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.     

Rehabilitation of the Aged

The rehabilitation needs and goals of older people differ in many respects from those of the young. In younger individuals a crippling condition may affect an otherwise healthy body, while in an older person it may be superimposed on other pre-existing degenerative diseases. Thus, in older patients the restorative or rehabilitative phase rarely can be separated from the phase of definitive medical treatment.The primary goal of rehabilitation of younger individuals is usually vocational. In the older group this goal or objective is, by and large, secondary. This does not minimize, however, the value of medical and social services in the rehabilitation of older persons. The simple ability to care for his own personal needs can do much to help the elderly disabled patient regain his dignity and self-respect and remove his fears of becoming a burden on his family or society.     

Differences in Nutrient Requirements Imply a Non-Linear Emergence of Leaders in Animal Groups

Collective decision making and especially leadership in groups are among the most studied topics in natural, social, and political sciences. Previous studies have shown that some individuals are more likely to be leaders because of their social power or the pertinent information they possess. One challenge for all group members, however, is to satisfy their needs. In many situations, we do not yet know how individuals within groups distribute leadership decisions between themselves in order to satisfy time-varying individual requirements. To gain insight into this problem, we build a dynamic model where group members have to satisfy different needs but are not aware of each other''s needs. Data about needs of animals come from real data observed in macaques. Several studies showed that a collective movement may be initiated by a single individual. This individual may be the dominant one, the oldest one, but also the one having the highest physiological needs. In our model, the individual with the lowest reserve initiates movements and decides for all its conspecifics. This simple rule leads to a viable decision-making system where all individuals may lead the group at one moment and thus suit their requirements. However, a single individual becomes the leader in 38% to 95% of cases and the leadership is unequally (according to an exponential law) distributed according to the heterogeneity of needs in the group. The results showed that this non-linearity emerges when one group member reaches physiological requirements, mainly the nutrient ones – protein, energy and water depending on weight - superior to those of its conspecifics. This amplification may explain why some leaders could appear in animal groups without any despotism, complex signalling, or developed cognitive ability.     

A practical guide to the study of social relationships

Behavioral ecologists have devoted considerable effort to identifying the sources of variation in individual reproductive success. 1 - 5 Much of this work has focused on the characteristics of individuals, such as their sex and rank. However, many animals live in stable social groups and the fitness of individuals depends at least in part on the outcome of their interactions with other group members. For example, in many primate species, high dominance rank enhances access to resources and reproductive success. 4 , 5 The ability to acquire and maintain high rank often depends on the availability and effectiveness of coalitionary support. 6 Allies may be cultivated and coalitions may be reinforced by affiliative interactions such as grooming, food sharing, and tolerance. 7 , 8 These findings suggest that if we want to understand the selective pressures that shape the social behavior of primates, it will be profitable to broaden our focus from the characteristics of individuals to the properties of the relationships that they form with others. The goal of this paper is to discuss a set of methods that can be used to quantify the properties of social relationships.     

Knowledge acquired and decisions made: triadic interactions during allogrooming in wild bonnet macaques,Macaca radiata.

The pressures of developing and maintaining intricate social relationships may have led to the evolution of enhanced cognitive abilities in many nonhuman primates. Knowledge of the dominance ranks and social relationships of other individuals, in particular, is important in evaluating one''s position in the rank hierarchy and affiliative networks. Triadic interactions offer an excellent opportunity to examine whether decisions are taken by individuals on the basis of such knowledge. Allogrooming supplants among wild female bonnet macaques (macaca radiata) usually involved the subordinate female of a grooming dyad retreating at the approach of a female dominant to both members of the dyad. In a few exceptional cases, however, the dominant member of the dyad retreated; simple non-cognitive hypotheses involving dyadic rank differences and agonistic relationships failed to explain this phenomenon. Instead, retreat by the dominant individual was positively correlated with the social attractiveness of her subordinate companion (as measured by the duration of grooming received by the latter from other females in the troop). This suggests that not only does an individual evaluate relationships among other females, but does so on the basis of the amount of grooming received by them. Similarly, the frequency of approaches received by any female was correlated with her social attractiveness when she was the dominant member of the dyad, but not when she was the subordinate. This indicated that approaching females might be aware of the relative dominance ranks of the two allogrooming individuals. In logistic regression analyses, the probability of any individual retreating was found to be influenced more by her knowledge of her rank difference with both the other interactants, rather than by their absolute ranks. Moreover, information about social attractiveness appeared to be used in terms of correlated dominance ranks. The nature of knowledge acquired by bonnet macaque females may thus be egotistical in that other individuals are evaluated relative to oneself, integrative in that information about all other interactants is used simultaneously, and hierarchical in the ability to preferentially use certain categories of knowledge for the storage of related information from other domains.     

