ALBERT

Notes: Abstract The partial differential equation of the random walk problem with persistence of direction and external bias is derived. By persistence of direction or internal bias we mean that the probability a particle will travel in a given direction need not be the same for all directions, but depends solely upon the particle's previous direction of motion. The external bias arises from an anisotropy of the medium or an external force on the particle. The problem is treated by considering that the net displacement of a particle arises from two factors, namely, that neither the probability of the particle traveling in any direction after turning nor the distance the particle travels in a given direction need be the same for all directions. A modified Fokker-Planck equation is first obtained using the assumptions that the particles have a distribution of travel times and speeds and that the average time of travel between turns need not be zero. The fional equation incopporating the assumption of a persistence of direction and an external bias is then derived. Applications to the study of diffusion and to long-chain polymers are then made.

Notes: Abstract The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.

Notes: Abstract A situation is considered in which a fluid containing a substance flows through a vessel at a constant rate, the substance being permeable to the vessel wall. In the region outside the vessel there is supposed to be rapid mixing in the direction perpendicular to the axis of the vessel but no mixing longitudinally. The solution for the spatial distribution at any time is given for the case of an arbitrary initial distribution along the vessel length in the absence of an input. The solution is also given for the case of a single impulsive input, the concentration being initially zero everywhere.

Notes: Abstract Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed.

Notes: Abstract Certain parameters are defined which roughly characterize the internal structure of networks. A given network structure uniquely determines the values of the parameters, but the reverse is not true. The parameters therefore define certain classes of networks. One of the parameters, thedispersion D(S) gives an indication of the “compactness” of the internal structure. Addition theorems and inequalities are derived relating the dispersions of sub-systems to the dispersion of the complete structure.

Notes: Abstract A mathematical theory is developed which permits the determination of certain parameters of an inhomogenous tissue, such as a nerve trunk without its epineurium. The parameters are the permeability coefficients for entrance into an exit of a substance from the nerve fibers, and the diffusion coefficient of the interstitial material. The experimental data required are the dimensions of the cross-section, the average diameter of the fibers, and the ratio of the cross-sectional are of the fibers to the total cross-section, as well as the time course of the decrease of the fraction of the substance left in the nerve trunk, when the trunk is immersed in a bathing solution containing none of it.

Notes: Abstract A model based on enzyme localization is developed which gives rise to an apparent active transport of a metabolite into or out of cells. The model is applied to three simple situations, using Fick's equation and the Rashevsky approximation. It is shown that the apparent efficiency can be made as large as desired if, for constant reaction, the outer cell region is made sufficiently small, or, for autocatalytic reaction, if the metabolite concentration in the outer region is sufficiently small. The physical limitations imposed by this mechanism are developed for all three situations.

Notes: Abstract A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.

Notes: Abstract The assumption of transitivity treated in part I is modified in various ways to describe an information-diffusion process, in which a certain amount of randomness of contact does occur. In one model a parameter is introduced which is indicative of a tendency to go beyond one's immediate vicinity to spread the information as the vicinity becomes saturated with knowers. In another model the randomness appears in the assumption that new knowers are uniformly distributed among the knowers. Two of the equations thus derived, each with two free parameters are in good agreement with experimental results.

Notes: Abstract A somewhat different approach to the principle of biotopological mapping, discussed in previous publications, is given. The organism is considered as a set of properties, each of which is in its turn a set of numerous subproperties which are logically included in the corresponding properties. Topology is introduced by an appropriate definition of neighborhoods, and four postulates are stated which concern the mapping of the spaces corresponding to higher organisms on those of lower ones. A number of conclusions are drawn from the postulates. Some of them correspond to well-known facts. For example, in man and some higher organisms appropriate emotional stimuli should produce gastrointestinal or cardiovascular disturbances; or some microorganisms should produce substances harmful to other microorganisms (antibiotics). Some other conclusions are still awaiting verification. One of them is, for example, that there must exist unicellular organisms which produce antibodies to appropriate antigens.

