Showing papers in "Bellman Prize in Mathematical Biosciences in 1979"
TL;DR: Two classes of two-event models for carcinogenesis are proposed in this paper, where mutations are associated with cell divisions, and the target tissue is allowed to grow in size, and both deterministic and stochastic elements are involved.
Abstract: Two classes of two-event models for carcinogenesis are proposed. These models differ from some other multistage theories of carcinogenesis in that mutations areassociated with cell divisions, and the target tissue is allowed to grow in size. The first class of models is entirely stochastic, whereas in the second class of models, both deterministic and stochastic elements are involved. The incidence curves for childhood and adult cancers generated by these models are exhibited. Tests of the models against data were not undertaken. However, with the proper choice of parameters, these models generate incidence curves with the qualitative features of the curves seen in many common human carcinomas. Moreover the model proposed here for embryonal tumors is not distinguishable from the two-mutation model of Knudson et al. [6, 10, 11] on the basis of available data.
481 citations
TL;DR: In this paper, the Sunflower seed pattern can be approximated by the method of Davis and Mathai, which has inspired the cover of this journal, and concise formulae are given for the points where visible arcs of different order begin and end, and concomitant locations of irregularity shaped junction seeds.
Abstract: The flower or seed pattern in a sunflower head can be approximately constructed by the method of Davis and Mathai [1], which has inspired the cover of this journal. This construction reflects, besides the well-known spiral arcs, two kinds of irregularities, one of which is also observed in natural heads, the other not. The latter kind of irregularities is shown to be due to the semiempirical construction employed and can be avoided by a more pertinent, wholly mathematical construction program, which predicts the whole pattern, including the irregularities that are seen in nature, and only these. In particular, concise formulae are given for the points where visible arcs of different order begin and end, and for the concomitant locations of irregularity shaped junction seeds.
306 citations
TL;DR: Improvements in a Newton-Euler approach to spatial open-chain mechanism analysis introduced by one of the authors are presented, which are shown to satisfy a number of necessary conditions, thereby validating to some extent both the methodology and the corresponding computer program.
Abstract: This paper presents some improvements in a Newton-Euler approach to spatial open-chain mechanism analysis introduced by one of the authors in an earlier publication. The improvements have to do both with the introduction of simplified notation and with more efficient computational procedures. The validity and utility of the method is illustrated by an application to the problem of calculating joint torques for the legs of a hexapod locomotion system. The results obtained agree well with experimental measurements and are also shown to satisfy a number of necessary conditions, thereby validating to some extent both the methodology and the corresponding computer program.
236 citations
TL;DR: In this article, a neurophysiological model is described which demonstrates the capacity to learn, to generalize, to compute multivariate mathematical functions, and to decompose input commands into sequences of output commands in a context-sensitive manner.
Abstract: Classical AND/OR goal, or task, decomposition techniques are generalized to deal with the problem of sensory-interactive goal-directed behavior in biological organisms. A neurophysiological model is described which demonstrates the capacity to learn, to generalize, to compute multivariate mathematical functions, and to decompose input commands into sequences of output commands in a context-sensitive manner. Evidence is presented that clusters of neurons with such properties are arranged in hierarchical structures in the brain so as to produce AND/OR task compositions. At the lowest levels in the motor system these clusters transform coordinates and compute servo functions. At the middle levels they decompose input commands into sequences of output commands which give rise to behavior patterns. Mechanisms by which feedback can alter these decomposition sequences to compensate for perturbations and uncertainties in the environment are described. At the highest levels of the hierarchy there are goal selecting and evaluating mechanisms. It is argued that in higher mammals these upper levels of the motor hierarchy are the mechanisms of planning and problem solving.
149 citations
TL;DR: In this article, the basic equations of the theory reduce to systems of ordinary differential equations, and they discuss certain qualitative aspects of these systems; in particular, they show that for many cases of interest periodic solutions are not possible.
Abstract: This paper presents two simple models for nonlinear age-dependent population dynamics. In these models the basic equations of the theory reduce to systems of ordinary differential equations. We discuss certain qualitative aspects of these systems; in particular, we show that for many cases of interest periodic solutions are not possible.
