Decision-making in variable environments-- a case of group selection and inter-generational conflict?

Suppose organisms need to engage in a particular action exactly once during some fixed period of time. Further suppose they can time this action to optimise their fitness based on the expected current payoff and the probability distribution of later payoffs. For an example we consider the timing of the annual nuptial flight in eusocial insects. Using two population genetics models, we ask whether stochasticity leads to evolutionary conflict between the queen and her offspring. We find that the winning phenotype is independent of who controls the timing. The best response to any non-equilibrium population strategy is the same in both control scenarios, a result that carries over to the diploid case. Although inter-generational conflict is therefore ruled out, the models support a previous observation that at equilibrium some of the offspring have a lower expected payoff than others. By measuring fitness in terms of relative reproductive success, we show that all individuals are in fact equally well off making group-selectionist arguments unnecessary. As such, the models should improve our understanding of the difficult conceptual problem of the unit of natural selection in stochastic environments.     

EVOLUTION IN FLUCTUATING ENVIRONMENTS: DECOMPOSING SELECTION INTO ADDITIVE COMPONENTS OF THE ROBERTSON–PRICE EQUATION

We analyze the stochastic components of the Robertson–Price equation for the evolution of quantitative characters that enables decomposition of the selection differential into components due to demographic and environmental stochasticity. We show how these two types of stochasticity affect the evolution of multivariate quantitative characters by defining demographic and environmental variances as components of individual fitness. The exact covariance formula for selection is decomposed into three components, the deterministic mean value, as well as stochastic demographic and environmental components. We show that demographic and environmental stochasticity generate random genetic drift and fluctuating selection, respectively. This provides a common theoretical framework for linking ecological and evolutionary processes. Demographic stochasticity can cause random variation in selection differentials independent of fluctuating selection caused by environmental variation. We use this model of selection to illustrate that the effect on the expected selection differential of random variation in individual fitness is dependent on population size, and that the strength of fluctuating selection is affected by how environmental variation affects the covariance in Malthusian fitness between individuals with different phenotypes. Thus, our approach enables us to partition out the effects of fluctuating selection from the effects of selection due to random variation in individual fitness caused by demographic stochasticity.     

Natural selection and social preferences

A large number of individuals are randomly matched into groups, where each group plays a finite symmetric game. Individuals breed true. The expected number of surviving offspring depends on own material payoff, but may also, due to cooperative breeding and/or reproductive competition, depend on the material payoffs to other group members. The induced population dynamic is equivalent with the replicator dynamic for a game with payoffs derived from those in the original game. We apply this selection dynamic to a number of examples, including prisoners' dilemma games with and without a punishment option, coordination games, and hawk-dove games. For each of these, we compare the outcomes with those obtained under the standard replicator dynamic. By way of a revealed-preference argument, our selection dynamic can explain certain "altruistic" and "spiteful" behaviors that are consistent with individuals having social preferences.     

Transforming the dilemma

How does natural selection lead to cooperation between competing individuals? The Prisoner's Dilemma captures the essence of this problem. Two players can either cooperate or defect. The payoff for mutual cooperation, R, is greater than the payoff for mutual defection, P. But a defector versus a cooperator receives the highest payoff, T, where as the cooperator obtains the lowest payoff, S. Hence, the Prisoner's Dilemma is defined by the payoff ranking T > R > P > S . In a well‐mixed population, defectors always have a higher expected payoff than cooperators, and therefore natural selection favors defectors. The evolution of cooperation requires specific mechanisms. Here we discuss five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity (or graph selection). Each mechanism leads to a transformation of the Prisoner's Dilemma payoff matrix. From the transformed matrices, we derive the fundamental conditions for the evolution of cooperation. The transformed matrices can be used in standard frameworks of evolutionary dynamics such as the replicator equation or stochastic processes of game dynamics in finite populations.     

Estimating fluctuating selection in age-structured populations

In age-structured populations, viability and fecundity selection of varying strength may occur in different age classes. On the basis of an original idea by Fisher of weighting individuals by their reproductive value, we show that the combined effect of selection on traits at different ages acts through the individual reproductive value defined as the stochastic contribution of an individual to the total reproductive value of the population the following year. The selection differential is a weighted sum of age-specific differentials that are the covariances between the phenotype and the age-specific relative fitness defined by the individual reproductive value. This enables estimation of weak selection on a multivariate quantitative character in populations with no density regulation by combinations of age-specific linear regressions of individual reproductive values on the traits. Demographic stochasticity produces random variation in fitness components in finite samples of individuals and affects the statistical inference of the temporal average directional selection as well as the magnitude of fluctuating selection. Uncertainties in parameter estimates and test power depend strongly on the demographic stochasticity. Large demographic variance results in large uncertainties in yearly estimates of selection that complicates detection of significant fluctuating selection. The method is illustrated by an analysis of age-specific selection in house sparrows on a fitness-related two-dimensional morphological trait, tarsus length and body mass of fledglings.     

