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  • 1
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    Springer
    In:  EPIC3Naturwissenschaften, Springer, 71(12), pp. 599-608, ISSN: 0028-1042
    Publication Date: 2014-06-04
    Repository Name: EPIC Alfred Wegener Institut
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    Bulletin of mathematical biology 46 (1984), S. 967-969 
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    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.
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    Bulletin of mathematical biology 46 (1984), S. 971-972 
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    Bulletin of mathematical biology 46 (1984), S. 973-974 
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    Bulletin of mathematical biology 43 (1981), S. 1-19 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 47-58 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 125-140 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 141-149 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 239-244 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 259-270 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 271-278 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 341-346 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 279-325 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 375-388 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 389-400 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 401-413 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 415-426 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 447-461 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 463-485 
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    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.
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    Bulletin of mathematical biology 43 (1981), S. 513-516 
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    Bulletin of mathematical biology 46 (1984), S. 11-17 
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    Notes: Abstract Linear birth and death processes are used to derive simple expressions for sequential extinction times and gene fixation probabilities in asexual populations.
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    Bulletin of mathematical biology 46 (1984), S. 1-10 
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    Notes: Abstract We are here concerned with the functionf which assigns to each pointP of an object the numberf(P) which is the shortest distance fromP to the border. This function appears in various guises in diverse biological studies. The functionf(P) is itself a measure of shape—or more precisely, an infinite set of measures, one for each point (and hence, in view of its geometric definition, usually in a form inconvenient for use). Thus in this paper we sought a reasonable representative of this infinite set of measures, namely themean of the numbersf(P) asP ranges over all points of the entity. Computability studies are developed for various classes of shapes. For example, (1) the mean for a lamina bounded by a polygon circumscribable about a circle of radiusr isr/3; (2) the mean for a domain bounded by a polyhedron circumscribable about a sphere of radiusr isr/4. The transition from pointwise to piecewisef(P), especially in the non-convex case, requires working with inequalities.
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    Bulletin of mathematical biology 46 (1984), S. 19-40 
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    Notes: Abstract A mathematical model for traveling bands of motile and chemotactic bacteria in the presence of cell growth and death is examined. It is found that asymptotic traveling wave solutions exist in the absence of chemotaxis, due to the balance of growth, death and random motility. Thus random motility confers the ecological advantage of population propagation through migration into nutrient-rich regions. The presence of chemotaxis amplifies this advantage by moving more cells into higher nutrient concentration regions, resulting in larger and faster bands. Therefore there seem to be two types of traveling bands that can be attained by chemotactic bacteria in the presence of growth and death: (1) these growth/death/motility bands; and (2) pure chemotactic ‘Keller-Segel'-type bands. Comparison to experimental observations by Chapman in 1973 indicate that the latter seem to be formed. The relationship between these two types of solution is at present uncertain. The growth/death/motility bands may have relevance on longer time or distance scales characteristic of microbial ecological systems.
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    Bulletin of mathematical biology 46 (1984), S. 115-125 
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    Notes: Abstract Based on the principle of minimum power, a mathematical model of the pathological functional state of the oxygen transport system is presented. The model is used to determine the optimal functional parameters of the oxygen transport system in hyperthyroidism, anemia and hypertension. Theoretical results are compared with clinical data.
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    Bulletin of mathematical biology 46 (1984), S. 139-153 
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    Notes: Abstract A modified SIRS model is developed as a framework for the study of epizootiological dynamics in an insect-pathogen system. Linearized stability analysis reveals that the system with one immune and one susceptible host class can exhibit stable, periodic or unstable behavior depending on model parameters. In general, high pathogenicity, short pathogen propagule lifespan and high host reproductive rate are stabilizing influences. Pathogen transmissibility and propagule production/host do not influence local stability. The effect of seasonal host reproduction is studied because most insect hosts are seasonal in temperate climates. The basic stability dependence on model parameters holds except as modified by the length of the reproduction interval. The results of this study are compared with the recent work of Anderson and May.
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    Bulletin of mathematical biology 46 (1984), S. 175-184 
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    Notes: Abstract The random predator-prey type interactions of the population species in a random varying environment have been investigated. The Fokker-Planck equation for the transition probability, corresponding to the stochastic differential equations established from Lotka-Volterra equations by the introduction of randomness and variability, has been integrated in the form of a path integral. The transition probabilities for extinction or survival of one or several species have been approximately evaluated and investigated.
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    Bulletin of mathematical biology 46 (1984), S. 155-174 
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    Notes: Abstract If two microbial populations compete for a single resource in a homogeneous environment with time invariant inputs they cannot coexist indefinitely if the resource competed for is not renewed by biological activity within the system. Mathematical studies have shown that in a predator-prey system, where the resource (prey) is self-renewing, the two competitors (predators) can coexist in a limit cycle. This suggests that if the resource competed for is renewed by biological activity within the system coexistence can occur in any microbial system provided that it exhibits the same features as, but without being, a predator-prey one. A food chain involving commensalism, competition and amensalism is presented here. Two subcases are considered. It is only when maintenance effects are taken into account that coexistence, in limit cycles, can occur for this system. Limit cycle solutions for the system are demonstrated with the help of computer simulations. Some necessary conditions for coexistence are presented, as are some speculations regarding the possible physical explanations of the results.
