Evolutionarily stable strategies and viability selection in mendelian populations

For various genetical structures, including haploid and diploid, one-locus n-alleles, and n-locus additive viability random mating models, natural selection resulting from intrapopulation conflicts between random individuals leads to exactly those genetical equilibria which determine a mixture of strategies evolutionarily stable according to the game theory definition of Maynard Smith and Price (1973).     

Evolutionarily stable strategies and short-term selection in Mendelian populations re-visited

This note concerns a one locus, two allele, random mating diploid population, subject to frequency-dependent viability selection. It is already known that in such a population, any evolutionarily stable strategies (ESS), if only accessible by the genotype-to-phenotype mapping, is the phenotypic image of a stable genetic equilibrium (Eshel, I. 1982. Evolutionarily stable strategies and viability selection in Mendelian populations. Theor. Popul. Biol. 22(2), 204-217; Cressman et al. 1996. Evolutionary stability in strategic models of single-locus frequency-dependent viability selection. J. Math. Biol. 34, 707-733). The opposite is not true. We find necessary and sufficient parametric conditions for global convergence to the ESS, but we also demonstrate conditions under which, although a unique, genetically accessible ESS exists, there is another, "non-phenotypic" genetically stable equilibrium.     

Evolutionarily stable resource allocation for production of wind-dispersed seeds

We developed a game-theoretic model for wind-dispersed seed production to examine the seed mass–dispersal ability relationship and the evolutionarily stable distance of seed dispersal in terms of exploitation of safe sites. We assumed trade-offs between masses of the embryo (including albumen) and the wind-dispersal structures per seed, and also between seed mass and number of seeds per parent. We showed that ESS wing-loading is independent of embryo mass; that is, heavy seeds are not poor dispersers if the cost of producing wind-dispersal structures per unit area is constant. The ESS embryo mass per seed depends only on the factors which determine the probability of a seedling being established from a seed. However, wing-loading was found to increase with embryo mass when the change in length was isometric and there was a negative correlation between seed mass and dispersal ability. Thus, the area–mass relationship in wind-dispersal structures may have large effects on the ESS production of wind-dispersed seeds. On the other hand, given that only a limited number of adults can be established at a safe site, the ESS seed dispersal distance depends on the relative degree of sib to non-sib competition. A parent disperses its seeds over a wide area to exploit many safe sites if sib competition is strong. However, it disperses its seeds within a narrow area if the mean number of parents per unit area is large, or if non-sib competition is strong. Thus, in addition to an upper limit on the number of adults per safe site, the degree of sib and non-sib competition may be important for the ESS dispersal distance in wind-dispersed seeds. This revised version was published online in July 2006 with corrections to the Cover Date.     

Evolutionarily stable dispersal strategies

Using the idea that life-history parameters are subject to natural selection and should approach values that are stable optima, with the population immune to invasion by mutant individuals, we derive an analytic expression for the evolutionarily stable dispersal rate in a stochastic island model with random site extinction. The results provide interesting contrasts between three different optimization criteria: species survival, individual fitness and gene fitness. We also consider the effects of sexual reproduction, and of localized migration (stepping-stone structure).     

Size-dependent leaf area ratio in plant twigs: implication for leaf size optimization

Background and Aims

Although many hypotheses have been proposed to explain variation in leaf size, the mechanism underlying the variation remains not fully understood. To help understand leaf size variation, the cost/benefit of twig size was analysed, since, according to Corner''s rule, twig size is positively correlated with the size of appendages the twig bears.

Methods

An extensive survey of twig functional traits, including twig (current-year shoots including one stem and few leaves) and leaf size (individual leaf area and mass), was conducted for 234 species from four broadleaved forests. The scaling relationship between twig mass and leaf area was determined using standardized major axis regression and phylogenetic independent comparative analyses.