Notes: Zusammenfassung Für die Praxis der Pflanzenernährung ist es wichtig, zu wissen, in welcher Weise die Ertragsbildung von der Konzentration eines mineralischen Nährstoffes in der Umgebung der Pflanze abhängt. Da nur diejenigen Nährstoffmengen das physiologische Geschehen in der Pflanze unmittelbar zu beeinflussen vermögen, die sich in der Pflanze befinden, wird angenommen, dass das Wachstum zum Zeitpunktt, d.h. die Geschwindigkeit der Trockensubstanzzunahme zu diesem Zeitpunkt, eine Funktion der zur Zeitt in der Pflanze enthaltenen Nährstoffmenge ist. Diese Nährstoffmenge wird natürlich im Intervall vor dem Zeitpunktt aufgenommen. Deshalb und auch noch aus anderen Gründen hängt das Wachstum zur Zeitt davon ab, wie die in der Umgebung der Pflanze herrschende Konzentration des betrachteten Nährstoffes in demjenigen Zeitintervall verläuft, das sich von der Aussaat bis zum Zeitpunktt erstreckt. Die angegebene Annahme fürhrt zusammen mit einigen weiteren naheliegenden Annahmen zu einem Ansatz, der Ergebnisse liefert, die in verschiedener Hinsicht gut mit der Erfahrung übereinstimmen. Jedoch gibt es auch noch Widersprüche zwischen Theorie und Erfahrung. Durch weitere Ausgestaltung der Theorie lassen sich diese Widersprüche beseitigen. Es wird angeregt, Versuche durchzuführen, deren Resultate Hinweise für die weitere Ausgestaltung der Theorie liefern.

Notes: Abstract The dynamics of cell multiplication and differentiation in tissues in asteady state and the kinetics of isotope incorporation into the DNA have been theoretically analyzed. Equations have been derived, with the aid of which thegeneration time, thelife span, and the distribution or rate of death of the cells can be obtained if the tissue is in asteady state, i.e., if the number of cells is maintained constant by constant, equal rates of cell division and cell death and if the mean DNA content per cell is also constant. An equation has also been derived which gives thegeneration time in the case of logarithmic multiplication of cells. Two special cases have been analyzed: InCase 1, the isotope is considered as being introduced into the metabolic system at zero time only; inCase 2, the specific activity of the DNA precursor is considered as being maintained constant. The use of the method has been illustrated by an example in which thegeneration time and themean, themedian, and themode life span, as well as the curve of the rate of death of leukocytes in a patient with chronic leukemic granulocytic leukemia, have been obtained from the rate of P32 incorporation into the DNA. The merits and the limitations of the method are discussed.

Notes: Abstract By means of the Laplace transform, the behavior of a simplified model of the cardiovascular system is mathematically formulated. This formulation allows mathematical expression of the periodicity of the cardiac output and the systemic response. With the cardiac output represented as half of a sine function cycle, the systolic aortic pressure becomes the sum of a sine term and exponential terms, while the sum of the exponential terms alone represents the diastolic pressure. The characteristics of the mathematical expressions for systole and diastole are analyzed, and some relationships of potentially practical value are derived. Variation in the parameters of the system yields mathematical results consistent with the expected physical ones.

Notes: Abstract A generalization of Landahl's approximation method (H. D. Landahl,Bull. Math. Biophysics,15, 49–61, 1953) for non-linear diffusion problems is suggested. The method is applied to sorption, desorption, and free diffusion problems involving concentration-dependent diffusion coefficients. With some limitations, the results compare favorably with those obtained by numerical methods.

Notes: Abstract Recently a theorem for representing current generators in a volume conductor by the superposition of a central dipole, quadrupole, octopole, etc., has been established by G. C. K. Yeh, J. Martinek, and H. de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958). This theorem makes possible the representation of any discrete or line, surface- or volume-distributed current source by a unique model which can be determined for each given case by surface potential measurements and closed form analysis. In this paper the multipole representations of an eccentric dipole and an eccentric double-layer are obtained in terms of the various parameters of the assumed singularities, and the contributions to surface potentials due to each of the multipoles are compared. Certain numerical results corresponding to those of E. Frank (Amer. Heart J.,46, 364–78, 1953) are carried out and compared. Furthermore, the multipole representation of a partially damaged double-layer is also determined and compared with that of an undamaged one. It is concluded that within the range of parameters corresponding to human subjects the higher-order multipoles can contribute significantly to the surface potentials compared with the dipole.

Notes: Abstract In line with a recent suggestion by the author (Bull. Math. Biophysics,20, 267–73, September, 1958) that not only does the organism as a whole map on the primordial, but that each organ can also be thus mapped, it is shown that the previously introduced abstract spaces, which represent an organism, contain subspaces which map continuously on the space of the primordial. Several theorems about those subspaces are proven.

