A comparative investigation of certain difference equations and related differential equations: Implications for model-building

Many mathematical models for physical and biological problems have been and will be built in the form of differential equations or systems of such equations. With the advent of digital computers one has been able to find (approximate) solutions for equations that used to be intractable. Many of the mathematical techniques used in this area amount to replacing the given differential equations by appropriate difference equations, so that extensive research has been done into how to choose appropriate difference equations whose solutions are “good” approximations to the solutions of the given differential equations. The present paper investigates a different, although related problem. For many physical and biological phenomena the “continuum” type of thinking, that is at the basis of any differential equation, is not natural to the phenomenon, but rather constitutes an approximation to a basically discrete situation: in much work of this type the “infinitesimal step lengths” handled in the reasoning which leads up to the differential equation, are not really thought of as infinitesimally small, but as finite; yet, in the last stage of such reasoning, where the differential equation rises from the differentials, these “infinitesimal” step lengths are allowed to go to zero: that is where the above-mentioned approximation comes in. Under this kind of circumstances, it seems more natural tobuild themodel as adiscrete difference equation (recurrence relation) from the start, without going through the painful, doubly approximative process of first, during the modeling stage, finding a differential equation to approximate a basically discrete situation, and then, for numerical computing purposes, approximating that differential equation by a difference scheme. The paper pursues this idea for some simple examples, where the old differential equation, though approximative in principle, had been at least qualitatively successful in describing certain phenomena, and shows that this idea, though plausible and sound in itself, does encounter some difficulties. The reason is that each differential equation, as it is set up in the way familiar to theoretical physicists and biologists, does correspond to a plethora of discrete difference equations, all of which in the limit (as step length→0) yield the same differential equation, but whose solutions, for not too small step length, are often widely different, some of them being quite irregular. The disturbing thing is that all these difference equations seem to adequately represent the same (physical or biological) reasoning as the differential equation in question. So, in order to choose the “right” difference equation, one may need to draw upon more detailed (physical or) biological considerations. All this does not say that one should not prefer discrete models for phenomena that seem to call for them; but only that their pursuit may require additional (physical or) biological refinement and insight. The paper also investigates some mathematical problems related to the fact of many difference equations being associated with one differential equation.     

Age of epiphyseal closure in tamarins and marmosets

Estimates of the chronological age for animals of unknown age provide useful information for medical, demographic, and evolutionary studies. Skeletal development, as indicated by epiphyseal closure, can be used to estimate an animal's chronological age or specify its stage of development. Many studies of Primate skeletal development have used animals of unknown age, with the order of epiphyseal closure providing a relative age for each animal. This study examines the age of epiphyseal closure at 22 epiphyseal sites using animals of known age at death in three calitrichid species (Saguinus fuscicollis, Saguinus oedipus, and Callithrix jacchus). The observed average age of epiphyseal closure is similar in these tamarins and marmosets. There is a significant difference in rate of development between the species. Regression equations can predict the age of unknown animals to within 4.8 months for S. fuscicollis, 8.6 months for S. oedipus, and 7.6 months for C. jacchus (twice the standard error of the estimate). These age estimates allow us to determine if an animal is relatively mature or immature, but are largely unacceptable for studies in which precise age estimates are necessary. The order of epiphyseal closure is similar across 11 monkey species (using additional data from published literature) and supports the suggestion of a general pattern in Primate skeletal development. Am. J. Primatol. 41:129–139, 1997. © 1997 Wiley-Liss, Inc.     

Bone age: its contribution to the prediction of maturational or biological age

The bone age of normal children was determined by both the method of Wilkins and the method of Greulich and Pyle. In boys, the mean of the difference between the bone age and the chronological age was significantly greater than zero with the Wilkins method but not with the Greulich and Pyle method. In normal children of both sexes, bone age determined by the Greulich and Pyle method proved to be an important component in equations designed to assess progress along the pathway of growth (Biological Age). These equations combine information concerning body growth, cell growth and body composition. Bone age determined by the Wilkins method, however, was not a significant component in such equations. Biological age equations that include the Greulich and Pyle bone age have been applied to children with a number of diseases that affect growth. In comparison to chronological age, biological age was advanced in obese girls and retarded in boys with both severe congenital heart disease and pituitary dwarfism. Following seven months of therapy with human growth hormone, the latter group showed evidence of catch up growth.     