Leading ladies: leadership of group movements in a pair-living, co-dominant, monomorphic primate across reproductive stages and fruit availability seasons

For gregarious species, individuals must maintain cohesion while minimizing the costs of coordinated travel. Leaders of group movements potentially influence energy expenditure, energy intake, and predation risk for individuals in the group, which can have important fitness consequences. Models of pair-living species predict that energetic asymmetries lead to an emergent leader, with those in greater need leading. We investigated sex differences in leadership in pairs of red-bellied lemurs, Eulemur rubriventer, a monomorphic species with bisexual dispersal and no discernible hierarchy, to determine whether higher energetic requirements by adult females lead to female leadership. We collected leadership data in Ranomafana National Park, Madagascar on six groups of habituated E. rubriventer for 13 consecutive months between 2004-2005. To determine whether females led group movements more than males, we examined the difference in leadership frequencies of progressions in adult males and adult females within each group (n = 1,346 progressions). We further investigated the behavioral context (i.e. travel followed by feeding or not) and seasonal contexts (fruit availability, reproduction) of leadership. Group leadership was distributed, with different individuals leading the group at different times. However, females led significantly more than males, a pattern which was consistent in both feeding and non-feeding contexts and throughout all fruiting seasons and reproductive stages. While disparities in energetic status among the sexes may impact leadership in this species, leadership did not differ with changes in food availability or reproductive stage, and thus we were unable to determine whether female leadership might be related to changes in energetic status. Females may have higher energetic needs than males at all times, not merely seasonally, or female leadership may be unrelated to immediate energetic need. Rather, female leadership may be a legacy of female dominance not currently expressed in other contexts.     

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.     

A theory of leadership in human cooperative groups

Two types of models aim to account the origins of rank differentiation and social hierarchy in human societies. Conflict models suggest that the formation of social hierarchies is synonymous with the establishment of relationships of coercive social dominance and exploitation. Voluntary or 'integrative' models, on the other hand, suggest that rank differentiation - the differentiation of leader from follower, ruler from ruled, or state from subject - may sometimes be preferred over more egalitarian social arrangements as a solution to the challenges of life in social groups, such as conflict over resources, coordination failures, and free-riding in cooperative relationships. Little formal theoretical work, however, has established whether and under what conditions individuals would indeed prefer the establishment of more hierarchical relationships over more egalitarian alternatives. This paper provides an evolutionary game theoretical model for the acceptance of leadership in cooperative groups. We propose that the effort of a leader can reduce the likelihood that cooperation fails due to free-riding or coordination errors, and that under some circumstances, individuals would prefer to cooperate in a group under the supervision of a leader who receives a share of the group's productivity than to work in an unsupervised group. We suggest, in particular, that this becomes an optimal solution for individual decision makers when the number of group members required for collective action exceeds the maximum group size at which leaderless cooperation is viable.     

Agonistic Support in Juvenile Japanese Macaques: Cognitive and Functional Implications

Agonistic support is generally considered a classical example of altruistic behaviour. It plays a critical role in the process of rank acquisition. This study investigates agonistic coalitions involving juvenile Japanese macaques (Macaca fuscata) living in a large captive group. Kinship was a good predictor of agonistic support received by juvenile macaques. Kinship also had a significant, though weaker, influence on agonistic support given by juvenile macaques. The latter was strongly influenced by the prospective rank of the individual against which the coalition was directed, i.e. by the rank relationship between the target of the coalition and the juvenile’s mother. Juvenile macaques intervened aggressively more against prospectively subordinate individuals than against prospectively dominant individuals. These results imply agonistic support is not necessarily altruistic, and support the hypothesis that primates may understand the rank relationship that exist among their group mates, i.e. that they are aware of third‐party social relationships. They also suggest that selective pressures associated with the process of rank acquisition may have favoured the evolution of higher cognitive abilities.     