Notes: Abstract The DNA-protein coding problem is given a general algebraic formulation, the consequences of which are then explored by standard mathematical methods. To keep the treatment self-contained, the mathematical techniques to be used are explained in detail. It is demonstrated that there exista priori a countably infinite number of different abstract DNA-protein codes, thereby showing that inductive attempts to construct such a code will most likely be fruitless. A notion of ergodicity is then introduced, which imposes a number of restrictions on the admissible codes, and, in fact, these considerations enable us toderive a small portion of a code which, if our hypothesis of ergodicity is correct, must occur in nature. Finally, we discuss briefly the problem as to whether there can exist more than one DNA-protein code in nature.

Notes: Abstract The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart.

Notes: Abstract In a preceding paper (Bull. Math. Biophysics 20, 71–93, 1958) the principle of biotopological mapping was formulated in terms of a continuous mapping of an abstract space, made from the set of biological properties which characterize the organism, by an appropriate definition of neighborhoods. In this paper it is shown that we may consider directly the mappings of the different sets of properties which characterize different organisms without taking recourse to abstract spaces. All the verificable conclusions made in the preceding paper remain valid. A serious difficulty mentioned previously is, however, avoided and the possibility of more general predictions is established.

Notes: Abstract The general Theory of Categories is applied to the study of the (M, R)-systems previously defined. A set of axioms is provided which characterize “abstract (M, R)-systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for anad hoc postulation of a replication mechanism. A time-lag structure is introduced into these abstract (M, R)-systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)-system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.

Notes: Abstract The transient stage of the random dispersal of logistic populations is investigated, using a Sturm-Liouville series leading to an infinite system of non-linear integral equations. These equations are then solved via a successive approximation scheme. R. A. Fisher's (steady-state) velocity of advance paradox is discussed. An illustrative example is worked to the second order of approximation.

Notes: Abstract An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.

Notes: Abstract Diffusion through a flat pore into a large open region is proportional to the linear dimension of the pore and not to its area. This was first explained by Brown and Escombe (1900) for a circular pore and is here generalized, by means of a dimensional argument, to include any type of regular opening. The problem is further generalized to include diffusion through pores of finite thickness, finite distance apart, and into finite regions. Since this problem cannot be solved exactly, an approximation method is introduced. Reasons for the credibility of the approximation are presented. It is then shown, by means of the approximation method, that the diffusive flow through a pore is equal to the total concentration difference divided by the resistance of the system. The resistance, in turn, is the sum of the resistances of all portions of the system, each of which is calculated. The result is compared with results which have been calculated exactly for limiting cases and found to agree very well. The results are then applied to a standard method of computing pore size in membranes, and it is shown that the correction factor is negligible.

Notes: Abstract V. S. Ivlev [Experimental Ecology of Nutrition of Fishes, 1955, Moscow (in Russia)] has shown that the food uptake by fishes during a fixed interval of time is an exponential function of the concentration of food. Ivlev's equation is derived here, and it is shown that it can hold only for non-stationary conditions, such as prevailed in Ivlev's experiments. For a stationary state, the rate of food uptake should tend asymptotically to a limiting value as the concentration increases, but the variation is not exponential. Different other aspects of the problem are investigated, and definite new experimental procedures suggested. The implications of Ivlev's findings on the effect of non-uniformity of food distribution upon the rate of food consumption are studied from a mathematical point of view. The conclusion is reached that whereas a fish does not, in the process of eating, move directly to an individual food particle which it perceives, it does move more or less directly to large aggregates of particles, if the latter are distributed nonuniformly.

Notes: Abstract Some relational aspects of the property of self-reproduction of biological systems are studied. If in addition to the requirement of the property of self-reproduction we add also the requirement of adaptability of the organism to changing environment, this imposes certain conditions on the topology of the graphs which represent such systems. A further study of the relational properties of such systems seems to offer the possibility of deriving the principle of biological mapping from the requirement of self-reproduction and adaptability. An examination of the problem of the original formation of such self-reproducing systems in connection with the established fact of impossibility of spontaneous generation leads to the conclusion that an organism must inhibit such processes which, in the absence of organisms, would lead to spontaneous generation.