Key Results

Leaf area was found to scale positively and allometrically with both stem and twig mass (stem mass plus leaf mass) with slopes significantly smaller than 1·0, independent of life form and habitat type. Thus, the leaf area ratio (the ratio of total leaf area to stem or twig mass) decreases with increasing twig size. Moreover, the leaf area ratio correlated negatively with individual leaf mass. The results of phylogenetic independent comparativeanalyses were consistent with the correlations. Based on the above results, a simple model for twig size optimization was constructed, from which it is postulated that large leaf size–twig size may be favoured when leaf photosynthetic capacity is high and/or when leaf life span and/or stem longevity are long. The model''s predictions are consistent with leaf size variation among habitats, in which leaf size tends to be small in poor habitats with a low primary productivity. The model also explains large variations in leaf size within habitats for which leaf longevity and stem longevity serve as important determinants.

Conclusions

The diminishing returns in the scaling of total leaf area with twig size can be explained in terms of a very simple model on twig size optimization.Key words: Allometry, leaf size, twig size, leaf area ratio, scaling relationship, broadleaved species     

Evolutionarily stable strategies of migration in heterogeneous environments

By means of a simulation model we are showing that the rates of migration can be related to avoidance of competition between relatives, especially in clonal organisms. This could result in a strong selective pressure for migration, even at a high cost. In addition, if the habitat is fragmented, migration can strongly affect local dynamics and result in a dramatic decrease of the densities in some places. In parthenogenetically reproducing organisms like aphids, the level of relatedness in local populations is expected to be very high and therefore they can serve as a good model group for testing these hypotheses. This revised version was published online in July 2006 with corrections to the Cover Date.     

Evolutionarily stable sex ratios and sex allocations

The paternal fitness of a sexual individual is equated with the fitness of those eggs of its potential mates which it is able to fertilize. This property enables the total sexual fitness of individuals to be expressed in terms of female gamete contributions in separate equations for a cosex (an individual in a population composed of a single sexual class which combines male and female functions) and for parents in a dioecious population. The general equations are used in phenotypic models of selection which examine conditions maximizing the fitness advantage of one phenotype over another with a different sex ratio or allocation. As an example, it is shown that finite population size confers full stability on the sexual allocations in a cosexual population and on the sex ratio in a dioecious population.The use of fitness advantages provides the outcome of selection for all frequencies of contrasted phenotypes. It is therefore possible to redefine an ESS to allow for persistent variability in a population. A phenotype is an ESS in a population if, from any initial frequency, it is protected from loss by its fitness advantage. The conditions for a rare mutant to spread invariably coincide with those for its fixation only if an individual of any phenotype affects the fitness of other individuals of all phenotypes in identical ways.     

Evolutionarily stable germination strategies with time-correlated yield

We investigate the effect of auto-correlated yield on the evolutionarily stable germination fraction of dormant seeds. By using both analytics and numerics, we first show that in a regime of small fluctuations a positive correlation reduces dormancy and a negative correlation enhances dormancy. By extending the numerical analysis we also show that in the regime of large fluctuations a more complex picture emerges where also negative correlations can reduce dormancy.     

Evolutionarily stable sets in symmetric extensive two-person games.

Evolutionarily stable (ES) sets are characterized for evolutionary games in extensive form. It is shown that, for the normal form of games involving informational symmetries or repeated play, the standard approach of determining evolutionarily stable strategies (ESSs) often fails to predict the evolutionary outcome. The dynamic stability of ES sets is proved in both the pure strategy and mixed strategy models. ES sets are shown to also generalize the notion of direct ESSs (an earlier attempt to apply ESS theory to extensive games). The theory is illustrated by three examples of biological games in extensive form.     

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 for localized dispersal in two dimensions

Evolutionarily stable dispersal strategies are calculated for a generalized two-dimensional stepping-stone model in which dispersal is not restricted to nearest neighbour sites. It is shown analytically that to an excellent approximation the ESS is independent of the dispersal pattern and is the same as the result of an earlier paper which used the simple but unrealistic island model (the dispersal survival rate of the previous paper must be replaced by an average value). When dispersal to different distances is under separate genetic control it is also possible to make general predictions concerning the fall-off of genetic correlation with distance. Two generalizations of the model are considered: random extinction of sites (creating empty sites which are opportunities for colonization), and perennation of existing occupants. These modifications are intended to bring the model closer to two real-world circumstances: dispersal between discrete colonies of animals living on a patchy resource, and seed and pollen dispersal in evenly distributed plant species. Simulations are required to validate any extensions of the analytic model; I present a relatively fast technique for determining the ESS, which relies in part on the analytic results.     

