Mathematical biology of social behavior: IV. Imitation effects as a function of distance

In previous publications, social groups have been studied in which each individual has a preference for one of two possible mutually exclusive activities. This preference is measured by a quantity ϕ. The value ϕ=0 corresponds to no preference; a preference for one activity is measured by a positive ϕ, the preference for the other by a negative ϕ. The quantity ϕ varies from individual to individual. It has been shown previously that, owing to effects of imitation, even when the average ϕ for the group is zero, one of the two behaviours will be chosen by the majority of the group. Whereas in previous studies the imitation effect was considered as independent of the distance between the imitating and imitated individuals, in the present study the case is considered in which the effect of imitation decreases with the distance between the individuals. It is found that under certain conditions a greater percentage near the center of the area occupied by the group, rather than near the periphery, exhibits the chosen behavior. The possible sociological meaning of this gradient of behavior is discussed.     

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.     

Mathematical biological of social behavior: II

A group of individuals is considered in which each individual has tendencies to exhibit one or another of two mutually exclusive behaviors. Neurobiophysically this may be described in terms of Landahl's reciprocally inhibited parallel reaction chains. The spontaneous excitations ε₁ and ε₂ at the central connections of each chain are a measure of the “natural” tendency of the individual toward one or the other of the two behaviors. According to equations derived by H. D. Landahl, the probability of one or the other behavior is determined by the difference ε₁ − ε₂. A population of individuals is considered in which ε₁ − ε₂ is distributed in some continuous way, and therefore in which the probability of a given behavior is distributed continuously between 0 and 1. The effect of other individuals exhibiting a given behavior is to increase the corresponding ε of the individual. Thus behavior of others affects the probability for a given behavior of each individual. It is shown that the equations describing the behavior of the population on the basis of this neurobiophysical picture reduce in the first approximation to the differential equations which were postulated by the author in his previous work on social behavior.     

On imitative behavior

Previous studies of effects of imitation on individuals in a population, in which the tendencies ϕ towards one or another of two mutually exclusive behaviors are distributed, are amplified by considering the distribution, not of ϕ directly, but of the excitations ɛ₀₁ and ɛ₀₂ of the two centers which mediate each of the two behaviors. It is shown how the distribution of ϕ is derived from those of ɛ₀₁ and ɛ₀₂. It is found that when both tendencies ɛ₀₁ and ɛ₀₂ are weak, the choice of one of the two behaviors not only is originally determined by pure chance, but that it is impossible to effect a change of the behaviour of a large population from one adopted behavior to a possible opposite one, by inhibiting the tendency towards the first behavior. Such a change by inhibition is possible only when the tendencies toward both mutually exclusive behaviors are sufficiently strong. A possible application to the persistency of irrelevant established behavior patterns, such as handshakes, is suggested.     

Contributions to the theory of imitative behavior

The imitation effects in a social group depend both on the size of the group and on the distribution of a certain psychobiological quantity ϕ which measures the tendency of an individual towards a given behavior. The distribution function of ϕ determines the ratio μ of the individuals in the society who adopt a given behavior. When the size of the social group is not too large, the actual distribution of ϕ will deviate from the most probable one, and therefore communities of the same size and having the same parameters may have different values of μ. Approximate equations are developed which give the probability of a given μ for a group of a given size. Possible effects of interactions of communities of different sizes are briefly discussed. A generalization of the theory of imitative behavior to any number of mutually exclusive behaviors is given, and its possible sociological implications are discussed.     

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.     

Mathematical theory of motivation interactions of two individuals: I

The behavior of two individuals is considered as consisting of an increase or a decrease of productive output. Motivation for increase is the derivative of a “satisfaction function”. This is an algebraic sum of the well-known Thurstone satisfaction curve and another essentially negative quantity, which is a product of a “reluctance parameter” and the “effort”. Each individual attempts to maximize his own total satisfaction. The resulting behavior is examined under a variety of conditions; namely, 1) equal sharing of produced without prescribed sharing of effort; 2) various contracts prescribing the sharing of effort; 3) situations in which one individual is more aware of the underlying motivations than the other. It is these latter situations which under the simplest assumptions of equal sharing, without prescribed sharing of effort, lead to parasitism, i.e. total cessation of effort on the part of one individual. This happens when one individual becomes aware of the other'sautomatic adjustment of his effort so as to bring about a total optimum output, which is a constant. Parasitism is prevented by various forms of contracts in which either the effort necessary for the total optimum output is shared according to a prescribed ratio or the effort of one individual is fixeda priori as a function of the effort of the other. In the latter case the respective efforts become a function of a single variable, and each of the satisfaction functions is maximized by a particular value or values of this variable. In general, these critical values do not coincide for the two satisfaction functions. The problem of finding forms of contract which will result in identical critical (maximizing) values of the variable for both satisfaction functions leads to a functional equation.     

