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PubMed1999

Multiple patterns of parental care.

Many animals show multiple patterns of parental care, where more than one of the four basic patterns (biparental care, uniparental care by males or females, or no care) is present within a single population during a single breeding season. We consider three reasons for the existence of multiple patterns of parental care: (1) mixed-strategy behaviours; (2) time-dependent behaviour with parents changing their care decision during the breeding season; and (3) quality differences between individuals leading to different care decisions being made depending on the qualities of both parents. The basic framework we use to investigate these is a two-stage game-theoretical model, and we highlight the importance of including feedback between the parental care decisions made by population members and the probability that a deserting individual will find a new mate. Including this feedback may introduce a nonlinear dependence of the fitness payoffs on the frequencies with which the pure strategies ('care' and 'desert') are played by each of the sexes. This can have important consequences for the existence of evolutionarily stable strategies (ESSs). For example, mixed-strategy ESSs may exist (an outcome forbidden if the feedback is not included) and, in one model, the feedback also prevents uniparental care by either sex from being evolutionarily stable. We also point out that decisions made by animals without dependent offspring can have important consequences for observed parental care behaviour. Copyright 1999 The Association for the Study of Animal Behaviour.

Journal Article↗
PubMed2005

Time-dependent animal conflicts: 1. The symmetric case.

Animal conflicts are often characterized by time-dependent strategy sets. This paper considers the following type of animal conflicts: a member of a group is at risk and needs the assistance of another member to be saved. As long as assistance is not provided, the individual which is at risk has a positive, time-dependent rate of dying. Each of the other group members is a potential helper. Assisting this individual accrues a cost, but losing him decreases the inclusive fitness of each group member. A potential helper's interval between the moment an individual finds itself at risk and the moment it assists is a random variable, hence its strategy is to choose the probability distribution for this random variable. Assuming that each of the potential helpers knows the others' strategies, we show that the ability to observe their realizations influences the evolutionarily stable strategies (ESS) of the game. According to our results, where the realizations can be observed ESS always exist: immediate assistance, no assistance and delayed assistance. Where the realizations cannot be observed ESS do not always exist, immediate assistance and no assistance are possible ESS, while delayed assistance cannot be an ESS. We apply our model to the n brothers' problem and to the parental investment conflict.

Animals↗
PubMed2006

Revisiting matrix games: the concept of neighborhood invader strategies.

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.

Animals↗
PubMed2001

Evolution of cooperation through indirect reciprocity.

How can cooperation through indirect reciprocity evolve and what would it be like? This problem has previously been studied by simulating evolution in a small group of interacting individuals, assuming no gene flow between groups. In these simulations, certain 'image scoring' strategies were found to be the most successful. However, analytical arguments show that it would not be in an individual's interest to use these strategies. Starting with this puzzle, we investigate indirect reciprocity in simulations based on an island model. This has an advantage in that the role of genetic drift can be examined. Our results show that the image scoring strategies depend on very strong drift or a very small cost of giving help. As soon as these factors are absent, selection eliminates image scoring. We also consider other possibilities for the evolution of indirect reciprocity. In particular, we find that the strategy of aiming for 'good standing' has superior properties. It can be an evolutionarily stable strategy and, even if not, it usually beats image scoring. Furthermore, by introducing quality variation among individuals into the model, we show that the standing strategy can be quality revealing, adding a new dimension to indirect reciprocity. Finally, we discuss general problems with currently popular modelling styles.

Animals↗
PubMed2001

Parasites and supernormal manipulation.

Social parasites may exploit their hosts by mimicking other organisms that the hosts normally benefit from investing in or responding to in some other way. Some parasites exaggerate key characters of the organisms they mimic, possibly in order to increase the response from the hosts. The huge gape and extreme begging intensity of the parasitic common cuckoo chick (Cuculus canorus) may be an example. In this paper, the evolutionary stability of manipulating hosts through exaggerated signals is analysed using game theory. Our model indicates that a parasite's signal intensity must be below a certain threshold in order to ensure acceptance and that this threshold depends directly on the rate of parasitism. The only evolutionarily stable strategy (ESS) combination is when hosts accept all signallers and parasites signal at their optimal signal intensity, which must be below the threshold. Supernormal manipulation by parasites is only evolutionarily stable under sufficiently low rates of parasitism. If the conditions for the ESS combination are not satisfied, rejector hosts can invade using signal intensity as a cue for identifying parasites. These qualitative predictions are discussed with respect to empirical evidence from parasitic mimicry systems that have been suggested to involve supernormal signalling, including evicting avian brood parasites and insect-mimicking Ophrys orchids.