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    Bulletin of mathematical biology 46 (1984), S. 127-137 
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    Notes: Abstract The branching structure of the mammalian arterial tree has been known to be close to that of an optimal conduit system of the minimum work model characterized as the branch system of constant wall shear rate. The physiological mechanism producing such construction was considered to be based on the local response of arterial caliber induced by the wall shear stress (shear rate × blood viscosity) and thereby maintaining this stress constant, which was previously observed at the canine common carotid artery shunted to the external jugular vein. The stress levels at various parts of the arterial system estimated from available data fell within ±50% of the mean (15 dyn/cm2), which was consistent with the value predicted from the model. Theoretical analyses on the cost function of the model indicated that the suspected variation of shear rate levels in the arterial tree due to the anomalous changes in blood viscosity which might bring about 3- to 4-fold differences between the minimum and maximum shear rates would cause less than 10% increase in the total energy cost. It was concluded that a local adaptive response to wall shear stress is the mechanism which effectively optimizes the design of the arterial tree.
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    Bulletin of mathematical biology 46 (1984), S. 185-185 
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    Bulletin of mathematical biology 46 (1984), S. 187-203 
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    Notes: Abstract The structure of the genetic code is related to a Gray code, which is a plausible theoretical model for an amino acid code. The proposed model implies that the most important factor in shaping the code was the effects of mistakes in translation, not effects of mutations. Another possible implication is that the preservation of stiffness and flexibility at appropriate places in a protein chain is as important in protein structure as the appropriate placement of hydrophilic (external) and hydrophobic (internal) residues. Other results are a simple conceptualization of the relationships among the 20 amino acids and their relations to their codons. The detailed relationships are summarized in the following ‘similarity alphabet’: ala, thr, gly, pro, ser; asp, asn, glu, gln, lys; his, arg, trp, tyr, phe; leu, met, ile, val, cys; (ATGPS DNEQK HRWYF LMIVC in the one-letter code). This alphabet falls into four groups of amino acids: small, external, large, internal. The approximate relation of the groups to their codons is expressed as: the first base of a codon controls size—a purine means a small amino acid, a pyrimidine means large; the middle base controls cloisterednes—purine means external, pyrimidine means internal. These relationships express the minimum change principle upon which the code appears to be founded.
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    Bulletin of mathematical biology 46 (1984), S. 269-282 
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    Notes: Abstract A theoretical study of the Brusselator model with non-uniform distribution of component A and a concentration-dependent diffusion coefficient has been performed. Numerical simulation reveals that a variable diffusion coefficient alters the bifurcation pattern and the stability properties of the steady-state as well as periodic solutions. A simple approximate method, based on one-point collocation, has been proposed to analyze the bifurcation phenomena for the case of fixed boundary conditions and low system size.
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    Bulletin of mathematical biology 46 (1984), S. 283-294 
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    Notes: Abstract In this paper the effects of the occurrence of cut trees in the topological analysis of branching patterns have been studied. It is assumed that branches are removed at random from the trees. We prove that, for both the segmental and terminal growth models, the probability distributions of the cut trees are identical to those of complete trees.
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    Bulletin of mathematical biology 46 (1984), S. 247-268 
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    Notes: Abstract The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL p[0,∞], (1〈-p〈-∞). A discussion is included of theL p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O K (relating extravascular concentration to intravascular) and K a, k (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O K ‖ p 〈-1, ‖K a, k ‖ p 〈-1), isotone, linear operators onL p . The uniform convergence relation $$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$ (as operators onL p) is derived, whereP n (a) is the Poisson probabilitye −a a n /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖Ok‖=‖Ka,k‖p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.
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    Bulletin of mathematical biology 46 (1984), S. 295-326 
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    Notes: Abstract One particular kind of structure offers possible explanations, for long-term memory, efficient consolidation of stored information from the environment, clustering of data strings and multimodal functioning. It is a possible model for pieces of neural structure and its use offers a uniform method for both studying and constructing an extensive class of mechanisms.
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    Bulletin of mathematical biology 46 (1984), S. 327-332 
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    Notes: Abstract Levenshtein dissimilarity measures are used to compare sequences in application areas including coding theory, computer science and macromolecular biology. In general, they measure sequence dissimilarity by the length of a shortest weighted sequence of insertions, deletions and substitutions required, to transform one sequence into another. Those Levenshtein dissimilarity measures based on insertions and deletions are analyzed by a model involving valuations on a partially ordered set. The model reveals structural relationships among poset, valuation and dissimilarity measure. As a consequence, certain Levenshtein dissimilarity measures are shown to be metrics characterized by betweenness properties and computable in terms of well-known measures of sequence similarity.
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    Bulletin of mathematical biology 46 (1984), S. 337-337 
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    Bulletin of mathematical biology 46 (1984), S. 333-336 
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    Notes: Abstract It is pointed out that the methane flux measured experimentally for certain ponds and swamps is quantitatively consistent with a commensal dependence of Methanobacteria on O2-chemotactic motile aerobic bacteria. The Methano species is thereby shielded from oxygen and provided with carbon dioxide for the anaerobic production of methane.
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    Bulletin of mathematical biology 46 (1984), S. 357-370 
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    Notes: Abstract A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra-type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of the random fluctuations are not too large.
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    Bulletin of mathematical biology 46 (1984), S. 371-377 
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    Notes: Abstract One of Bobisud's models for the evolution of cannibalism is reanalyzed by applying the method of finding evolutionarily stable strategies (or ESS's). It is demonstrated that ‘no cannibalism’ never will be an ESS if the initial rate of cannibalism is too large. It is further demonstrated that individual selection may even result in the evolution of cannibalism during food abundance. Some empirical case studies are briefly discussed in relation to this model.