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.     

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.     

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.     

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.     

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.     

Benefiting friends or dominants: prosocial choices mainly depend on rank position in long-tailed macaques (<Emphasis Type="Italic">Macaca fascicularis</Emphasis>)

Long-term observational studies in a number of animal species suggest that exchange patterns of social acts depend on long-term emotional bonds. Therefore, it is expected that the frequency of prosocial behavior will depend on the strength of such a bond. In this study we tested whether variation in relationship quality among unrelated individuals, i.e., “friends” and “nonfriends,” is predictive of the prosocial behavior of long-tailed macaques in two experiments. First, we related relationship quality to prosociality in a dyadic prosociality test, and second, we gave subjects the choice to give to either a friend or a nonfriend in a triadic choice test. We show that prosocial behavior of long-tailed macaques in the dyadic test is not related to relationship quality. When given the choice to give to either a friend or a nonfriend in the triadic test, there is a minor indication that long-tailed macaques show a preference to give to their friends, yet this indication is neither significant nor consistent. In contrast, subordinate long-tailed macaques make a more “competitive” choice and avoid giving to the individual closest in rank. Therefore, in the short-term situation of experimental tests, prosocial behavior of long-tailed macaques seems unaffected by the relationship quality of the dyad/triad tested, and the relative dominance position of these dyads/triads seems to have a much stronger effect on their prosocial behavior.     

Signals found in the grooming interactions of wild Japanese monkeys of the koshima troop

Signaling behaviors appearing in grooming interactions of wild Japanese monkeys were analysed. Vocal signals found in the grooming interactions had the content of asking the objective animal “if the vocal signaler may groom the recipient animal.” They could be divided into two categories of vocal sounds, VG-1 and VG-2. The former was uttered in common by all the troop members. The latter was uttered just before grooming by the groomer and is considered to have deeper connection with grooming. Each individual uttered mainly one kind of vocal sound out of VG-2, and the preferred vocal sounds for each individual differed. Furthermore, VG-2 differed in different troops. Behavioral signals had the content of showing “the acceptance of grooming” or showing “the request to be groomed.” The appearance of these signaling behaviors was closely related to the inter-individual relationships of grooming partners, especially as to whether or not they had blood relationships. Basically the monkeys have a system in which they must avoid each other, except in the case of mothers and their offspring, and if they had to approach too closely against this basic system, as in grooming interactions, there appeared signaling behaviors as mentioned above.     

The dynamic behavior of phycobilisome movement during light state transitions in cyanobacterium <Emphasis Type="Italic">Synechocystis</Emphasis> PCC6803

Light state transition is a physiological function of oxygenic organisms to balance the excitation of photosystem II (PSII) and photosystem I (PSI), hence a prerequisite of oxygen-evolving photosynthesis. For cyanobacteria, phycobilisome (PBS) movement during light state transition has long been expected, but never observed. Here the dynamic behavior of PBS movement during state transition in cyanobacterium Synechocystis PCC6803 is experimentally detected via time-dependent fluorescence fluctuation. Under continuous excitation of PBSs in the intact cells, time-dependent fluorescence fluctuations resemble “damped oscillation” mode, which indicates dynamic searching of a PBS in an “overcorrection” manner for the “balance” position where PSII and PSI are excited equally. Based on the parallel model, it is suggested that the “damped oscillation” fluorescence fluctuation is originated from a collective movement of all the PBSs to find the “balance” position. Based on the continuous fluorescence fluctuation during light state transition and also variety of solar spectra, it may be deduced that light state transition of oxygen-evolution organisms is a natural behavior that occurs daily rather than an artificial phenomenon at extreme light conditions in laboratory.     