Adaptation, Physiological↗
PubMed2002

Group formation games with reappraisal.

The formation of a social group, such as the group of individuals sharing a territory, depends on the interaction between choices made by individuals to stay or disperse. The process can be modelled as a multi-player variant of the well-known War of Attrition in evolutionary game theory, as shown by Blackwell (1997; J. Theor. Biol.189, 175-181). In this paper, we extend the set of strategies defined there by allowing reappraisal during the game. We give a formal analysis of the evolutionarily stable strategy, where one exists, and illustrate it with an example based on badger (Meles meles) territoriality. The results predict that group size will be well adapted to, and very sensitive to, the precise conditions under which the game is played, and give an indication of the potential for parent-offspring conflict.

Animals↗
PubMed2001

Arms races and the evolution of big fierce societies.

The causes of biological gigantism have received much attention, but only for individual organisms. What selection pressures might favour the evolution of gigantic societies? Here we consider the largest single-queen insect societies, those of the Old World army ant Dorylus, single colonies of which can have 20 million workers. We propose that colony gigantism in Dorylus arises as a result of an arms race and test this prediction by developing a size-structured mathematical model. We use this model for exploring and potentially explaining differences in colony size, colony aggression and colony propagation strategies in populations of New World army ants Eciton and Old World army ants Dorylus. The model shows that, by determining evolutionarily stable strategies (ESSs), differences in the trophic levels at which these army ants live feed forwards into differences in their densities and collision rates and, hence, into different strategies of growth, aggression and propagation. The model predicts large colony size and the occurrence of battles and a colony-propagation strategy involving highly asymmetrical divisions in Dorylus and that Eciton colonies should be smaller, non-combative and exhibit equitable binary fission. These ESSs are in excellent agreement with field observations and demonstrate that gargantuan societies can arise through arms races.

Aggression↗
PubMed1999

Reproductive skew and indiscriminate infanticide.

In communally breeding animals, there is an evolutionary conflict over the partitioning of reproduction within the group. If dominant group members do not have complete control over subordinate reproduction, this conflict may favour the evolution of infanticidal behaviour (by either subordinates or dominants or both). Elimination of offspring, however, is likely to be constrained by the difficulty of discriminating between an individual's own progeny and those of cobreeders. Here, we develop an evolutionarily stable strategy (ESS) model of reproductive partitioning, which demonstrates that killing of young can be favoured, even if such discrimination is not possible. The model predicts that infanticide will typically be associated with elevated levels of offspring production, and is most likely to prove evolutionarily stable when the coefficient of relatedness between cobreeders is low, and offspring are cheap to produce. The effect of infanticide is to release subordinates from the reproductive restraint they would otherwise be forced to exercise, leading to reduced reproductive skew. When infanticide is possible, addition of numerous young to the joint brood will not lower overall productivity, because progeny in excess of the most productive brood size are eliminated. Subordinates are thus free to contribute more young to the brood than would otherwise be the case. In addition, we show that the possibility of infanticide may influence the pattern of reproduction within a group even if no offspring are actually killed at equilibrium. Copyright 1999 The Association for the Study of Animal Behaviour.

Journal Article↗
PubMed1979

Game theory and the evolution of behaviour.

How far can game theory account for the evolution of contest behaviour in animals? The first qualitative prediction of the theory was that symmetric contests in which escalation is expensive should lead to mixed strategies. As yet it is hard to say how far this is borne out, because of the difficulty of distinguishing a 'mixed evolutionarily stable strategy' maintained by frequency-dependent selection from a 'pure conditional strategy'; the distinction is discussed in relation to several field studies. The second prediction was that if a contest is asymmetric (e.g. in ownership) then the asymmetry will be used as a conventional cue to settle it. This prediction has been well supported by observation. A third important issue is whether or not information about intentions is exchanged during contests. The significance of 'assessment' strategies is discussed.

Aggression↗
PubMed2002

Evolutionary stability of vigilance coordination among social foragers.