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    Bulletin of mathematical biology 46 (1984), S. 379-387 
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    Notes: Abstract A computer algorithm is presented which equiprobably generates any member of the set of all directed trees withk labeled terminal nodes and unlabeled interior nodes. The algorithm requires roughlyk 2 /2 storage locations. The one-time initialization requiresO(k 2 ) time, while generating each tree requiresO(k) time.
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    Bulletin of mathematical biology 46 (1984), S. 515-527 
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    Notes: Abstract The comparison of several sequences is central to many problems of molecular biology. Finding consensus patterns that define genetic control regions or that determine structural or functional themes are examples of these problems. Previously proposed methods, such as dynamic programming, are not adequate for solving problems of realistic size. This paper gives a new and practical solution for finding unknown patterns that occur imperfectly above a preset frequency. Algorithms for finding the patterns are given as well as estimates of statistical significance.
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    Bulletin of mathematical biology 46 (1984), S. 501-514 
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    Notes: Abstract A new development is introduced here in the use of dynamic programming in finding pattern similarities in genetic sequences, as was first done by Needleman and Wunsch (1969). A condition of pattern similarity is defined and an algorithm is given which scans any set of similarities and screens out those which fail to meet the condition. When the set to be scanned contains every pair of segments, one from each of two given sequences of lengthsm andn (i.e. every possible location for a pattern similarity), then it completes the scan in a number of computational steps proportional tom·n, leaving those pairs of segments which satisfy the similarity condition. The algorithm is based on the concept of match density, as suggested by Goad and Kanehisa (1982).
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    Bulletin of mathematical biology 46 (1984), S. 529-543 
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    Notes: Abstract This paper concerns sequences of letters in which certain “distinguished” words are of interest. Such sequences arise as data in numerous fields including genetics and neuroscience. A probability distribution is given for the number of occurrences of a chosen word in a randomized sequence of letters. Such words are considered “favored” if they occur more than expected at random. Favored words have been discovered in nerve impulse trains and may reflect a neural coding scheme.
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    Bulletin of mathematical biology 46 (1984), S. 545-552 
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    Notes: Abstract As the volume of protein sequence data grows, rapid methods for searching the protein sequence database become of primary importance. Rigorous comparison of sequences is obtained with the well-known dynamic programming algorithms. However, these algorithms are not rapid enough to use for routinely searching the entire database. In this paper we discuss some methods that can be used for rapid searches.
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    Bulletin of mathematical biology 46 (1984), S. 553-566 
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    Notes: Abstract We give algorithms for computing the extent of similarity between two or three sequences of letters. The similarity measures we consider include a penalty for inserting gaps within the sequence in order to enhance similarity. The magnitude of the penalty for gaps is assumed to be independent of their size in order to accommodate certain biological applications. Our algorithm for three sequence comparisons, which is based on solving a system of recursive equations, improves upon the efficiency of existing methods. Although the system of recursive equations utilized by the algorithm is quite complicated as it stands, it has none the less been simplified by appeal to combinatorial considerations.
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    Bulletin of mathematical biology 46 (1984), S. 567-577 
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    Notes: Abstract Well-known dynamic programming algorithms exist for comparing two finite sequences inO(N 2) time and storage, whereN is the common sequence length. Extensions to the comparison ofM finite sequences requireO((2N) M) time and storage, making such algorithms difficult even forM=3. A simple generalization of the sequences makes it possible to obtain some results about the geometry of sequence alignments. These ideas suggest heuristic approaches to problems of comparing several sequences. IfM sequences are known to be related by a binary tree, they can be aligned inO(MN 2) time andO(N 2+NM) storage.
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    Bulletin of mathematical biology 46 (1984), S. 661-672 
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    Notes: Abstract Protein sequences of the Dayhoff databank of 1984 have been analyzed to evaluate the occurrences of the 400 dipeptides and 8000 tripeptides. Expected values and standard deviations for the di- and tripeptides were determined by Monte Carlo and binomial approximation. A condensed format containing this information, labeled a uniqueness diagram, is presented and made available in the form of a microfiche.
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    Bulletin of mathematical biology 46 (1984), S. 827-844 
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    Notes: Abstract In two-state sliding filament models of muscle contraction a partial differential equation must be solved to find the cross-bridge distribution functionn(x, t). In this paper the analytical form of this function is obtained by integration along the characteristic line and special cases are presented in which the explicit expression forn(x, t) can be completely determined. These analytical solutions provide a direct mathematical connection between the microscopic contraction parameters contained in the kinetic theories and macroscopic muscle dynamics and are thus used to investigate what parameters influence the transient contractile tension in typical experimental conditions. The results of this investigation are consistent with relevant aspects of muscle physiology.
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    Bulletin of mathematical biology 46 (1984), S. 845-857 
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    Notes: Abstract Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP〉2 nets is examined briefly.
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    Bulletin of mathematical biology 46 (1984), S. 869-877 
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    Notes: Abstract The influence of topographical situation on the spread of infection is studied. The investigation is based on a multigroup model. The population under consideration is thought to be divided into subpopulations living in regions that are separated from each other by natural barriers (mountains). Infection is carried from one region to another by migrating infectives. Migration is possible only along the river system so that the structure of the epidemiological network is that of a symmetric tree. The results allow comparison of the velocity of propagation of the epidemic for different geographical situations and allow quantification of the “channel-effect”, according to which mountainous regions are channels rather than barriers to the spread of an epidemic.