Life history evolution and demographic stochasticity

Summary Can demographic stochasticity bias the evolution of life history traits? Under a neutral version of the Cole-Charnov-Schaffer model, variance in offspring number for both annuals and perennials depends on the precise values of fitness components. Either annuals or perennials may have the larger variance (for equal ), depending on the importance of random survivalversus fixed reproduction. By extension, the variance in offspring number should generally depend on whether is mainly composed of highly variable elements or elements with limited variation. Thus, data about the variability of demographic parameters may be as important as data about their mean values.This result concerns only one source of demographic stochasticity, the probabilistic nature of demographic processes like survival. The other source of demographic stochasticity is the fact that populations are composed of whole numbers of individuals (integer arithmetic). Integer arithmetic without probabilistic demography (or environmental variation) can make it difficult for rare invaders to persist in populations even when selection would favour the invaders in a deterministic model. Integer arithmetic can also cause population coexistence when the equivalent deterministic model leads to exclusion. This effect disappears when demography is probabilistic, and probably also when there is environmental variation. Thus probabilistic demography and environmental variation may make some population patterns more, rather than less, understandable.     

Intraindividual variability in behavior shapes fitness landscapes

Although intraindividual variability (IIV) in behavior is fundamental to ecological dynamics, the factors that contribute to the expression of IIV are poorly understood. Using an individual‐based model, this study examined the effects of stochasticity on the evolution of IIV represented by the residual variability of behavior. The model describes a population of prey with nonoverlapping generations, in which prey take refuge upon encountering a predator. The strategy of a prey is characterized by the mean and IIV (i.e., standard deviation) of hiding duration. Prey with no IIV will spend the same duration hiding in a refuge at each predator encounter, while prey with IIV will have variable hiding durations among encounters. For the sources of stochasticity, within‐generation stochasticity (represented by random predator encounters) and between‐generation stochasticity (represented by random resource availability) were considered. Analysis of the model indicates that individuals with high levels of IIV are maintained in a population in the presence of between‐generation stochasticity even though the optimal strategy in each generation is a strategy with no IIV, regardless of the presence or absence of within‐generation stochasticity. This contradictory pattern emerges because the mean behavioral trait and IIV do not independently influence fitness (e.g., the sign of the selection gradient with respect to IIV depends on the mean trait). Consequently, even when evolution eventually leads toward a strategy with no IIV (i.e., the optimal strategy), greater IIV may be transiently selected. Between‐generation stochasticity consistently imposes such transient selection and maintain high levels of IIV in a population.     

Evolutionary rescue under demographic and environmental stochasticity

Populations suffer two types of stochasticity: demographic stochasticity, from sampling error in offspring number, and environmental stochasticity, from temporal variation in the growth rate. By modelling evolution through phenotypic selection following an abrupt environmental change, we investigate how genetic and demographic dynamics, as well as effects on population survival of the genetic variance and of the strength of stabilizing selection, differ under the two types of stochasticity. We show that population survival probability declines sharply with stronger stabilizing selection under demographic stochasticity, but declines more continuously when environmental stochasticity is strengthened. However, the genetic variance that confers the highest population survival probability differs little under demographic and environmental stochasticity. Since the influence of demographic stochasticity is stronger when population size is smaller, a slow initial decline of genetic variance, which allows quicker evolution, is important for population persistence. In contrast, the influence of environmental stochasticity is population-size-independent, so higher initial fitness becomes important for survival under strong environmental stochasticity. The two types of stochasticity interact in a more than multiplicative way in reducing the population survival probability. Our work suggests the importance of explicitly distinguishing and measuring the forms of stochasticity during evolutionary rescue.     

