Showing papers in "Bellman Prize in Mathematical Biosciences in 1981"
TL;DR: The metric presented in this paper makes possible the comparison of the many nonbinary phylogenetic trees appearing in the literature, and provides an objective procedure for comparing the different methods for constructing phylogenetics trees.
Abstract: A metric on general phylogenetic trees is presented. This extends the work of most previous authors, who constructed metrics for binary trees. The metric presented in this paper makes possible the comparison of the many nonbinary phylogenetic trees appearing in the literature. This provides an objective procedure for comparing the different methods for constructing phylogenetic trees. The metric is based on elementary operations which transform one tree into another. Various results obtained in applying these operations are given. They enable the distance between any pair of trees to be calculated efficiently. This generalizes previous work by Bourque to the case where interior vertices can be labeled, and labels may contain more than one element or may be empty.
2,519 citations
TL;DR: In this article, it was shown that for a collection of individuals from several species residing in a region R which is the union of a number of nonoverlapping subregions, the mean s and variance σ 2 of the number of species from C represented in r are given by explicit functions of α, provided the number n i are known.
Abstract: Consider a collection C of individuals from several species residing ina region R which is the union of a number of nonoverlapping subregions. Let n i be the number of individuals from species i which belong to C and hence reside somewhere in R . The simplest hypothesis about the spatial distribution throughout R of the members of C is that their placement in dwelling sites is random and noninteractive, with the probability of a given individual of C residing in a particular subregion r of R equal to the ratio α ofthe area of r to the area of R . It is shown here that when this hypothesis of random placement holds, the mean s and variance σ 2 of the number of species from C represented in r are given by explicit functions of α, provided the number n i are known. Thus, if all the species in C have been censured, species-area data permit a test of the hypothesis of random placement. The nature of the dependence of s and σ 2 on α is discussed in detail for special cases in which the numbers n i are given by such theoritical abundance relations as the logarithmic series distribution, the “broken stick” distribution, the lognormal distribution, the Poisson lognormal distribution, and the gamma distribution. The arguments employed here to deduce consequences of the hypothesis of random choice are rigorous and exact. No use is made of the assumption, commonly made heretofore (but not in general correct, even under the hypothesis of random placement), that a species-area curve (giving the number of species expected to be found in a sample of known area ) must have the same form as the corresponding collector's curve (giving the number of species expected in a sample of a known number of individuals ). Nor is it assumed in advance, as is often done in the theory of island biogeography, that the distribution of individuals throughout the subregions of R is such thatthe species abundance relations for subregions of different areas must be of a preassigned type, i.e., must share a common form, such as that associated with a truncated lognormal distribution.
401 citations
TL;DR: In this article, a simplified Keller-Segel model for the chemotactic movements of cellular slime mold is reconsidered, in particular, under the circumstances under which the cell distribution can autonomously develop a δ-function singularity.
Abstract: A simplified Keller-Segel model for the chemotactic movements of cellular slime mold is reconsidered. In particular, we ask for the circumstances under which the cell distribution can autonomously develop a δ-function singularity. By the use of suitable differential inequalities, we show that this cannot happen in the case of one-dimensional aggregation. For three or more dimensions, we produce time developments which do become singular, while in the important special case of two-dimensional motion, we advance arguments that the possibility of chemotactic collapse requires a threshold number of cells in the system.
359 citations
TL;DR: In this article, a rational mathematical procedure is proposed for approximating solutions to the partial differential equations of cross-bridge kinetics in theories of muscular contraction of the type first proposed by A. F. Huxley.
Abstract: A rational mathematical procedure is proposed for approximating solutions to the partial differential equations of cross-bridge kinetics in theories of muscular contraction of the type first proposed by A. F. Huxley. The essence of the procedure is to approximate the exact bond-distribution functions by distributions of prescribed form, and this leads to a set of first-order ordinary differential equations on the low-order moments of the approximate distributions. Thus the procedure effectively results in a lumped-parameter model of muscle approaching the structural simplicity of the classic two-element model, but one which exhibits more realistic behavior. The approximation is worked out in detail for Huxley's original (1957) two-state model (modified slightly to produce a more realistic stretch response), compared with exact solutions of the model, and used to predict muscle behavior under various conditions. It is anticipated that this approximation, with its attendant conceptual and computational simplifications, will make recent theoretical advances in molecular contraction mechanics more accessible for applications in macroscopic muscle dynamics. Generalizations of the procedure to the case of length-dependent behavior are discussed.