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    Bulletin of mathematical biology 43 (1981), S. 59-67 
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    Notes: Abstract The theory of computational complexity and certain explicitly-stated hypotheses imply limitations on the information processing power of biological systems. Parallelism, special purpose organization, and analog mechanisms may provide speedup critical for life processes, but have little power in the face of exponential growth. We show that “polynomially simulatable” biological systems cannot exhibit dynamic behavior which produces the solution of an intractable problem. The argument implies that parallelism does not allow biological systems to defeat the exponential explosion, but rather is important because it allows polynomial time algorithms to be used more efficiently.
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    Bulletin of mathematical biology 43 (1981), S. 81-88 
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    Notes: Abstract A correlation matrix analysis is applied to the base sequence of MS2 and ϕX174 in comparison with sets of simulated sequences with different degrees of constaint Significant differences between a codified sequence, and a statistical one in terms of the “correlation matrix” for sets of different length cannot be found. This result is analysed in terms of nucleotide sequences with different levels of informational content.
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    Bulletin of mathematical biology 43 (1981), S. 101-109 
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    Notes: Abstract A method of calculating the volume of a tree distal to a cut at the origin of a branch, using branching, diameter and length ratios, has been developed. The method was applied to bronchial tree casts from human, dog, sheep, hamster, and rat lungs. It was found that the exponenta in the equation weight=k×diameter a is approximately equal to 3.0 in sheep lung casts, as found by Hooper (1977), but it is greater than 3.0 in casts from the other four species.
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    Bulletin of mathematical biology 43 (1981), S. 111-116 
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    Notes: Abstract In this note we examine a continuous time version of a compartmental model introduced in a discrete time setting by S. R. Bernard. The model allows for more than one particle to leave the system at any time. This introduces additional randomness into the system, over the pure death system and this is reflected in the variance function.
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    Bulletin of mathematical biology 43 (1981), S. 89-99 
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    Notes: Abstract The mean first passage time for free diffusion can be derived directly by solving a simple analogue steady state problem. In this problem the diffusion starting region is considered as a time independent source of diffusing particles and the diffusion target assumes the behaviour of a perfectly absorbing sink. It is shown here that the transit time between the source and the sink, which in this particular problem is equal to the ratio between the holdup of the system and the total flux, is identical to the Brownian movement concept of the mean first passage time for free diffusion. This established identity considerably facilitates the derivation and investigation of the timing of diffusion in complicated structures such as those commonly found in living organisms.
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    Bulletin of mathematical biology 43 (1981), S. 121-123 
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    Bulletin of mathematical biology 43 (1981), S. 117-120 
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    Bulletin of mathematical biology 43 (1981), S. 201-211 
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    Notes: Abstract In this paper three stochastic models are developed for a class of two-compartment systems to analyse the randomness of the leaving process of the particles in the system. Results in closed form for the distribution of the leaving process of the particles in the system are given both for general and exponential sojourn time distributions and also in association with forward recurrence time distributions with and without Poisson input.
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    Bulletin of mathematical biology 43 (1981), S. 213-232 
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    Notes: Abstract Two simple models are proposed and analysed, in which it is shown that the formation of a new polymer, resulting from an “error” in the template action mechanism of production of an old polymer, may compromise the stability of the initial system under specific conditions, in the context of prebiotic evolution. Autocatalysis is shown to be a “selective advantage”, enabling the “mutant” to dominate in concentration and even replace the initial polymer. The addition of a third molecule playing the role of a catalyst causes hysteresis effects.
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    Bulletin of mathematical biology 43 (1981), S. 165-181 
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    Notes: Abstract The problem of extinction of the prey population in a microbial predator-prey interaction in a chemostat has been examined. Usual deterministic lumped parameter models were used for the dynamics of the chemostat for large numbers of the two populations; the generalized birth and death stochastic process was employed for the description of the random variations at small prey numbers. Extinction probabilities of the prey population were calculated for different holding times and chemostat volumes, and their dependence upon the growth parameters of the two populations was studied. It was found that extinction was possible when the Monod model was used for the specific growth rate of the predators as a function of the prey number density. On the other hand, the decrease of the feeding activity of the predators at low prey densities predicted by the multiple saturation model acts as a regulatory factor that prevents extinction of the prey. In view of the fact that extinction of the prey has never been observed in the laboratory, the latter model seems more appropriate to describe the dynamics of microbial predation.
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    Bulletin of mathematical biology 43 (1981), S. 233-238 
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    Notes: Abstract During exposures of the eye to light, the choroidal circulation may have a regulatory influence on the retinal temperature. This is investigated using a mathematical model and a finite-difference technique. It is predicted that the choroidal blood flow a small effect on retinal temperature, which may be important.
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    Bulletin of mathematical biology 43 (1981), S. 427-446 
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    Notes: Abstract A probabilistic model of a spatially localized, mutually exitatory (inhibitory) population of neurons is formulated to help explain average evoked potential and post-stimulus time histogram measurements. The model is based on the stochastic activity of single neurons within interactive masses of neurons which exhibit co-operative behavior. Macrostate variables corresponding to the above measurements are related through the model to features of neural operation at the individual and ensemble level. Steady-state solution are obtained and their physiological implications are discussed.
