathematical model for the slow wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected relaxation oscillators. The method of harmonic balance is used here to obtain analytical results for the entrained frequencies and amplitudes of two oscillators coupled with a parallelRLC network. By perturbations and linearisation about these values the conditions for stable limit-cycles are found and regions in theRLC parameter space which give one or two stable limit-cycle conditions are derived. These analytical results are compared with simulated results and found to creelate well for a waveshape factor of ε=0.1 and fairly well for ε=1.0. The single limit-cycle region corresponds to the requirement for a single mode having a frequency higher than the uncoupled value in small-intestinal data, while the double limit-cycle region corresponds to the two rhythms found in human large-intestinal activity.
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    Bulletin of mathematical biology 39 (1977), S. 391-396 
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    Notes: Abstract Discrete particle simulation techniques developed for problems in plasma physics have been adapted to investigate one-dimensional dissipative structures. The results of the model are found to be consistent with bifurcation analysis of nonlinear reaction-diffusion equations. Two distinct type of behavior are observed corresponding to a localized steady-state solution and a time-dependent solution.
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    Bulletin of mathematical biology 39 (1977), S. 587-595 
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    Notes: Abstract Isobaric inert gas counterdiffusion has been demonstrated to produce gas lesions in man (Lambertsen and Idicula, 1975) and lethal gas embolism in animals (Lambertsen, Cunnington and Cowley, 1975). Equations have been derived for the stable-state supersaturation pressures developing at interfaces during inert gas counterdiffusion (Graveset al., 1973). The present analysis is a mathematical treatment of the kinetics of the isobaric counterdiffusion of a pair of gases through a membrane consisting of two layers composed of substances with different diffusion coefficients and solubilities for each of the gases involved. The time to reach the stable supersaturation state due to isobaric counterdiffusion, even when circulatory transport and pulmonary washout times are included, is found to be at least an order of magnitude smaller than the time required for visible bubble formation and tissue distortion.
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    Bulletin of mathematical biology 39 (1977), S. 607-612 
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    Notes: Abstract Suppose that a collimated beam of light traveling along the axis of the eye deposits energy, and that the rate at which energy is deposited by the beam is known at every point of the eye. The thermal response in the chrioretinal region of the eye is predicted by convolving this rate of deposition per unit volume with the fundamental solution of the associated heat operator.
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    Bulletin of mathematical biology 39 (1977), S. 619-622 
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    Notes: Abstract Exact solutions for the two-species Volterra prey-predator equations and for the Volterra equations for competition are obtained in the special case that the rate of increase of both species in the absence of interaction is the same.
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    Bulletin of mathematical biology 39 (1977), S. 623-627 
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    Bulletin of mathematical biology 39 (1977), S. 629-632 
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    Bulletin of mathematical biology 39 (1977), S. 633-633 
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    Bulletin of mathematical biology 39 (1977), S. 635-636 
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    Bulletin of mathematical biology 39 (1977), S. 637-641 
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    European journal of wildlife research 23 (1977), S. 219-219 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    European journal of wildlife research 21 (1975), S. 15-34 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    European journal of wildlife research 21 (1975), S. 81-81 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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    European journal of wildlife research 21 (1975), S. 82-82 
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    European journal of wildlife research 21 (1975), S. 139-144 
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    European journal of wildlife research 21 (1975), S. 192-194 
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    European journal of wildlife research 21 (1975), S. 194-194 
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    European journal of wildlife research 21 (1975), S. 182-190 
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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
    Description / Table of Contents: Summary The results are presented in a survey of peeling by mouflon in spruce stands in the former mouflon research district Padberg, county of Hochsauerland, of the Research Station for Hunting Lore and Wildlife Damage Prevention. Observations were made from 1958 to 1970. Peeling in winter reached a peak in the extreme winter of 1962/63 when 8% of the trees were peeled. Summer peeling was considerably less, usually about 10% of the total peeling. Based on total peeling, root peeling was about 15% to 25% in winter, while the value for the dry summer of 1959 was 60%, based on the summer peeling per trunk. Mouflon peeled spruce stands over 30 years old more than those under 30. Further, little peeling of beech by Mouflon was proven. Of the tested anti-peeling preparations, chemical ones were effective on beech and spruce and mechanical protection in the form of a dry band for spruce. The protection was effective over a period of five or more years. The mechanical-biological protection, a sort of smoothing, reduced peeling up to 80% and more, compared to untreated trees, as far as the experiments are evaluated in which unprotected trees were peeled 3% and more. One stand of spruce, 18 years old, was badly damaged by the Flammigersche protective method; 12 trees died. The smoothing method caused only minor bark damage, which can be tolerated since this method is quite inexpensive.