211 citations
TL;DR: In this paper, a model of a myelinated axon is constructed, and some threshold conditions are derived using Lyapunov and comparison theorem techniques, and the model takes the form of a nonlinear differential-difference system.
Abstract: A model of a myelinated axon is constructed, and some threshold conditions are derived using Lyapunov and comparison theorem techniques. The model takes the form of a nonlinear differential-difference system, and the threshold conditions show the rest state to be locally, but not globally, stable.
122 citations
TL;DR: In this article, a method called exhaustive modeling is proposed to obtain the whole set of output indistinguishable linear compartmental models using standard decomposition of a minimal representation into four mutually exclusive parts.
Abstract: Structures are unidentifiable if there exist several parameter values which lead to the same input-output relations. This paper presents a method, called exhaustive modeling, to obtain the whole set of output indistinguishable linear compartmental models. The method uses standard decomposition of a minimal representation into four mutually exclusive parts, namely: observed-controlled, observed-uncontrolled, unobserved-controlled, unobserved- uncontrolled. The connection of this standard decomposition with Kalman's canonical form is developed. Several simple examples are included to illustrate the method.
117 citations
TL;DR: In this article, a model for age-dependent populations with diffusion in one-dimensional environments is presented, in which three simple birth-death mechanisms are used to avoid crowding.
Abstract: This paper discusses some models for age-dependent populations with diffusion in one-dimensional environments. Three simple birth-death mechanisms are used. There are two diffusion processes: one represents random movement and the other movement to avoid crowding. The resulting models are reduced to systems of partial differential equations. These are sufficiently simple that an analysis of the interaction of diffusion and the birth-death processes is possible. An analysis is presented of the large-time behavior of populations in limited environments.
76 citations
TL;DR: A criterion is developed which indicates that the theory can break down at high degrees of cross-linking for receptors which have two or more combining sites, and in such cases intramolecular reactions need to be considered.
Abstract: The interaction of multivalent ligand with cell surface receptors is the first step in triggering a number of biological responses. Here I present and analyze a mathematical model for the binding of a multivalent ligand, containing v identical reactive sites, to monovalent or multivalent cell surface receptors. The initial binding of such ligands to a cell is followed by further reactions on the surface which cross-link receptors and lead to the formation of ligand-receptor clusters. In the presence of monovalent haptens, I consider the possibility that clustering can be competitively inhibited. By introducing the concept of an effective valence for the ligand, which is the number of reactive groups that can simultaneously bind to surface receptors, one can take into account steric hindrance and the inability of some ligand determinants to participate in cross-linking reactions because they are oriented incorrectly. When the ligand's effective valence is greater than two, a number of different criteria can be used to measure receptor clustering: (1) the fraction of receptor sites occupied by multiply bound ligand, (2) the concentration of cross-links holding distinct receptors together, and (3) the concentration of cell surface aggregates. The dependence of these three aggregation measures on ligand and monovalent hapten concentration is studied. A number of theorems are proven about the general shape and characteristics of cross-linking curves. For ligands which can simultaneously bind more than two cell surface receptors the cross-linking curve is not symmetric, although as in the case of bivalent ligands, the curve generated using criteria (1) or (2) has a single maximum. Throughout the analysis intramolecular reactions and ring formation are ignored. A criterion is developed which indicates that the theory can break down at high degrees of cross-linking for receptors which have two or more combining sites. In such cases intramolecular reactions need to be considered. As an application, the theory is used to interpret previously published experiments on the release of histamine by basophils.
75 citations
TL;DR: In this paper, a density dependent population that is subject to small environmental and demographic randomness is studied and the notion of persistence, as measured by time to extinction, is compared with other qualitative measures of stability.
