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.     

The problem of exchange between two or more individuals,motivated by hedonistic considerations

Let two or more individuals each possess different quantities of two objects of satisfaction. Under certain conditions they may agree to exchange part of the objects if this leads to an increase of each one's satisfaction. The equations which govern this process have been derived by G. E. Evans (1930) for the case of two individuals. A different proof of these equations is given here and the equations are generalized to the case of more than two individuals.     

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 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.     

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.     

Time variable osmotic pressures produced by coupled reactions as a possible cause of cell division

It has been shown in a previous paper that coupled reactions may produce diffusion phenomena which are characterized by highly asymmetric distributions of concentrations in a spherical cell. Under certain conditions such diffusion fields are unstable, the asymmetries in concentrations tending to increase indefinitely. This results in an increase of asymmetrically distributed osmotic pressures which may eventually result in a division of a cell. Constriction without elongation, as in cleavage, is studied. The process of division is brought about in this case by ordinary osmotic pressure, and not by the diffusion drag forces.     

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.     

Relatedness and the evolution of mechanisms to divide labor in microorganisms

Division of labor occurs when cooperating individuals specialize to perform different tasks. In bacteria and other microorganisms, some species divide labor by random specialization, where an individual''s role is determined by random fluctuations in biochemical reactions within the cell. Other species divide labor by coordinating across individuals to determine which cells will perform which task, using mechanisms such as between‐cell signaling. However, previous theory, examining the evolution of mechanisms to divide labor between reproductives and sterile helpers, has only considered clonal populations, where there is no potential for conflict between individuals. We used a mixture of analytical and simulation models to examine nonclonal populations and found that: (a) intermediate levels of coordination can be favored, between the extreme of no coordination (random) and full coordination; (b) as relatedness decreases, coordinated division of labor is less likely to be favored. Our results can help explain why coordinated division of labor is relatively rare in bacteria, where groups may frequently be nonclonal.     

On the importance of evolving phenotype distributions on evolutionary diversification

Evolutionary branching occurs when a population with a unimodal phenotype distribution diversifies into a multimodally distributed population consisting of two or more strains. Branching results from frequency-dependent selection, which is caused by interactions between individuals. For example, a population performing a social task may diversify into a cooperator strain and a defector strain. Branching can also occur in multi-dimensional phenotype spaces, such as when two tasks are performed simultaneously. In such cases, the strains may diverge in different directions: possible outcomes include division of labor (with each population performing one of the tasks) or the diversification into a strain that performs both tasks and another that performs neither. Here we show that the shape of the population’s phenotypic distribution plays a role in determining the direction of branching. Furthermore, we show that the shape of the distribution is, in turn, contingent on the direction of approach to the evolutionary branching point. This results in a distribution–selection feedback that is not captured in analytical models of evolutionary branching, which assume monomorphic populations. Finally, we show that this feedback can influence long-term evolutionary dynamics and promote the evolution of division of labor.     

Seasonally Variable Eusocially Selected Traits in the Paper Wasp, Mischocyttarus mexicanus

The expression of alternative traits that benefit eusocial individuals but are not directly involved in reproductive differences among those individuals, which I call ‘eusocially selected traits’, may vary in response to environmental changes if this increases an individual's inclusive fitness. In this study, I describe traits that separate individuals within the reproductive division of labor of Mischocyttarus mexicanus, a eusocial paper wasp, and determine whether observed eusocially selected traits vary across seasons. I examined M. mexicanus because females initiate new nests throughout most of the year where they experience different conditions depending on the season. Findings from this study suggest two main conclusions: (1) phenotypic differences among M. mexicanus females are mixed, showing specialized, generalized, and context‐dependent eusocially selected traits and (2) a female's position within the reproductive division of labor may be influenced by its state. The presence of context‐dependent traits, e.g. large females initiated solitary nests in the spring and grouped nests during the summer, suggests that the payoff for pursuing different positions within the reproductive division of labor changes across seasons. The expression of context‐dependent eusocially selected traits also suggests that, roles, instead of castes, may better reflect the reproductive division of labor among individuals of eusocial species like M. mexicanus.     