A contribution to the mathematical biology of social behavior: Riots by oppressed groups

The author’s theory of imitation or mass behavior (N. Rashevsky:Mathematical Biology of Social Behavior, chapter xii, revised edition. Chicago: The University of Chicago Press, 1959; also Rashevsky:Looking at History through Mathematics, Cambridge, Mass.: The MIT Pres, 1968), when society chooses one of two mutually exclusive behaviors, is applied to the interaction of two social groups, an oppressor group and an oppressed one. Using crude approximations, conditions are derived as to when the oppressed group will revolt or riot, when the revolt will be suppressed, and when the oppressors will completely give in and oppression will end. Even in the simple approximation used, the situation depends on 14 parameters showing that a simplistic view on riots such as mere strong punishments is utterly inadequate. It is also shown that situations may exist in which revolution-like changes from one type of behavior of a society to another cannot be prevented by any measures.     

Estimate of complexity of behavioral patterns in ants: analysis of hunting behavior in <Emphasis Type="Italic">Myrmica rubra</Emphasis> (Hymenoptera,Formicidae) as an example

A method for estimating the complexity of behavioral patterns of ants based on the Kolmogorov complexity is considered. Behavioral sequences are presented as “texts” compressed with the KGB Archiver (v. 1.2). The elements of behavior (a total of 10) singled out from video records served as an alphabet. The comparison of “successful” and “incomplete” hunting behaviors in Myrmica rubra showed that successful hunting patterns were characterized by less complexity than “incomplete” ones. It was assumed that complete patterns had less redundancy and better predictability. The smallest complexity was revealed in complete hunting patterns of naive (laboratory reared) ants in comparison with members of a natural colony. In perspective, quantitative evaluation of complexity of behavioral patterns will help to evaluate the level of discrete variability within ant colonies.     

Reproductive patterns in mantled howler monkeys: Estrus,mate choice and copulation

The present study was undertaken to evaluate non-random mating patterns in two groups of mantled howler monkeys in two tropical dry forest habitats. Sexual dimorphism, female estrus stage, male dominance rank, sexual solicitations and copulations were assessed. Males are significantly larger than females, but female weight varies more than male weight. The length of female estrus cycles is comparable in both habitats, but females in the more strongly seasonal habitat demonstrate greater estrus synchrony relative to their numbers. Males solicit potential mates more frequently than females, a pattern explained by the relatively high rate of sexual solicitation by high-ranking males. Females in “peak” estrus solicit “alpha” males, while females in other stages of estrus solicit males equally by rank. Intersexual aggression occurs rarely, and “forced copulations” are attempted but, apparently, are unsuccessful. Sexual solicitations by “alpha” males and “peak” estrus females are most likely to lead to copulation, and “alpha” males are more likely to copulate than “gamma” males. In general, latencies from first solicitation to copulation are expensive in time, especially for high-ranking males. Estimated annual reproduction success favors high-ranking males, and results indicate that male and female mating behavior is mutually coordinated and controlled.     

Biosemiosis and the Cellular Basis of Mind

The human brain is a complex organ made up of neurons and several other cell types, and whose role is processing information for use in elicitation of behaviors. To accomplish this, the brain requires large amounts of energy, and this energy is obtained by the oxidation of glucose (Glc). However, the question of how the oxidation of Glc by individual neurons in brain results in their collective ability to rapidly generate feats of cognition that allow them to recognize the nature of the universe in which they live and to communicate this information remains unclear. In this article, insights into this process are provided by first considering the brain’ s homeostatic “operating system” for supply of energy to stimulated neurons, and how this system defines the basic unit of brain “structure”. This is followed by consideration of the brain’s “two-cell” neuronal communication mechanism which defines the basic unit of brain “function”. Finally, an analysis of the nature of frequency-encoded “neuronal languages” that enable ensembles of neurons to translate energy derived from the oxidation of Glc into a collective “mind”, the aggregate of all brain processes including those involving perception, thought, insight, foresight, imagination and behavior.     

Patterns of orientation behavior in marsh frogs (Rana ridibunda Pall.) of southern populations

The behavior of frogs from southern areas with an arid climate released during the reproductive period between their “home” pond and a less distant river is shown to be independent of weather conditions. The experiments were performed on frogs living near the village of Dosang, Astrakhan oblast. The frogs were released 60 to 150 m from their “home” breeding waterbody and 60 to 80 m from an “alien” waterbody. Four experiments were performed on 27 individuals. The movements of the frogs were traced by the method of “tracking by a thread.” The results revealed no preference in the frogs to return to their own breeding ponds, rather than to the nearby river. This behavior is peculiar to frogs of southern populations.