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.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.     

Evolutionarily Stable Relocation Strategy in an Antlion Larva

Antlion pits are often spatially aggregated even though there are potential negative effects associated with the aggregation (e.g., heightened competition and predation risk). This study investigated the possibility that a strategy leading to aggregation can be an evolutionarily stable strategy (ESS). In particular, the strategy considered was ‘decreasing relocation tendency when there are neighbors’. An individual based model showed that the strategy can be the unique ESS when the spatial distribution of prey is not completely random and antlions can learn it from their past foraging experiences. A laboratory experiment was conducted to examine the effects of the presence of neighbors and foraging success on the relocation behavior of antlion larvae. Antlions reduced their relocation tendency with respect to these factors, consistent with the predicted ESS. The results suggest that pit aggregations are formed because antlions reduce their relocation tendency when neighbors exist, and this strategy is an ESS.     

Simulating natural selection as a culling mechanism on finite populations with the hawk-dove game

The behaviors of individuals and species are often explained in terms of evolutionary stable strategies (ESSs). The analysis of ESSs determines which, if any, combinations of behaviors cannot be invaded by alternative strategies. Two assumptions required to generate an ESS (i.e., an infinite population and payoffs described only on the average) do not hold under natural conditions. Previous experiments indicated that under more realistic conditions of finite populations and stochastic payoffs, populations may evolve in trajectories that are unrelated to an ESS, even in very simple games. The simulations offered here extend earlier research by employing truncation selection with random parental selection in a hawk-dove game. Payoffs are determined in pairwise contests using either the expected outcome, or the result of a random variable. In each case, however, the mean fraction of hawks over many generations and across many independent trials does not conform to the expected ESS. Implications of these results and philosophical underpinnings of ESS theory are offered.     

Allocation of resources to male and female functions in hermaphrodites

The question of"how a self-fertile hermaphrodite will distribute the resources that it allocates to reproduction is studied by means of the ESS approach. Different models of the relations between allocation to male function, the male and female fertilities, and the selfing rate, yield different conclusions about how much resource should be allocated to male function. Values below a half are obtained with one model, while another can give values greater than a half. Even with no selfing, values other than a half are usually obtained; with both models studied, the values decrease with increasing selling. If the selfing rate is assumed to be independent of the fraction of resources allocated to male function, it can be shown that the ESS allocation to male function always decreases as selling increases. The types of relations that might be expected in species with different types of breeding biology, and some data on allocation to male function, are reviewed.
The implications for the fitness of male- and female-sterility mutations are discussed. It is argued that the concavity or convexity of the curve relating female fertility to male fertility is not a good guide to when hermaphroditism should exist when there is some selfing. Even with a concave relation, male-sterility mutants can have a higher fitness than hermaphrodites, if there is some selling and inbreeding depression. Also, when the selfing rate depends on allocation to male I unction, an hermaphrodite ESS does not always exist when the function is concave (as it does when there is no selfing), and such an ESS may exist when the relation is convex. The fitness of male- or female-sterility mutants may also depend on the existence of 'fixed costs'. It is shown that these do not ailed the ESS allocation of resources.     

Survivor's dilemma: Defend the group or flee?

We consider a survival game of gregarious individuals, in which the aim of the players is survival to reproductive age under predator attacks. The survivor’s dilemma (shortly: SVD) game consists in the following: a group member either surely survives alone by fleeing, while its defensive mate may be killed; or tries to save its mate’s life, risking to get killed. The dilemma is that, in every single attack, fleeing ensures maximal survival probability, but if its mate survives by fighting both, and they remain together, its risk to be killed at the next attack will be lower. We show that, if defense is successful enough, then the one-attack game is a prisoner’s dilemma (PD), where fleeing is the strict ESS. We have additively decomposed the SVD game, according to the survival of the group mate of the focal prey, into two games: the aim of the “collective game” is survival of the group of prey. Counter-wise, the aim of the “hostile game” is survival alone (focal prey survives and its mate is killed by the predator). We obtain the following results: if the attack number is large enough, the multi-attack SVD game is dominated by the “collective game” in the sense that each individual can ensure its own maximal survival probability by maximizing the group survival probability in each attack. In the hostile game, the only strict ESS is the fleeing strategy. In the collective game there are two different cases: either defense is a unique strict ESS, or the collective game is bistable, i.e. fleeing and defense are local strict ESS’s. If defense is the only strict ESS in the collective game, and the attack number is large enough, defense replaces fleeing strategy in the multi-attack SVD game. However, in the bistable case, defense cannot invade into the fleeing population. It is shown that, if the interaction between relatives is frequent enough, than defense can replace fleeing strategy, in spite of the fact that in the well-mixed population the collective game is bistable.     

On the stochastic theory of compartments: I. A single-compartment system

A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.     

