On the utilization of food by planktophage fishes

Making some plausible assumptions about the over-all mechanism of food catching and consumption by fishes and evaluating in the light of those assumptions some available experimental data, it is possible to calculate from those data the variation of several important factors with the concentration of food. The factors considered are: total rate of metabolism, total diurnal energy expenditure in the process of feeding, average number of hours per day during which the fish feeds, average length of path traveled by a fish per day, and the so-called “energetic coefficient of growth.” A possible relation with the work of N. Rashevsky (Bull. Math. Biophysics,20, 299–308, 1959) is discussed.     

Contribution to the mathematical theory of mass behavior: I. The propagation of single acts

The propagation of a single act in a large population is supposed to depend on some external circumstance and on an “imitation component”, where encounters with individuals who are performing or have already performed the act contribute to the tendency of an individual to perform it. The “tendency” to perform is supposed to be measured by the average frequency of stimuli, randomly distributed in time, impinging on the individual. The deduced equation is a relation between the fraction of the population who have performed the act and time, provided the time course of the “external circumstance” and the way in which the imitation component contributes are known. Several special cases are studied, in particular, cases without the imitation component, cases with imitation only, and various mixed cases. Examples are given of social situations in which such factors may operate and general suggestions are made for the systematization of observations and/or experiments to test the assumptions of the theory.     

An ant’s eye view of culture: propagation of new traditions through triggering dormant behavioural patterns

We consider a previously unknown way of propagation of behavioural traditions in animal communities using hunting in ants as an example. We experimentally revealed that common litter dwelling ants Myrmica rubra effectively hunt jumping prey and the way the hunting behavioural pattern is distributed within ant colonies is rather sophisticated. Comparison of our results with those obtained on vertebrates enables us to suggest that “distributed social learning” plays an important role in spreading new traditions in animal communities: initial performances by a few carriers of an “at once and entirely” available behavioural pattern propagate this pattern among specimens which have only dormant “sketches” of it. Spread of these behaviours in populations is based on relatively simple forms of social learning such as social facilitation which underlies species’ predisposition to learn certain sequences of behavioural acts. To be triggered, carriers of dormant “sketches” of a relevant behavioural pattern should encounter performances of this pattern with sufficient frequency. We call this strategy triggering of dormant behavioural patterns. Integration of behaviour thus takes place not only at the individual level but at the population level as well.     

Some theoretical consequences of the allometric growth equations

The rate of growth of an organism may be described as a function of both its mass and of the time. In particular, if the rate of growth is described as a function of mass alone, one derives the so-called principle of “equifinality”, according to which the final mass achieved is independent of any interruption of the growth process. Since this phenomenon is observed in many instances, the assumption that the rate of growth is a function of mass alone has some theoretical justification. On the other hand, it is obvious that in many instances irreversible effects can be achieved by interrupting the growth process, thus indicating that the rate of growth may depend on variables other than mass. The consequences of some of these alternative assumptions are examined. A growth factorG is postulated, which is supposed to decay with time, and whose concentration determines the over-all rate of growth per unit mass. The rate of decay ofG is calculated from the postulated growth equation under two alternative assumptions, (1) thatG is a constituent of all the cells of the organism and (2) thatG is produced separately and distributed through the organism. It is shown that in each case the concentration ofG satisfies a simple differential equation. While in normal growth the results deduced from these hypotheses are indistinguishable from those deduced from the assumption that the rate of growth depends on mass alone, widely different results are obtained in the case of interrupted growth. In particular, the rate at which growth is resumed after interruption and the final mass achieved depend on the time and on the duration of interruption. It appears, therefore, that various assumptions about the nature of the “aging” of the organism lead to various quantitative relations between growth and the times of interruption and resumption of growth.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

Resolving the limitation – regulation debate

Many recent ecological studies have demonstrated that animal populations are limited by their food. Examples are presented here to refute the view that natural populations are regulated by negative feedback mortality factors. Additionally, several incorrect statements in a recent publication are discussed, specifically (1) that there is no difference between the concepts of regulation and limitation; (2) that the debate is about what causes the time it takes a population to reach the carrying capacity of its habitat, not what sets that carrying capacity; (3) that the results of a laboratory experiment using a closed population with fixed amounts of food represents what happens in natural open populations with varying supplies of food; (4) that a thermostat analogy can be used, assuming that an “equilibrium” is controlling natural populations “from above” instead of the original steam analogy which says the varying input of a resource “from below” is the controlling factor.     

