Evolutionarily stable strategy in a sex- and frequency-dependent selection model

In this paper, a sex-dependent matrix game haploid model is investigated. For this model, since the phenotypes of female and male individuals are determined by alleles located at a single locus and are sex dependent, any given genotype corresponds to a strategy pair. Thus, a strategy pair is an ESS if and only if the allele corresponding to this strategy pair cannot be invaded by any mutant allele. We show that an ESS equilibrium must be locally asymptotically stable if it exists.     

Evolutionarily stable age at first reproduction in a density-dependent model.

We develop a new model of life history evolution to investigate the evolution of age at first reproduction. Density dependence is taken into account. For a given "species", age of maturity, offspring survival, immature survival, adult survival, fecundity, immature age-classes entering in competition with adults and immature competitive ability are traits adjustable by natural selection, and constitute a particular strategy. On the contrary, the type of intraspecific competition (scramble or contest), strength of competition and inherent net reproductive rate Ro(inh) are fixed (specific) characteristics. As a consequence of fixing Ro(inh), the evolution of any trait will affect trade-offs between others. Evolutionarily stable strategies are determined numerically by using the mathematical concept of Lyapunov exponents. Altogether, we consider 960 different hypothetical "species" (i.e. different combinations of fixed traits). Corresponding ESSs are analyzed with respect to their age at first reproduction, adult survival and immature competitive ability components. They appear to be gathered in three groups. One is intuitive and characterized by a reduction of immature competitive ability and a correlation of age of maturity with adult survival; populations reach mainly equilibria. The two other groups respectively include "species" with low age of maturity but high adult survival, and "species" close to semelparity with delayed maturity; immature competitive ability may not be minimized, and populations possibly exhibit complex dynamics. In conclusion, the hypothesis that the evolution of a demographic parameter modifies trade-offs between others turns out to have important consequences. We argue that life history theory cannot ignore the source and mode-of-operation of density dependence and must regard potential short-term instability as essential.     

The truthful signalling hypothesis: an explicit general equilibrium model

In mating competition, the truthful signalling hypothesis (TSH), sometimes known as the handicap principle, asserts that higher-quality males signal while lower-quality males do not (or else emit smaller signals). Also, the signals are "believed", that is, females mate preferentially with higher-signalling males. Our analysis employs specific functional forms to generate analytic solutions and numerical simulations that illuminate the conditions needed to validate the TSH. Analytic innovations include: (1) A Mating Success Function indicates how female mating choices respond to higher and lower signalling levels. (2) A congestion function rules out corner solutions in which females would mate exclusively with higher-quality males. (3) A Malthusian condition determines equilibrium population size as related to per-capita resource availability. Equilibria validating the TSH are achieved over a wide range of parameters, though not universally. For TSH equilibria it is not strictly necessary that the high-quality males have an advantage in terms of lower per-unit signalling costs, but a cost difference in favor of the low-quality males cannot be too great if a TSH equilibrium is to persist. And although the literature has paid less attention to these points, TSH equilibria may also fail if: the quality disparity among males is too great, or the proportion of high-quality males in the population is too large, or if the congestion effect is too weak. Signalling being unprofitable in aggregate, it can take off from a no-signalling equilibrium only if the trait used for signalling is not initially a handicap, but instead is functionally useful at low levels. Selection for this trait sets in motion a bandwagon, whereby the initially useful indicator is pushed by male-male competition into the domain where it does indeed become a handicap.     

Evolutionarily stable sets in the single-locus frequency-dependent model of natural selection

Recent developments in the static theory of evolutionarily stable sets (ESSets) are applied to the single-locus frequency-dependent model of natural selection. Particular emphasis is paid to the ESSet properties of the preimage of an ESS (or ESSet) under the genotype-phenotype map. When an ESS is realized in genetic equilibrium with redundancy in a diploid sexual population, the basic problem in biological terms is whether the corresponding set of allele frequencies is an evolutionarily stable set. The interesting question of the dynamic stability of this preimage is also discussed and a geometric condition developed which implies its evolutionary and dynamic stability.The authors appreciate detailed suggestions for improvement made by the reviewers of the original version of this paper. Financial assistance from the Natural Sciences and Engineering Research Council of Canada and from the Hungarian National Scientific Research Fund (OTKA Projects T029320 and T037271) is also gratefully acknowledged.     

Evolutionarily stable nesting strategy in a digger wasp.