A twelve-step program for evolving multicellularity and a division of labor

The volvocine algae provide an unrivalled opportunity to explore details of an evolutionary pathway leading from a unicellular ancestor to multicellular organisms with a division of labor between different cell types. Members of this monophyletic group of green flagellates range in complexity from unicellular Chlamydomonas through a series of extant organisms of intermediate size and complexity to Volvox, a genus of spherical organisms that have thousands of cells and a germ-soma division of labor. It is estimated that these organisms all shared a common ancestor about 50 +/- 20 MYA. Here we outline twelve important ways in which the developmental repertoire of an ancestral unicell similar to modern C. reinhardtii was modified to produce first a small colonial organism like Gonium that was capable of swimming directionally, then a sequence of larger organisms (such as Pandorina, Eudorina and Pleodorina) in which there was an increasing tendency to differentiate two cell types, and eventually Volvox carteri with its complete germ-soma division of labor.     

Outline of a mathematical biology of leadership

Leadership, whether executive, political or any other type, is connected with the achievement of some goal by the social group through an appropriate organization of that group. From this point of view different leadership ranks in a group would be assigned to individuals according to their ability to organize the group for the purpose of reaching a specific goal. The situation is actually complicated by the circumstance that an individual may have the necessary ability but may not like the responsibility connected with the leadership, or vice versa. Also, he may not be interested in the goal. The suggested mathematical approach is to consider that the satisfaction of an individual is a function of his leadership rank, of the goal, and of several other parameters. If each individual tends to adjust his position in society so as to maximize his own satisfaction, this condition gives us the equations which determine the leadership rank of each individual. It is found that, in general, the rank of an individual depends not only on his ability, but on the abilities of all other individuals. The method enables us to calculate the distribution function of abilities among individuals of a given rank, and leads to results which allow, in principle, experimental verification.     

Three-player social parasitism games: implications for resource defense and group formation

Abstract: Individuals that produce resources are often exploited by several individuals; such exploitation may or may not be tolerated. We modeled the decision of a resource owner to accept one scrounger (the "satellite") and of both of these to accept being joined by another (the "floater"). In general, satellites tolerated floaters when competition between them was low, while owners tolerated satellites when facilitation between satellite and floater was high. When floaters were likely to find resources without joining, owners were more likely to resist satellites. In some cases, Nash equilibria were also mutually beneficial for two of the three individuals. Our model makes the counterintuitive prediction that mutually beneficial coalitions between satellites and floaters can only arise when the net benefits arising from the other's presence are low. When facilitation between satellites and floaters is high, satellites and owners may form mutually beneficial groups and groups with division of labor, or alternatively, owners may benefit from encouraging floaters to join. Finally, our model suggests there must be differences in competitive ability or some benefit of familiarity for owners to tolerate satellites but not floaters. We discuss empirical evidence for these and other predictions of the model.     

Worker policing limits the number of reproductives in a ponerine ant

Reproductive division of labour is an essential feature of insect sociality, but the regulation of sterility among colony members remains incompletely understood. Ant workers and queens are morphologically divergent and workers are only capable of producing males in a colony, although they usually do not do so. Worker policing is one mechanism proposed for their infertility and it can be expressed as either aggressive inhibition of ovarian activity among workers or destruction of worker-laid eggs. A few studies have shown that workers with developed ovaries are preferentially attacked by nest-mates, but adequate demonstration of worker policing also requires evidence that these attacks result in the suppression of ovarian activity or death. We investigated worker policing in the ponerine ant Harpegnathos saltator in which workers are able to mate and replace the founding queen. Five colonies were each divided into two groups, one of which consisted exclusively of infertile workers. Some individuals in the orphaned groups began laying eggs during the three-week separation and upon reunification these were vigorously attacked by infertile workers of the other groups. The ovarian activity of these new egg layers became inhibited, as revealed by subsequent dissection of marked individuals. Worker policing in H. saltator appears to function primarily in preventing an excess of reproductive workers.     