Abstract: Persistence, as measured by time to extinction, is studied in a density dependent population that is subject to small environmental and demographic randomness. Diffusion processes are formally derived from branching processes in constant and random environments. The moment generating function of the extinction time, which satisfies a second order ordinary differential equation, is found asymptotically in the limit of small diffusion and is related to the diffusion limit of the Galton-Watson process and the Ornstein-Uhlenbeck process. Extinction occurs with probability one, though the mean and variance of the extinction time are found to be exponentially large and suggest the extinction time is exponentially distributed. The notion of persistence is compared with other qualitative measures of stability. Four examples are studied and compared.
69 citations
TL;DR: In this article, the authors develop an optimal harvesting policy for age-structured populations using a model for which the basic equations reduce to a pair of ordinary diffential equations for the total population and the per-capita birth-rate.
Abstract: In this paper we develop optimal harvesting policies for age-structured populations using a model for which the basic equations reduce to a pair of ordinary diffential equations for the total population and the per-capita birth-rate. Our assumptions insure the existence of a critical size Pc(t) which maximizes the instantaneous growth-rate at time t. We study the infinite-horizon problem, using the overtaking criterion of optimality, and show that: for a large population with ample per-capita birth-rate the optimal policy is to instantly reduce the stock to the critical value Pc(0), and then to harvest along the path Pc(t); for a sufficiently small population it is optimal to refrain from harvesting until the population reaches Pc(t), and then to harvest along Pc(t) for all subsequent time; for a large population with a small per-capita birth-rate, it is generally best to initially remove a given amount of stock, then to refrain from harvesting until the population reaches Pc(t), and finally to harvest alongPc(t).
67 citations
TL;DR: It is found that sensitivity to noise increases monotonically as exploitation increases, and this is primarily determined by the behavior of the dominant eigenvalue rather than the subdominant modes of the response.
Abstract: The sensitivity of a periodically reproducing population with overlapping generations to fluctuations in its environment is examined with a model that preserves the essential features of the age structure of the population. The effect of fluctuations is introduced by perturbing the recruitment of young individuals. This is appropriate for many populations of interest. It is found that the return time is in very poor correspondence with the population's sensitivity to noise and, except for determining whether a system is either stable or unstable, is of little use. The results are affected very little by the preservation or not of age structure within the adult population, and only in detail by the use of discrete rather than continuous-time models. The determining factors are the form of the stock-recruitment relationship (particularly the degree of density dependence) and the way in which noise is introduced. For a specific example based on the North Sea herring, we find that sensitivity to noise increases monotonically as exploitation increases, and this is primarily determined by the behavior of the dominant eigenvalue rather than the subdominant modes of the response. Such an increase in sensitivity is undesirable in managed populations. The variance of the population and yield depend upon the density-dependent nature of the noise, and the form of this is particularly important for overexploited stocks. In general there can be a tradeoff between yield and stability of yield.
TL;DR: A mathematical model for a two-competing-prey, one-predator system is analyzed and the conclusion is that the outcomes depend critically on the prey species' capability of invading the complementary subcommunity formed by predator species and other prey.
Abstract: In this paper we analyze a mathematical model for a two-competing-prey, one-predator system and discuss the effect of predation on the two competing prey. Our basic assumption is that the predator can survive on either of the two competing prey in a one-predator, one-prey subcommunity. Conditions for the coexistence of the two competing prey and the extinction of either prey are completely determined and discussed. The conclusion is that the outcomes depend critically on the prey species' capability of invading the complementary subcommunity formed by predator species and other prey. Our mathematical analysis is rigorous and global.
TL;DR: This paper gives a review of mathematical models of cell populations containing a description of the cycle or even of the biochemical events controlling cell growth.
Abstract: This paper gives a review of mathematical models of cell populations. Attention is focused on the models containing a description of the cycle or even of the biochemical events controlling cell growth. An effort is made to present the material in a consistent framework, and to bring into evidence possible connections. The applications of the models to the analysis of experimental data from tumor cell populations are particularly stressed.