Notes: Abstract In the human, the antagonistic, extensor-flexor system of the leg is an example of a common type of neurophysiological feedback system. After a brief introduction to the neuroanatomy and physiology of this feedback system, the paper formulates transfer functions from temporal response data available in the literature. A feedback stability analysis, based on the extension of Nyquist's stability criteria to multiple-loop systems and utilizing flow-graph techniques, demonstrates the stable behavior of the system. Expressions are given relating the sensitivity of the system to variations in muscle response and Golgi tendon organ (tension receptor) response. By considering the events taking place at synapses and end-plates during “isometric tension-small knee angle excursion” conditions as stationary stochastic processes, an external “noise” input to the system is given, whose spectrum is derived from the statistics of a shot-process representation of these events. The paper concludes with some correlations between the analytical results and clinical syndromes.

Notes: Abstract In this continuation of a previous report it is shown how the Volterra population dynamics, which underlies the statistical theory, can be based on a variational principle; how the dynamics can be generalized as regards both the behavior of total populations and migration phenomena; and how many directly observable data, such as amplitudes and frequencies of oscillation of a population, fit into the statistical theory and can test it. Such a test is carried out in some detail using the fox-catch data of Elton, with a clear indication that the theory is capable of comprehending the major statistical properties of population-time curves. A final section sketches an extension of the theory to cover secular variations of external conditions such as temperature of the environment.

Notes: Abstract It is observed that a dynamical continuity equation for biomass distribution yields the asymptotic steady-state exponential dependencen=A exp( $$ - m/\bar m$$ ) exhibited by certain fishery data, wherem is the biomass of an individual,n is the number of individuals per unit biomass interval, andA, $$\bar m$$ are positive constants. This dynamical approach to biomass distribution is an alternative to the global maximization principle proposed recently by Lurié and Wagensberg.

Notes: Abstract Biological tree-like structures, such as mammalian tracheobronchial airways, are complicated branching systems. One problem in modeling such systems is the reassignment of the number of segments at a given generation in the model being constructed. A hypothesis is proposed which has successfully been used in modeling mammalian lung airways.

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).

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.

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.

Notes: Abstract By studying the behavior of various tracer species in the lungs, one can assess many important characteristics which distinguish normal and abnormal function. Quantitative evaluation of function depends on the use of an appropriate model in conjunction with experimental data. A multi-compartment model is derived from mass balances to describe dynamic as well as (breath-averaged) steady-state transport processes between the environment and pulmonary capillary blood. The breathing cycle is divided into three time periods (inspiration, expiration, and pause) so that the model equations are discrete in time. No other model of tracer species transport in the lungs deals simultaneously with species dynamics, variable breathing pattern, distribution inhomogeneities, and non-equilibrium between alveolar gas and capillary blood. Models currently in the literature are shown to be special cases of the model presented here.

Notes: Abstract Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.

Notes: Abstract The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.

Notes: Abstract It was hypothesized in an earlier work that sensory perception can occur only when the perceiving system is uncertain about the nature of the event being perceived. In the absence of any uncertainty, perception will not take place. The response of the sensory afferent neuron (impulse transmission rate) was calculated using Shannon's measure of uncertainty or entropy. It will now be shown that when the event being perceived is the position and momentum of a particle, Shannon's measure of uncertainty leads to the Heisenberg Uncertainty relationship.

Notes: Abstract It is not unusual for several classifications to be given for the same collection of objects. We present a method, called majority rule, which can be used to define a consensus of these classifications. We also discuss some mathematical properties of this consensus tree.

Notes: Abstract The dependence of the spatial concentration profiles of morphogens on a characteristic dimension is obtained by continuation techniques for Gierer and Meinhardt's activator-inhibitor model of morphogenesis. The study of the behaviour of the system during growth, where the linear and exponential increase of the characteristic dimension is considered, revealed that more complex patterns of morphogen spatial concentrations appear regularly in a reproducible way.

Notes: Abstract Computer models have been used by various authors to simulate both the growth of normal cellular tissue and the development of cancerous cells within normal tissue. As these models were the result of considerable idealization, the purpose of the present paper is to propose a model in which the degree of simplification is relaxed: the features of simultaneous growth, and cell growth whose rate depends on the free absorbing periphery of the cell are introduced. Simulation experiments have been conducted using the model, and the results are presented.