The different limits of weak selection and the evolutionary dynamics of finite populations

Evolutionary theory often resorts to weak selection, where different individuals have very similar fitness. Here, we relate two ways to introduce weak selection. The first considers evolutionary games described by payoff matrices with similar entries. This approach has recently attracted a lot of interest in the context of evolutionary game dynamics in finite populations. The second way to introduce weak selection is based on small distances in phenotype space and is a standard approach in kin-selection theory. Whereas both frameworks are interchangeable for constant fitness, frequency-dependent selection shows significant differences between them. We point out the difference between both limits of weak selection and discuss the condition under which the differences vanish. It turns out that this condition is fulfilled by the popular parametrization of the prisoner's dilemma in benefits and costs. However, for general payoff matrices differences between the two frameworks prevail.     

Lifetime reproductive output: individual stochasticity,variance, and sensitivity analysis

Lifetime reproductive output (LRO) determines per-generation growth rates, establishes criteria for population growth or decline, and is an important component of fitness. Empirical measurements of LRO reveal high variance among individuals. This variance may result from genuine heterogeneity in individual properties, or from individual stochasticity, the outcome of probabilistic demographic events during the life cycle. To evaluate the extent of individual stochasticity requires the calculation of the statistics of LRO from a demographic model. Mean LRO is routinely calculated (as the net reproductive rate), but the calculation of variances has only recently received attention. Here, we present a complete, exact, analytical, closed-form solution for all the moments of LRO, for age- and stage-classified populations. Previous studies have relied on simulation, iterative solutions, or closed-form analytical solutions that capture only part of the sources of variance. We also present the sensitivity and elasticity of all of the statistics of LRO to parameters defining survival, stage transitions, and (st)age-specific fertility. Selection can operate on variance in LRO only if the variance results from genetic heterogeneity. The potential opportunity for selection is quantified by Crow’s index \(\mathcal {I}\), the ratio of the variance to the square of the mean. But variance due to individual stochasticity is only an apparent opportunity for selection. In a comparison of a range of age-classified models for human populations, we find that proportional increases in mortality have very small effects on the mean and variance of LRO, but large positive effects on \(\mathcal {I}\). Proportional increases in fertility increase both the mean and variance of LRO, but reduce \(\mathcal {I}\). For a size-classified tree population, the elasticity of both mean and variance of LRO to stage-specific mortality are negative; the elasticities to stage-specific fertility are positive.     

Threshold Games and Cooperation on Multiplayer Graphs

ObjectiveThe study investigates the effect on cooperation in multiplayer games, when the population from which all individuals are drawn is structured—i.e. when a given individual is only competing with a small subset of the entire population.MethodTo optimize the focus on multiplayer effects, a class of games were chosen for which the payoff depends nonlinearly on the number of cooperators—this ensures that the game cannot be represented as a sum of pair-wise interactions, and increases the likelihood of observing behaviour different from that seen in two-player games. The chosen class of games are named “threshold games”, and are defined by a threshold, M > 0, which describes the minimal number of cooperators in a given match required for all the participants to receive a benefit. The model was studied primarily through numerical simulations of large populations of individuals, each with interaction neighbourhoods described by various classes of networks.ResultsWhen comparing the level of cooperation in a structured population to the mean-field model, we find that most types of structure lead to a decrease in cooperation. This is both interesting and novel, simply due to the generality and breadth of relevance of the model—it is likely that any model with similar payoff structure exhibits related behaviour. More importantly, we find that the details of the behaviour depends to a large extent on the size of the immediate neighbourhoods of the individuals, as dictated by the network structure. In effect, the players behave as if they are part of a much smaller, fully mixed, population, which we suggest an expression for.

Highlights

• Observed behaviour depends on the size of each player’s immediate interaction neighbourhood.
• When the number of players is much larger than the number of required cooperators, average payoff decreases.
• Most network structures lead to a decrease in cooperation compared to the fully mixed case.

     

Does evolution lead to maximizing behavior?

A long‐standing question in biology and economics is whether individual organisms evolve to behave as if they were striving to maximize some goal function. We here formalize this “as if” question in a patch‐structured population in which individuals obtain material payoffs from (perhaps very complex multimove) social interactions. These material payoffs determine personal fitness and, ultimately, invasion fitness. We ask whether individuals in uninvadable population states will appear to be maximizing conventional goal functions (with population‐structure coefficients exogenous to the individual's behavior), when what is really being maximized is invasion fitness at the genetic level. We reach two broad conclusions. First, no simple and general individual‐centered goal function emerges from the analysis. This stems from the fact that invasion fitness is a gene‐centered multigenerational measure of evolutionary success. Second, when selection is weak, all multigenerational effects of selection can be summarized in a neutral type‐distribution quantifying identity‐by‐descent between individuals within patches. Individuals then behave as if they were striving to maximize a weighted sum of material payoffs (own and others). At an uninvadable state it is as if individuals would freely choose their actions and play a Nash equilibrium of a game with a goal function that combines self‐interest (own material payoff), group interest (group material payoff if everyone does the same), and local rivalry (material payoff differences).     