146 citations
TL;DR: In this paper, a set of conditions for structural identifiability of compartmental systems is given that can be tested in an easy way directly on the compartmental diagram, and their relation with other structural properties is discussed.
Abstract: The paper deals with the relations between a priori identifiability and other structural properties and with identifiability conditions for compartmental systems. Reference is made to the notions of fixed structure system and of structural property. Structural controllability and observability and input and output connectability of compartmental systems are defined, and their relation discussed. Formal definitions are given for structural identifiability and unique structural identifiability. A newly formulated set of conditions for structural identifiability of compartmental systems is given that can be tested in an easy way directly on the compartmental diagram. Some observations reported in recent literature are discussed.
95 citations
TL;DR: In this paper, the authors considered the problem of finding the canonical solution of the nonautonomous logistic equation in the limit of large t by each solution obeying an initial condition of the form x(t0) = x0> 0.
Abstract: For each choice r and K as functions mapping (−∞, ∞) into a compact subset of (0, ∞), the nonautonomous logistic equation, x (t)=r(t)x(t) 1− x(t) K(t) , (a) possesses a special solution x∗:(−∞, ∞)→(0, ∞), here called the canonical solution, which is approached, in the limit of large t, by each solution obeying an initial condition of the form x(t0) = x0> 0. At each time t, the value x∗(t) of x∗ is given by a functional F of the histories of r and K up to t, i.e., x ∗ (t)= F (r t ,K t ) , (b) where rt(s)=r(t−s) and Kt(s)=K(t−s) for all s⩾0. The functional F , although nonlinear has a simple form and possesses regularity properties of the type assumed in the theory of “fading memory”. Results from that theory are applied to obtain asymptotic approximations to F appropriate for those cases in which the “carrying capacity of the environment,” K, varies slowly in time and for those cases in which K fluctuates at an arbitrary rate, but remains close to a constant value. For each choice of r, K, t0, and x0, the solution x of (a) obeying the initial condition x(t0)=x0>0, is given by x(t)= F (r t ,T (t−t 0 ,x 0 ) K t ) , where T(t−t0,x0) is an appropriate “leveling operator.” For this reason, the results obtained for the canonical solution (b) hold with only minor modifications for all other positive solutions. Toward the end of the paper it is pointed out that the negative of the functional F enters into relations analogous to those which restrict the stress functional in the thermodynamics of materials with fading memory; here K plays the role of the “strain,” and an appropriate “free-energy functional” is constructed. It is shown that, on solutions x of (a), the value ϕ(t) of the free-energy functional obeys the relations ϕ(t)= 1 2 x(t) 2 −K(t)x(t) and ϕ (t) ⩽ − x(t) K (t) .
89 citations
TL;DR: It is suggested that age-selective predation could be a more effective means of extinction than age-indiscriminate predation.
Abstract: This paper combines previously developed models for nonlinear age-dependent population dynamics with the classical Volterra-Lotka model of interacting predator and prey populations. For the resulting models the basic evolution equations of the theory reduce to systems of ordinary differential equations. The limiting dynamics of the predator and prey populations are shown to depend substantially on what ages of prey are eaten by predators. Two cases in particular are studied: where the predator eats all ages of prey indiscriminately, and where the predator eats only eggs (or newborns, equivalently). Indiscriminate eating is found to lead to stable periodic oscillations in numbers of predator and prey, such as occur in the Volterra-Lotka equations, while egg eating leads to oscillations which increase rapidly in amplitude and result, ultimately, in the extinction of both predator and prey. It is suggested that age-selective predation could be a more effective means of extinction than age-indiscriminate predation.
70 citations
TL;DR: In this article, the authors studied the dynamics of a chemostat in which two microbial populations grow and compete for a common substrate and showed that the two populations cannot coexist in a spatially uniform environment which is subject to time invariant external influences unless the dilution rate takes on one of a discrete set of special values.