TL;DR: This paper presents a generalization of the classical compartmental models using the concept of "pipes" between the compartments and taking into account the time needed for the material to pass through these pipes.
Abstract: This paper presents a generalization of the classical compartmental models using the concept of “pipes” between the compartments and taking into account the time needed for the material to pass through these pipes. A general linear delay differential system describing such models is derived and some properties of its solutions—such as nonnegativity, boundedness, stability—are established. For stationary and asymptotically stationary systems the asymptotic solution is obtained, too. The theory presented generalizes several earlier results concerning more special equations.
TL;DR: In this article, structural equivalence of linear parametric systems is introduced, which is a generalization of the conventional concept of system equivalence and is closely related to the notion of structural identifiability.
Abstract: In this paper we introduce the concept of structural equivalence, which is a generalization of the conventional concept of system equivalence and is closely related to the notion of structural identifiability. We develop conditions for structural equivalence of linear parametric systems which are based on the known criteria for structural identifiability. The particularization of the results to the class of linear compartmental models offers a new means for tackling the a priori model selection problem in this area. Some examples are given to illustrate the application of the presented theory.
TL;DR: This paper develops and analyzes a mathematical model for a generalized inflammatory response to bacterial invasion of a tissue region assumed to be homogeneous on a macroscopic scale, in order to study the dynamical behavior of the system.
Abstract: The inflammatory response to bacterial invasion of tissue is a complex combination of chemical and physical events, and the outcome of a particular challenge is dependent upon the interaction of these. Thus a wide variety of behavior can be observed for the operation of this response. In this paper we develop and analyze a mathematical model for a generalized inflammatory response to bacterial invasion of a tissue region assumed to be homogeneous on a macroscopic scale, in order to study the dynamical behavior of the system. Our analysis allows interpretation of the outcome of a challenge in terms of key parameters representing the rate processes involved. It demonstrates how abnormalities in these processes can lead to pathological behavior. Numerical values for the parameters are estimated from experimental literature sources, and used in example computations to illustrate the predictions of the model under physiological conditions.
TL;DR: In this article, a direct method of determining whether unknown parameters of a dynamical system can be determined uniquely from an input-output experiment is presented, relying on the fact that a completely controllable and completely observable system is parameter identifiable if and only if it is not algebraically equivalent to any other system.
Abstract: A direct method of deciding whether or not the unknown parameters of a dynamical system can be determined uniquely from an input-output experiment is presented. The method relies on the fact that a completely controllable and completely observable system is parameter identifiable if and only if it is not algebraically equivalent to any other system.
TL;DR: It is proved that two undirected binary cladistic characters are compatible iff their smaller states are disjoint or one is a subset of the other.
Abstract: It is proved that two undirected binary cladistic characters are compatible iff their smaller states are disjoint or one is a subset of the other. The concept of a cladistic character as an ordered tree of subsets is defined. Cladistic characters that have the same number of elements in their corresponding states are defined to be “nesting equivalent.” The equivalence classes of this relation are called “nestings.” A certain class of n-tuples is shown to have a biunique correspondence with the n!-membered set of all nestings of n binary characters. The model of randomness proposed is that all characters that are nesting equivalent are equally likely. The probability that a pair of undirected binary characters is compatible is derived under this model. This result is extended to collections of undirected binary characters, to collections of directed binary characters, and finally to collections that may include multistate characters. Some proofs are presented which allow a more efficient use of the n-tuple representation of ordered trees of subsets.
TL;DR: In this article, the existence of a family of traveling wave solutions which are trains of infinitely many pulses has been shown, and the set of integer sequences which identify the spacing between individual pulses in a solution serve as a natural index for the family.