Scalable rules for coherent group motion in a gregarious vertebrate

Individuals of gregarious species that initiate collective movement require mechanisms of cohesion in order to maintain advantages of group living. One fundamental question in the study of collective movement is what individual rules are employed when making movement decisions. Previous studies have revealed that group movements often depend on social interactions among individual members and specifically that collective decisions to move often follow a quorum-like response. However, these studies either did not quantify the response function at the individual scale (but rather tested hypotheses based on group-level behaviours), or they used a single group size and did not demonstrate which social stimuli influence the individual decision-making process. One challenge in the study of collective movement has been to discriminate between a common response to an external stimulus and the synchronization of behaviours resulting from social interactions. Here we discriminate between these two mechanisms by triggering the departure of one trained Merino sheep (Ovis aries) from groups containing one, three, five and seven naïve individuals. Each individual was thus exposed to various combinations of already-departed and non-departed individuals, depending on its rank of departure. To investigate which individual mechanisms are involved in maintaining group cohesion under conditions of leadership, we quantified the temporal dynamic of response at the individual scale. We found that individuals'' decisions to move do not follow a quorum response but rather follow a rule based on a double mimetic effect: attraction to already-departed individuals and attraction to non-departed individuals. This rule is shown to be in agreement with an adaptive strategy that is inherently scalable as a function of group size.     

Vigilance and social organization in two species of primates

Animals that live in groups may accrue anti-predator benefits by virtue of their association with other individuals. While a number of factors (e.g. habitat quality, group geometry, dominance rank) have been shown to affect the relationship between group living and individual rates of visual vigilance, sociality and vigilance have not been studied specifically in terms of the two competing demands they impose upon an individual's visual time. Two captive groups each of tamarins, Saguinus labiatus, and squirrel monkeys, Saimiri sciureus, were used in three related experiments to determine whether they looked at the social or non-social environment while they engated in a simulated foraging task. The tamarins, whose social behaviour is quite pacific and cooperative relative to the more hierarchical squirrel monkeys, looked at the non-social environment when they visually interrupted their foraging. Squirrel monkeys, however, looked more often at group mates. Detection of predators is probably more likely when an individual directs its attention not to conspecifics, but to the environment. Hence, social organizations in which individuals must pay social attention in order to monitor threats and avoid aggression may reduce individual rates of vigilance for predators. Failure to consider various targets of attention may well overestimate vigilance for predators by including other sorts of attentional phenomena in the measurement of vigilance.     

A note on a possible mathematical approach to the theory of individual freedom

As suggested in previous publications, freedom may be defined quantitatively as a restriction upon the choice of a number of activities. If the choice is determined by maximizing the satisfaction function, it is suggested that freedom may be defined in terms of the satisfaction function. If an individual is isolated and no physical restrictions limit his choice of activities, he is free to choose any activity in an amount which maximizes his satisfaction. This isolated state may be considered therefore as that of maximum freedom. If the individual interacts with another, he will choose different amounts of his object of satisfaction depending on whether he behaves egoistically or altruistically. But in either case the value chosen will not maximize his satisfaction function considered alone. A simple analytical expression is suggested as a measure of freedom in this case, and some problems which arise from this suggestion are mentioned.     

Promiscuous Mating among Blue Monkeys in the Kakamega Forest,Kenya

13 females and 12 of 19 males present in a group of blue monkeys (Cercopithecus mitis stuhlmanni) mated promiscuously during an intense bout of sexual activity. Several aspects of their behavior are considered as mechanisms for determining mating patterns. Estrus synchrony coincided with a multi-male influx, allowing promiscuity. The length of stay and rank of an incoming male were correlated with his number of copulations in the group, while the original resident and highest ranking male was never seen to mate. Thus, although agonistic behavior, especially among males, affected mating success, it did not guarantee copulations. Adjustments of position relative to other individuals also affected who mated with whom. These and earlier observations suggest that there may be more than one way to be successful in mate competition, and lead us to question whether the kinds of behavior we observed have merely a reproductive function.     

Individuality and the existence of species through time

The individuality of species provides the basis for linking practical taxonomy with evolutionary and ecological theory. An individual is here defined as a collection of parts (lower-level entities) that are mutually connected. Different types of species individual exist, based on different types of connection between organisms. An interbreeding species is a group of organisms connected by the potential to share common descendants, whereas a genealogical species is integrated by the sharing of common ancestors. Such species definitions serve to set the limits of species at a moment of time and these slices connect through time to form time-extended lineages. This perspective on the nature of individuality has implications that conflict with traditional views of species and lineages: (1) Several types of connections among organisms may serve to individuate species in parallel (species pluralism); (2) each kind of species corresponds to a distinct kind of lineage; (3) although lineage branching is the most obvious criterion to break lineages into diachronic species, it cannot be justified simply by reference to species individuality; (4) species (like other individuals) have fuzzy boundaries; (5) if we wish to retain a species rank, we should focus on either the most- or least-inclusive individual in a nested series; (6) not all organisms will be in any species; and (7) named species taxa are best interpreted as hypotheses of real species. Although species individuality requires significant changes to systematic practice and challenges some preconceptions we may have about the ontology of species, it provides the only sound basis for asserting that species exist independently of human perception.     