Inbreeding coefficients for stochastically varying small population sizes-bias of calculation based on effective numbers

The commonly used procedure to calculate inbreeding coefficients by effective population numbers (Ne) by the harmonic mean of generation-by-generation population sizes involves a computational bias. If the individual population sizes are considered as realizations of a binomially distributed random variable with sample size N and probability p, this bias can be investigated for the two cases p = constant and p = variable (Markov chain). The bias is of practical relevance only for small probabilities p, short period of initial successive generations, and small population sizes. The largest values for this computational bias are in the range of 0.05-0.06. It is concluded that for most practical purposes the approximate procedure is appropriate.     

Group control of multi-car elevator systems without accurate information of floor stoppage time

A multi-car elevator system is an elevator system that has more than one car installed in each elevator shaft. This system enables us to improve the transportation capability without increasing the occupied floor space. The primary purpose of this study is to consider the group control problem of operating cars efficiently without collision nor reversal in a multi-car elevator system, based on a detailed and realistic model where floor stoppage time of cars cannot be known in advance. In the context of elevator systems, reversal means that a car travels in the direction opposite to the desired direction of on-board passengers, which is prohibited because it makes passengers uncomfortable. We first propose an optimization-based collision and reversal avoidance method to operate cars in the same shaft. Next, we construct simple methods to allocate calls to individual cars under immediate and delayed guidance policies. Under the immediate guidance policy a call is allocated to a car immediately after it is registered, while under the delayed guidance policy the allocation may be changed until it is actually served. The effectiveness of the proposed group control method is examined by computer simulation.     

An asymmetric parental investment conflict with continuous strategy sets

In the parental investment conflict each of the sexes decides how much to invest in its brood, where its decision influences both sexes' fitness. In nature, each species is usually characterized by a common parental care pattern, male-only care, female-only care or biparental care. A possible way for understanding the factors that have led each species to adopt its unique parental care pattern is to analyse a male's and a female's decision process using a game-theoretical model. This paper suggests a two-stage game-theoretical model with two types of players, male and female. During the game each parent makes three decisions. The interval between the beginning of the game, i.e. after mating and having offspring, and the moment a parent starts to care for them is a random variable. Thus, in the first stage a parent chooses the cumulative probability distribution of this interval, and its amount of parental care. In the second stage the other parent chooses its probability for cooperation. It is assumed that as long as parental care is not provided the offspring are at risk, and that parental caring accrues a different cost for each sex. We compute the Evolutionary Stable Strategies (ESS) under payoff-relevant asymmetry, and show that uniparental and biparental care are possible ESS. We also characterize cases where the sex having the lower cost "forces" the sex having the higher cost to care and vice versa.     

Fighting for food: a dynamic version of the Hawk-Dove game

Summary The Hawk-Dove game (Maynard Smith, 1982) has been used to analyse conflicts over resources such as food. At the evolutionarily stable strategy (ESS), either a proportionp* of animals always play Hawk, or each animal has a probabilityp* of playing Hawk. We modify the standard Hawk-Dove game to include a state variable,x, that represents the animal's level of energy reserves. A strategy is now a rule for choosing an action as a function ofx and time of day. We consider a non-reproductive period and adopt the criterion of minimizing mortality over this period. We find the ESS, which has the form play Hawk if reserves are belowc* (t) at timet, otherwise play Dove. This ESS is very different from the ESS in the standard Hawk-Dove game. It is a pure ESS that depends on the animal's state and on time. Furthermore, it is characterized by the strong condition that any single mutant that does not adopt the ESS suffers a reduction in fitness. The standard Hawk-Dove game assumes pay-offs that are related to fitness; our approach starts from a definition of fitness and derives the pay-offs in the process of finding the ESS. When the environment becomes worse (e.g. food becomes less reliable or energy expenditure increases) the ESS changes in such a way as to increase the proportion of animals that will play Hawk.     

The ESS in an asymmetric generalized war of attrition with mistakes in role perception

We derive the ESS for the generalized asymmetric war of attrition, where payoffs to contestants may vary in time and may depend on some characteristic, called the "role" of an individual. We use the same approach as Hammerstein & Parker (1982), who examined an asymmetric war of attrition. We consider two roles, A and B. Role A is assumed to be favoured with respect to payoffs. It is assumed that there is always a true asymmetry, so in each contest one individual has role A and the other has role B. It is assumed that roles are assigned to contestants at random and that they can make mistakes in role perception. It is shown that, under certain assumptions about shapes of payoff functions and probabilities of making mistakes, there is an ESS which can be characterized by two probability distributions with non-overlapping support. Individuals who perceive their role as A should choose larger persistence times. This ESS structure is similar to that in the asymmetric war of attrition. In that model, the resource values and the cost rates are constant. We consider situations where all these values may change in time and where rewards and costs may be equal after some finite time. Such shapes of payoff functions arise naturally in competitive patch depletion (Sjerps & Haccou, 1994a,b). As a result, the probability density functions that specify the conditional strategies are no longer necessarily negative exponentials (as in the war of attrition), but may have very different shapes. Furthermore, under some conditions there is a maximum persistence time, at which there can be an atom of probability. We give explicit expressions as well as numerical approximations for the ESS.     