Abstract: Some nerve axon equations, in addition to admitting a traveling wave solution in the form of a solitary pulse, also admit a family of double pulse solutions. In this case, application and extension of the results of Silnikov further show the existence of an uncountable discrete family of traveling wave solutions which are trains of infinitely many pulses. All solutions in this family travel with the velocity of the solitary pulse solution. Included are nonperiodic doubly infinite trains, trains which are infinite in only one direction, and a countable family of periodic solutions. The set of integer sequences which identify the spacing between individual pulses in a solution serve as a natural index for the family. A proof and computational results are provided for the piecewise linear FitzHugh- Nagumo equations.
TL;DR: A high-efficiency mechanochemical model of contraction for striated muscle is developed via the formalism of statistical mechanics that describes the time evolution of ultrastructural (macromolecular) phenomena and their ensemble (macroscopic) averages.
Abstract: A high-efficiency mechanochemical model of contraction for striated muscle is developed via the formalism of statistical mechanics. The myosin cross-bridges of the half-sarcomere ensemble cycle through five biochemical states. The structural components of the sliding-filament system include extensible myosin S2-units and indefinite arrays of equivalent actin sites. ATP ligand exchanges provide the far-from-equilibrium thermodynamic driving potential for cross-bridge cycling. The strain-dependent rate constants obey the self-consistency requirements of detailed balance. The mathematical solution for the linear time-varying system of equations of cross-bridge biokinetics employs discrete-time state transition matrices. Computer simulations describe the time evolution of ultrastructural (macromolecular) phenomena and their ensemble (macroscopic) averages. Modest fidelity for mechanical transients of muscle under many varied protocols, including conditions of fatique, is obtained between the simulations and their experimental counterpart.
TL;DR: It is found that stable predator-prey coexistence can be achieved by introducing either self-limitation of prey or mutual interference between predators.
Abstract: This paper studies predator-prey interactions where the predators eat only newborn prey or, equivalently, eggs. Such an egg-eating model has previously been shown numerically to lead to instability in the size of both populations. Various modifications of the egg-eating model are discussed, and studied for stability. It is found that stable predator-prey coexistence can be achieved by introducing either self-limitation of prey or mutual interference between predators. Spatial diffusion of both populations and unequal hunting capacities for different ages of predators have no effect on the stability of the system.
TL;DR: In this article, a Kolmogorov-type predator-prey system with periodic coefficients was considered, and conditions under which periodic solutions exist, using bifurcation techniques.
Abstract: We consider a Kolmogorov-type predator-prey system with periodic coefficients, and derive conditions under which periodic solutions exist, using bifurcation techniques. The results are applied to a system with a periodic carrying capacity.
TL;DR: Prospective topologies of phylogenetic trees can be tested as hypotheses using statistics based on parsimony if the unknown branch lengths of the trees are different for different characters, and when the transition probabilities between the states of characters are unequal in a known or unknown degree.
Abstract: Prospective topologies of phylogenetic trees can be tested as hypotheses using statistics based on parsimony. If the unknown branch lengths of the trees are different for different characters, the method still works. When the transition probabilities between the states of characters are unequal in a known or unknown degree, the method still works. Hybridization or horizontal gene transfer in the history of a group can never be rejected; whether it can be confidently detected is problematical. Only four species are treated here and only binany character.
TL;DR: It is shown that realized genotypic fitness is frequency-independent only for exceptional cases, and it was possible to find necessary and sufficient conditions for one genotype to be superior in realized fitness to another genotype for all genotyping frequencies (frequency-independent ranking).
Abstract: The meaning of genotypic fitness in Mendelian populations consisting of unisexual and/or bisexual individuals is discussed for a model which is applicable toplant populations but not only to these. In order to encourage general application the model is formulated for arbitrary numbers of loci, sex-linked or autosomal, and arbitrary numbers of alleles per locus. The necessity of expressing fitness values in terms of the number of successful gametes is demonstrated. To prevent confusion with other frequently used fitness concepts, the term “realized genotypic fitness” is introduced. Such fitness values have the general advantage that they determine directly the gametic frequencies in the next generation. It is shown that realized genotypic fitness is frequency-independent only for exceptional cases. Necessary and sufficient conditions for the occurrence of such casesare derived. The mode of dependence of realized fitness on genotypic frequencies is illustrated: along straight lines passing through the state space of genotypic structures, realized genotypic fitness changes monotonically. Hence the conclusion that extreme fitness values are not assumed when all genotypic frequencies are positive. It was possible to find necessary and sufficient conditions for one genotype to be superior in realized fitness to another genotype for all genotypic frequencies (frequency-independent ranking).
