Forms of output distribution between two individuals motivated by a satisfaction function

Motivations of two individuals governed by a satisfaction function are assumed to determine their respective “efforts”, which result in the production of “output”, i.e., objects of satisfaction. In previous papers the sharing of output was prescribed in advance. In the present article, however, the sharing formula itself is determined to a certain extent by the satisfaction function. The rate of remuneration per unit of output for each individual is taken to be proportional to the derivative of the satisfaction of the other individual with respect to the effort of the first. The formulation of this condition leads to a partial differential equation whose solutions determine the sharing formula. Sharing determined in this way is referred to as sharing according to the Condition of Mutual Need (C.M.N.). Satisfaction resulting from five different situations are the computed and compared: (1) an individual producing and consuming alone; (2) two individuals sharing equally and neither taking the “initiative” to determine the optimum output; (3) sharing determined by C.M.N. with optimum output determined as in (2); (4) equal sharing but with one individual taking “initiative” in determining optimal output; and (5) sharing determined by C.M.N. and optiml output by the “initiative” of one individual. further considerations concern conditions imposed on the arbitrary function occurring in the solution of the above-mentioned partial differential equation.     

A problem in the mathematical biophysics of interaction of two or more individuals which may be of interest in mathematical sociology

The behavior of an individual may be discussed in terms of a “satisfaction function”. An individual may be considered to always behave in such a way as to make his satisfaction function a maximum. The interaction of two individuals may consist in a cooperation in the production of any kind of objects of satisfaction. Those objects may be either material goods or anything else. The satisfaction of each individual is determined by his share in the total output as well as by the effort he makes. It is shown that for a prescribed method of sharing a behavior in which each individual attempts to maximize the total satisfaction of both individuals results in a greater output than a behavior in which each individual attempts to maximize his own satisfaction.     

Mathematical theory of motivation interactions of two individuals: II

The behavior of two individuals, consisting of effort which results in output, is considered to be determined by a satisfaction function which depends on remuneration (receiving part of the output) and on the effort expended. The total output of the two individuals is not additive, that is, together they produce in general more than separately. Each individual behaves in a way which he considers will maximize his satisfaction function. Conditions are deduced for a certain relative equilibrium and for the stability of this equilibrium, i.e., conditions under which it will not “pay” the individual to decrease his efforts. In the absence of such conditions “exploitation” occurs which may or may not lead to total parasitism. Some forms of the inverse problem are considered, where the form of behavior is given and forms of the satisfaction function are deduced which lead to it.     

Coexistence of congeneric spiny lobsters on coral reefs: differences in conspecific aggregation patterns and their potential antipredator benefits

Den sharing by conspecific spiny lobsters (aggregation) is modulated by chemical attraction but may confer several, not necessarily mutually exclusive, antipredator byproduct benefits: a “guide effect”, which only benefits the individual attracted to a sheltered conspecific; a “dilution effect”, which reduces per-capita risk of predation simply through aggregation; or active “group defense”. Each potential benefit has a different set of predictors (relationships between aggregation and conspecific or predator densities), but conflicting results could suggest the simultaneous operation of more than one benefit. These predictions were tested for coexisting Panulirus guttatus (a reef-obligate) and Panulirus argus (a temporary reef-dweller) using data collected during 11 surveys on fixed sites over a coral reef in Mexico. P. guttatus greatly outnumbered P. argus, but P. argus showed a greater tendency to aggregate. All three benefits of den sharing operated for the more social P. argus, with “group defense” being of the most benefit for larger individuals, and the “guide” and “dilution” effects for smaller individuals recently immigrating into the reef habitat and sharing dens with larger conspecifics. P. guttatus did not display “group defense” and its aggregations appeared to be modulated by the interplay between attraction and aggressive behaviors. This species relied more on solitary crypticity, especially at larger sizes, but appeared to benefit from a “guide effect” at high conspecific densities. In experimental tanks, each species tended to aggregate when tested separately, but when tested jointly, aggregation among P. guttatus was significantly reduced. The experimental results reflect the differential patterns of aggregation between the fore-reef, where P. guttatus dominated, and the back-reef, where coexistence of both species was greater.     

A suggestion for a new approach to the mathematical theory of imitative behavior

The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.     