Dietary ratio of protein to carbohydrate induces plastic responses in the gastrointestinal tract of mice

Some vertebrates change the size of their digestive system in response to quantity and fibre content of ingested food, but the effects of dietary nutrients on gut structure remain poorly understood. Here we investigate how the protein to carbohydrate ratio of diets affects the mass of the gastrointestinal tract in mice. We fed 6-week-old male mice one of five isocaloric diets differing only in protein to carbohydrate ratio (the “no-choice” treatments), while a further four treatment groups received nutritionally complementary food pairings from which they could self-select a diet (the “choice” treatments). After 32 days, we measured the resulting dry mass of stomachs, intestines, caeca and colons. In the no-choice treatments, the stomachs were heavier in the mice fed diets containing more protein and less carbohydrate, indicating that larger stomachs may be needed for efficient digestion of the protein-rich food. In contrast, intestines, caeca and colons were heavier when diets contained more carbohydrates and less protein. This response may function to increase the digestive rate of carbohydrates when the dietary content of this macronutrient increases, but it may also indicate a compensatory response to increase amino acid uptake from a protein-deficient food. Mice in the choice treatments self-selected a diet with a protein to carbohydrate ratio of 0.46, and had gut dimensions similar to the expectation derived from no-choice treatments for this diet composition. Our results provide an example of plasticity in the differential allocation of resources to organ function, which is triggered by variation in resource quality.     

Response time and threshold of a random net

The response time of a random net is defined as the expected time (measured in the number of synaptic delays) required for the excitation in the net (measured by the fraction of neurons firing per unit time) to reach a certain level. The response time is calculated in terms of the net parameters as a function of the intensity of the outside stimulation. Two principal types of cases are studied, 1) an instantaneous initial stimulation, and 2) continuously applied stimulation. It is shown that for a certain type of net where the required level of excitation is small, the response time-intensity equation reduces to the one derived on the basis of the “one-factor” theory applied to a neural connection. More general assumptions, however, give different types of equations. The concept of the “net threshold” is defined, and its calculation indicated. The net threshold for instantaneous stimulation is, in general, greater than that for continuous stimulation. The results are discussed with reference to existing theories of reaction times.     

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.     

Word organization in coding DNA: A mathematical model

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps' law, which relates vocabulary to text-length. Our analysis shows that Heaps' law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning. IM and AS contributed equally to this work.     

Two types of using of space in the resident common shrews <Emphasis Type="Italic">Sorex araneus</Emphasis> L.

The home range of resident animals is considered as “familiar area” including a “foraging area.” It has been revealed that the activity of an average animal unit in the “foraging area” could be approximated by normal distribution. Estimation of activity distribution in the “familiar area” (beyond the “foraging area”) was impeded by means of marking since it might be difficult to record distant movements, and the method does not provide an essential body of data. In the case of the common shrew Sorex araneus, the “familiar area” was estimated using pitfall as animals evade them in the known areal. The “foraging area” radius of the average shrew was taken to be 30 m (95% of the animal unit activity), the radius of “familiar area” was within the range of 180–240 m. The “foraging area” was expected to provide the animal with vital resources, and the “familiar area” reflects its need for exploratory activity.     

Influence of body composition on the metabolic rate of nestling European shags (Phalacrocorax aristotelis)

During the early development of avian nestlings, their mass-specific resting metabolic rate (RMR) changes in a biphasic pattern with the peak value often being much higher than that expected for an adult bird of similar body mass. In the present study we examined the possible influence of variations in the size of internal organs in “setting” the high RMR and peak metabolic rate (PMR) during development in a large altricial species, the European shag (Phalacrocorax aristotelis). Thermoneutral RMR and cold-exposure induced PMR were measured in nestlings 15 days old, the age at which the highest RMR occurred during development. Body mass averaged 414 g. Mean values of RMR and PMR were 5.75 W and 9.08 W, respectively; the RMR value corresponds to approximately 250% of the expected value for an adult non-passerine bird of similar body mass. The masses of all the organs measured (breast and leg muscles, heart, liver, intestine, and kidney) varied isometrically with total body mass. However, large chicks had a significantly lower fractional water content than small chicks, suggesting that the former had achieved a higher level of functional maturity. In contrast to what has been suggested for adult birds in general, the heart and kidney masses of shag nestlings were not significantly correlated with the metabolic rates. The intestine length, in contrast, was highly and positively correlated with both the RMR and the PMR, i.e. intestine length was a better predictor of RMR and PMR than was total body mass. In addition, liver mass was positively correlated with RMR. The results of the present study suggest that the liver in particular may play a key role in establishing the high, mass-specific RMR which is attained during development in bird chicks. Our results also support previous suggestions that early in their development, altricial chicks mainly allocate energy to the growth of `energy-processing' organs (such as the intestine and liver) rather than to `energy-consuming' organs. Accepted: 3 March 1999     

Some theoretical considerations relating to the meaning and measurement of chloroplast reducing potential

The meaning of “chloroplast reducing potential” is examined in order to find out whether determinations of it might not allow one to circumscribe possible mechanisms of photosynthesis. One can define a physical situation where a measure of “chloroplast reducing potential” could allow one to draw an important conclusion about the mechanism of photosynthesis. However, in order to achieve this, it is necessary to make very severe assumptions and to suppose that some very serious experimental difficulties can be overcome.     

