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  • Articles  (11,507)
  • Springer  (11,507)
  • American Geophysical Union (AGU)
  • Annual Reviews
  • 1995-1999  (7,137)
  • 1980-1984  (4,370)
  • 1935-1939
  • 1995  (7,137)
  • 1981  (4,370)
  • Mathematics  (11,507)
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  • Articles  (11,507)
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  • 1995-1999  (7,137)
  • 1980-1984  (4,370)
  • 1935-1939
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  • 1
    Electronic Resource
    Electronic Resource
    Multibreath tracer species dynamics in the lung (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 1-19 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460935
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  • 2
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    Global stability of population models (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 47-58 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460938
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  • 3
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    The logistic equation with a diffusionally coupled delay (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 125-140 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459438
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  • 4
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    Uncertainty in physiology and physics (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 141-149 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459439
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  • 5
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    Consensusn-trees (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 239-244 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459446
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  • 6
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    Morphogen pattern formation and development in growth (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 259-270 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02462200
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  • 7
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    Simulation experiments with a model for cellular growth (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 271-278 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02462201
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  • 8
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    Complementary extremum principles for a nonlinear model of heat conduction in the human head (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 341-346 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of complementary variational principles is used to obtain maximum and minimum principles for 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462204
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  • 9
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    On Zeeman's equations for the nerve impulse (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 279-325 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02462202
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  • 10
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    Asymptotic analysis of the Michaelis-Menten reaction equations (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 375-388 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459428
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  • 11
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    Pulse evolution on coupled nerve fibres (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 389-400 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459429
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  • 12
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    Field-induced forces at dielectric interfaces as a possible mechanism of RF hearing effects (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 401-413 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459430
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  • 13
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    The construction of homogeneous nets (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 415-426 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459431
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  • 14
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    Contractile analysis and kinetic energy of the myocardium (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 447-461 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459433
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  • 15
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    Limit cycles in periodically perturbed population systems (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 463-485 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459434
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  • 16
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    On a unified approach to pattern generation (1981)
    Springer
    Bulletin of mathematical biology 43 (1981), S. 513-516 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02459437
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  • 17
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    Hypnosis modelling in neural networks (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 1-20 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In the framework of the neural network theory effects similar to hypnotic displays are constructed. They are based on the associative paradigm involving non-linear interaction of excitatory and inhibitory channels with synaptic memory. The non-linearity of long-term memorizing processes may cause effects exhibited by blind spots, which are interpreted as the first stage of hypnosis. More complicated phenomena are discussed in terms of a two-layer network.
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    URL: http://dx.doi.org/10.1007/BF02458314
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  • 18
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    Dynamics of small autocatalytic reaction networks—II. Replication, mutation and catalysis (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 21-61 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Mutation is introduced into autocatalytic reaction networks. The differential equations obtained are neither of repliator-type nor can they be transformed straightway into a linear equation. Examples of low dimensional dynamical systems —n=2, 3 and 4 — are discussed and complete qualitative analysis is presented. Error thresholds known from simple replication-mutation kinetics with frequency independent replication rates occur here as well. Instead of cooperative transitions or higher order phase transitions the thresholds appear here as supercritical or subcritical bifurcations being analogous to first-order phase transitions.
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    URL: http://dx.doi.org/10.1007/BF02458315
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  • 19
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    Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 63-76 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The non-linear behavior of a differential equations-based predator-prey model, incorporating a spatial refuge protecting a consant proportion of prey and with temperature-dependent parameters chosen appropriately for a mite interaction on fruit trees, is examined using the numerical bifurcation code AUTO 86. The most significant result of this analysis is the existence of a temperature interval in which increasing the amount of refuge dynamically destabilizes the system; and on part of this interval the interaction is less likely to persist in that predator and prey minimum population densities are lower than when no refuge is available. It is also shown that increasing the amount of refuge can lead to population outbreaks due to the presence of multiple stable states. The ecological implications of a refuge are discussed with respect to the biological control of mite pests.
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    URL: http://dx.doi.org/10.1007/BF02458316
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  • 20
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    Optimal root/shoot-partitioning of carbohydrates in plants (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 99-107 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In many applications of control theory on plant growth models biomass maximization is postulated to avoid analytically unsolvable problems while fruit maximization is commonly considered to be a more realistic criterion. In a special case, we are able to compare these criteria. Iwasa and Roughgarden (1984,Theor. Pop. Biol. 25, 78–105) have investigated a certain class of plant growth models using a fruit maximization criterion. They proved that, in the vegetative growth period, the organs follow a certain path of balanced growth. We show that this path remains optimal when biomass maximization is postulated. This underlines the importance of the balanced growth path found by Iwasa and Roughgarden. Furthermore, our result suggests that in the vegetative growth period the biomass maximization criterion is a good approximation of fruit maximization. In another theoretical control investigation, Schultzeet al. (1983,Oecologia 58, 169–177) derived a different type of balanced growth path. We apply the theory of Iwasa and Roughgarden to an improved version of the model of Schulzeet al. This leads to a new description of balanced growth between root and shoot that reflects non-linearities in the water uptake process and constitutes an interesting hypothesis for further experimental testing.