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    Bulletin of mathematical biology 43 (1981), S. 503-512 
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    Notes: Abstract We consider a one-compartment system with stochastic transfer rate characterized either by Gaussian or by two-level jump process and study the time evolution of the (statistical) moments of the (random) amount of the substance present in the system. The effect of the coloured as well as of the white noise is investigated and it is found that the presence of stochasticity in the transfer rate parameter increases the relaxation time of the system. Finally, we obtain the conditions for the stability of the system in the moment sense.
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    Bulletin of mathematical biology 43 (1981), S. 487-501 
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    Notes: Abstract A model is described in which damage to a single intracellular locus can lead to a tumorigenic transformation. Assuming a large number of independent intracellular loci to be at risk and assuming that damage to a locus sufficient to cause a tumorigenic transformation occurs with probability greater than zero for all doses greater than zero, leads to the use of the Weibull distribution to characterize the probability of a nonspontaneous tumorigenic cellular transformation occurring after exposure to a given dose of carcinogen. The excess lifetime tumor incidence (i.e., the proportion of tumor bearers) above the spontaneous incidence is used as an estimate of the non-spontaneous incidence and is characterized by a tumor incidence function that represents the probability of occurrence of one or more non-spontaneous tumorigenic cellular transformations amongN(D) independent surviving cells per individual, after exposure to a doseD of carcinogen. The tumor incidence function is fitted to published data for the excess tumor incidence after exposure of animals or humans to ionizing radiation and after exposure of animals to chemical carcinogens.
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    Bulletin of mathematical biology 43 (1981), S. 549-561 
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    Notes: Abstract This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.
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    Bulletin of mathematical biology 43 (1981), S. 563-577 
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    Notes: Abstract This paper deals with the pulsatile blood flow in the lung alveolar sheets by idealizing each of them as a channel covered by porous media. As the blood flow in the lung is of low Reynolds number, a creeping flow is assumed in the channel. The analytical and numerical results for the velocity and pressure distribution in the porous medium are presented. The effect of an imposed slip condition is also studied. Comparisons with the corresponding results for the steady-state case are made at the end.
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    Bulletin of mathematical biology 43 (1981), S. 579-591 
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    Notes: Abstract The relationships that define the structure of a given ecosystem, social system, or even a physiological function can only exist if certain parameters are confined to a certain range of values. As the values change and exceed this given range the relationships are forced to change, and so produce a new pattern of relationships. The concept of a dynamic structure captures this potential for structural change in relation to a set of parameters. The precise definition of structure and allowable transformation constitutes the definition of a category. The total range of parameters associated with all the relevant structures provides a parameter space which is assumed to be a manifold. Maps with extra structure from the manifold to the category define dynamic structures. The domain of differential dynamic systems is the manifold, and a flow or movement across the manifold is associated with a series of structural transformations in the category. In some cases a structure outruns its parameter range, to be faced with an obstruction—an absence of possible transformations. Ways of studying such “obstructions” are considered along with the related problem of extending a dynamic structure beyond a previously given set of parameters. The cost or resistance of transformations is also studied. The concepts of dynamic structures are illustrated by the structural change of food webs and they are used in a necessarily qualitative fashion to study dominance structures of social orders and finally to speculate on the qualitative nature of evolutionary change of functional aspects of organisms.
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    Bulletin of mathematical biology 43 (1981), S. 705-715 
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    Notes: Abstract The preceding paper (Thorn, 1981) has shown that in a linear pharmacokinetic system with a multimodal impulse response the peak drug level may sometimes be smaller with slower rates of injection. This paper presents two theorems on this paradoxical injection rate effect where the injection is a constant infusion of finite duration. The first theorem establishes a graphical method for determining whether a given impulse response will give a paradoxical injection rate effect; and the second establishes that the maximum paradoxical increase in peak drug level is by a factor of two. It is further shown that in order to approach this maximum paradoxical increase the impulse response must contain two isolated, sharp, narrow pulses of approximately equal area. Some examples of bimodal arterial dye-dilution curves from the literature are discussed as impulse responses; and there is also a discussion of the behavior of drug level maxima and minima at different injection rates.
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    Bulletin of mathematical biology 43 (1981), S. 693-703 
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    Notes: Abstract This paper presents three theorems on the peak drug levels that result from injection into a linear pharmacokinetic system. As a preliminary, the “rate of injection” is defined in terms of time expansion or time contraction of the injection function (input). The first theorem then states that the peak drug level will not be greater when the rate of injection is slow than when it is fast, if the impulse response is unimodal. The second theorem sets limits for the time of the maximum drug level, in relation to the time of the maximum of the (unimodal) impulse response and the duration of the input. The third theorem defines conditions which assure a definitely lower peak drug level if the rate of injection is slower. A graphical method is suggested for determining the times and magnitudes of the peak drug levels that result from constant infusions of a fixed dose at different rates. An example is provided to show that if the impulse response is multimodal then the peak drug level may sometimes increase with a decrease in the injection rate.
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    Circuits, systems and signal processing 3 (1984), S. 267-294 
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    Notes: Abstract Pipeline techniques have been successfully applied to speeding up processing in both general- and special-purpose digital computers. Application of these techniques to nonrecursive (FIR) filters has been suggested and is quite straightforward. Application to recursive (IIR) filters has not previously been shown. In this paper, the technique for applying pipeline techniques to recursive filters is shown and the advantages and disadvantages of the technique are discussed. Using these techniques, recursive digital filters operating at hitherto impossibly high rates can be designed.