Abstract: The dynamics of a chemostat in which two microbial populations grow and compete for a common substrate is examined. It is shown that the two populations cannot coexist in a spatially uniform environment which is subject to time invariant external influences unless the dilution rate takes on one of a discrete set of special values. The dynamics of the same system are next considered in the stochastic environment created by random fluctuations of the dilution rate about a value that allows coexistence. The information needed for the description of the random process of the state of the chemostat is obtained from the transition probability density function. By modeling the system as a Markov process continuous in time and space, the transition probability density is obtained as solution of the Fokker-Planck equation. Analytical and numerical solutions of this equation show that extinction of either one population or the other will ultimately take place. The time required for extinction, the evolution of the mean composition with time, the steady states of the latter and the dependence of all the above on the intensity of the random noise are also calculated using constants appropriate to the competition of E. coli and Spirillum sp. The question of making predictions as to which population is the more likely to become extinct is treated finally, and the probabilities of extinction are calculated as solutions of the steady state version of the backward Fokker-Planck (Kolmogorov) equation.
67 citations
TL;DR: In this paper, structural identifiability analysis has been performed for closed and almost closed catenary and mamillary systems, and previously available results have been improved and extended for some specially constrained compartmental systems.
Abstract: Explicit results are given concerning structural identifiability and unique structural identifiability for some specially constrained compartmental systems. Identifiability analysis has been performed for closed and almost closed catenary and mamillary systems, and previously available results have been improved and extended. New classes of constrained compartmental systems (radial and tree-type systems) have been considered, their structural identifiability proved and the number of different solutions explicitly evaluated. The case has also been analyzed where noncompartmental appendices are connected to the considered compartmental systems.
62 citations
TL;DR: Extinction in a three species Lotka-Volterra competitive system is classified in terms of the model parameters to study persistence of the system assuming all pairwise interactions between species are known.
Abstract: Extinction in a three species Lotka-Volterra competitive system is classified in terms of the model parameters. Necessary and sufficient conditions are given for one and two species extinction; these are, alternatively, conditions for two and one species persistence. Persistence of the system is studied assuming all pairwise interactions between species are known. An intransitive species arrangement is the only case of persistence where pairwise interactions are, by themselves, sufficient to govern persistence. No persistent arrangement can contain a pair of species that interacts in an unstable manner.
TL;DR: Results are presented here that can enable comparative biologists to proceed with cladistic character analysis without a priori estimates of the direction of evolutionary trends.
Abstract: Methods for cladistic character compatibility analysis have required a priori estimates of the direction of evolutionary trends among the states of a character, in order to ensure consistency. Results are presented here that can enable comparative biologists, if they choose, to proceed with cladistic character analysis without a priori estimates of the direction of evolutionary trends. Specifying only undirected character state trees is sufficient to ensure the compatibility of a collection of characters each pair of which is compatible. The results have been implemented in a computer program, for which a FORTRAN source deck is available.
TL;DR: In this paper, it was shown that for a given intrinsic growth rate r and the environmental carrying capacity K, the optimal solution x ∗ with period p is a positive p -periodic function whose average value is unity.
Abstract: For each choice of the intrinsic growth rate r and the environmental carrying capacity K as positive, bounded, periodic functions with period p , the nonautonomous logistic equation, x (t)=r(t)x(t) 1− x(t) K(t) , possesses an asymptotically stable, positive, periodic solution x ∗ with period p . If K is not a constant function, but is piecewise continuous on [0, p ], then the minimum and maximum values of x ∗ are related as follows to the extrema of K : K inf ∗ min ∗ max sup . The following question is treated here: For each specification of the function K , which functions r come close to maximizing x ∗ min ? It is shown that if r is expressed in the form r ( t )= δγ ( t ) with δ>0 and γ a positive p -periodic function whose average value is unity, then, for δ small enough, x ∗ ( t ) is approximately equal to a number x (γ, K ) which is independent of t ; in fact, for each t , lim δ→0 x ∗ (t)=p ∫ 0 p γ(τ) K(τ) dτ −1 = x (γ,K) . By an appropriate choice of γ, the number x (γ, K ) can be made arbitrarily close to K sup . The appropriate choices are those for which γ is approximately a “Dirac function” with its support concentrated at times for which K is close to K sup .
TL;DR: If the allele A 1 is initially rare and the population large, the underlying Markov chain can be approximated by a nonsingular positively regular and critical multitype branching process.