Nutritional asymmetries are related to division of labor in a queenless ant

Eusocial species exhibit pronounced division of labor, most notably between reproductive and non-reproductive castes, but also within non-reproductive castes via morphological specialization and temporal polyethism. For species with distinct worker and queen castes, age-related differences in behavior among workers (e.g. within-nest tasks versus foraging) appear to result from physiological changes such as decreased lipid content. However, we know little about how labor is divided among individuals in species that lack a distinct queen caste. In this study, we investigated how fat storage varied among individuals in a species of ant (Dinoponera australis) that lacks a distinct queen caste and in which all individuals are morphologically similar and capable of reproduction (totipotent at birth). We distinguish between two hypotheses, 1) all individuals are physiologically similar, consistent with the possibility that any non-reproductive may eventually become reproductive, and 2) non-reproductive individuals vary in stored fat, similar to highly eusocial species, where depletion is associated with foraging and non-reproductives have lower lipid stores than reproducing individuals. Our data support the latter hypothesis. Location in the nest, the probability of foraging, and foraging effort, were all associated with decreased fat storage.     

Effects of gonadectomy and hormone replacement on a spontaneous novel object recognition task in adult male rats

Recent studies in adult male rats have shown that gonadal hormones influence performance on certain working memory and other types of cognitive tasks that are sensitive to lesions of the medial and/or orbital prefrontal cortices. This study asked whether gonadal hormone modulation of prefrontal cortical function in males also extends to the perirhinal division of the rat prefrontal cortex. Specifically, sham-operated control, gonadectomized, and gonadectomized rats supplemented with testosterone propionate or estradiol were tested on a spontaneous novel object recognition task, a paradigm where performance has been shown to be impaired by perirhinal cortical lesions. Using analyses of variance, regression analyses and post-hoc testing to evaluate group differences, it was found that during both the sample and test trials of the task all four groups spent similar absolute and proportional amounts of time ambulating, rearing, stationary, and exploring the two objects present. All groups also explored each of the two identical objects present during sample trials equally. However, during the test trials, only the control and gonadectomized rats given testosterone showed the expected increase in exploration of the novel objects presented, whereas the gonadectomized and gonadectomized, estradiol-supplemental groups continued to explore the novel and familiar objects equally. That regression analyses also identified significant correlations between low bulbospongiosus muscle weight and impaired novel vs. familiar object discrimination further indicates that gonadectomy in adult male rats adversely affects spontaneous novel object recognition in an androgen-sensitive, estrogen-insensitive manner.     

The dynamics of two viral infections in a single host population with applications to hantavirus

An SI epidemic model for a host with two viral infections circulating within the population is developed, analyzed, and numerically simulated. The model is a system of four differential equations which includes a state for susceptible individuals, two states for individuals infected with a single virus, one which is vertically transmitted and the other which is horizontally transmitted, and a fourth state for individuals infected with both viruses. A general growth function with density-dependent mortality is assumed. A special case of this model, where there is no coinfection and total cross immunity, is thoroughly analyzed. Several threshold values are defined which determine establishment of the disease and persistence at equilibrium for one or both of the infections within the host population. The model has applications to a hantavirus and an arenavirus that infect cotton rats. The hantavirus is transmitted horizontally whereas the arenavirus is transmitted vertically. It is shown through analysis and numerical simulations that both diseases can be maintained within a single host population, where individuals can be either infected with both viruses or with a single virus.     

Emergence and Consequences of Division of Labor in Associations of Normally Solitary Sweat Bees

Division of labor is a pervasive feature of animal societies, but little is known about the causes or consequences of division of labor in non-eusocial cooperative groups. We tested whether division of labor self-organizes in an incipient social system: artificially induced nesting associations of the normally solitary sweat bee Lasioglossum ( Ctenonomia ) NDA-1 (Hymenoptera: Halictidae). We quantified task performance and construction output by females nesting either alone or with a conspecific. Within pairs, a division of labor repeatedly arose in which one individual specialized on excavation and pushing/tamping while her nestmate guarded the nest entrance. Task specialization could not be attributed to variation in overall activity, and the degree of behavioral differentiation was greater than would be expected due to random variation, indicating that division of labor was an emergent phenomenon generated in part by social dynamics. Excavation specialists did not incur a survival cost, in contrast to previous findings for ant foundress associations. Paired individuals performed more per capita guarding, and pairs collectively excavated deeper nests than single bees – potential early advantages of social nesting in halictine bees.