The PhyloCode is fatally flawed, and the “Linnaean” System can easily be fixed

Promoters of the PhyloCode have mounted an intensive and deceptive publicity campaign. At the centerpiece of this campaign have been slogans such as that the Linnaean System will “goof you up,” that the PhyloCode is the “greatest thing since sliced bread,” and that systematists are “afraid” to propose new names because of “downstream consequences.” Aside from such subscientific spin and sloganeering, proponents of the PhyloCode have offered nothing real to back up claims of greater stability for their new system. They have also misled many into believing that the PhyloCode is the only truly phylogenetic system. The confusion that has been fostered involves several discrete arguments, concerning: a new “method” of “designating” names, rank-free taxonomy, uninomial nomenclature, and issues of priority. Claims that the PhyloCode produces a more stable nomenclature are false, as shown with the example of “paleoherbs.” A rank-free system of naming requires an annotated reference tree for even the simplest exchanges of information. This would be confusing at best and would cripple our ability to teach, learn, and use taxonomic names in the field or in publications. We would be confronted by a mass of polynomial names, tied together only by a tree graphic, with no agreed name (except a uninomial, conveying no hierarchy) to use for any particular species. The separate issue of stability in reference to rules of priority and rank can be easily addressed within the current codes, by implementation of some simple changes, as we will propose in this article. Thus there is no need to “scrap” the current Linnaean codes for a poorly reasoned, logically inconsistent, and fatally flawed new code that will only bring chaos.     

A theory of steady-state activity in nerve-fiber networks II: The simple circuit

It is found that for a simple circuit of neurons, if this contains an odd number of inhibitory fibers, or none at all, or if the product of the activity parameters is less than unity, then the stimulus pattern always determines uniquely the steady-state activity. For circuits not of one of these types, it is possible to classify exclusively and exhaustively all possible activity patterns into three types, here called “odd”, “even”, and “mixed”. For any pattern of odd type and any pattern of even type there always exists a stimulus pattern consistent with both, but in no other way can such an association of activity patterns be made.     

The visual ecology of Puerto Rican anoline lizards: habitat light and spectral sensitivity

The visual ecology of six closely related species of Puerto Rican anoline lizards was investigated and they were found to occupy four distinct habitat types in terms of light conditions: “full shade”, “partial shade”, “no shade”, and “forest canopy.”The habitats differed substantially in total radiance and irradiance as well as in the shape of the irradiance spectrum. The shape of the radiance spectrum was similar in all of the habitats. We used electroretinogram (ERG) flicker photometry to measure spectral sensitivity and found the curves for all six species to be similar. The spectral sensitivity peaked in the range 550–560 nm, which matched the peak in spectral radiance for all of the habitats. The shape of the spectral-sensitivity curve was similar to those of a number of other terrestrial vertebrates. We suggest that the convergence of the shape of the photopic ERG-determined spectral-sensitivity curve in many terrestrial vertebrates may, in part, be due to the fact that the background radiance of many terrestrial habitats is dominated by the reflectance spectrum of green vegetation which peaks at 550 nm. Accepted: 14 May 1997     

Theory of gastric function

On the basis of some assumptions concerning gastric emptying a theory of gastric function is developed which explains the known facts —chiefly the emptying of liquids according to an exponential law. On the basis of this theory an equation is derived which relates the rate of secretion to the quantity of liquid retained in the stomach after a certain time. A study is made of the general characteristics of this equation. Some assumptions concerning different formulae describing the rate of secretion are found useful in experimental investigations, particularly in Hollander’s dilution indicator method.     

Outline of a probabilistic approach to animal sociology: II

Under certain assumptions concerning the probabilities of “mutations,” i.e. changes of structure of bird societies, it is shown that the probability distribution for all possible structures of a society ofN individuals approaches a limit independent of the initial probability distribution. A formula for the limiting distribution is derived.     

Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population

An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.     

On the mathematical theory of rumor spread

The applicability of the theory of random nets to the theory of rumor spread is shown. In particular the “weak connectivity” of the net appears as the saturation fraction of “knowers” in a thoroughly mixed population through which a message diffuses where each knower tells the message to a finite average number of individuals. Further it is shown how the time course equation of rumor spread, where time is measured by the number of “removes” from the starters, can be translated into an ordinary continuous time course equation if the distribution of the telling intervals is known.