Reproductive potential of the earthworm Eisenia foetida

Summary Regression equations are provided for the earthworm Eisenia foetida with respect to age at which 50% of the population became clitellate at 25° C in relation to population density in activated sludge and in horse manure. Regression equations are provided for progeny per cocoon versus weight of cocoon, and weight of cocoon in relation to weight of parent; from these an equation is derived for progeny per cocoon relative to worm weight. Regression equations are given on (a) number of cocoons produced per adult in relation to age and population density from onset of adulthood to median peak production of cocoons, age 10 weeks, and from age 10 weeks to age 27 weeks, and (b) weight of worm in relation to population density and age between ages 5 and 27 weeks. From (a) and (b) a family of equations (c) are derived giving progeny per cocoon in relation to age of adult and population density. From equations (a) and (c) two families of equations are generated giving progeny per adult in relation to ascent to, and descent from, the median week of peak cocoon production in relation to population density. Data also are provided on age at which reproduction terminates in relation to population density, optimum population density for reproduction, and hatchability.     

Prediction of normal values for lactate threshold estimated by gas exchange in men and women

Lactate threshold (LT) is an index of exercise capacity and can be estimated from the gas exchange consequences of a metabolic acidosis (LT_GE). In recent years, it has emerged as a diagnostic tool in the evaluation of subjects with exercise limitation. The purpose of this study was to develop LT_GE prediction equations on a relatively large sample of adults and to cross-validate each equation. A total of 204 healthy, sedentary, nonsmoking subjects (103 men and 101 women), aged 20–70 years, underwent graded exercise testing on a cycle ergometer. The V-slope technique was used to detect LT_GE as the oxygen uptake (V˙O₂) at the breakpoint of the carbon dioxide output versus V˙O₂ relationship. Multiple linear regression was used to develop 12 equations with combinations of the following predictor variables: age, height, body mass, and fat-free mass. Eight of the equations are gender-specific and four are generalized with gender as a dummy variable. The equations were cross-validated using the predicted residual sum of squares (PRESS) method. The results demonstrate that the equations had relatively high multiple correlations (0.577–0.863) and low standard errors of the estimate (0.123–0.228 1 · min⁻¹). The PRESS method demonstrated that the equations are generalizable, i.e., can be used in future studies without a significant loss of accuracy. Since we tested only healthy, sedentary subjects, our equations can be used to predict the lower limit of normal for a given subject. Using individual data for healthy and diseased subjects from the literature, we found that our gender-specific equations rarely miscategorized subjects unless they were obese and mass was a predictor variable. We conclude that our equations provide accurate predictions of normal values for LT_GE and that they are generalizable to other subject populations. Accepted: 13 February 1997     

The interrelation between the number of deciduous teeth and the morphological maturity of a child

The process of the first dentition is part of a very short period during human development characterised by an extraordinary intensity of growth processes. The main purpose of the present study is to provide information on the relationship between three criteria of biological maturity of children: body height, body weight and the number of milk teeth. The present study is based on a cross-sectional material collected in 30 nurseries in ?ód? (Poland). The analysis was carried out on 1901 children (981 boys and 920 girls) aged from 2 to 36 months. All coefficients of Pearson correlation and partial correlations between the analysed variables are statistically significant at the 0.01 level. The highest value of coefficient correlation was noticed for the relation between dental and chronological age, as well as between dental and morphological age based on body height (about 0.90 and 0.85, respectively). Through an analysis of regression (backward) equations for predicting the number of teeth from chronological age, body weight and body height were calculated. These mathematical equations were confirmed in practice in about 80% of the children. Coefficients of determinations between the analysed variables are high for boys and girls (about 72% and 74%, respectively). This observations demonstrate conclusively that there exists a highly significant relation between deciduous tooth age and morphological age.     

The systematic formulation of population models for insects with dynamically varying instar duration

We develop a formalism for insect population dynamics which covers the situation where maturation from one instar to its successor is triggered by weight gain and not by chronological age. We specify assumptions which result in the instantaneous “subpopulations” of various instars obeying delay-defferential equations with time delays (representing instar duration) which are themselves dynamic variables, changing in response to the availability of food. We demonstrate the stabilizing potential of variable time delays by studying an idealised two-stage model in which maturation to the adult stage occurs after absorption of a given (fixed) quantity of food.     

Linearized oscillations in population dynamics

A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.     