Mathematical biology of division of labor between two individuals or two social groups

The concept of the satisfaction function is applied to the situation in which two individuals may each produce two needed objects of satisfaction, or may each produce only one of the objects and then make a partial exchange. It is shown that with a logarithmic satisfaction function there is no advantage in a division of labor, unless such a division materially increases the purely physical efficiency of production. This result appears to be connected with the particular choice of the form of satisfaction function (logarithmic). While the problem has not been solved for other forms, it is made plausible that satisfaction functions which have an asymptote will lead to a different result. Next the case is studied in which division of labor occurs between two groups of individuals. It is shown that in this case the relative sizes of the two groups are determined from considerations of maximum satisfaction. Possible applications to problems of urbanization are suggested.     

Multilocus genotypes, a tree of individuals, and human evolutionary history.

Our goal is to infer, from human genetic data, general patterns as well as details of human evolutionary history. Here we present the results of an analysis of genetic data at the level of the individual. A tree relating 144 individuals from 12 human groups of Africa, Asia, Europe, and Oceania, inferred from an average of 75 DNA polymorphisms/individual, is remarkable in that most individuals cluster with other members of their regional group. In order to interpret this tree, we consider the factors that influence the tree pattern, including the number of genetic loci examined, the length of population isolation, the sampling process, and the extent of gene flow among groups. Understanding the impact of these factors enables us to infer details of human evolutionary history that might otherwise remain undetected. Our analyses indicate that some recent ancestor(s) of each of a few of the individuals tested may have immigrated. In general, the populations within regional groups appear to have been isolated from one another for <25,000 years. Regional groups may have been isolated for somewhat longer.     

An empirical test of Lanchester's square law: mortality during battles of the fire ant Solenopsis invicta

Lanchester's models of attrition describe casualty rates during battles between groups as functions of the numbers of individuals and their fighting abilities. Originally developed to describe human warfare, Lanchester's square law has been hypothesized to apply broadly to social animals as well, with important consequences for their aggressive behaviour and social structure. According to the square law, the fighting ability of a group is proportional to the square of the number of individuals, but rises only linearly with fighting ability of individuals within the group. By analyzing mortality rates of fire ants (Solenopsis invicta) fighting in different numerical ratios, we provide the first quantitative test of Lanchester's model for a non-human animal. Casualty rates of fire ants were not consistent with the square law; instead, group fighting ability was an approximately linear function of group size. This implies that the relative numbers of casualties incurred by two fighting groups are not strongly affected by relative group sizes and that battles do not disproportionately favour group size over individual prowess.     

Use of Social and Ecological Information in Tamarin Foraging Decisions

A major consequence of group living is that foragers may rely on social information in addition to ecological information to locate feeding sites. Although conspecifics can provide cues as to the spatial location of food patches, individual foraging decisions also must include some assessment of the likelihood of obtaining access to a resource other group members seek. This likelihood differs in the 2 models generally proposed to explain intragroup social foraging: the information-sharing model and the producer-scrounger model. We conducted an experimental field study on wild groups of emperor (Saguinus imperator) and saddleback (S. fuscicollis) tamarins to determine the foraging strategies adopted by individual group members and their relationship to social rank, food intake, and the ability to use ecological and social information in making intra-patch foraging decisions. Individual tamarins applied different behavioral strategies compatible with a finder-joiner paradigm to solve foraging problems. About half of the individuals in each study group initiated 74%–90% of all food searches and acted as finders. Most alpha individuals adopted a joiner strategy by monitoring the activities of others' to obtain a reward. The individual arriving first at a reward platform enjoyed a finder's advantage. Despite differences in search effort, both finders and joiners presented similar abilities in learning to associate ecological cues with the presence of food rewards at our experimental feeding stations. We conclude that within a group foraging context, tamarins integrate social and ecological information in decision-making.