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    URL: http://dx.doi.org/10.1007/BF02458318
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  • 21
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    Ion induced deformation of soft tissue (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 77-98 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper the effects of changing the ion concentration in and around a sample of soft tissue are investigated. The triphasic theory developed by Laiet al. (1990,Biomechanics of Diarthrodial Joints, Vol. 1, Berlin, Springer-Verlag) is reduced to two coupled partial differential equations involving fluid ion concentration and tissue solid deformation. These equations are given in general form for Cartesian, cylindrical and spherical geometries. After solving the two equations quantities such as fluid velocity, fluid pressure, chemical potentials and chemical expansion stress may be easily calculated. In the Cartesian geometry comparison is made with the experimental and theoretical work of Myerset al. (1984,ASME J. biomech. Engng,106, 151–158). This dealt with changing the ion concentration of a salt shower on a strip of bovine articular cartilage. Results were obtained in both free swelling and isometric tension states, using an empirical formula to acount for ion induced deformation. The present theory predicts lower ion concentrations inside the tissue than this earlier work. A spherical sample of tissue subjected to a change in salt bath ion concentration is also considered. Numerical results are obtained for both hypertonic and hypotonic bathing solutions. Of particular interest is the finding that tissue may contract internally before reaching a final swollen equilibrium state or swell internally before finally contracting. By considering the relative magnitude, and also variation throughout the time course of terms in the governing equations, an even simpler system is deduced. As well as being linear the concentration equation in the new system is uncoupled. Results obtained from the linear system compare well with those from the spherical section. Thus, biological swelling situations may be modelled by a simple system of equations with the possibility, of approximate analytic solutions in certain cases.
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    URL: http://dx.doi.org/10.1007/BF02458317
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  • 22
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    Memory capacity in large idiotypic networks (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 109-136 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Many models of immune networks have been proposed since the original work of Jerne [1974,Ann. Immun. (Inst. Pasteur) 125C, 373–389]. Recently, a limited class of models (Weisbuchet al., 1990,J. theor. Biol. 146, 483–499) have been shown to maintain immunological memory by idiotypic network interactions. We examine generalizations of these models when the networks are both large and highly connected to study their memory capacity, i.e. their ability to account for immunization to a large number of random antigens. Our calculations show that in these minimal models, random connectivities with continuously distributed affinities reduce the memory capacity to essentially nil.
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    URL: http://dx.doi.org/10.1007/BF02458319
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  • 23
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    A kinetic model of coronary reactive hyperemic response to transient ischemia (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 137-156 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A kinetic model is proposed to delineate the factors that determine the coronary reactive hyperemic response (RHR) to transient ischemia. The model comprises of myocardial-interstitial (M) and vascular (V) compartments. Vasodilator metabolites (VM) are produced in the M compartment during the interval of coronary occlusion. The rate of VM production is dependent on the flow rate during the ischemic period, the ratio of excess flow above the control level (R) to the loss of flow during occlusion period (D), the amount of oxygen stored and the degree of vasodilation in the V compartment prior to occlusion. Following a complete release of occlusion, VM are transported from the M to V compartment and are washed out or degraded with time. The time course of RHR is determined by the coronary patency which is proportional to VM concentration in the V compartment. Based on a set of numerical constants, the model is tested by simulating RHR to the various occlusion manoeuvres: a pair of 10 sec occlusions separated by brief release, a 15 sec release followed by a second brief occlusion, a brief release of an occlusion followed by restriced inflow and a period of restricted inflow after occlusion. The simulated results fit the experimental R/D and RH durations data of canine hearts. Factors that determine the impairment of RH capacity in coronary stenosis are suggested in terms of the model scheme.
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    URL: http://dx.doi.org/10.1007/BF02458320
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  • 24
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    Kinetic study of an enzyme-catalysed reaction in the presence of novel irreversible-type inhibitors that react with the product of enzymatic catalysis (1995)
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    Bulletin of mathematical biology 57 (1995), S. 157-168 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In the present paper a kinetic study is made of the behaviour of a Michaelis-Menten enzyme-catalysed reaction in the presence of irreversible inhibitors rendered unstable in the medium by their reaction with the product of enzymatic catalysis. A general mechanism involving competitive, non-competitive, uncompetitive and mixed irreversible inhibition with one or two steps has been analysed. The differential equation that describes the kinetics of the reaction is non-linear and computer simulations of its dynamic behaviour are presented. The results obtained show that the systems studied here present kinetic co-operativity for a target enzyme that follows the simple Michaelis-Menten mechanism in its action on the substrate, except in the case of an uncompetitive-type inhibitor.
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    URL: http://dx.doi.org/10.1007/BF02458321
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  • 25
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    Book reviews (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 169-173 
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  • 26
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    On the estimation of alternative pathways of fatty acid oxidation in the liverIn vivo (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 191-203 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The relative contributions of mitochondrial β-oxidation and peroxisomal β-oxidation and peroxisomal ω-oxidation to the oxidation of a given fatty acidin vivo can be quantitated by an isotopic method. The approach requires infusion of a fatty acid labelled on two specific carbon atoms (e.g. [1-14C] and [11-14C] palmitate) to an isotopic steady state, with subsequent isolation and degradation of an acetylated conjugate as a product of the liver cytosolic acetyl CoA pool and of ketone bodies as a product of the liver mitochondrial acetyl CoA pool.