Evolution of a plastic quantitative trait in an age-structured population in a fluctuating environment

We analyze weak fluctuating selection on a quantitative character in an age-structured population not subject to density regulation. We assume that early in the first year of life before selection, during a critical state of development, environments exert a plastic effect on the phenotype, which remains constant throughout the life of an individual. Age-specific selection on the character affects survival and fecundity, which have intermediate optima subject to temporal environmental fluctuations with directional selection in some age classes as special cases. Weighting individuals by their reproductive value, as suggested by Fisher, we show that the expected response per year in the weighted mean character has the same form as for models with no age structure. Environmental stochasticity generates stochastic fluctuations in the weighted mean character following a first-order autoregressive model with a temporally autocorrelated noise term and stationary variance depending on the amount of phenotypic plasticity. The parameters of the process are simple weighted averages of parameters used to describe age-specific survival and fecundity. The "age-specific selective weights" are related to the stable distribution of reproductive values among age classes. This allows partitioning of the change in the weighted mean character into age-specific components.     

Shadows of predation: habitat-selecting consumers eclipse competition between coexisting prey

Ecologists have made substantial progress evaluating the influences of adaptive behaviors on population dynamics and communities. But no-one has examined the joint influences of stochastic variation, predators, and density-dependent habitat selection on our interpretations of species coexistence. I begin the search with simulation models of habitat isodars (lines along which the fitness of individuals is identical in two or more habitats) assuming ideal-free habitat selection by two prey species exploited by a habitat-selecting parasitoid predator. The models include both regulating and non-regulating stochasticity. The intriguing results include the following: (1) all three species often achieved a true ideal-free distribution; (2) predators reduced prey population sizes and increased the frequency of local habitat extinctions; (3) despite the predator's differential reduction of prey densities, there was no evidence of apparent competition; (4) all species exhibited pulses of dispersal associated with donor–receiver population dynamics; (5) isodars produced valid estimates of competition between prey only in constant environments lacking habitat-selecting predators; (6) habitat-selection by predators forced prey into their preferred habitats; (7) the resulting ghost of competition obscured the prey species' competitive interaction; (8) isodars correctly revealed density-dependent habitat selection by the predator. Overall, the results appeared to depend primarily on the predator's habitat choice, rather than on prey trade-offs between competitive ability and reduced value (handling time) to the predator. Habitat selection theory, and its revelation via isodars, can thus provide considerable insight into processes affecting real communities, and most especially if ecologists assess carefully the constraints for their analysis and interpretation. Nevertheless, isodars designed to measure competition are likely to be most reliable in donor-controlled or experimental systems where regulating stochasticity has relatively little influence on prey dynamics.     

The evolution of payoff matrices: providing incentives to cooperate

Most of the work in evolutionary game theory starts with a model of a social situation that gives rise to a particular payoff matrix and analyses how behaviour evolves through natural selection. Here, we invert this approach and ask, given a model of how individuals behave, how the payoff matrix will evolve through natural selection. In particular, we ask whether a prisoner's dilemma game is stable against invasions by mutant genotypes that alter the payoffs. To answer this question, we develop a two-tiered framework with goal-oriented dynamics at the behavioural time scale and a diploid population genetic model at the evolutionary time scale. Our results are two-fold: first, we show that the prisoner's dilemma is subject to invasions by mutants that provide incentives for cooperation to their partners, and that the resulting game is a coordination game similar to the hawk-dove game. Second, we find that for a large class of mutants and symmetric games, a stable genetic polymorphism will exist in the locus determining the payoff matrix, resulting in a complex pattern of behavioural diversity in the population. Our results highlight the importance of considering the evolution of payoff matrices to understand the evolution of animal social systems.     

Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure

We describe a mathematically exact method for the analysis of spatially structured Markov processes. The method is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity. As an example, we consider a spatial version of the Levins metapopulation model, in which the habitat patches are distributed in the d-dimensional landscape Rd in a random (but possibly correlated) manner. Assuming that the dispersal kernel is characterized by a length scale L, we examine how the behavior of the metapopulation deviates from the mean-field model for a finite but large L. For example, we show that the equilibrium fraction of occupied patches is given by p(0)+c/L(d)+O(L(-3d/2)), where p(0) is the equilibrium state of the Levins model and the constant c depends on p(0), the dispersal kernel, and the structure of the landscape. We show that patch occupancy can be increased or decreased by spatial structure, but is always decreased by stochasticity. Comparison with simulations show that the analytical results are not only asymptotically exact (as L-->infinity), but a good approximation also when L is relatively small.     