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    Circuits, systems and signal processing 3 (1984), S. 295-314 
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    Notes: Abstract The long-standing problem of reconstructing a function from its samples is considered again. Assuming a sequence of oversampled values, a set of appropriate idealized reconstruction filters can be defined, which do not suffer from instability or slow convergence. The realization — a cascade of a nonrecursive digital filter, D/A-converter, and a fixed/analog smoothing filter — demands the design of the digital filter for the increase of the sampling rate. The design of this nonrecursive filter is the purpose of this paper. Approximations in the frequency as well as in the time domain are presented.
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    Circuits, systems and signal processing 3 (1984), S. 105-122 
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    Notes: Abstract In this paper we formulate power systems as nonlinear nearly Hamiltonian systems. Using the invariance principle for ordinary differential equations, necessary and sufficient conditions for asymptotic stability are established and a new method of estimating the domain of attraction of the stable equilibrium point is developed. The present results constitute a novel approach to stability analysis and involve the following three steps: a. Given a system with dissipation, the stability of its equilibrium is ascertained by determining the stability of the associated conservative system. b. Attractivity of the stable equilibrium of the entire system (with dissipation) is determined from the system topology. c. An estimate of the domain of attraction of the asymptotically stable equilibrium is obtained by making use of results obtained in (a) and (b). The stability criterion developed in this paper sheds new light on the mechanism of instability in power systems and it provides analytical verification to the concept of the potential-energy boundary surface (PEBS). The PEBS is a hypersurface which makes up a part of the boundary of the domain of attraction of the stable equilibrium in a power system. The existence and properties of the PEBS have thus far been deduced primarily via simulations and heuristic methods.
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    Circuits, systems and signal processing 3 (1984), S. 161-176 
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    Notes: Abstract In conventional television systems, picture scanning in vertical and temporal directions is usually very defective with regard to the sampling theorem. In this paper some deficiencies such as aliasing, line-structure distortion, line flickering, and large-area flickering are investigated with regard to their dependence on the inter-lacedpicture-scanning process. The three-dimensional reconstruction filtering of the sampled picture is especially analyzed with respect of the viewer's perception. Furthermore, it will be shown that in connection with a new concept of picture scanning published earlier [1], [2], a flat-field reproduction without any 25/30-Hz flicker can be achieved by vertical filtering only. This is true even though the final reproduction by the monitor is performed with interlace. The vertical filtering can then be optimized in the sense of maximum picture sharpness and resolution with negligible ringing as well. Practical results are given in this paper.
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    Circuits, systems and signal processing 3 (1984), S. 177-191 
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    Notes: Abstract In this paper a new type of “velocity-selecting/rejecting filter” which passes or stops a particular event in a seismic signal is proposed. The velocity-selecting filter is based on a time-space band-pass filter with sharp passband for a particular direction, and similarly, the velocity-rejecting filter is based on a time-space band-stop filter. A technique for designing such filters, in terms of an infinite-impulse-response (IIR) filter, is presented, in which a rotated version of separable filter is used. Finally, numerical examples are included to illustrate the design theory.
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    Circuits, systems and signal processing 3 (1984), S. 347-359 
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    Circuits, systems and signal processing 3 (1984), S. 315-325 
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    Notes: Abstract A special type of factorization for operators defined on partially ordered Hilbert resolution spaces is considered. The main result includes, as a particular case, the classical Schur-Coleski triangular factorization. Connections with stochastic optimization and image-processing problems are established.
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    Circuits, systems and signal processing 3 (1984), S. 409-417 
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    Notes: Abstract In a recent paper on nonlocal expansions necessary and sufficient conditions are given under whichf −1 has a generalized power series expansion, whenf is an invertible locally Lipschitz map between certain general subsets of a complex Banach space. Here we establish the validity of a conceptually interesting algorithm for obtaining the expansion. Basically, we show that a certain contraction mapping iteration generates iteratesℐ 1,ℐ 2,... such that eachℐ k yields all of the terms of the generalized power series expansion off −1 up to order (k + 1), assuming merely that the expansion off −1 exists. An earlier different result along related lines is mentioned.
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    Circuits, systems and signal processing 3 (1984), S. 447-475 
    ISSN: 1531-5878
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    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract The representation of functions in a basis function expansionz(t)= ∑k=1/∞=,a k〉 x k (t) is straightforward when the basis functionsx k (t) are orthogonal. There has been very little work up to this time in determining how to use nonorthogonal bases in signal representation. On the other hand, applications in data compression and signal synthesis often require using specific tailor-made bases. Presented here is a method for constructing very general nonorthogonal bases. Orthogonality has often been used to show that a basis spans the set of functions of interest and to calculate the coefficients of the representation. In this paper, both of these fundamental aspects are addressed for nonorthogonal bases. A new basis {y k (t)} is obtained by performing a linear transformation on a known existing basis {x k (t)}. This transformation is constructed such that the coefficients of signal representation on the new basis are readily found. Then, a useful and sufficient condition is placed upon the new basis such that representations converge. The fundamental methods are applied to the standard examples of signal representation. The complex sinusoids, the Rademacher functions, the orthogonal polynomials, and the decaying exponentials are used as the original basis {x k (t)} from which a new basis {y k (t)} is generated. Two examples are given to illustrate general applications: one in signal synthesis and one in signal analysis.