Parasitism and symbiosis in an N-person non-constant-sum continuous game

A non-constant-sum continuous game analyzed in a previous paper by one of the authors (A. Rapoport,Bull. Math. Biophysics,18, 15–30, 1956) is extended from two toN players with special emphasis onN=3. It is shown that the concept of stability becomes in this case one of “pairwise stability,” and depends on the so-called distribution matrix of rewards. The distribution matrix and the collusion structure jointly determine the end states of the game. Conditions which lead to the emergence of one, two, or no “parasites” are derived. An apparatus is described which provides a physical analogue of the game, making possible the isolation of behavioral variables under the prescribed conditions of the game.     

Contribution to the probabilistic theory of neural nets: I. Randomization of refractory periods and of stimulus intervals

Aggregates of neurons are considered in which the frequency of occurrence of neurons with a specified value of the refractory period follows certain probability distributions. Input-output functions are derived for such aggregates. In particular, if input and output intensities are defined in terms of stimulus frequencies and firing frequencies per neuron respectively, it is shown that a rectangular distribution of refractory periods leads to a logarithmic input-output curve. If input and output are defined in terms of the total number of stimuli and firings in the aggregate, it is shown how the “mobilization” picture leads to the logarithmic input-output curve. By randomizing the intervals between stimuli received by a single neuron and by introducing an inhibitory neuron a very simple “filter net” can be constructed whose output will be sensitive to a particular range of the input, and this range can be made arbitrarily small.     

An analysis of the 2o and 10o field color-matching functions

The 2^o and 10^o field color-matching functions are independent: one specification is not a linear transformation of the other, even after correcting for macular pigment effects. Therefore, the “true” color-matching functions which directly describe the linear responses of the eye must be different for the two field sizes. This means that a given stimulus will, in general, have a different chromaticity depending upon the field size, regardless of the choice of any one colorimetric co-ordinate system for all field sizes. However, in spite of these chromaticity differences, a large uniform field usually appears nearly uniform. Such color uniformity implies that even though chromatic differences occur as a function of retinal position or field size, these differences are small. If this is the case, then the underling “true” color-matching functions determining the observed color-matching functions must be nearly, but not quite, identical. These differences vanish as identity between the sets of color-matching functions is approached. This property suggests a method of calculating the “true” color-matching functions. The “true” color-matching functions must approximate those obtained by minimizing the chromaticity differences between two independent sets of data. This can be done by assuming that the coefficients of transformation should be adjusted so as to produce as nearly identical chromaticities for spectrum stimuli as possible. In this paper, it is also assumed that the “true” color-matching functions have no negative values, as if they were based on actual absorption spectra. This article describes the calculation of the “true” 2^o and 10^o field color-matching functions satisfying these two conditions. For both field sizes, the maxima of the three functions are near 435, 540, and 585 mμ, after correcting for the filtering effects of the ocular media and macular pigment.     

Allometric Effects of <Emphasis Type="Italic">Agriophyllum squarrosum</Emphasis> in Response to Soil Nutrients,Water, and Population Density in the Horqin Sandy Land of China

We monitored the allometric effects for greenhouse-grown Agriophyllum squarrosum plants in response to variations in population density and the availability of soil nutrients and water. Biomass allocations were size-dependent. The plasticity of roots, stems, leaves, and reproductive effort was “true” in response to changes in nutrient content. At a low level of soil minerals, plants allocated more resources to the development of roots and reproductive organs than to leaves, but data for stem allocations were consistent for tradeoffs between the effects of nutrients and plant size. The plasticities of leaf allocation and reproductive effort were “true” whereas those of root and stem allocations were “apparent” in response to fluctuations in soil water, being a function of plant size. Decreasing soil water content was associated with higher leaf allocation and lower reproductive effort. Except for this “apparent” plasticity of leaf allocation, none was detected with population density on biomass allocation. Architectural traits were determinants of the latter. For roots, the determining trait was the ratio of plant height to total biomass; for stems and reproduction, plant height; and for leaves, the ratio of branch numbers to plant height.     

The probabilistic approach to the effects of radiations and variability of sensitivity

The calculation of the size of the “sensitive volume” or “control center” in biological effects of radiations is discussed from the viewpoint of the probabilistic theory of these phenomena based on the concept of random “effective events”. On the bases of that theory, the resistivity of a microorganism to radiation is defined as its “mean life” under a radiation of one roentgen per minute. This mean is calculated for processes with and without recovery. The case of variable sensitivity, as it occurs for instance during mitosis, is discussed in detail. Methods are given to calculate this variability from survival curves or similar experimental data. The theory is applied to experiments of A. Zuppinger on irradiation ofAscaris eggs with X-rays.     