Correlates and predictors of the age of menarche

In a cross-sectional study of 452 girls between 10 and 16 years of age 36 indices of physical and 50 of mental development were tested for their correlation with age at menarche and chronological age, as well as for their predictive power for estimating menarche by multiple regression analysis. Indices of physical maturity and body weight when adjusted for chronological age showed the highest partial correlation coefficients with age at menarche. Among mental characters which show lower intercorrelations with menarche occurred the highest correlation coefficients for a handmotor factor "Spurennachzeichnen" and a factor "Gruppenabh?ngigkeit" (which indicates a type of social motivation). In general physical and mental factors correlate higher with chronological age than with age at menarche. By multiple regression analysis we determined 14 physical and 25 mental predictors explaining 21% and 17% respectively of the variance of age at menarche. The error of the estimate predicting menarche on body weight without knowledge of onset of menarche is +/- 1 year. Using chronological age in a sample of girls before menarche the error of the estimate only is +/- 6 months. To compare the predictive power of chronological age combined with body weight or with skeletal age the time interval is calculated within which 95% of girls attain menarche. The range of prediction extends from 4.3 to 1 year on chronological age (11-16 years); using mean body weight it can be improved by 1.8 to 6 months, while using mean skeletal age an improvement of 0.2 to 3.9 months is possible compared with body weight. The correlations between age at menarche and physical and mental variables are attributed to a common hormonal influence on rate of development.     

Oxygen solubility in evaporated seawater as a function of temperature and salinity

Most studies of oxygen solubility values for high salinity conditions have used synthetic solutions. The object of this study is therefore to propose an equation, valid for high salinity conditions, based on the analysis of oxygen saturation in evaporated seawater. In this study, the solubility of oxygen in evaporated seawater has been determined over a temperature range of 8–35°C and with salinity values of up to 133‰. Based on experimental data, an equation is proposed that introduces a S ² (salinity) term, at 1 atm pressure, giving increased importance to salinity. The equation provides a valid means of predicting the amount of dissolved oxygen in this range of temperatures and salinities. In addition, for high salinity conditions, with this equation there is no need to extrapolate other established equations, which are less accurate at salinities higher than 40‰. The use of the proposed equation offers a more precise way of calculating oxygen solubility in seawater at high salinity values (up to 133‰), and small deviations from experimental values, of the order of 2 μmol kg⁻¹, are obtained. Handling editor: J. Melack     

Ageing in a variable habitat: environmental stress affects senescence in parasite resistance in St Kilda Soay sheep

Despite widespread empirical evidence for a general deterioration in the majority of traits with advancing age, it is unclear whether the progress of senescence is chronologically determined, or whether factors such as environmental conditions experienced over the lifespan are more important. We explored the relative importance of ‘chronological’ and ‘environmental’ measures of age to changes in parasite resistance across the lifespan of free-living Soay sheep. Our results show that individuals experience an increase in parasite burden, as indicated by gastrointestinal helminth faecal egg count (FEC) with chronological age. However, chronological age fails to fully explain changes in FEC because a measure of environmental age, cumulative environmental stress, predicts an additional increase in FEC once chronological age has been accounted for. Additionally, we show that in females age-specific changes are dependent upon the environmental conditions experienced across individuals'' life histories: increases in FEC with age were greatest among individuals that had experienced the highest degree of stress. Our results illustrate that chronological age alone may not always correspond to biological age, particularly in variable environments. In these circumstances, measures of age that capture the cumulative stresses experienced by an individual may be useful for understanding the process of senescence.     

Skeletal age determination by nandibular histomorphometry

50 undecalcified cross sections of mandible of 50 individuals of known sex and age (32 males and 18 females) were examined. The sample age range was 18–97 years. The avnumber of secondary osteons (X1) and average number of fragmented osteons (X2) were computed to correlate them with chronological age at death (Y). Linear regression analyses showed that in this age group, age at death could be estimated with a S.E. of ±6.42 years for the variable X1 and a S.E. of ±11.45 years for the variable X2. Some remarks about the reliability of bone methods for age at death estimation are also made.     

Simplified approach for developing the rate expressions for enzyme-catalyzed reactions

A Simplified version of rate equations for enzyme-catalyzed reactions has been developed in which the rate equation is analyzed with two different mechanisms: ordered bi–ter and Ping-Pong bi–bi mechanisms. This procedure is able to develop the rate expressions accurately. Random sequential order mechanism can be effectively explained by this approach.     

On the Fokker-Planck equation in the stochastic theory of mortality: II

This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.     

Studies on deoxyribonucleic acid polymerase from normal and down’s syndrome skin fibroblasts in vitro

Summary This investigation showed that: (a) hypotonic pH 5.0 extracts of normal and Down’s syndrome fibroblasts cells (DS) exhibit DNase activity which appears to be very similar in nature and extent; (b) DNA polymerase activity was generally increased in DS cells when compared with normal cells; (c) DNA polymerase activity was increased with increased passage age for both normal and DS fibroblasts, but the normal fibroblast was never found to overlap with DS cell activity; and (d) normal cells appear to have a similar DNA polymerase activity at all chronological ages, whereas DS cells exhibit a decrease in activity with increased age of the donor.     