Notes: Abstract The theory of complementary variational principles is used to obtain maximum and minimum principles for a nonlinear model of heat conduction in the human head. Accurate variational solutions are obtained in illustrative calculations. The effect of nonlinearity is seen to be significant from a comparison with the linearized model.

Notes: Abstract A model for the nerve impulse due to Zeeman (1972) and based on catastrophe theory is compared with alternative models and criticisms of Zeeman's model by Sussmann and Zahler (1977, 1978) are assessed. The criticisms of Zeeman's motivation for his model are found to carry some weight. Sussmann and Zahler (1977, 1978) list numerous features of Zeeman's model which, they state, are not in agreement with experiment. These statements as they stand are largely erroneous, and the model still remains to be tested by a critical series of experiments. However, a detailed analysis reveals defects in Zeeman's model, not among those claimed by Sussmann and Zahler, showing that the explicit equations of the model cannot be correct. The possibility of a modified approach along similar lines and its ultimate adoption remains open.

Notes: Abstract The irreversible Michaelis-Menten reaction is studied by the use of the method of multiple scales. Three stages of the reaction are identified, one of which is studied in detail. The results are compared with those of two earlier analyses.

Notes: Abstract A numerical study of the coupled nerve fibre problem is given which verifies and extends the perturbation theory of Luzader. Pulses on adjacent fibres can couple together with two possible stable pulse separations.

Notes: Abstract A possible mechanism for effects of microwave radiation on the auditory system is the generation of field-induced forces at interfaces that divide materials of dissimilar electrical properties. A general expression for these “Maxwell stresses” is derived and then used to calculate the approximate magnitude of field-induced force within the organ of Corti during microwave exposure. Comparison of the results with data on the force needed to excite cochlear hair cells indicates auditory responses could be evoked by this mechanism at power densities near the threshold of rf hearing sensations.

Notes: Abstract A definition of homogeneity for neural networks is given which permits their construction as group quotients. The significance of this for neural dynamics is discussed.

Notes: Abstract The left ventricle is represented as a cylinder contracting both radially and longitudinally. A simple method is indicated to derive an expression for the rate of change of the kinetic energy of this three-dimensional model, which quantity can be used as an index for the study of the contractile behaviour of the myocardium. An application to the study of muscle mechanics is also indicated.

Notes: Abstract A perturbation method is proposed to calculate approximately the limit cycle type nonequilibrium steady-state resulting from periodic perturbation of coefficients of stable population systems; the two species Lotka-Volterra competition system is explicity studied and the results are formulated for general multi-species population systems. Avoidance of competitive or other types of exclusion of species in a periodic environment is indicated.

Notes: Abstract A theorem is proved, concerning expected values of a multitype branching process in a varying environment. The consequence of the theorem is that the branching process can be treated (in the sense of expected values) as a dynamical system with control terms. This is of importance in situations where the process serves as an abstract model of the dynamics of malignant cells for use in chemotherapy. A simple example of this kind is given.

Notes: Abstract A method is developed for a full nonlinear evaluation of all velocities and stresses represented in the Navier-Stokes equations and in the general stress tensor. The information required is essentially that for solution of linearized forms. The solution is analytical except for the calculation of the axial velocity, which requires computer assistance to step through time and space. The treatment of the problem, although directed towards solutions involving fluid flow in elastic vessels, is also adaptable to solid deformations (strain vs rate of strain) where the general stress tensor applies. A special case for the distorting ellipse is presented as well as a limited, spatially analytic, solution.

Notes: Abstract The stability characteristics and dynamical behavior of a system of mutually excitatory neurons in close spatial proximity are investigated with a mathematical model. The model predicts the existence of uniform, intermediate levels of activity other than those of no activity and maximal activity. The model also, yeilds a good explanation of data obtained from periglomerular neurons in the olfactory bulb of the cat.

Notes: Abstract A multicompartmental model in which particles enter the system from the environment and reproduce according to a Markov branching process has been considered. Explicit expressions have been obtained for the mean vector and the correlation structure for the numbers of particles in different compartments in different time points of the system. Growth rates of the mean vector and some special cases of the system are also discussed.