Carbohydrate tolerance,serum albumin and protein values of normal and “waster” marmosets (Callithrix jacchus)

The major objective of this study was to establish standard glucose and lactose tolerance curves for the common marmoset (Callithrix jacchus). These data were utilized to establish criteria for detection of abnormal glucose tolerance and characterization of some aspects of the “marmoset wasting syndrome” which has been observed in this species. Glucose and lactose tolerance tests were performed on healthy animals and typical “marmoset wasters.” Eighteen normal animals were 18 to 36 months old and weighed 194–280 g. Six “wasters” were in the age range of 24 to 84 months and weighed 163–253 g. Seven experiments were carried out for each glucose tolerance test. In each trial it was observed that the serum glucose concentration (SGC) of the healthy animal after 90 min was two times higher than the pre-administration concentrations. The SGC returned to the pre-administration concentration within 150–300 min in animals administered glucose at dosages of 2 g/kg and 1 g/kg of body weight. However, at the dosage level of 5 g/kg body weight, the SGC of the animals tripled after 30 min and required 300 min to return to the pre-administration level. The 2 g/kg dosage level was chosen as typical. When similar experiments were conducted with animals identified as “chronic wasters,” all of the animals except one were observed to be inefficient in the absorption of glucose. When lactose was administered at a level of 4 g/kg, similar results were obtained. Normal and “waster” marmosets were also subjected to serum total protein, albumin and electrophoresis determinations in an effort to establish additional criteria that may be utilized in the identification of the “marmoset wasting” syndrome. Serum albumin was significantly higher in the “waster” marmosets 30 min following an oral administration of glucose than was observed in normal animals. Total protein values were not significantly lower in the “wasters” when subjected to the same tolerance test. The albumin level in normal animals was not affected by similar glucose tolerance tests. The electrophoretic patterns of serum protein for normal animals exhibited more bands than was observed in patterns of serum protein for “waster” marmosets. From these data, it seems logical that these diagnostic tests may be useful in developing a profile for the early detection of the “wasting” syndrome in marmosets.     

Restudy and reassignment ofDentalium antiquum Goldfuss, 1841 (Middle Devonian)

Dentalium antiquum is one of two moderately well-known Devonian “scaphopods” in the German literature. Examination of the type material and a few specimens in other institutions indicates more individual variability than is to be expected in species of the molluscan class Scaphopoda. The species is transferred with question toColeolus Hall, a fossil presumed to be a calcareous “worm” tube.      

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.     

Analysis of the exponential decay model of the neuron showing frequency threshold effects

The exponential decay model of a neuron has been analyzed using the “random walk” approach of stochastic processes and an “absorbing barrier” solution is obtained forg ^T (s)—the Laplace transform of the output pulse interval density function. An expression for the mean output frequency is derived from this and a variety of input-output curves plotted which show frequency threshold effects in single neurons. Our results are compared with those of other authors obtained by computer simulation techniques, and the significance of these results discussed with reference to the possible behavior of networks constructed of such neuron units.     

Gain and diversity in advanced generation coastal Douglas-fir selections for seed production populations

Sublines are used in the third-generation breeding and testing of coastal Douglas-fir in British Columbia, with the original intent of selecting only one genotype per subline for production populations (e.g., seed orchards) to eliminate relatedness among parents (therein called “1/SL”). We evaluated three additional selection scenarios that did not consider the subline structure. One of the scenarios strictly selected on the basis of the highest breeding values of the trees (“TOP”); another scenario used the TOP selections, but assigned the number of ramets per selection proportionally to the selection breeding value (“LIND”); lastly, a simulated annealing technique was applied to maximize gain under explicit constraints on coancestry (“OPTS”). All three alternative selection scenarios resulted in some relatedness and coancestry among selections, but the last two provided increases in average breeding values compared to those obtained by the 1/SL scenario. Effective population sizes (and consequently inbreeding coefficients) varied among the three selection scenarios. Effects of the various selections on merchantable volume at rotation age were determined using a linear regression model based on an individual tree model (TASS), which was first run to determine the relationship between merchantable volume and inbreeding (f). LIND and TOP selections yielded the highest breeding values but, due to the increased coancestry among selections, paid a penalty in the merchantable volume determination. OPTS maximized merchantable volume at rotation age 60 after including more than 13 selections with an increase of around 3% over that obtained by the 1/SL selection scenario, with an associated increase in Ne of 50%. Other implications of the three alternative selection scenarios are discussed.     