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    URL: http://dx.doi.org/10.1007/BF02460615
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  • 27
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    Correlations of rates of insulin release from islets and plateau fractions for β-cells (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 229-246 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Pancreatic β-cells in intact islets of Langerhans perfused with various glucose concentrations exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. The fraction of the time spent in the active phase is called the plateau fraction and appears to be strongly correlated with the rate of release of insulin from islets as glucose concentration is varied. Here this correlation is quantified and a theoretical development is presented in detail. Experimental rates of insulin release are correlated with “effective” plateau fractions over a range of glucose concentrations. There are a number of different models for BEA in pancreatic β-cells and a method is developed here to quantify the dependence of a glucose dependent parameter on glucose concentration. As an example, the plateau fractions computed from the Sherman-Rinzel-Keizer model are matched with experimental plateau fractions to obtain a relationship between the model's glucose-dependent parameter, β, and glucose concentration. Knowledge of the relationships between β and glucose concentration and between experimental measurements of rates of insulin release and plateau fractions permits the determination of theoretical rates of insulin release from the model.
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    URL: http://dx.doi.org/10.1007/BF02460617
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  • 28
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    The development of concentration gradients in a suspension of chemotactic bacteria (1995)
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    Bulletin of mathematical biology 57 (1995), S. 299-344 
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    ISSN: 1522-9602
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    Notes: Abstract When a suspension of bacterial cells of the speciesBacillus subtilis is placed in a chamber with its upper surface open to the atmosphere complex bioconvection patterns are observed. These arise because the cells: (1) are denser than water; and (2) usually swim upwards, so that the density of an initially uniform suspension becomes greater at the top than the bottom. When the vertical density gradient becomes large enough, an overturning instability occurs which ultimately evolves into the observed patterns. The reason that the cells swim upwards is that they are aerotactic, i.e. they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell (N) and oxygen (C) concentrations, and these are solved in the pre-instability phase of development whenN andC depend only on the vertical coordinate and time. Numerical results are obtained for both shallow- and deep-layer chambers, which are intrinsically different and require different mathematical and numerical treatments. It is found that, for both shallow and deep chambers, a thin boundary layer, densely packed with cells, forms near the surface. Beneath this layer the suspension becomes severely depleted of cells. Furthermore, in the deep chamber cases, a discontinuity in the cell concentration arises between this cell-depleted region and a cell-rich region further below, where no significant oxygen concentration gradients develop before the oxygen is fully consumed. The results obtained from the model are in good qualitative agreement with the experimental observations.
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    Topological and phenomenological classification of bursting oscillations (1995)
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    Bulletin of mathematical biology 57 (1995), S. 413-439 
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    Notes: Abstract We describe a classification scheme for bursting oscillations which encompasses many of those found in the literature on bursting in excitable media. This is an extension of the scheme of Rinzel (inMathematical Topics in Population Biology, Springer, Berlin, 1987), put in the context of a sequence of horizontal cuts through a two-parameter bifurcation diagram. We use this to describe the phenomenological character of different types of bursting, addressing the issue of how well the bursting can be characterized given the limited amount of information often available in experimental settings.
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    Book reviews (1995)
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    Bulletin of mathematical biology 57 (1995), S. 499-506 
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    Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions (1995)
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    Bulletin of mathematical biology 57 (1995), S. 461-486 
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    Notes: Abstract To ensure its sustained growth, a tumour may secrete chemical compounds which cause neighbouring capillaries to form sprouts which then migrate towards it, furnishing the tumour with an increased supply of nutrients. In this paper a mathematical model is presented which describes the migration of capillary sprouts in response to a chemoattractant field set up by a tumour-released angiogenic factor, sometimes termed a tumour angiogenesis factor (TAF). The resulting model admits travelling wave solutions which correspond either to successful neovascularization of the tumour or failure of the tumour to secure a vascular network, and which exhibit many of the characteristic features of angiogenesis. For example, the increasing speed of the vascular front, and the evolution of an increasingly developed vascular network behind the leading capillary tip front (the brush-border effect) are both discernible from the numerical simulations. Through the development and analysis of a simplified caricature model, valuable insight is gained into how the balance between chemotaxis, tip proliferation and tip death affects the tumour's ability to induce a vascular response from neighbouring blood vessels. In particular, it is possible to define the success of angiogenesis in terms of known parameters, thereby providing a potential framework for assessing the viability of tumour neovascularization in terms of measurable quantities.
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    Nonlinear Approximation by Trigonometric Sums (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 29-48 
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    Notes: Abstract We investigate the $L_p$ -error of approximation to a function $f\in L_p({\Bbb T}^d)$ by a linear combination $\sum_{k}c_ke_k$ of $n$ exponentials $e_k(x):= e^{i\langle k,x\rangle}=e^{i(k_1x_1+\cdots+k_dx_d)}$ on ${\Bbb T}^d,$ where the frequencies $k\in {\Bbb Z}^d$ are allowed to depend on $f.$ We bound this error in terms of the smoothness and other properties of $f$ and show that our bounds are best possible in the sense of approximation of certain classes of functions.