Two alternative “strategies” will not coexist in a population unless on average they are equally successful. The most likely way for such an equilibrium to be maintained is through something equivalent to frequency-dependent selection. Females of the digger wasp Sphex ichneumoneus (Sphecidae) nest in underground burrows. They usually dig and provision these by themselves but occasionally a nest is jointly occupied. The two wasps fight whenever they meet and in the end only one of the two females lays an egg in the shared nest. Two models based on the theory of mixed evolutionarily stable strategies were developed and tested on comprehensive field data from two North American populations of these wasps. The first model proposes two strategies called founding and joining. Founders start burrows alone, but they are more successful when they are joined by a joiner. At equilibrium founders and joiners are equally successful, which amounts to an amicable, sharing relationship. The predictions of this amicable model are decisively rejected by the data. The second model proposes two strategies called digging and entering. Diggers dig their own burrows but they often have to abandon these burrows because of temporary unsuitability. Enterers move in later, thereby exploiting abandoned burrows as a valuable resource. They do not distinguish an adandoned burrow from one that is still occupied. Therefore sharing of burrows arises as an unfortunate by product of selection for entering abandoned burrows, and Model 2 is not an amicable model. Its quantitative predictions are impressively fulfilled in one population, though not in another population. This is one of the only examples yet known of a mixed evolutionarily stable strategy in nature. Yet the word strategy itself can confuse, and this paper tries the experiment of substituting “decision”, defined as a moment at which the animal commits future time to a course of action.     

Evolutionarily stable strategies of mutual help between relatives having unequal fertilities

The evolutionarily stable strategy of mutual help between relatives having unequal fertilities is studied in a kin selection model, which also takes into account competition between kins and the possibility of reciprocation. It turns out that competition and reciprocation can establish ESSs which are completely different from those expected by Hamilton's basic theory.     

Evolutionarily stable mutation rate in a periodically changing environment

Evolution of mutation rate controlled by a neutral modifier is studied for a locus with two alleles under temporally fluctuating selection pressure. A general formula is derived to calculate the evolutionarily stable mutation rate μ(ess) in an infinitely large haploid population, and following results are obtained. (I) For any fluctuation, periodic or random: (1) if the recombination rate r per generation between the modifier and the main locus is 0, μ(ess) is the same as the optimal mutation rate μ(op) which maximizes the long-term geometric average of population fitness; and (2) for any r, if the strength s of selection per generation is very large, μ(ess) is equal to the reciprocal of the average number T of generations (duration time) during which one allele is persistently favored than the other. (II) For a periodic fluctuation in the limit of small s and r, μ(ess)T is a function of sT and rT with properties: (1) for a given sT, μ(ess)T decreases with increasing rT; (2) for sT </= 1, μ(ess)T is almost independent of sT, and depends on rT as μ(ess)T & 1.6 for rT << 1 and μ(ess)T & 6/rT for rT >> 1; and (3) for sT >/= 1, and for a given rT, μ(ess)T decreases with increasing sT to a certain minimum less than 1, and then increases to 1 asymptotically in the limit of large sT. (III) For a fluctuation consisting of multiple Fourier components (i.e., sine wave components), the component with the longest period is the most effective in determining μ(ess) (low pass filter effect). (IV) When the cost c of preventing mutation is positive, the modifier is nonneutral, and μ(ess) becomes larger than in the case of neutral modifier under the same selection pressure acting at the main locus. The value of c which makes μ(ess) equal to μ(op) of the neutral modifier case is calculated. It is argued that this value gives a critical cost such that, so long as the actual cost exceeds this value, the evolution rate at the main locus must be smaller than its mutation rate μ(ess).     

Evolutionarily [corrected] stable strategies: a review of basic theory

Widely successful in applied population biology, the Evolutionarily Stable Strategy concept remains controversial because of the severe restrictions present in its original formulation. We review theory which explores and relaxes these restrictions, finding the concept to be quite robust and adaptable, incorporating considerations such as genetics, population diversity, environmental variability, and mutation.     