Dentition and age determination of the giraffe Giraffa camelopardalis

Thirteen tooth eruption stages and their corresponding chronological ages are descri bed from a series of giraffe jaws. These can be used for age determination in giraffes with immature dentition. Significant correlations of the lingual crown height ( r =0-957; P < 0001) and lingual occlusal surface width ( r =0-959; P < 0001) with the number of dark staining incremental lines in the cementum of thin decalcified sections of the maxillary first molar were found. The regression equations derived from these relationships provide a further method for determining the age of a giraffe. A composite plate showing maxillary first molar wear patterns provides a means of roughly assigning an age to a particular specimen. Thin sections of undecalcified teeth, mandible measurements, various other indices of tooth wear and eye lens mass were investigated and found unsuitable for age determination in this     

Changes in the Growth Capacity of the Developing Wheat Embryo: 1. THE INFLUENCES OF THE ENVELOPING TISSUES11

Grains stimulated to germinate by removal of the outer pericarpshow three types of embryo growth according to their physiologicalage. The first type gives rise only to callus tissue, the secondpredominantly to growth of the coleoptile, which fails to penetratethe testa, and the third, to normal germinative growth. Thesecond type of growth (coleoptile only) corresponds to a conditionreferred to as ‘non-visible’ sprouting, occasionallyfound in prematurely sprouted grains in the field. The phasesof germination capacity are related to total grain moistureand not to the chronological age of the grain, i.e. days post-anthesis.The effects of drying upon grain germinability are also discussedand it is shown that drying can cause both qualitative and quantitativechanges in germination capacity. Key words: Wheat embryo, Germination capacity, Germinability     

Biomechanical analysis of movement strategies in human forward trunk bending. I. Modeling

 Two behavioral goals are achieved simultaneously during forward trunk bending in humans: the bending movement per se and equilibrium maintenance. The objective of the present study was to understand how the two goals are achieved by using a biomechanical model of this task. Since keeping the center of pressure inside the support area is a crucial condition for equilibrium maintenance during the movement, we decided to model an extreme case, called “optimal bending”, in which the movement is performed without any center of pressure displacement at all, as if standing on an extremely narrow support. The “optimal bending” is used as a reference in the analysis of experimental data in a companion paper. The study is based on a three-joint (ankle, knee, and hip) model of the human body and is performed in terms of “eigenmovements”, i.e., the movements along eigenvectors of the motion equation. They are termed “ankle”, “hip”, and “knee” eigenmovements according to the dominant joint that provides the largest contribution to the corresponding eigenmovement. The advantage of the eigenmovement approach is the presentation of the coupled system of dynamic equations in the form of three independent motion equations. Each of these equations is equivalent to the motion equation for an inverted pendulum. Optimal bending is constructed as a superposition of two (hip and ankle) eigenmovements. The hip eigenmovement contributes the most to the movement kinematics, whereas the contributions of both eigenmovements into the movement dynamics are comparable. The ankle eigenmovement moves the center of gravity forward and compensates for the backward center of gravity shift that is provoked by trunk bending as a result of dynamic interactions between body segments. An important characteristic of the optimal bending is the timing of the onset of each eigenmovement: the ankle eigenmovement onset precedes that of the hip eigenmovement. Without an earlier onset of the ankle eigenmovement, forward bending on the extremely narrow support results in falling backward. This modeling approach suggests that during trunk bending, two motion units – the hip and ankle eigenmovements – are responsible for the movement and for equilibrium maintenance, respectively. Received: 1 July 1999 / Accepted in revised form: 23 October 2000     

Age and terminal reproductive attempt influence laying date in the thorn‐tailed rayadito

Age‐specific variation in reproductive effort can affect population dynamics, and is a key component of the evolution of reproductive tactics. Late‐life declines are a typical feature of variation in reproduction. However, the cause of these declines, and thus their implications for the evolution of life‐history tactics, may differ. Some prior studies have shown late‐life reproductive declines to be tied to chronological age, whereas other studies have found declines associated with terminal reproduction irrespective of chronological age. We investigated the extent to which declines in late life reproduction are related to chronological age, terminal reproductive attempt or a combination of both in the thorn‐tailed rayadito Aphrastura spinicauda, a small passerine bird that inhabits the temperate forest of South America. To this end we used long‐term data (10 years) obtained on reproductive success (laying date, clutch size and nestling weight) of females in a Chilean population. Neither chronological age nor terminal reproductive attempt explained variation in clutch size or nestling weight, however we observed that during the terminal reproductive attempt older females tended to lay later in the breeding season and younger females laid early in the breeding season, but this was not the case when the reproductive attempt was not the last. These results suggests that both age‐dependent and age‐independent effects influence reproductive output and therefore that the combined effects of age and physiological condition may be more relevant than previously thought.