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    Bulletin of mathematical biology 43 (1981), S. 33-45 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A two urn Polya-type scheme is considered in whichr black balls (corresponding to the stable form of an element) are added to urn one at every stage and the same number of balls are removed at random at every stage from the same urn. In between these two operations, which form a stage or iteration, a fixed number of balls is exchanged at random between urns one and two. Urn one has a given initial number of white balls (corresponding to a radioactive form of the same element). The problem of interest is to study the stochastic aspect of the number of white balls remaining in urn one (and/or urn two) aftern iterations.
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    Bulletin of mathematical biology 43 (1981), S. 21-32 
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    Topics: Biology , Mathematics
    Notes: Abstract We obtain within the action-angle variable approach new expressions, involving the Dirac delta function, for time periods and time averages of dynamical variables which are useful for nonlinear biological oscillator problems. We combine these with Laplace transformation techniques for evaluating the required perturbation expansions. The radii of convergence of these series are determined through a complex variable approach. The method is powerful enough to yield explicit results for such systems as the two species Volterra model, Goodwin's model of protein synthesis etc. and as an illustration, is applied here to Cowan's model of neuroelectric activity. We also point out the usefulness of the action integral in the case where parameters occurring in dynamics have slow time variations.
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    Bulletin of mathematical biology 43 (1981), S. 69-79 
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    Notes: Abstract Zwanzig and Mori's projection-operator method is used in order to derive a generalized nonlinear Fokker-Planck equation for one “relevant” species in the many species conservative Volterra model. The deterministic, autonomous, Markovian equations of motion, when averaged over a suitable ensemble of initial conditions in general give rise to a non-autonomous, non-Markovian stochastic process for the evolution of this relevant species. Moreover, this relevant species may show irreversible damping, although self-interaction terms are absent in the many species model.
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    Bulletin of mathematical biology 43 (1981), S. 151-163 
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    Notes: Abstract The hydrodynamical problem of flow in proximal renal tubule is investigated by considering axisymmetric flow of a viscous, incompressible fluid through a long narrow tube of varying cross-section with reabsorption at the wall. Two cases for reabsorption have been studied (i) when the bulk flow,Q, decays exponentially with the axial distancex, and (ii) whenQ is an arbitrary function ofx such thatQ-Q 0 can be expressed as a Fourier integral (whereQ 0 is the flux atx=0). The analytic expressions for flow variables have been obtained by applying perturbation method in terms of wall parameter ε. The effects of ε on pressure drop across the tube, radial velocity and wall shear have been studied in the case of exponentially decaying bulk flow and it has been found that the results are in agreement with the existing ones for the renal tubules.
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    Bulletin of mathematical biology 43 (1981), S. 183-199 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Voltage clamp experiments, which determine the kinetic parameters of calcium conductance of cardiac muscle, (d ∞,f ∞, τ d and τ f ) are analyzed with a generally accepted expression for slow inward currentI s=g sdf (E-E R). Activation (d) and inactivation (f) reach the final valuesd ∞ andf ∞ with time constants τ d and τ f respectively. The analysis indicates that the measuredf ∞ agrees with the theoreticalf ∞, but the measuredd ∞ differs from the theoreticald ∞ by a factor which depends on τ d . The peak tension can be made to correlate closely with the theoreticald ∞ after a correction factor is applied to the raw measurements of activation. It can be shown that experiments designed to measure τ f can also be used to determine τ d with greater accuracy.
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    Bulletin of mathematical biology 43 (1981), S. 245-247 
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    Bulletin of mathematical biology 43 (1981), S. 249-257 
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    Topics: Biology , Mathematics
    Notes: Abstract A deterministic model for an SIR epidemic with silent infections is investigated. It is shown for the model studied that the extent to which silent infections are present may be determined from data concerning only those individuals with symptomatic infection.
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    Bulletin of mathematical biology 43 (1981), S. 327-340 
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    Notes: Abstract An urn contains balls of different colors. Specified numbers of each color are added and form a reinforcement. The total reinforcement is randomly removed, forming a depletion. The process, not necessarily with the same reinforcements, is performed a number of times. The factorial moment generating function of the urn configurations at any stage is given in terms of multivariate difference operators. Cases when the reinforcement vector is defined as a stochastic variable are considered. The problem is a generalization of an urn model associated with radioactive atoms and stable atoms proposed by S. R. Bernard. The solutions given here have a definite application to the problem of modelling tracers in compartmental systems.
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    Bulletin of mathematical biology 43 (1981), S. 347-360 
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    Notes: Abstract A multi-compartmental model with particles producing offspring according to the Markov branching process has been studied. Explicit results are given for the two-compartmental system and for irreversible general multicompartmental systems. The known models in stochastic compartmental analysis are shown to be particular cases of this model and applications are cited.
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    Bulletin of mathematical biology 43 (1981), S. 371-372 
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    Bulletin of mathematical biology 43 (1981), S. 361-370 
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    Notes: Abstract The irreversible Michaelis-Menten scheme may be reduced to a pair of autonomous first-order differential equations. The phase-plane behaviour of these is investigated.
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    Bulletin of mathematical biology 43 (1981), S. 372-373 
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    Bulletin of mathematical biology 43 (1981), S. 517-548 
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    Notes: Abstract A discrete one-dimensional model of convection-diffusion in branching alveolar ducts is described and it is shown that, for a suitable choice of effective axial dispersion, the solution closely approximates that for an axially symmetric representation, at least for Peclet numbers Pe〈1. Following earlier work a composite model of a uniform lung is formed by matching such a respiratory pathway (now having the more convenient one-dimensional form) onto a trumpet representation of the conducting airways. Enhanced mixing due to heart action, and isotropic volume changes of trumpet (in addition to the pathway) during breathing are additional factors included. Calculations are made of O2 concentrations during steady-state breathing and of the concentration of inert gas during single breath wash-out of a gas mixture containing it. Predicted alveolar levels in each case agree extremely well with published data, although no alveolar slope is obtained for the inert gas.