Evolutionarily stable kleptoparasitism: consequences of different prey types

We present two elaborations of the model of Broom and Ruxtonthat found evolutionarily stable kleptoparasitic strategiesfor foragers. These elaborations relax the assumption that thedistribution of times required to handle discovered food itemsis exponential. These changes increase the complexity of themodel but represent a significant improvement in biologicalrealism. In one elaboration, handling takes a fixed interval,t_h, at the end of which the whole value of the food item isobtained. We liken this to peeling then consuming a small orange.The other elaboration also assumes that handling takes a fixedinterval, t_h, but this time the reward from the food item isextracted continuously throughout the handling period. We likenthis to eating an apple. Both models predict that increasingfood density, the ease with which food items can be discovered,or the length of aggressive contests all act to make kleptoparasitismless common. The difference between the evolutionarily stablestrategy solutions of the apple and orange models provides aclear prediction of our theory. When prey items require handlingbefore yielding a lump sum at the end, then kleptoparasiticattacks will be focused on prey items near the end of theirhandling period. However, if prey items yield reward continuouslyduring handling, then attacks should be biased toward newlydiscovered food items. Another key difference between the modelpredictions is that kleptoparasitism increases with foragerdensity in the apple model, but decreases in the orange model.     

Evolutionarily stable stealing: game theory applied to kleptoparasitism

We present an individual-based model of a group of foraginganimals. Individuals can obtain food either by discovering itthemselves or by stealing it from others (kleptoparasitism).Given that challenging another individual for a discovered fooditem costs time (which could otherwise be spent searching foran undiscovered item), attempting to steal from another maynot always be efficient We show that there is generally a uniquestrategy that maximizes uptake rate—always or never challengingothers. For any combination of parameter values, we can identifywhich strategy is appropraite. As a corollary to this, we predictthat small changes in ecolgical conditions can, under some circumstances,cause a dramatic change in the aggressive behavior of individuals.Further, we investigate situations where searching for undiscoveredfood and searching for potential opportunities for stealingare mutually exclusive activities (i.e., success at one canonly be improved at the expense of the other). Using game theory,we are able to find the evolutionarily stable strategy for investmentin these two activities in terms of the ecological parametersof the model.     

Evolutionarily stable mixed mating in a variety of genetic systems

Many species display a mixture of close inbreeding and outbreeding which is referred to as mixed mating. For selfing species, models predict that such mixed mating systems can be stable. Conversely, models considering separate sex species have not been able to explain mixed mating systems. This failure may be a result of the unrealistic assumption that recurrent inbreeding does not increase the inbreeding coefficient. Here we show that mixed mating is expected in separate sex systems when recurrent inbreeding is taken into account. A female that allows her brother to sibmate with her gives an extra mating opportunity to said brother. This kin selective advantage should be strongest in genetic systems where the male is more related to the female. In support of this idea, we find that inbreeding evolves most easily in selfers, followed by diploid sibmating, followed by haplodiploid sibmating. Consideration of published values for the regression of fitness on inbreeding coefficient suggests that many species fall in a range where some selfing/sibmating is optimal.     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.     

Evolutionarily stable dispersal of a waterstrider in a temporally and spatially heterogeneous environment

Summary Evolutionary stable dispersal and wing muscle histolysis strategies are studied in the waterstriderGerris thoracicus. These strategies relate to spreading reproductive risk. Overwintering individuals have the choice of dispersing to either a brackish sea bay or a rock pool habitat. The former is reproductively more favorable than the latter during warm dry years and less favorable during cool wet years. After spring migration, individuals may histolyse their flight muscles and lay all their eggs in one pool or they may retain their flight ability and lay fewer eggs in total but spread them in several pools. We use a simple two-habitat model to examine the question of habitat dispersal. Our results indicate that, although the value of the evolutionary stable dispersal depends on the degree of variability in the environment and on the probability of local extinctions in either habitat, the population always disperses to both habitats as a consequence of density dependent growth. We use a more detailed multiple-rockpool habitat model to examine the question of wing muscle histolysis as a response to density dependence. Our results indicate that a wing muscle histolysis response to population density is an evolutionarily stable strategy when compared with the two alternatives of females always histolysing or never histolysing their flight muscles. The application of evolutionarily stable theory to stochastic problems presents a number of difficulties. We discuss these difficulties in the context of computing evolutionarily stable strategies for the problems at hand.     