The theory of games and the evolution of animal conflicts

The evolution of behaviour patterns used in animal conflicts is discussed, using models based on the theory of games. The paper extends arguments used by Maynard Smith &; Price (1973) showing that ritualized behaviour can evolve by individual selection. The concept of an evolutionarily stable strategy, or ESS, is defined. Two types of ritualized contests are distinguished, “tournaments” and “displays”; the latter, defined as contests without physical contact in which victory goes to the contestant which continues longer, are analyzed in detail. Three main conclusions are drawn. The degree of persistence should be very variable, either between individuals or for the same individual at different times; a negative exponential distribution of persistence times is predicted. Individuals should display with constant intensity, independent of how much longer they will in fact continue. An initial asymmetry in the conditions of a contest can be used to settle it, even if it is irrelevant to the outcome of a more protracted conflict if one were to take place.     

Co-evolution of seed size and seed predation

Using the evolutionarily stable strategy (ESS) approach in a model for the co-evolution of seed size and seed predation, I show that seed size variation within individual plants is favoured if there is a trade-off in the predator's attack rate for different seed sizes. A single seed size is not evolutionarily stable because a predator that is optimally adapted to one particular seed size cannot prevent invasion by plants with a different seed size. The model generates the following predictions. The ESS consists of a continuous range of seed sizes. Small seeds tend to be attacked more frequently than big seeds. Plants with many resources and plants with low (frequency-independent) juvenile mortality have more variable seeds than plants with few resources and a high juvenile mortality. Seed size variation is higher in fluctuating populations regulated by seed predation alone than in stable populations (partially) regulated by seedling competition. Predator searching behaviour does not directly affect the ESS seed size range, but may have an indirect effect by affecting population stability or the significance of seedling competition as a population regulating mechanism. Moreover, seed size distributions are found to be more skewed in favour of small seeds if predation is spatially non-uniform than if predation is more even. Application of the model to systems of several co-evolving plant and predator species is discussed.     

Evolutionary limits to the frequency of aggression between related or unrelated conspecifics in diploid species with simple mendelian inheritance

Game theory has been used by some authors to analyse evolutionary limits to the expression of aggression in theoretical haploid parthenogenetic species. Others have examined frequency dependent selection, of which aggression may be a case, by applying population genetic models to diploid species. A model is presented which attempts to combine these two approaches. Game theory is used to determine evolutionarily stable strategies and corresponding stable polymorphisms for a two-strategy game played by members of a diploid sexual species, when choice of strategy is determined by two alleles at a single locus. Results are given for dominant, co-dominant and recessive determination of choice of the more aggressive of two strategies, for two levels of relationship: unrelated players and sibs. It is found that for a range of models of single locus inheritance the evolutionarily stable strategy (ESS) determined for haploid species remains the stable population strategy for diploid sexual species, when players are unrelated. In sibling contestants aggression is reduced. The mixed strategy haploid ESS underestimates, but the pure strategy haploid ESS provides a good indication of the degree to which relatedness lessens aggression in diploid species. For both haploid and diploid species there may be a considerable advantage to confining conflicts to kin.     

Critical Social Learning: A Solution to Rogers's Paradox of Nonadaptive Culture

Alan Rogers (1988) presented a game theory model of the evolution of social learning, yielding the paradoxical conclusion that social learning does not increase the fitness of a population. We expand on this model, allowing for imperfections in individual and social learning as well as incorporating a "critical social learning" strategy that tries to solve an adaptive problem first by social learning, and then by individual learning if socially acquired behavior proves unsatisfactory. This strategy always proves superior to pure social learning and typically has higher fitness than pure individual learning, providing a solution to Rogers's paradox of nonadaptive culture. Critical social learning is an evolutionarily stable strategy (ESS) unless cultural transmission is highly unfaithful, the environment is highly variable, or social learning is much more costly than individual learning. We compare the model to empirical data on social learning and on spatial variation in primate cultures and list three requirements for adaptive culture.     