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    Uniformly Elliptic PDEs with Bounded, Measurable Coefficients (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 237-259 
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    Notes: Abstract Let $L[\,\cdot\,]$ be a nondivergent linear second-order uniformly elliptic partial differential operator defined on functions with domain $\Omega.$ Consider the question, "When is a function u a solution of $L[u] = 0$ on $\Omega$ ?" The naive answer, "u is a solution of $L[u] = 0$ on $\Omega$ if $u\in C^2(\Omega)$ and $L[u](x) = 0$ for all $x\in\Omega,$ " is clearly too limited. Indeed, if the coefficients of L are in $W^{1,2}\cap L^{\infty},$ then L can be rewritten in divergence form for which the notion of a "weak" solution can be applied. In this case there could be infinitely many functions that are "weak" but not classical solutions. More importantly, even if the coefficients of L are just bounded and measurable, the recent results of Krylov permit us to construct "solutions" of $L[u] = 0$ on $\Omega,$ and these "solutions" are generally no better than continuous; the "weak" solutions previously mentioned can be obtained by this construction, too. The preceding discussion provides us with an adequate extrinsic definition of solution (i.e., given a function u we either prove that it is or is not the result of such a construction) that has been used by several authors, but one that is not particularly satisfying or illuminating. Our major contribution in this paper is to show the following. I. There is an intrinsic definition of solution that is equivalent to the extrinsic one. II. Furthermore, the intrinsic definition is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients.
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    A Weighted Tauberian Theorem (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 397-406 
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    Notes: Abstract We prove a Tauberian theorem of the form $\phi * g (x)\sim p(x)w(x)$ as $x \to \infty,$ where p(x) is a bounded periodic function and w(x) is a weighted function of power growth. It can be used to study the weighted average of the form $(T^\alpha (\hbox {ln }T)^\beta)^{-1}\int _0^T h(t) \, dt.$
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    Limits on the computing power of biological systems (1981)
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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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    A topological study of the base sequences of MS2 and ϕX174 genoma (1981)
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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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    Volume of the conducting airways calculated from morphometric parameters (1981)
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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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    Variability in a single compartment system: A note on S. R. Bernard's model (1981)
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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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    The diffusion transit time; A simple derivation (1981)
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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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    Nonlinear oscillations in biology (1981)
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    Bulletin of mathematical biology 43 (1981), S. 121-123 
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    A comment on Lin's paper on nonlinear diffusion in biological systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 117-120 
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    On the stochastic theory of compartments: The leaving process of the two-compartment systems (1981)
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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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    Models on prebiotic polymer competition: A deterministic approach (1981)
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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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    Extinction probabilities in microbial predation: A birth and death approach (1981)
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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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    Prediction of the effect of the choroidal circulation on the retinal temperature during illumination (1981)
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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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    Spatially localized interactive neural populations—I. A mathematical model (1981)
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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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    A one-compartment model with stochastic parameter (1981)
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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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    A dose-response model for estimating lifetime tumor risks when cell killing occurs (1981)
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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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    A stochastic model for then-compartment irreversible system (1981)
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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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    Pulsatile blood flow in a rigid pulmonary alveolar sheet with porous walls (1981)
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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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    Dynamic structure theory: A structural approach to social and biological systems (1981)
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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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    Peak drug levels in linear pharmacokinetic systems—II. Conditions for a paradoxical injection rate effect with rectangular input (1981)
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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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    Peak drug levels in linear pharmacokinetic systems—I. Effect of rate of injection (1981)
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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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    Neuronal growth via hybrid system of self-growing and diffusion based grammar rules: I (1995)
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    Bulletin of mathematical biology 57 (1995), S. 205-227 
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    Notes: Abstract The formation of neuronal networks requires axonal growth towards target neutons. A simple set of grammar rules is introduced to describe axonal growth towards target cells situated both at short and long distances from the growing neuron. Growth for short distances is descrbed by growth following the highest gradient of a chemical compound (which is spread by diffusion from the targets). This approach fails to describe long-distance growth, which is addressed by adopting a graph grammar theory for growing trees. With these rules a flexible tool to draw network of neurons by computer can be developed.
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    Interlocking triads of growth control in tumors (1995)
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    Bulletin of mathematical biology 57 (1995), S. 345-366 
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    Notes: Abstract A pair of growth control triads are used to describe coincident tumor growth and liver regeneration after partial hepatectomy. The models are extensions of previous growth control models which describe tumor growth in an unperturbed host (Michelson and Leith, 1991,Bull. math. Biol. 53, 639–656; idem, 1992, Proceedings of the Third International Conference on Communications and Control, Vol. 2, pp. 481–490; idem, 1992,Bull. math. Biol. 55, 993–1011; idem,J. theor. Biol. 169, 327–338). The linkage between the two triads depends upon systemic signals carried by soluble factors, and mathematical descriptors based upon biological first principals are proposed. The sources of the growth factors, their targets and the processing of their signals are investigated. Analyses of equilibrium in the constant coefficients case and simulated growth curves for the dynamic system are presented, and the effects of growth factor-induced mitogenesis and angiogenesis are discussed in particular. A case is made for early and late responses in the coupled control system. The biology of the signal processing paradigm is placed within a new theoretical context and discussed with regard to tumor adaptation, liver differentiation and the development of a tumor hypoxic fraction.