Abstract: We consider a finite dioecious random mating population that is observed at times 0,1,… Let there be age groups 0,1,…, K 1 among males and age groups 0,1,…, K 2 among females, so that the population consists of K 1 + K 2 +2 parts, called age-sex classes. It is assumed that the numbers of individuals in the various age-sex classes do not change with time and that there is no mutation, selection, or migration. One locus, with an allele A 1 that is initially rare, is studied. A general result obtained by Pollak (1979) is then used to obtain expressions for the effective population number, whether the locus under consideration is autosomal or sex-linked. Another result in the same paper is used to derive expressions for the mean time to extinction of a line of individuals with A 1 , which are descended from a single ancestor in age-sex class ( ai ), where i =1,2 and a =1,…, K i .
TL;DR: In this paper, the classical approach to the solution for a Markov transition probability matrix to the calculation of increment-decrement life tables is described and applied to compute a marital status life table for United States females with 1970 census and vital statistics data on population and vital events.
Abstract: The classical approach to the solution for a Markov transition probability matrix to the calculation of increment-decrement life tables is described. The method is applied to compute a marital status life table for United States females with 1970 census and vital statistics data on population and vital events. Comparison with the table for 1960 by Schoen and Nelson (1974) reflects the changing pattern of marital status over the period 1960–70.
TL;DR: Spectral power densities and autocorrelation functions have been computed for several types of discrete dynamical systems: both chaotic and noisy as mentioned in this paper, which can be distinguished from other types of dynamic systems by their appearance in the frequency domain.
Abstract: Spectral power densities and autocorrelation functions have been computed for several types of discrete dynamical systems: both chaotic and noisy. In some cases, chaotic dynamical systems can be distinguished from other types of dynamical systems by their appearance in the frequency domain.
TL;DR: In this paper, a general harvesting model is presented that allows the dynamics of the population to be divided into two distinct phases, viz, the harvesting-season dynamics, modeled by an n-dimensional system of ordinary nonlinear differential equations, and the spawning season, modelled by n difference equations.
Abstract: A general harvesting model is presented that allows the dynamics of the population to be divided into two distinct phases, viz, the harvesting-season dynamics, modeled by an n-dimensional system of ordinary nonlinear differential equations, and the spawning season, modeled by n difference equations. A maximum principle for this type of system is presented. The concept of “maximum sustainable yield” for periodic forms of such systems is introduced and discussed. The model is then simplified to exhibit linear-bilinear dynamics and in this form is shown to be a natural extension of the Beverton-Holt model in fisheries management. A method for deriving maximum-sustainable-yield solutions is presented, using this formulation and its corresponding maximum principle. By considering the solution in the limit as the control constraint set [0, b] becomes unbounded above, the concept of the “ultimate” sustainable yield is introduced. Finally, models including only scalar harvesting are introduced as being of practical value. The question is explored as to how maximum-sustainable-yield solutions are to be constructed from the maximum principle.
TL;DR: In this article, an age-structured population model is formulated in which the age-specific fertility rate depends on the past history of the population, and a priori estimates for the solutions are obtained.
Abstract: In this paper an age-structured population model is formulated in which the age-specific fertility rate depends on the past history of the population. Existence and uniqueness results and a priori estimates for the solutions are obtained. Moreover the dependence of the population upon the past history is investigated.
TL;DR: In this article, a mathematical theory of behavior which is intended to describe some features of animal behavior is developed, and individual and collective behavioral laws invariant with respect to the class of retardable elements are found.
Abstract: A mathematical theory of behavior which is intended to describe some features of animal behavior is developed. Individual and collective behavioral laws invariant with respect to the class of retardable elements are found. Theorems are proved, which make it possible to know the requirements for the collective to be “sensible”.
TL;DR: In this paper, the authors discuss the possibility of removing (inadvertently or intentionally) decay terms in the process of forming integral transforms, and they also consider the problem of deconvolution.
Abstract: Structural identifiability in compartmental systems deals with the map fromimpulse-response parameters to model parameters. If the data are analyzed in terms of integral transforms s k (Fourier, moments, etc.), then we may study also the map from the s k to impulse-response parameters. This paper is mainly concerned with the latter correspondence. In other words, we discuss the possibility of removing (inadvertently or intentionally) decay terms in the process of forming integral transforms.