Notes: Abstract The study of bi-directionally coupled oscillators is relevant in biological modelling of such systems as gastro-intestinal electrical activity, cardiac pacemarkers, cardiovascular and respiratory interactions and circadian rhythms. Interconnecting pathways in biological systems often exhibit pure time-delay characteristics. In this paper the multiple-mode limit-cycle behaviour of such systems is analysed using the method of harmonic blance. It is shown that the coupling time delay radically affects the number, frequency and amplitudes of entrained limit-cycles.

Notes: Abstract The effect of keeping all the parameters constant, except the diffusion coefficients, in a pair of reaction-diffusion equations is studied. It is shown that the stability of the constant solution and the bifurcation points can be easily established by constructing a simple stability diagram. The possible qualitatively different diagrams are enumerated.

Notes: Abstract We have numerically examined more than one million Large Complex Systems (LCS) of interacting variables (interpretable as interacting populations) governed by Generalized Lotka-Volterra Equations (GLV), with self-regulation term. The scope was to have some insight on the stability-complexity relationship. We considered systems of prey-predator type, and we gave appropriate rules for constructing the model systems, rules that specify the behaviour of model systems in order to put them near the biological reality. The results show, among other things, a strict correlation between the stability and the prey-predator ratio (which, in our model, uniquely determines the connectedness of the system).

Notes: Abstract The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O2 in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.

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.

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.

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.

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.

Notes: Abstract We examine certain mathematical structures presented in Part I. The most important of these are the energy structures determined by the couple (ω×E, ψ) the space of causality defined by ψ-1(0) and the notion of collapsibility, i.e., the descent of a species from a higher to a lower equilibrium configuration as a result of energy loss.

Notes: Abstract This paper discusses a general stochastic model for a two-compartment reversible system with non-homogeneous Poisson inputs, arbitrary residence times at each of the compartments and time-dependent transition probabilities. The probability distributions of the number of particles in each compartment and in the system are obtained together with the number of particles which depart from the system. In addition, various covariance functions with a time lag are obtained. Some of the above obtained results are deduced for time-independent arrivals, exponential residence times and time-independent transition probabilities. Fluctuations of the particles present in the system are also analysed. Similar analysis is provided for the model into which some particles are initially introduced at the system. Some possible applications are discussed at the end.

Notes: Abstract The general multispecies prey-predator system with Gompertz's antisymmetric interactions is nonlinearly stable in the absence of dispersion and continues to remain stable with dispersion under both homogeneous reservoir and zero flux boundary conditions in a region containing the equilibrium state. It is proved that a general multispecies food-web model without antisymmetric interactions is stable in the absence of dispersion and remains stable with dispersion in the above-mentioned region.

Notes: Abstract We introduce a graphical approach in the study of the qualitative behavior ofm species predator-prey systems. We prove that tree graphs imply global stability for Volterra models and local stability for general models; furthermore, we derive sufficient conditions so that loop graphs imply stability and boundedness of the solutions.

Notes: Abstract A system of integro-differential equations is derived to describe epizootics of a fungal pathogen in an insect population. Because of piecewise continuous behavior under some parametric conditions, it is concluded that standard phase orbits can be misleading. Using a different analytic approach yields a simple system of finite difference equations. Both the continuous and discrete versions are compared to classical forms. The continuous version differs from a classical one in possessing a second derivative dependent on population density. The discrete version differs in maintaining positive, non-zero populations of both infectives and susceptibles in finite time.

Notes: Abstract This note is an attempt to demonstrate that hypothalamic pulsatile GnRH secretion is not the result of a short-term, negative steroid hormone feedback. Clarification of this point is of importance for further modelling the control of gonads.

Notes: Abstract A modern theory of the calculus of variations is used to form necessary and sufficient conditions for the existence of a Lagrangian representation of a system of first-order ordinary differential equations. There exists a theorem to the effect that when a system of ordinary differential equations is variationally self-adjoint, the fulfillment of such conditions is guaranteed. In addition, self-adjointness, allows establishement of an algorithm by which a Lagrangian for the system may be explicitly constructed. Examples in mathematical biology are given to illustrate the use of the stated theorem.

Notes: Abstract Engineering optimal control theory is applied to equations describing insulin and glucose interactions. The nature of the optimal controller is established. It is shown how the results can be utilized in a closed loop feedback control system.

Notes: Abstract An attempt is made to compare the conditions for the general error-optimality of linear systems developed by Kalman with the conditions for feasibility of linear models of neuromuscular and physiological control systems. Models of three actual physiological systems are tested for both the above criteria. Theoretical analysis presented here shows that there are no simple relationships between the two sets of conditions. Analysis carried out on the physiological systems models suggests the need for a general set of conditions for other optimality criteria, such as time and energy minimization, similar to Kalman's condition for error minimization.