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    Optimal population stabilization and control using the Leslie matrix model (1995)
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    Bulletin of mathematical biology 57 (1995), S. 381-399 
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    Notes: Abstract We consider the problem of optimal stabilization and control of populations which follow the Leslie model dynamics, within state space and control systems theory and methodology. Various types of culling strategies are formulated and introduced into the Leslie model as control inputs, and their effect on global asymptotic stability is investigated. Our new approach provides answers to several unexplored problems. We show that in general it is possible to achieve a desired stable equilibrium population level, through the design of a class ofshifted-proportional stabilizing culling policies. Further, we formulate general non-linear constrained opitmization problems, for obtaining the cost-optimal policy among this generally infinite class of such stabilizing policies. The theoretical findings are illustrated through the solution of the problem over an infinite planning horizon for a numerical example. A comparative study of the costs and dynamic effects of various culling strategies also supports the mathematical results.
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    A data assimilation technique applied to a predator-prey model (1995)
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    Bulletin of mathematical biology 57 (1995), S. 593-617 
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    Notes: Abstract A new approach for data assimilation, which is based on the adjoint method, but allows the computer code for the adjoint to be constructed directly from the model computer code, is described. This technique is straightforward and reduces the chance of introducing errors in the construction of the adjoint code. Implementation of the technique is illustrated by applying it to a simple predator-prey model in a model fitting mode. A series of identical twin numerical experiments are used to show that this data assimilation approach can successfully recover model parameters as well as initial conditions. However, the ease with which these values are recovered is dependent on the form of the model equations as well as on the type and amount of data that are available. Additional numerical experiments show that sufficient coefficient and parameter recoveries are possible even when the assimilated data contain significant random noise. Thus, for biological systems that can be described by ecosystem models, the adjoint method represents a powerful approach for estimating values for little-known biological parameters, such as initial conditions, growth rates, and mortality rates.
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    Building activity and longevity of queenless groups of the Oriental hornet,Vespa orientalis (1995)
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    Bulletin of mathematical biology 57 (1995), S. 441-460 
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    Notes: Abstract The effect of group size on behavioral parameters of the Oriental hornet,Vespa orientalis, was assessed experimentally under laboratory conditions. Hornet groups of various sizes (ranging from 1 to 100 individuals per group) comprised of young individuals (0–24 hr of age) devoid of a queen were placed in artificial breeding boxes (ABBs). The following three quantitative parameters were evaluated: the amount and rate of building as a function of the number of hornets in the group, the rate of oviposition as, related to group size and the longevity of hornets as a function of their group size. The probability for the occurrence of these events was similarly considered and additional behavioral parameters were only assessed qualitatively. Results of this investigation revealed a relation between the three mentioned quantitative behavioral parameters and the number of hornets per group. The number of hornets per group was positively related to the extent of building, the number of cells built by a group is $$2\pi \sqrt {group size} $$ , but negatively related to the rate of building. As for the delay of building, a non-monotone relation was found. The relation between number of hornets per group and the oviposition delay was found to be non-monotone; the number of hornets per group and their longevity were found to be inversely related. Discrepanices were recorded on the very small (1–2 individuals) or very large (100 individuals) hornet groups.
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    Fuzzy classification of nucleotide sequences and bacterial evolution (1995)
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    Bulletin of mathematical biology 57 (1995), S. 527-537 
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    Notes: Abstract A new method for reconstructing evolutionary relationship among bacteria by use of rRNA sequence data is proposed. The method is based on the concept of fuzzy classification of probabilitiesp(i), p(i/j) andp(i/j*) (i=A,G,C,U) of each sequence. The resulting partition tree shares common features of previous works but has some new peculiarities.
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    Book reviews (1995)
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    Bulletin of mathematical biology 57 (1995), S. 619-630 
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    Gyrotaxis: Interaction between algae and flagellates (1995)
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    Bulletin of mathematical biology 57 (1995), S. 631-650 
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    Notes: Abstract We describe the behaviour of motile microorganisms (e.g. flagellates) attracted by “gyrotaxis” to a sinking, non-motile particle (e.g. an algal cell). The model is based on the application of Stokes' solution for the flow field around the settling cell. The volume within which the flagellate is attracted to the sinking particle is determined from the trajectories of the flagellate. The model of gyrotaxis has several applications; these include the colonization of sinking marine snow particles with motile microoganisms and suspension feeding by protozoa.
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    Effect of dimensionality on Lotka-Volterra predator-prey dynamics: Individual based simulation results (1995)
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    Bulletin of mathematical biology 57 (1995), S. 507-526 
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    Notes: Abstract The effect of varying habitat dimensionality on the dynamics of a model predator-prey system is examined using an individual-based simulation. The general results are that in one dimension fluctuations in abundance of prey and predators occur over a large range of spatial scales (extinctions occur over many spatial scales). In two dimensions (and low mobilities of prey and predators) the dynamics become more predictably periodic at local scales and constant at larger scales due to statistical stabilization. In three dimensions, the model can become “phase-locked” with prey and predators displaying oscillations in abundance over large spatial scales.