TL;DR: In this paper, a stratified random sampling technique is proposed to estimate the Laplace transform of a given parameter, where the population proportion is estimated at the random time T where T has an exponential distribution, and the probability of an individual having a desired characteristic at time t, can only be estimated by a population proportion at time T having that characteristic.
Abstract: Parameter estimation can be complex when populations change in time and p(t), the probability of an individual having a desired characteristic at time t, can only be estimated by the population proportion at time t having that characteristic. We consider sampling at the random time T where T has an exponential distribution; the population proportion then estimates the Laplace transform of p(t). This reduces the complications of parameter estimation in many common situations—for example, when p(t) is a convolution of functions, each involving different parameters. We show that a stratified random sampling technique gives a more efficient parameter estimation method, and that this lends itself to a relatively straightforward parameter estimation technique based on ordinary fixed sampling times. The efficiency of this method is explored in two ways: analytically in a simple case where p(t) is the probability of transition by time t of an organism in a two-stage model with Erlangian transition times; and by simulation in a model where p(t) is the probability of a biological organism being in a given stage of development and the parameters are the death rate and the maturing rates in each stage. Finally, data on the hatching of parasitic nematodes O.circumcincta are fitted by least squares and by transform methods. Particularly in the complex simulation context the transform method has several desirable properties; explicit parameter estimators can be found, and they compare well with other estimation methods.
TL;DR: It is concluded that the behavior of this predator-prey system is more complicated than previously thought and that caution should be exercised in interpreting any conclusions of species packing that are based solely upon a graphical analysis of the MacArthur model.
Abstract: This paper concerns the exploitative competition of two predators for two prey species. We analyze the model proposed by MacArthur with more general parameters. MacArthur used the model to address questions of species packing on resources, but he did not completely determine analytically under what conditions will neither, one, or both predator species and one or both prey species, survive. Our analysis is global for all cases except one, for which we performed a numerical analysis. We conclude that the behavior of this predator-prey system is more complicated than previously thought. In particular, we analyze cases in which the two-predator, two-prey system catastrophically collapses to a one-predator, two-prey system, or even to one-predator, one-prey system. We also show that there are cases in which the initial numbers of the two predators determine the pattern of this collapse. These conclusions suggest that caution should be exercised in interpreting any conclusions of species packing that are based solely upon a graphical analysis of the MacArthur model.
TL;DR: In this article, it was shown that the differential-equation model for compartmental systems is consistent with a stochastic description, and that the fractional transfer coefficients can be determined from the corresponding set of differential parameters and vice versa.
Abstract: This paper shows that the differential-equation model for compartmental systems is consistent with a stochastic description. Consequently, we may employ either a differential-equation or a stochastic formulation, either for parameter identification or for physical interpretation, as best suits the purpose. The differential-equation parameters, the so-called fractional transfer coefficients, may be determined from the corresponding set of stochastic parameters and vice versa.
TL;DR: In this paper, the authors present a set of biochemically reasonable conditions for the rate laws which imply the existence of observable oscillations, which were obtained by linear stability analysis and computer simulation.
Abstract: Many models of open enzyme regulated systems are given by two first order differential equations of a certain type. We present a set of biochemically reasonable conditions for the rate laws which imply the existence of observable oscillations. A simple positive feedback mechanism is discussed as an example. Results on the allosteric model of glycolytic oscillations, which were formerly obtained by linear stability analysis and computer simulation, are extended and verified by mathematical proofs.
TL;DR: In this paper, a model for a population that is growing exponentially and at certain instants of time undergoes mass emigration due to population size pressure is proposed, where the numbers emigrating are independently and identically distributed and the intervals of time between any two emigration times have distributions which are functionals of the history of the population evolution since the last emigration.
Abstract: A model is proposed for a population that is growing exponentially and at certain instants of time undergoes mass emigration due to population size pressure. The numbers emigrating are independently and identically distributed, and the intervals of time between any two emigration times have distributions which are functionals of the history of the population evolution since the last emigration. Some results are given for the time to extinction of the population and its first moment, and also of the evolution of the population level. More details are given when the emigration size distribution is negative exponential.
TL;DR: In this paper, the authors considered the development of many chronic conditions is characterized by stages and diseases advance with time from mild through intermediate stages to severe to death, and the process is irreversible, but a patient may die while being in any one of the stages.