TL;DR: The empirical application indicated that the within-household and between- households transmission rates differ by a factor of the order of 10 5 for each type of susceptible, thus lending support to the postulate that a contagious disease epidemic effectively consists of the summation of smaller, almost independent outbreaks.
Abstract: A discrete-time model and a continuous-time model, each allowing for two types of susceptibles, are constructed for studying the spread of an infectious disease. The estimation of four parameters is considered for the purpose of assessing the potential (measure of opportunity) that an individual has for provoking the disease in a given susceptible from the same, or from a different, household when the susceptible was, or was not, previously vaccinated. In order to avoid generalities and to support the reasonableness of the basic assumptions, the approach is presented with continuing reference to a detailed data set for an epidemic of variola minor (Brazil, 1956). The empirical application indicated that the within-household and between-households transmission rates differ by a factor of the order of 10 5 for each type of susceptible, thus lending support to the postulate that a contagious disease epidemic effectively consists of the summation of smaller, almost independent outbreaks (occuring, in this case, in the households). In the study epidemic, previous vaccination seemed to reduce the between-households transmission rate much more than the within-household transmission rate. Moreover, on average, the chance that a previously vaccinated susceptable acquires variola minor from an infected person of the same household is about one-fifth of the corresponding chance of an unvaccinated susceptible.
TL;DR: In this paper, a set of parameters which measure the mean time a random particle resides in individual compartments in response to a given load distribution is presented. But these parameters are related to other time parameters and to each other.
Abstract: This paper is concerned with a set of parameters which measure the mean time a random particle resides in individual compartments in response to a given load distribution. These parameters are related to other time parameters and to each other.
TL;DR: In this paper, an analytic nonconstant periodic solution for predator-prey systems with continuous time delay is presented, in the case that the integral kernel is the function ae −at by making use of the fixed-point theorem.
Abstract: This paper combines previously developed models and results for predator-prey systems with continuous time delay by Volterra, Bownds, Cushing, MacDonald, Worz- Busekros, and others. We present an analytic nonconstant periodic solution in the case that the integral kernel is the function ae −at by making use of the fixed-point theorem.
TL;DR: In this paper, the authors derived mean residence times for certain populations of particles, such as those initially in a given compartment or those which exit from a compartment, and the washout curve was interpreted and analyzed in terms of these concepts.
Abstract: Mean residence times for certain populations of particles, such as those initially in a given compartment or those which exit from a given compartment, are derived. The previously given mean residence time for all particles is expressed in terms of those constituent mean times. These same concepts are developed for a system under steady input. The washout curve is interpreted and analyzed in terms of these concepts.
TL;DR: In this article, a set-labeled tree is used to derive a recurrence equation giving the number t ( n, r ) of cladistic characters on n species having r states.
Abstract: A cladistic character can be viewed as a type of set-labeled tree. This representation is used to derive a recurrence equation giving the number t ( n,r ) of cladistic characters on n species having r states. Values for t ( n,r ) are given for r up to 5 and n up to 30.
TL;DR: In this paper, a necessary and sufficient condition for local identifiability, based solely on a consideration of the structure of the model, the characteristics of the observations, and the possibilities of injection into the system, was proposed.
Abstract: When the experimental results are insufficient in relation to the complexity of the model under consideration, there exist an infinity of solutions of the problem of numerical identification of the model parameters. In such cases, the majority of analogue or numerical calculation procedures yield a particular solution of no real value. By the study of local identifiability, such a situation may be revealed. This paper gives a necessary and sufficient condition for local identifiability, based solely on a consideration of the structure of the model, the characteristics of the observations, and the possibilities of injection into the system. Being based purely on structural characteristics, the condition has the great advantage over previous techniques of requiring no calculation and thus being very easy to apply.