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    Dicarboxylic acid fluxes during gluconeogenesis. No channelling of mitochondrial oxalacetate (1995)
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    Bulletin of mathematical biology 57 (1995), S. 557-568 
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    Notes: Abstract A rather complete model of the gluconeogenic pathway was used, with the known separate pools of mitochondrial and cytosolic oxalacetate, malate and aspartate. The fumarase, malate dehydrogenase and glutamate oxalacetate transaminase reactions were assumed to be isotopically actively reversible, but none at isotopic equilibrium. Malate was assumed to exchange actively between the mitochondrial and cytosol, while aspartate exchange was more limited, in agreement with the known electrogenic nature of aspartate export from the mitochondria. This model was fit to14C data obtained in hepatocyte studies, and to the whole rat14C data obtained by Heath and Rose (Biochem J. 227, 851–876, 1985). The latter data were easily fit to our model, when a single mitochondrial oxalacetate pool was assumed. However, invoking two mitochondrial oxalacetate pools, as proposed by Heath and Rose, with the oxalacetate formed via pyruvate carboxylase preferentially channelled to gluconeogenesis, could not be fit with the known differences in scrambling in glucose and glutamate produced from L[3-14C]lactate.
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    Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells (1995)
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    Bulletin of mathematical biology 57 (1995), S. 569-591 
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    Notes: Abstract Oscillatory secretion of insulin has been observed in many different experimental preparations ranging from pancreatic islets to the whole pancreas. Here we examine the mathematical features underlying a possible model for oscillatory secretion from the perifused, insulin-secreting cell line, HIT-15. The model includes the kinetics of uptake of glucose by GLUT transporters, the rate of glucose metabolism within the cell, and the effect of glucose on the rate of insulin secretion. Putative feedback by insulin on the rate of glucose transport into the cells is treated phenomenologically and leads to insulin oscillations similar to those observed experimentally in HIT cells. The resulting set of ordinary differential equations is simplified by time-scale analysis to a two-variable set of ordinary differential equations. Because of this simplification we can explore, in great detail, the characteristics of the oscillations and their sensitivity to parameter variation using phase plane analysis.
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    Sperm motility in the presence of boundaries (1995)
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    Bulletin of mathematical biology 57 (1995), S. 679-699 
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    Notes: Abstract The fluid dynamics of sperm motility near both rigid and elastic walls is studied using the immersed boundary method. Simulations of both single and interacting organisms are presented. In particular, we find that nearby organisms originally undulating with a 90° phase shift may adjust their relative swimming velocities and phase-lock. Comparisons with previous analytical results are also discussed. The tendency of a near-wall to attract organisms is demonstrated.
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    Contributions of memory B cells to secondary immune response (1995)
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    Bulletin of mathematical biology 57 (1995), S. 713-731 
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    Notes: Abstract The secondary immune response is one of the most important features of immune systems. During the secondary immune response, the immune system can eliminate the antigen, which has been encountered by the individual during the primary invasion, more rapidly and efficiently. Both T and B memory cells contribute to the secondary response. In this paper, we only concentrate on the functions of memory B cells. We explore a model describing the memory contributed by the specific long-lived clone which is maintained by continued stimulation with a small amount of antigens sequestered on the surfaces of the follicular dendritic cells (FDC). The behavior of the secondary response provided by the model can be compared with experimental observations. The model shows that memory B cells indeed play an important role in the secondary response. It is found that a single memory cell in a long-lived clone may not be long-lived. In the present note, the influences of relevant parameters on the secondary response are also explored.
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    Reproduction rate, feeding process, and liebig limitations in cell populations—II (1995)
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    Bulletin of mathematical biology 57 (1995), S. 749-782 
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    Notes: Abstract A biological system consisting of a population of cells suspended in a liquid substrate is considered. The general problem addressed in the paper is the derivation of the kinetic pattern of population growth as a statistical effect of a very large number of elementary interactions between a single cell and the molecules of nutrient in substrate. Solution of the problem is obtained in the form of equation expressing the population growth ratec as a function of substrate concentration,C s. The analytical expression derived is applied to a real bacterial population (Escherichi coli) and kinetic patterns are theoretically computed. The major findings, expressed roughly, without nuances, are: (i) the concentration of nutrient at the cell membrane,C c, can only be equal to either 0 (for theC s below some threshold valueC *) orC s (forC s〉C *); (ii) the Michaelis-Menten-Monod kinetics observed in experiments is an artifact: the pure (not contaminated by foreign factors) dependence ofc onC s is actually such that the functionc=c(C s) has practically linear increase whenC s〈C *, and is constant,c=c(C *)=const, whenC s〉C *; (iii) the Liebig principle is strictly fulfilled: up to a feasible accuracy of observation, under no circumstances can population growth be limited (controlled) by more than one substrate component—replacement of a limiting component for another one is an instant event rather than a gradual process.