Diversity in times of adversity: probabilistic strategies in microbial survival games

Population diversification strategies are ubiquitous among microbes, encompassing random phase-variation (RPV) of pathogenic bacteria, viral latency as observed in some bacteriophage and HIV, and the non-genetic diversity of bacterial stress responses. Precise conditions under which these diversification strategies confer an advantage have not been well defined. We develop a model of population growth conditioned on dynamical environmental and cellular states. Transitions among cellular states, in turn, may be biased by possibly noisy readings of the environment from cellular sensors. For various types of environmental dynamics and cellular sensor capability, we apply game-theoretic analysis to derive the evolutionarily stable strategy (ESS) for an organism and determine when that strategy is diversification. We find that: (1) RPV, effecting a sort of Parrondo paradox wherein random alternations between losing strategies produce a winning strategy, is selected when transitions between different selective environments cannot be sensed, (2) optimal RPV cell switching rates are a function of environmental lifecycle asymmetries and environmental autocorrelation, (3) probabilistic diversification upon entering a new environment is selected when sensors can detect environmental transitions but have poor precision in identifying new environments, and (4) in the presence of excess additive noise, low-pass filtering is required for evolutionary stability. We show that even when RPV is not the ESS, it may minimize growth rate variance and the risk of extinction due to 'unlucky' environmental dynamics.     

Fighting costs stabilize aggressive behavior in intersexual conflicts

We analyze the evolution of aggressive behavior in intersexual conflicts, with a special reference to mate guarding behavior in crustaceans. An analysis of a discrete-strategy game shows that an ESS with only one of the sexes being aggressive prevail if fighting costs or fitness values of winning are asymmetric. Non-aggressiveness of both sexes is stable if fighting behavior is very costly for females and if the cost is at least partly paid independent of the strategy of the opponent. Most interestingly, the solutions of both sexes being aggressive prevails only if both sexes have some probability of winning, and if fighting costs are small. Second, we solve for the expected levels of aggressiveness in a game with continuous strategies. The form of the fighting cost function largely determines the stability of the solution. When fighting cost increases linearly with aggressiveness, mutual aggressiveness fluctuates cyclically instead of stabilizing at an ESS. However, if there is an asymmetry in fitness payoffs, a solution with only the sex having most to lose being aggressive alone is possible. With quadratically increasing fighting costs an ES combination of mutual aggressiveness may exist. It is predicted that fights between the sexes should be hardest when payoffs are symmetric, and that an overt behavioral conflict will always take place as long as there is a fitness loss to each of the sexes if losing the conflict and both sexes have a chance to win. We discuss the models in the context of fights preceding precopulatory guarding, but the models offer a general frame for analyzing any intersexual conflict. This revised version was published online in July 2006 with corrections to the Cover Date.     

Future Bloom and Blossom Frost Risk for Malus domestica Considering Climate Model and Impact Model Uncertainties

The future bloom and risk of blossom frosts for Malus domestica were projected using regional climate realizations and phenological ( = impact) models. As climate impact projections are susceptible to uncertainties of climate and impact models and model concatenation, the significant horizon of the climate impact signal was analyzed by applying 7 impact models, including two new developments, on 13 climate realizations of the IPCC emission scenario A1B. Advancement of phenophases and a decrease in blossom frost risk for Lower Saxony (Germany) for early and late ripeners was determined by six out of seven phenological models. Single model/single grid point time series of bloom showed significant trends by 2021–2050 compared to 1971–2000, whereas the joint signal of all climate and impact models did not stabilize until 2043. Regarding blossom frost risk, joint projection variability exceeded the projected signal. Thus, blossom frost risk cannot be stated to be lower by the end of the 21st century despite a negative trend. As a consequence it is however unlikely to increase. Uncertainty of temperature, blooming date and blossom frost risk projection reached a minimum at 2078–2087. The projected phenophases advanced by 5.5 d K⁻¹, showing partial compensation of delayed fulfillment of the winter chill requirement and faster completion of the following forcing phase in spring. Finally, phenological model performance was improved by considering the length of day.     

Evolutionary stability: States and strategies

Different aspects and modifications of the definition of an evolutionarily stable (ES) strategy that have been considered in the literature can be incorporated in a unifying concept which regards the population context. This concept of evolutionary stability will generally characterize population states in both pure- and mixed-strategist models. In particular, it includes ES strategies, represented as a phenotype unique in an ES population. For an important class of mixed-strategist models, no strict ESS can exist. This will be the case whenever the success of an individual strategy is considered to follow as an average from the successes of its behavioural components. Instead, ESS results may be obtained from what will be called a “degenerate” form of the model, which is simply an ESS model on the level of elementary actions. Then, however, the correct interpretation of an ESS is not an individual phenotype but rather a population mixture of elementary actions. If an ES state exists in a mixed-strategist model it may be determined by an equilibrium condition; if there is an ES strategy, a different approach—mainly maximum considerations—is needed for finding it. An equilibrium condition does not hold for the components of an ES strategy straightforwardly; but it can be derived in terms of an auxiliary ESS model that considers first-order effects of the components. Several examples illustrate the significance of these results. Particularly, two models on “Games between Relatives” are reconsidered in order to display both their formal interrelation and the different meaning of their results in the context of mixed-strategist models.