Abstract: Development of many chronic conditions is characterized by stages. Diseases advance with time from mild through intermediate stages to severe to death. Generally, the process is irreversible, but a patient may die while being in any one of the stages. Suppose there are k stages in a disease process S 1 ,…, S k , and a death state R . Given an individual in stage S i at time τ, for 0⩽τ d τ) possible transitions are S i → S i +1 with intensity v i,i +1 ϑ(τ) and S i → R with intensity μ i ϑ(τ). The density function of survival time T of an individual who is initially in stage S 1 at time 0 is given by f T (t) = ∑ j=1 k v 12… v j−1 , j μ j θ(t) ∑ i=1 j 1 ∏ l=1 l≠1 (v ii −v ll e v ii ∫ t O θ (τ) dτ where v ii =-( v i,i +1 +μ i ). Particular functions for θ(τ), moments and estimation of parameters are discussed.
TL;DR: It is shown from the solution of the partial differential equation that the size distribution will at first broaden and then eventually become narrow again as the cohort grows in average size.
Abstract: The growth in size of organisms in a population cohort and the change through time of the number of organisms in the cohort can be modeled in a unified way by means of a partial differential equation. This equation can be solved analytically for reasonable assumptions concerning growth and mortality rates. A cohort will have an initial distribution in sizes both because all offspring are not produced at exactly the same time and because there is some variation in the sizes of organisms at the time of reproduction. Assuming typical organism growth patterns, we show from the solution of the partial differential equation that the size distribution will at first broaden and then eventually become narrow again as the cohort grows in average size. This conclusion is in agreement with growth data on some marine and freshwater organisms. Genetic variations that cause the growth rate to be distributed normally among organisms in the cohort population are also shown to lead to predictable changes in the size distribution through time.
TL;DR: The under-agarose migration assay of Nelson et al. and others is modeled and analyzed mathematically and the relationship between the experimental cell migration-distance data and the random motility and chemotactic coefficients is determined.
Abstract: The correlation between leukocyte motility properties and the effective function of the inflammatory defense response has stimulated efforts to characterize random motility and chemotaxis using in vitro assays In order that in vitro measurements may be useful in predicting in vivo behavior, the experimental data must be used to determine quantitative parameters upon which the in vivo results will depend The key parameters for leukocyte migration are the random motility coefficient and chemotactic coefficient In this study, the under-agarose migration assay of Nelson et al and others is modeled and analyzed mathematically The relationship between the experimental cell migration-distance data and the random motility and chemotactic coefficients is determined
TL;DR: In this article, the problem of determining the density function of the distribution from the input-output relation observed over a large range of concentrations was formulated, with a kernel given by a transcendental equation.
Abstract: Enzymatic elimination of substrates carried by the blood through the liver occurs in a large ensemble of anatomically defined elements arranged in parallel. A common measured input concentration of the substrate is changed by elimination into an output concentration with values distributed over the elements, of which only the flow-weighted mean is measured. The output concentration from each element is determined in the steady state by the input concentration, the blood flow and the kinetic parameters of the enzymatic reaction according to a previously established model based on Michaelis-Menten kinetics. Assuming a distribution of blood flow and of the enzymatic elimination capacity over the elements, the problem is formulated of determining the density function of the distribution from the input-output relation observed over a large range of concentrations. The density function satisfies an integral equation of the first kind, with a kernel given by a transcendental equation. Results are obtained on the uniqueness and existence of the density function, on its determination in closed form and from a series expansion, and on the relation of the problem to the problem of moments.
TL;DR: P precise formulation and proof of this statement are given using basic mathematical properties of disease incidence, and an approximation relates incidence trends with infectee numbers.
Abstract: Trends in the incidence of a communicable disease are related to the number of new cases caused by each infective (the infectee number, analogous to the net reproductive rate in population dynamics). In [1] Yorke, Hethcote, and Nold stated that if an endemic communicable disease approaches equilibrium, then the average infectee number approaches 1, and sketched a simple proof. Here, precise formulation and proof of this statement are given using basic mathematical properties of disease incidence. Also, an approximation relates incidence trends with infectee numbers.