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    Idiotypic regulation of B cell differentiation (1995)
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    Bulletin of mathematical biology 57 (1995), S. 841-881 
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    Notes: Abstract We study the equilibrium properties of idiotypically interacting B cell clones in the case where only the differentiation of B cells is affected by idiotypic interactions. Furthermore, we assume that clones may recognize and be stimulated by self antigen in the same fashion as by antiantibodies. For idiotypically interacting pairs of non-autoreactive clones we observe three qualitatively different dynamical regimes. In the first regime, at small antibody production an antibody-free fixed point, the virgin state, is the only attractor of the system. For intermediate antibody production, a symmetric activated state replaces the virgin state as the only attractor of the system. For large antibody production, finally, the symmetric activated state gives way to two asymmetric activated states where one clone suppresses the other clone. If one or both clones in the pair are autoreactive there is no virgin state. However, we still observe the switch from an almost symmetric activated state to two asymmetric activated states. The two asymmetric activated states at high antibody production have profoundly different implications for a self antigen which is recognized by one of the clones of the pair. In the attractor characterized by high autoantibody concentration the self antigen is attacked vigorously by the immune system while in the opposite steady state the tiny amount of autoantibody hardly affects the self antigen. Accordingly, we call the first state the autoimmune state and the second the tolerant state. In the tolerant state the autoreactive clone is down-regulated by its anti-idiotype providing an efficient mechanism to prevent an autoimmune reaction. However, the antibody production required to achieve this anti-idiotypic control of autoantibodies is rather large.
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    The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models (1995)
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    Bulletin of mathematical biology 57 (1995), S. 899-929 
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    Notes: Abstract It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI A parameters given by Connor and Stevens for the crab axon. When theI A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg A decreases and as inactivation rate decreases. We also study howI A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.
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    Fractals in science (1995)
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    Bulletin of mathematical biology 57 (1995), S. 939-941 
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    Coping with chaos: Analysis of chaotic data and the exploitation of chaotic systems (1995)
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    Bulletin of mathematical biology 57 (1995), S. 945-946 
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    On Decomposition Theorems for Hardy Spaces on Domains in ℂn and Applications (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 65-107 
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    Notes: Abstract Hardy spaces of analytic functions are studied both on strongly pseudoconvex domains in ℂn and on domains of finite type in ℂ2. Duality theorems, atomic decompositions, and factorization of functions are treated. Mapping properties of certain Hankel operators are studied.
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    Irregular Sampling in Wavelet Subspaces (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 181-189 
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    Notes: Abstract As a particular wavelet subspace, the Paley-Wiener space $B_{\pi}$ has both regular and irregular sampling theorems. A regular sampling theorem in general wavelet subspaces has been established for several years. In this paper, we discuss the irregular sampling problem in wavelet subspaces.
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    Autocorrelation Functions as Translation Invariants in L1 and L2 (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 217-225 
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    Notes: Abstract The Adler-Konheim theorem [Proc. Amer. Math. Soc. 13 (1962), 425-428] states that the collection of nth-order autocorrelation functions ${\cal M} = \{M^n(\cdot): n=1,2,\dots\}$ is a complete set of translation invariants for real-valued L1 functions on a locally compact abelian group. It is shown here that there are proper subsets of ${\cal M}$ that also form a complete set of translation invariants, and these subsets are characterized. Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions. In addition, any infinite subset of $\cal M$ is complete up to a sign. While stated here for functions on $\cal R,$ the proofs presented hold for functions on any locally compact abelian group that is not compact, in particular, on ${\cal R}^n$ and the integer lattice ${\cal Z}^n.$
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    Fourier Inequalities and Moment Subspaces in Weighted Lebesgue Spaces (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 49-64 
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    Notes: Abstract For $k \in {\Bbb N}, k \not= 0,$ define ${\cal F}_kf(\gamma) = \int_{{\Bbb R}^n} f(t)R_k(-2i \pi \gamma.t) \, dt, n\geq 1,$ where $R_k(i\lambda) = e^{i\lambda} - \sum^{k-1}_{j=0} \left(i \lambda \right)^{j} / \left(j~!\right).$ Pointwise estimates and weighted inequalities describing the local Lipschitz continuity of ${\cal F}_kf$ are established. Sufficient conditions are found for the boundedness of ${\cal F}_k$ from $L^p_v$ into $L^q_\mu,$ and a spherical restriction property is proved. A study of the moment subspaces of $L^p_v$ is next developed in the one-variable case, for $1 〈 p 〈 \infty, v$ locally integrable, $v 〉 0$ a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in $L^p_v.$ Characterizations are also given for each class. Applications related to the approximation and decomposition of ${\cal F}_k$ are discussed.
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    Designing Multiresolution Analysis-type Wavelets and Their Fast Algorithms (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 135-159 
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    Notes: Abstract Often, the Dyadic Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets, or the splines wavelets, whereas in continuous-time wavelet decomposition a much larger variety of mother wavelets is used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods for constructing wavelets ψwanted, with some desired shape and properties and which are associated with semi-orthogonal multiresolution analyses. We explain in detail how to design any desired wavelet, starting from any given multiresolution analysis. We also explicitly derive the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design techniques with examples that we have programmed with Matlab routines.
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    Weil Multipliers (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 191-215 
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    Notes: Abstract This paper studies a class of linear operators on spaces of functions of one real variable, which correspond to multiplication by a measurable function under the Weil transform $\Theta.$ These operators are called Weil multipliers, and arise out of the authors' study of Gabor series and radar ambiguity functions. Representation theory provides a natural class of Weil multipliers: the set of doubly periodic functions with absolutely convergent Fourier series, ${\bf A}({\bf T}^2).$ It will be proved that functions in ${\bf A}({\bf T}^2)$ are $L^p$ multipliers for all $1 \leq p \leq 2$ and, therefore, define bounded linear endomorphisms of ${\bf L}^p({\bf R}).$ Also, we record the fact that the Wiener lemma tells us something about the orbit structure of these multipliers acting on function spaces on the Heisenberg nilmanifold. Linear maps that correspond to multiplication by a function under a unitary conjugacy have a particularly simple spectral decomposition, which yields an approximation theory for these operators and provides insight into the foundation of the authors' previous work on approximate orthonormal bases. Finally, the problem of inversion of a multiplier will be analyzed for smooth functions that have a specified structure near their zeros.