    Abstract: Résumé Il est rendu compte de relevés d'écorcements causés par le Mouflon dans l'ancien territoire de chasse expérimental (Station de Recherches Cynégétiques et de Prévention des Dégâts de Gibier) de Padberg (Haut Sauerland). La période d'observation s'étendit de 1958 à 1970. L'écorcement d'hiver atteignit sa valeur la plus élevée au cours de l'hiver exceptionnel de 1962–63 avec un taux de 8% de pieds touchés. L'écorcement d'été, par contre, ne présenta généralement pas un caractère aigu en se situant aux environs ou en-dessous de 10% de la totalité des pieds écorcés au niveau du tronc. Par rapport à l'ensemble des écorcements causés aux troncs, l'écorcement causé aux racines traçantes au cours de l'hiver s'est situé entre 15 et 25% tandis qu'au cours de l'été sec de 1959, ce type d'écorcement s'est élevé à 60% des écorcements d'été causés aux troncs. Le Mouflon écorça plus intensivement les peuplements d'Epicéa de plus de trente ans que ceux de moins de 30 ans. Quelques écorcements d'été au Hêtre furent également signalés. Parmi les répulsifs expérimentés, les produits chimiques (pour l'Epicéa et le Hêtre) et la protection mécanique au moyen d'un manchon de ramilles sèches (pour l'Epicéa) se sont avérés pleinement efficaces tout au long d'une période de cinq ans et plus. Le traitement mécanique-biologique au moyen d'un rabot réduisit l'écorcement des pieds traités par rapport aux pieds non-traités dans une proportion de 80% et plus, l'écorcement concernant 3% des pieds non-traités. Un peuplement d'Epicéa traité à l'âge de 18 ans au moyen du grattoir de Flammig fut fortement endommagé par le traitement en question et 12 arbres succombèrent. Le traitement au rabot ne causa que des dommages faibles à modérés à l'écorce; ils ne portèrent pas à conséquence en raison du coût avantageux du traitement mécanique-biologique.
    Notes: Zusammenfassung Mitgeteilt werden Auszählergebnisse für das Schälen durch Muffelwild in Fichtenbeständen in dem ehemaligen Muffelwildversuchsrevier Padberg, Hochsauerlandkreis, der Forschungsstelle für Jagdkunde und Wildschadenverhütung. Der Beobachtungszeitraum erstreckte sich von 1958 bis 1970. Die Winterschäle erreichte mit 8% geschälter Stämme in dem Extremwinter 1962/63 den Höchstwert. Die Sommerschäle trat demgegenüber meist zurück und bewegte sich um oder unter 10% der Gesamtstammschäle. Bezogen auf die Gesamtstammschäle lag die Wurzelschäle im Winter im Bereich von 15% bis 25%, der Wert für den trockenen Sommer 1959 lag bei 60% bezogen auf die Sommerschäle am Stamm. Das Muffelwild schälte über 30 Jahre alte Fichtenbestände stärker als unter 30 Jahre alte. Nachgewiesen wurden weiterhin geringe Sommerschälschäden durch Muffelwild an Buche. Von den erprobten Schälschutzmitteln zeigten die chemischen Präparate bei der Buche und Fichte und der mechanische Schälschutz in Form des Trockeneinbandes bei der Fichte über einen Zeitraum von fünf und mehr Jahren eine volle Abwehrwirkung. Der mechanisch-biologische Schälschutz, als Hobelverfahren durchgeführt, minderte die Muffelwildschäle gegenüber unbehandelt um 80% und mehr, soweit die Versuche gewertet werden, bei denen an ungeschützten Stämmen 3% und mehr Schäle auftrat. Ein im Alter von 18 Jahren mit Flammigerschen Schutzkratzer behandelter Fichtenbestand wurde durch das Schutzverfahren stark geschädigt, 12 Stämme starben ab. Das Hobelverfahren brachte nur geringe bis mäßige Rindenschäden, die zum Teil wegen der Billigkeit des mechanisch-biologischen Schutzes in Kauf genommen werden könnten.