Coordination can greatly improve the efficiency of anti-predatory vigilance scans by increasing predator detection for a constant proportion of time spent vigilant. However, it has been rarely found in nature and most studies have detected or assumed independent scanning by group members. In this study, we analysed the functional consequences of the coordinated alternation of vigilance scanning by group foragers. We introduce coordination by assuming that interscan intervals (ISIs) follow a modified gamma distribution. Depending on the parameters of the distribution, successive scans can be evenly spaced (coordinated scanning) or may present a high overlap (uncoordinated scanning). Comparing evolutionarily stable strategies for animals that do not coordinate their scanning with animals that do coordinate their anti-predator behaviour shows that coordination has a marked effect on survival probability. Moreover, the coordinating strategy is quite robust against mutants that scan independently with exponential distributions of ISIs. However, coordination breaks down when animals can continuously adjust their level of coordination by deciding the proportion of time they spend monitoring the behaviour of other group members. In this case, coordination is only evolutionarily stable if it can be very easily achieved.

Animals↗
PubMed1997

The logic of contrition.

A highly successful strategy for the Repeated Prisoner's Dilemma is Contrite Tit For Tat, which bases its decisions on the "standings" of the two players. This strategy is as good as Tit For Tat at invading populations of defectors, and much better at overcoming errors in implementation against players who are also using it. However, it is vulnerable to errors in perception. In this paper, we discuss the merits of Contrite Tit For Tat and compare it with other strategies, like Pavlov and the newly-introduced Remorse. We embed these strategies into an eight-dimensional space of stochastic strategies which we investigate by analytical means and numerical simulations. Finally, we show that if one replaces the conventions concerning the "standing" by other, even simpler conventions, one obtains an evolutionarily stable strategy (called Prudent Pavlov) which is immune against both mis-perception and mis-implementation.

Animals↗
PubMed2002

Sexually transmitted disease and the evolution of mating systems.

Sexually transmitted diseases (STDs) have been shown to increase the costs of multiple mating and therefore favor relatively monogamous mating strategies. We examine another way in which STDs can influence mating systems in species in which female choice is important. Because more popular males are more likely to become infected, STDs can counteract any selective pressure that generates strong mating skews. We build two models to investigate female mate choice when the sexual behavior of females determines the prevalence of infection in the population. The first model has no explicit social structure. The second model considers the spatial distribution of matings under social monogamy, when females mated to unattractive males seek extrapair fertilizations from attractive males. In both cases, the STD has the potential to drastically reduce the mating skew. However, this reduction does not always happen. If the per contact transmission probability is low, the disease dies out and is of no consequence. In contrast, if the transmission probability is very high, males are likely to be infected regardless of their attractiveness, and mating with the most attractive males imposes again no extra cost for the female. We also show that optimal female responses to the risk of STDs can buffer the prevalence of infection to remain constant, or even decrease, with increasing per contact transmission probabilities. In all cases considered, the feedback between mate choice strategies and STD prevalence creates frequency-dependent fitness benefits for the two alternative female phenotypes considered (choosy vs. randomly mating females or faithful vs. unfaithful females). This maintains mixed evolutionarily stable strategies or polymorphisms in female behavior. In this way, a sexually transmitted disease can stabilize the populationwide proportion of females that mate with the most attractive males or that seek extrapair copulations.

Choice Behavior↗
PubMed1999

Stochastic effects in LMC models.

LMC (local mate competition) was first introduced by W. D. Hamilton to explain extraordinary female-biased sex ratios observed in a variety of insects and mites. In the original model, the population is subdivided into an infinite number of colonies founded by a fixed number of inseminated females producing the same very large number of offspring. The male offspring compete within the colonies to inseminate the female offspring and then these disperse at random to found new colonies. An unbeatable sex ratio strategy is found to be female-biased. In this paper, the effects of having colonies of random size and foundresses producing a random finite number of offspring are considered. The exact evolutionarily stable strategy (ESS) sex ratio is deduced and comparisons with previous approximate or numerical results are made. As the mean or the variance of brood size increases, the ESS sex ratio becomes more female-biased. An increase in the variance of colony size increases the ESS proportion of males when the mean brood size and colony size are both small, but decreases this proportion when the mean brood size or the mean colony size is large.

Animals↗
PubMed2002

Evolution of flowering strategies in Oenothera glazioviana: an integral projection model approach.