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    On the Uniqueness in the Inverse Conductivity Problem (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 227-235 
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    Notes: Abstract Let $\Omega$ be a smooth domain in R2 containing a polygon D. The inverse conductivity problem to the the elliptic equation ${\rm div}((1+(k-1)\chi_D)\nabla u)=0\ {\rm in }\ \Omega$ is considered. We show that D is uniquely determined from boundary measurements corresponding two appropriately chosen Neumann datas.
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    Haar Type Orthonormal Wavelet Bases in R2 (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 1-14 
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    Notes: Abstract K.-H. Grochenig and A. Haas asked whether for every expanding integer matrix A ∈ Mn(ℤ) there is a Haar type orthonormal wavelet basis having dilation factor A and translation lattice ℤn. They proved that this is the case when the dimension n = 1. This article shows that this is also the case when the dimension n = 2.
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    On the Reproducing Formula of Calderon (1995)
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    The journal of Fourier analysis and applications 2 (1995), S. 15-28 
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    Notes: Abstract Let $1 〈 p 〈 \infty, h \in L_p ({\Bbb R}),$ and $f \in L_{p^\prime} ({\Bbb R}).$ Under certain conditions on $h,$ we shall prove that $\int_y^\infty (h_t \ast f)(x)\,dt/t$ converges nontangentially to $f(x_0)$ at $(x_0,0)$ for $a.a.\, x_0.$
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    URL: http://dx.doi.org/10.1007/s00041-001-4020-9
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    On a two urn model of Polya-type (1981)
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    Bulletin of mathematical biology 43 (1981), S. 33-45 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    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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    URL: http://dx.doi.org/10.1007/BF02460937
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    Nonlinear biological oscillators: Action-angle variable approach with an application to Cowan's model of neuroelectric activity (1981)
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    Bulletin of mathematical biology 43 (1981), S. 21-32 
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    ISSN: 1522-9602
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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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    URL: http://dx.doi.org/10.1007/BF02460936
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    A generalized Fokker-Planck equation in the case of the Volterra model (1981)
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    Bulletin of mathematical biology 43 (1981), S. 69-79 
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    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02460940
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    A hydrodynamical study of the flow in renal tubules (1981)
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    Bulletin of mathematical biology 43 (1981), S. 151-163 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02459440
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    Some comments on the conductance of slow inward current in cardiac muscle (1981)
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    Bulletin of mathematical biology 43 (1981), S. 183-199 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    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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    On the relevance of stochastic compartmental models to pharmacokinetic systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 245-247 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02459447
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    Identification of silent infections in SIR epidemics (1981)
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    Bulletin of mathematical biology 43 (1981), S. 249-257 
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    ISSN: 1522-9602
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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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    URL: http://dx.doi.org/10.1007/BF02462199
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  • 88
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    A reinforcement-depletion urn problem—I. Basic theory (1981)
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    Bulletin of mathematical biology 43 (1981), S. 327-340 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    Stochastic compartmental model with branching particles (1981)
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    Bulletin of mathematical biology 43 (1981), S. 347-360 
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    Topics: Biology , Mathematics
    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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    On Wei's fundamental equation for nerve excitation (1981)
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    Bulletin of mathematical biology 43 (1981), S. 371-372 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02462207
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    Phase-plane analyses of the Michaelis-Menten reaction equations (1981)
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    Bulletin of mathematical biology 43 (1981), S. 361-370 
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    Topics: Biology , Mathematics
    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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    Reply to Chyuan-Yih Lee (1981)
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    Bulletin of mathematical biology 43 (1981), S. 372-373 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
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    URL: http://dx.doi.org/10.1007/BF02462208
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    Further considerations in a theoretical description of gas transport in lung airways (1981)
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    Bulletin of mathematical biology 43 (1981), S. 517-548 
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    Topics: Biology , Mathematics
    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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    Effects of convective and dispersive interactions on the stability of two species (1981)
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    Bulletin of mathematical biology 43 (1981), S. 593-610 
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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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    Role of the hypergeometric distribution in the transmission of excitation through sympathetic ganglia (1981)
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    Bulletin of mathematical biology 43 (1981), S. 611-618 
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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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    The uniqueness of protein sequences: A Monte Carlo analysis (1981)
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    Bulletin of mathematical biology 43 (1981), S. 619-639 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    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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    Tolerance nets (1981)
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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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    Relation between diameter and flow in branches of the bronchial tree (1981)
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    Bulletin of mathematical biology 43 (1981), S. 681-691 
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    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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    Stochastic analysis of a nonlinear model for selection of biological macromolecules (1981)
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    Bulletin of mathematical biology 43 (1981), S. 665-680 
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    Topics: Biology , Mathematics
    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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    Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering (1981)
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    Bulletin of mathematical biology 43 (1981), S. 651-664 
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    ISSN: 1522-9602
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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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    URL: http://dx.doi.org/10.1007/BF02458415
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