Twins in free-ranging Hanuman langurs (Presbytis entellus)

Two pairs of twins were observed in free-ranging Hanuman langurs (Presbytis entellus) at Jodhpur, India. Each twin had a preferred nipple for suckling, the preference of which developed during the first 5 weeks of life. The behaviour of the twins was mostly synchronized. In more than 60% of all activity bouts both showed the same behaviour, with “nipple contact” and “playing” prevailing. Differences in physical strength and development were apparent within each pair. While the total amount of allomothering was almost the same, individual weekly scores differed considerably.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

Phase change-related variations of dome shape in Eucalyptus urophylla × Eucalyptus grandis shoot apical meristems

Shoot apical meristem (SAM) domes derived from five different outdoor and in vitro sources of juvenile and mature Eucalyptus urophylla × Eucalyptus grandis akin genotypes were compared. Overall measurements of SAM dome height H and diameter D ranged from 2 to 35 μm and 20 to 80 μm, with significant differences according to the various physiological origins of plant material investigated. SAM domes from the mature trees “Mat” were taller than those from the rejuvenated ministock plants “Rej”; from the in vitro microcuttings “IVM” of the same clone and also from the in vitro juvenile seedlings “IVJ”, whereas outdoor seedlings “Juv” exhibited intermediate SAM dome height. SAM domes from the rejuvenated material “Rej”, from the in vitro mature “IVM” and juvenile “IVJ” origins were also narrower than those from the outdoor seedlings “Juv” and to lesser extent than those from the mature trees “Mat”. Overall, the mature source “Mat” displayed bigger and somehow sharper hemispherical domes than those from “Rej” and “Juv”, physiologically more juvenile, or those from the in vitro origins “IVM” and “IVJ” which looked flatter and smaller. SAM dome height, diameter D and H/D values varied also significantly according to the plastochron. More specifically, H, D, and H/D SAM differences between the five origins were not significant during the early plastochron phase corresponding to leaf initiation, to become more salient as leaf structures started to elongate and to differentiate. This was particularly obvious for mature tree “Mat” SAM dome shapes which showed at this stage much higher H/D values than the other SAM sources. A shape index S used for characterizing more accurately dome shape confirmed these trends. These observations provide additional arguments to the view that juvenility in trees becomes more and more time- and shoot-tip restricted as ageing increases in the course of time during the ontogenetical process and could be ultimately confined to the most organogenic phase of SAM, from which shoot characteristics derive.     

Mathematical biology of social behavior

When an individual grows up in a society, he learns certain behavior patterns which are “accepted” by that society. He may in general have a tendency toward behavior patterns other than those which are “accepted” by the society. This tendency toward such unaccepted behavior may be due to a process of cerebration which results in doubt as to the “correctness” of the accepted behavior. Thus, on the one hand, the individual learns to follow the accepted rules almost automatically; on the other hand, he may tend to consciously break those rules. Using a neural circuit, suggested by H. D. Landahl in his theory of learning, a neurobiophysical interpretation of the above situation is outlined. Mathematical expressions are derived which describe the social behavior of an individual as a function of his age, social status, and some neurobiophysical parameters.     

Voiceprint identification and its application to sociological studies of wild Japanese monkeys (Macaca fuscata yakui)

Japanese monkeys often exchange the particular vocal sound, “coo,” especially when they feed or move as a group. It was considered that the “coo” sound had no positive social meaning, perhaps because the “coo” sound network and its function were hidden behind other behavioral observations. For identification of the vocalizer only from hearing the “coo” sound, three phonetic values, i.e., the “fundamental,” “duration,” and “formants,” plus other characteristics were used as indices of voiceprints. The results indicated that these were effective for identifying the vocalizer in two-thirds of the adults in the study troop which was composed of 12 adults and 16 immature members. The “coo” sound exchange network among the troop members (adults) was drawn on the basis of the voiceprint identification. The network showed three characteristics as follows: (1) matriarchs of the kin-groups frequently exchanged “coo” sounds with each other; (2) the other females exchanged “coo” sounds mostly within their own kin-groups; and (3) males seldom participated in the “coo” sound exchange. This suggests that “coo” sound exchange plays a central role for the matriarch of kin-groups in binding each kin-group and, ultimately, in binding all members together into an organized troop.