Indirect genetic effects and kin recognition: estimating IGEs when interactions differ between kin and strangers

Social interactions among individuals are widespread, both in natural and domestic populations. As a result, trait values of individuals may be affected by genes in other individuals, a phenomenon known as indirect genetic effects (IGEs). IGEs can be estimated using linear mixed models. The traditional IGE model assumes that an individual interacts equally with all its partners, whether kin or strangers. There is abundant evidence, however, that individuals behave differently towards kin as compared with strangers, which agrees with predictions from kin-selection theory. With a mix of kin and strangers, therefore, IGEs estimated from a traditional model may be incorrect, and selection based on those estimates will be suboptimal. Here we investigate whether genetic parameters for IGEs are statistically identifiable in group-structured populations when IGEs differ between kin and strangers, and develop models to estimate such parameters. First, we extend the definition of total breeding value and total heritable variance to cases where IGEs depend on relatedness. Next, we show that the full set of genetic parameters is not identifiable when IGEs differ between kin and strangers. Subsequently, we present a reduced model that yields estimates of the total heritable effects on kin, on non-kin and on all social partners of an individual, as well as the total heritable variance for response to selection. Finally we discuss the consequences of analysing data in which IGEs depend on relatedness using a traditional IGE model, and investigate group structures that may allow estimation of the full set of genetic parameters when IGEs depend on kin.     

Evolutionarily stable transition rates in a stage-structured model. An application to the analysis of size distributions of badges of social status

This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete population model with a general transition matrix. In the model, individuals are categorized into n dominance classes, newborns lie in the subordinate class, and it is considered as evolutionary trait a vector eta of probabilities of transition among classes. For this trait, we obtain the evolutionary singular strategy and prove its neutral evolutionary stability. Finally, we obtain conditions for the invading potential of such a strategy, which is sufficient for the convergence stability of the latter. With the help of the previous results, we provide an explanation for the bimodal distribution of badges of status observed in the Siskin (Carduelis spinus). In the Siskin, as in several bird species, patches of pigmented plumage signal the dominance status of the bearer to opponents, and central to the discussion on the evolution of status signalling is the understanding of which should be the frequency distribution of badge sizes. Though some simple verbal models predicted a bimodal distribution, up to now most species display normal distributions and bimodality has only been described for the Siskin. In this paper, we give conditions leading to one of these two distributions in terms of the survival, fecundity and aggression rates in each dominance class.     

Evolutionarily stable growth of a sapling which waits for future gap formation under closed canopy

Summary A model was developed to examine the ESS sapling growth waiting for future gap formation under closed canopy. Assumptions are: a sapling has two parts, a trunk and a photosynthetic part, and allocates annual photosynthates to these two parts; and a sapling with a larger photosynthetic part has a larger production rate, but a sapling with a larger trunk is more successful in competition after gap formation. The ESS growth schedule of a sapling typically consists of three phases: (1) the sapling first allocates all annual photosynthates to the photosynthetic part, then (2) it allocates annual photosynthates both to its trunk and to photosynthetic part, and both parts grow simultaneously, and finally (3) it also allocates annual photosynthates to both parts, but the size of the photosynthetic part stays constant due to annual loss, and only the trunk size increases. A sapling should allocate photosynthates more to the trunk if mortality or probability of gap formation is large. However, a sapling should allocate photosynthates more to the photosynthetic part if large trunks are strongly advantageous in competition after gap formation.     

Evolutionarily stable infection by a male-killing endosymbiont in Drosophila innubila: molecular evidence from the host and parasite genomes

Maternally inherited microbes that spread via male-killing are common pathogens of insects, yet very little is known about the evolutionary duration of these associations. The few examples to date indicate very recent, and thus potentially transient, infections. A male-killing strain of Wolbachia has recently been discovered in natural populations of Drosophila innubila. The population-level effects of this infection are significant: approximately 35% of females are infected, infected females produce very strongly female-biased sex ratios, and the resulting population-level sex ratio is significantly female biased. Using data on infection prevalence and Wolbachia transmission rates, infected cytoplasmic lineages are estimated to experience a approximately 5% selective advantage relative to uninfected lineages. The evolutionary history of this infection was explored by surveying patterns of polymorphism in both the host and parasite genomes, comparing the Wolbachia wsp gene and the host mtDNA COI gene to five host nuclear genes. Molecular data suggest that this male-killing infection is evolutionarily old, a conclusion supported with a simple model of parasite and mtDNA transmission dynamics. Despite a large effective population size of the host species and strong selection to evolve resistance, the D. innubila-Wolbachia association is likely at a stable equilibrium that is maintained by imperfect maternal transmission of the bacteria rather than partial resistance in the host species.