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.

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.

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.

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.

Notes: Abstract Following arteriolar occlusion, tissue oxygen concentration decreases and anoxic tissue eventually develops. Although anoxia first appears in the region most distal to the capillary at the venous end, it eventually spreads throughout the entire region of supply. In this paper the changing oxygen concentration, from the time of occlusion until the tissue is entirely anoxic, is examined mathematically. The equations governing oxygen transport to tissue are solved by iterating a nonlinear integral equation. This solution is valid until anoxia first appears. After anoxia develops it is necessary to solve a moving boundary problem. This is done using the method of matched asymptotic expansions, and accurate solutions are obtained for a wide range of physiological conditions.

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.

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.

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.

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.

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).

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.

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.

Notes: Abstract In order to evaluate the effect of anatomic asymmetries on the gas concentration distribution in the pulmonary airways, a Monte Carlo simulation of combined bulk flow and molecular diffusion was carried out in a realistic distal airway model (Parkeret al., 1971). This airway model, composed of branches distal to the 0.5-ram diameter airways, contained an upper symmetric segment consisting of four generations of conducting airways and a lower asymmetric segment of alveolar ducts and sacs arranged in five transport paths of varying lengths. In accounting for the volume increases of these ducts and sacs occurring during normal respiration, uniform alveolar filling rates and a fixed length-to-diameter ratio of all airways were assumed. For a pulse injection of inert tracer gas, the simulation was employed to determine the longitudinal concentration profiles in the conducting airways. In the alveolated airways, not only were the longitudinal profiles determined along each path, but radial transport from the core to the periphery of the airways was considered. The results of the simulations indicate that geometric asymmetries alone contribute substantially to regional concentration variations in the distal airways. For example, when a gas bolus is injected at mid*inspiration, there are concentration differences as great as 40% between two points along different transport paths located equi-distant from the proximal end of the model. As viewed from the terminal end of the model (acinus), average concentration differences as large as 6-to-1 exist between the longest and shortest transport paths respectively for gas boli introduced near the end of inspiration. The results further indicate because of large radial diffusion rates, no significant concentration differences exist between the periphery a-ld the central core of alveolated airways. Simulation of the expired concentration profiles indicate that boll injected very late during inspiration exhibit a sloping tail, unlike the earlier injected boll whose tails are virtually horizontal. Through the use of superposition teehniqnes, it was found that these sloping tails correspond to an alveolar slope of 1.5 vol% between 750 and 1250 ml expired for a continuous washing of tracer. This result is in disagreement with other transport analyses which did not directly account for the effect of geometric asymmetries.

Notes: Abstract Assuming a spherical geometry for the left ventricle, passive elastic stiffness-stress relations have been obtained on the basis of linear elasticity theory and large deformation theory. Employing pressure-volume aata taken from rat hearts of various age groups, it is shown that young rat heart muscle (1 month) is stiffer than either adult (7 months) or old rat heart muscle (17 months). Although the qualitative results are similar for both elasticity theories, the large deformation theory gave results in closer agreement with those obtained from papillary muscle studies. These results imply that stiffness of muscleper se can be assessed from left ventricular pressure-volume data.

Notes: Abstract Deliberate evaluation of the quantum theory of nerve excitation is made by comparing it with Hill's theory in fitting the experimental data on threshold-frequency relation, optimum frequency (v0) for nerve excitation and strength-duration relation. Decrease of v0 and increase of all the time constants (Hill's λ andk, Wei'sT 2 and spike durationw) with decreasing temperature are interpreted on the basis of the dipole relaxation timeT 2 but inexplicable from Hill's theory or any other existing theory. The closeness ofk,T 2 andw values is explained. A variety of experimental results obtained by others is discussed. Finally, a comparison is made between the Hodgkin-Huxley equations and the quantum theory. Most of the facts (electrical and non-electrical) tend to support the thesis that nerve excitation is a macroscopic expression of quantum transitions of dipoles between energy states.

Notes: Abstract In the periodic Leslie model the asymptotic period of total population is a divisor of the asymptotic period of the population vector. Under reasonable circumstances these periods are identical.