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    Bulletin of mathematical biology 43 (1981), S. 593-610 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The evolution and local stability of a system of two interacting species in a finite two-dimensional habitat is investigated by taking into account the effects of self- and cross-dispersion and convection of the species. In absence of cross-dispersion, an equilibrium state which is stable without dispersion is always stable with dispersion provided that the dispersion coefficients of the two species are equal. However, when the dispersion coefficients of the two species are different, the possibility of self-dispersive instability arises. It is also pointed out that the cross-dispersion of species may lead to stability or instability depending upon the nature and the magnitude of the cross-dispersive interactions in comparison to the self-dispersive interactions. The self-convective movement of species increases the stability of the equilibrium state and can stabilize an otherwise unstable equilibrium state. The effect of cross-convection (in absence of self-dispersion and self-convection) is to stabilize the equilibrium state in a prey-predator model with positive cross-dispersion coefficients for the prey species. Finally, it is shown that if the system is stable under homogeneous boundary conditions it remains so under non-homogeneous boundary conditions.
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    Bulletin of mathematical biology 43 (1981), S. 611-618 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Various observations suggest that sympathetic ganglia act as local integrative centers redistributing preganglionic excitation (i.e. the information issued by the central nervous system) to the postganglionic fibers (and effector organs). In order to support this concept a simple mathematical model of the elementary integration process, treating the case of a single preganglionic compound action potential, has been developed. This quantitative description, based on a few elementary assumptions, shows a possible way of processing preganglionic excitation in the ganglion. It is shown that on a particular ganglion cell the probability distribution of the number of activated synapses obeys hypergeometric distribution and hence, the postganglionic compound action potential is built up of several compound action potentials occurring at different times. The former correspond to different groups of firing cells. The model discloses modes of structural and temporal pattern generation performed by the sympathetic ganglion.
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    Bulletin of mathematical biology 43 (1981), S. 619-639 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Amino acid sequences have already been examined in some detail in order to relate them to structural aspects, homology and gene duplication. This report introduces the concept of internal uniqueness of tripeptides within protein sequences and uses the Monte Carlo method to study this property. Some idea of internal uniqueness may be obtained from such an analysis using only a single sequence if the probability of the random occurrence is about 0.001 or less. This method of analysis is similar to that used in quantitative evaluations of homology. When the probability of the random occurrence is larger than 0.001 a homologous group of sequences is required and the random probabilities may be compared with the real occurrences within the group. From such an examination insulin and cytochrome c are identified as protein sequences with high internal uniqueness. A comparison of data from internal uniqueness and gene duplication analyses shows that these two properties need not be related. Results of the analysis point to internal uniqueness as an additional parameter for inclusion in speculations on why twenty amino acids are coded in protein structure.
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    Bulletin of mathematical biology 43 (1981), S. 641-650 
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    Topics: Biology , Mathematics
    Notes: Abstract The concept of a tolerance net formalises simultaneously spatial closeness and nearness of neuronal activity. A method of constructing tolerance group nets is presented, leading to a means of construction of all very homogenous tolerance nets as group quotients. The dihedral group of order eight is taken as an illustrative example.
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    Bulletin of mathematical biology 43 (1981), S. 681-691 
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    Topics: Biology , Mathematics
    Notes: Abstract Two methods are described for calculating the value of the exponentx in the equation flow =k×diameter x , as pertaining to a branch of the bronchial tree. In the lungs from three humans, two dogs, one hamster, and one rat mean values ofx between 2.419 and 2.903 were found. They lie within the range of 2.333 to 3.0 predicted by the analysis of Uylings (Bull. Math. Biol. 39, 501–519, 1977).
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    Bulletin of mathematical biology 43 (1981), S. 665-680 
    ISSN: 1522-9602
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    Notes: Abstract A stochastic analysis of a nonlinear selection model is presented. The model, based on Eigen and Schuster's theory of selection and evolution of biological macromolecules, considers the effects of fluctuations on the individual concentrations of macromolecules as well as the total population numbers in constrained systems. Our analysis shows that one of the models most often treated deterministically (referred to as constant organization in the literature) becomes unstable when fluctuations in the total population number are considered. An alternative model which apparently has built in self-regulating properties is analyzed and proves to be stable except for some special cases of degeneracy.
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    Bulletin of mathematical biology 43 (1981), S. 651-664 
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    Notes: Abstract Previous compartmental models have introduced variability either at the particle or at the replicate level. This paper integrates both types of variability through the concept of clustering. The paper develops two different, general clustered models, each with time-dependent hazard rates for the clusters and for the particles within the clusters, and each with random initial number and sizes of clusters. The coefficient of variation of the total number of particles,CV[X(t)], for either model is shown to be bounded below, under very broad conditions, by the coefficient of variation of the initial number of clusters,CV[c(0)]. This high relative variability of the clustered models makes them potentially very useful in kinetic modeling. In many applications, binding and clustering are common phenomena, and two applications of the models to such phenomena are breifly outlined.
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    Bulletin of mathematical biology 43 (1981), S. 717-717 
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    Bulletin of mathematical biology 43 (1981), S. 719-719 
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