Evolution of generosity in the demand game

An ecological system often requires moderate, and sometimes even generous interactions among constituents for achieving a sustainable coexistence. Here we propose a configuration individual-based model for demonstrating the evolution of generosity in an evolutionary demand game. In the game, two players, proposers and responders, simultaneously make their demands. If the sum of demands is no more than the total amount of the available resource at each game, each player obtains its own demand. However, both players get nothing if the sum exceeds the total amount. We incorporated generosity by discounting players demands. In every generation, random pairs were formed, and each pair played the demand game. For the next generation, individuals left a number of offspring proportional to their total payoff. Demand and generosity levels of an individual were inherited by its offspring with slight modification by a small random mutation. When only proposers were allowed to discount their demands, distribution of generosity levels had its mode at zero, and hence generosity did not evolve. However, when both proposers and receivers were allowed to discount their demands, the mean generosity level rose from zero. The resultant populations were not homogeneous, but were made of heterogeneous individuals with high and low generosity levels. Mean demand and generosity levels fluctuated greatly because of the neutral selection for the demand and generosity combinations that equally maximized the payoff of the demand game. Spatially limited interaction increased generosity levels even if only proposers discounted their demands.     

The evolution of dispersal under demographic stochasticity

Temporal and spatial variations of the environment are important factors favoring the evolution of dispersal. With few exceptions, these variations have been considered to be exclusively fluctuations of habitat quality. However, since the presence of conspecifics forms part of an individual's environment, demographic stochasticity may be a component of this variability as well, in particular when local populations are small. To study this effect, we analyzed the evolution of juvenile dispersal in a metapopulation model in which habitat quality is constant in space and time but occupancy fluctuates because of demographic stochasticity. Our analysis extends previous studies in that it includes competition of resources and competition for space. Also, juvenile dispersal is not given by a fixed probability but is made conditional on the presence of free territories in a patch, whereas individuals born in full patches will always disperse. Using a combination of analytical and numerical approaches, we show that demographic stochasticity in itself may provide enough variability to favor dispersal even from patches that are not fully occupied. However, there is no simple relationship between the evolution of dispersal and various indicators of demographic stochasticity. Selected dispersal depends on all aspects of the life-history profile, including kin selection.     

Alternative reproductive tactics and status-dependent selection

The status-dependent selection model on alternative reproductivetactics predicts a single switch-point in status: usually allplayers above some status (e.g., competitive ability) shouldpractice the tactic with the higher average payoff, while thosebelow that point should make the "best of a bad job" by practicingthe alternative, lower payoff tactic. Many empirical studiesindeed show a relationship between status and tactic choice,but they do not conform to this single switch-point prediction.I modify the status-dependent selection model by consideringstatus-dependent fitness that is mediated, at least in part,by resource acquisition (e.g., status-based competition forterritories or nuptial gifts). With variation in resource quality,predicted tactic-choice distributions change: a high-statusmale may be territorial on a high-quality territory, a lowerstatus male may practice an alternative tactic, and an evenlower status male may be territorial on a low-quality territory.Tactic choice thus alternates as in many empirical studies andcan appear to be but is not actually stochastic. As the numberof theoretically predicted switch-points increases, however,mixed or mixed-conditional strategies should become more prevalent.While alternative tactics will likely usually differ in meanpayoff, viewing alternative reproductive tactics as inherently"better" or "worse" (e.g., viewing cuckoldry as "worse"—thebest of a bad job) is misleading if not tempered with awarenessthat payoff can vary greatly within tactics and overlap betweentactics.     

Fixation probabilities of random mutants under frequency dependent selection

Evolutionary game dynamics describes frequency dependent selection in asexual, haploid populations. It typically considers predefined strategies and fixed payoff matrices. Mutations occur between these known types only. Here, we consider a situation in which a mutation has produced an entirely new type which is characterized by a random payoff matrix that does not change during the fixation or extinction of the mutant. Based on the probability distribution underlying the payoff values, we address the fixation probability of the new mutant. It turns out that for weak selection, only the first moments of the distribution matter. For strong selection, the probability that a new payoff entry is larger than the wild type's payoff against itself is the crucial quantity.