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    Bulletin of mathematical biology 39 (1977), S. 1-22 
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    Topics: Biology , Mathematics
    Notes: Abstract A theory which is based on a set of assumptions different from those of the sodium theory is developed. Here the mobile ions are considered to be distributed at Donnan equilibrium and the axoplasm is regarded as an analog of a cation-exchanger. Following the spirit of the Debye-Hückel theory, some important features of the ionic distribution and electrical potential of the nerve fiber have been calculated. The results appear to be in better agreement with the experimental observations than the Goldman-Hodgkin-Katz equation.
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    Bulletin of mathematical biology 39 (1977), S. 43-58 
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    Notes: Abstract A negative resistance piece-wise linear model, and one which is the sum of two sine terms are used to solve the nerve axon problem for leading edge waveshape, pulse velocity, maximum rate of rise, and rise time for the “Hodgkin-Huxley axon.” The results are compared analytically and numerically to experiment and the calculated results of Hodgkin and Huxley, a fuse model, and a cubic solution.
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    Bulletin of mathematical biology 39 (1977), S. 23-42 
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    Notes: Abstract Mathematical models of the phenomena of the sorting out of embryonic cells are derived which predict multiple clustering of like cell types into many small islands. The experiments of Trinkaus and Lentz, and other essentially two-dimensional cell sorting results, provide examples. Two continuous models are introduced, both based on the Smoluchowski theory of Brownian motion. A mass-interaction model predicts local clumping of the internally segregating cell type within a continuum of the other type. A diffusion model simulates the suspension-aggregate experiments of Roth and Weston. A discrete model of sorting out, the exchange model, is re-derived in a clearer manner than before, and certain recent criticisms are responded to. Results in recent two-dimensional cell sorting experiments are discussed and a stability criterion for patterns of multiple clusters is introduced. The site frequency model is re-examined and criticized in terms of its application to the Roth-Weston experiments.
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    Bulletin of mathematical biology 39 (1977), S. 157-166 
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    Notes: Abstract Ifconstancy is a measure of an ecosystem's (in) variability through time andstability is a measure of the system's ability to damp and recover from environmental perturbations, then constancy depends not only on stability but also on the frequency and amplitude of perturbations—theenvironmental “noise level”. The stability of an ecosystem reflects its texture, extent, and viscosity (fine-scale structure); the noise level experienced by the system (“effective” noise level) reflects the level at any point (“ambient” noise level), the spectrum of stochastic scale (regional distribution of stochasticity), and the system's spatial extent (size, or number of patches included). The coefficient of variation of a limiting stochastic variate is a measure of the effective noise level. Ifp is the total number of patches in the system (its extent) andn is the number of contiguous patches with noise signals correlated through time (its stochastic scale), then the coefficient of variation is directly proportional to $$\sqrt {(n/p)} $$ whenevern〈p. Thus ecosystems of small stochastic scalen or large sizep damp out environmental noise by “spreading the risk” in space, thereby reducing their variability in time.
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    Bulletin of mathematical biology 39 (1977), S. 385-390 
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    Notes: Abstract We report here an analysis of pumping of blood by means of a non-invasive circulatory-assist device using the principle of Magnetohydrodynamics. This study shows that such a pump would require the application of a slowly moving axial magnetic field of strength of about 108 amp turns/m (approximately 106 oersteds). The results indicate that the temperature rise on account of induced currents is within permissible range.
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    Bulletin of mathematical biology 39 (1977), S. 373-383 
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    Notes: Abstract Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.
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    Bulletin of mathematical biology 39 (1977), S. 397-406 
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    Notes: Abstract A newM γ categorical representation of biological systems is developed which is an extension of previously reported work (Leguizamón, 1975a, b; 1976). In this representation, each material physical nature going to a component or produced by it, is given by a set whose cardinality corresponds to the quantity of the material physical nature. Biological and environmental diagrams, respectively represented byM γ andE categories, are used in order to obtain a theory of transfers between both systems. From the mathematical representation of this theory, the transfers are classified as: material-energetic, exclusively energetic and theoretical material-theoretical energetic. These concepts are incorporated into a diagram, and later, a set of axioms is given, in order to have the previous developments as a new categoryT.
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    Bulletin of mathematical biology 39 (1977), S. 415-433 
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    Notes: Abstract A mathematical model of the dynamic (periodic) heat exchange from the respiratory tract of a chicken is postulated and solved analytically. The model expresses the periodic respiratory heat loss as a function of respiration rate, respiratory air velocity, ambient temperature and humidity ratio, and body (trachea) temperature. It is unique in that previous models have been formulated for steady state heat transfer. The processes of sensible and latent heat exchange are considered as uncoupled processes.
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