The timing of reproduction is a key determinant of fitness. Here, we develop parameterized integral projection models of size-related flowering for the monocarpic perennial Oenothera glazioviana and use these to predict the evolutionarily stable strategy (ESS) for flowering. For the most part there is excellent agreement between the model predictions and the results of quantitative field studies. However, the model predicts a much steeper relationship between plant size and the probability of flowering than observed in the field, indicating selection for a 'threshold size' flowering function. Elasticity and sensitivity analysis of population growth rate lambda and net reproductive rate R(0) are used to identify the critical traits that determine fitness and control the ESS for flowering. Using the fitted model we calculate the fitness landscape for invading genotypes and show that this is characterized by a ridge of approximately equal fitness. The implications of these results for the maintenance of genetic variation are discussed.

Biological Evolution↗
PubMed2005

Time-dependent animal conflicts: 2. The asymmetric case.

This paper presents an asymmetric game-theoretical model to the following type of animal conflicts: a member of a group is at risk and needs the help of another member to be saved. As long as assistance is not provided, this individual has a positive, time-dependent rate of dying. Assisting the individual which is at risk accrues a cost, but losing it decreases each member's inclusive fitness. A potential helper's interval between the moment a group member gets into trouble and the moment it assists is a random variable, hence its strategy is to choose the distribution of this random variable. In the asymmetric conflict all the potential helpers have identical strategy sets, but each plays a different role. For example, male or female and young or old. We consider both payoff-irrelevant asymmetry and payoff-relevant asymmetry and characterize each role's stable replies. The evolutionarily stable strategies (ESS) are computed, and the model is applied to the n brothers' problem. According to our results immediate assistance and no assistance are possible ESS both under payoff-relevant asymmetry and under payoff-relevant asymmetry.

Animals↗
PubMed1993

Chaos and the evolution of cooperation.

The "iterated prisoner's dilemma" is the most widely used model for the evolution of cooperation in biological societies. Here we show that a heterogeneous population consisting of simple strategies, whose behavior is totally specified by the outcome of the previous round, can lead to persistent periodic or highly irregular (chaotic) oscillations in the frequencies of the strategies and the overall level of cooperation. The levels of cooperation jump up and down in an apparently unpredictable fashion. Small recurrent and simultaneous invasion attempts (caused by mutation) can change the evolutionary dynamics from converging to an evolutionarily stable strategy to periodic oscillations and chaos. Evolution can be twisted away from defection, toward cooperation. Adding "generous tit-for-tat" greatly increases the overall level of cooperation and can lead to long periods of steady cooperation. Since May's paper [May, R. M. (1976) Nature (London) 261, 459-467], "simple mathematical models with very complicated dynamics" have been found in many biological applications, but here we provide an example of a biologically relevant evolutionary game whose dynamics display deterministic chaos. The simulations bear some resemblance to the irregular cycles displayed by the frequencies of host genotypes and specialized parasites in evolutionary "arms races" [Hamilton, W. D., Axelrod, R. & Tanese, R. (1990) Proc. Natl. Acad. Sci. USA 87, 3566-3573; Seger, J. (1988) Philos. Trans. R. Soc. London B 319, 541-555].

Biological Evolution↗
PubMed2000

An ESS maximum principle for matrix games.

Previous work has demonstrated that for games defined by differential or difference equations with a continuum of strategies, there exists a G-function, related to individual fitness, that must take on a maximum with respect to a virtual variable v whenever v is one of the vectors in the coalition of vectors which make up the evolutionarily stable strategy (ESS). This result, called the ESS maximum principle, is quite useful in determining candidates for an ESS. This principle is reformulated here, so that it may be conveniently applied to matrix games. In particular, we define a matrix game to be one in which fitness is expressed in terms of strategy frequencies and a matrix of expected payoffs. It is shown that the G-function in the matrix game setting must again take on a maximum value at all the strategies which make up the ESS coalition vector. The reformulated maximum principle is applicable to both bilinear and nonlinear matrix games. One advantage in employing this principle to solve the traditional bilinear matrix game is that the same G-function is used to find both pure and mixed strategy solutions by simply specifying an appropriate strategy space. Furthermore we show how the theory may be used to solve matrix games which are not in the usual bilinear form. We examine in detail two nonlinear matrix games: the game between relatives and the sex ratio game. In both of these games an ESS solution is determined. These examples not only illustrate the usefulness of this approach to finding solutions to an expanded class of matrix games, but aids in understanding the nature of the ESS as well.

Biological Evolution↗