Evolutionarily stable strategy and invader strategy in matrix games

In this paper, we consider the concepts of evolutionarily stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in single species with frequency-dependent interactions. We find some general relationships among the three concepts in matrix games. The main conclusion is that ESS and NIS are equivalent to each other and are both equivalent to local superiority; a strategy with global superiority must be a GIS; a GIS may not be equivalent to its global superiority in games with more than two players; and in any two-player matrix game a GIS is just equivalent to its global superiority. In two-player games, globally asymptotic stability in the replicator dynamics has also been shown. Equivalent conditions for the three concepts stated by payoff comparisons are given and are applied to examples involved.     

Evolutionarily stable strategies in differential and difference equation models

Summary A definition for an evolutionarily stable strategy (ESS) is given which is applicable to a general differential equation population model and two difference equation analogs. With the introduction of a fitnessgenerating function, it is possible to develop necessary conditions for the determination of an ESS for each of these systems. In most situations, an ESS for one system will also be an ESS for the other. Necessary conditions for an ESS are obtained. Under certain restrictions, they are shown to be valid, even under an unstable equilibrium in population density. the results are illustrated with an example which has the same ESS solution whether a continuous or discrete model is used. The behavior of the ESS for the discrete model is then examined under unstable equilibrium conditions in population density.     

Investigating successive Australian barley breeding populations for stable resistance to leaf rust

Key message

Genome-wide association studies of barley breeding populations identified candidate minor genes for pairing with the adult plant resistance gene Rph20 to provide stable leaf rust resistance across environments.

Abstract

Stable resistance to barley leaf rust (BLR, caused by Puccinia hordei) was evaluated across environments in barley breeding populations (BPs). To identify genomic regions that can be combined with Rph20 to improve adult plant resistance (APR), two BPs genotyped with the Diversity Arrays Technology genotyping-by-sequencing platform (DArT-seq) were examined for reaction to BLR at both seedling and adult growth stages in Australian environments. An integrated consensus map comprising both first- and second-generation DArT platforms was used to integrate QTL information across two additional BPs, providing a total of four interrelated BPs and 15 phenotypic data sets. This enabled identification of key loci underpinning BLR resistance. The APR gene Rph20 was the only active resistance region consistently detected across BPs. Of the QTL identified, RphQ27 on chromosome 6HL was considered the best candidate for pairing with Rph20. RphQ27 did not align or share proximity with known genes and was detected in three of the four BPs. The combination of RphQ27 and Rph20 was of low frequency in the breeding material; however, strong resistance responses were observed for the lines carrying this pairing. This suggests that the candidate minor gene RphQ27 can interact additively with Rph20 to provide stable resistance to BLR across diverse environments.

     

Evolutionarily stable defence and signalling of that defence

We examine the evolution and maintenance of defence and conspicuousness in prey species using a game theoretic model. In contrast to previous works, predators can raise as well as lower their attack probabilities as a consequence of encountering moderately defended prey. Our model predicts four distinct possibilities for evolutionarily stable strategies (ESSs) featuring maximum crypsis. Namely that such a solution can exist with (1) zero toxicity, (2) a non-zero but non-aversive level of toxicity, (3) a high, aversive level of toxicity or (4) that no such maximally cryptic solution exists. Maximally cryptic prey may still invest in toxins, because of the increased chance of surviving an attack (should they be discovered) that comes from having toxins. The toxin load of maximally cryptic prey may be sufficiently strong that the predators will find them aversive, and seek to avoid similar looking prey in future. However, this aversiveness does not always necessarily trigger aposematic signalling, and highly toxic prey can still be maximally cryptic, because the increased initial rate of attack from becoming more conspicuous is not necessarily always compensated for by increased avoidance of aversive prey by predators. In other circumstances, the optimal toxin load may be insufficient to generate aversion but still be non-zero (because it increases survival), and in yet other circumstances, it is optimal to make no investment in toxins at all. The model also predicts ESSs where the prey are highly defended and aversive and where this defence is advertised at a cost of increased conspicuousness to predators. In many circumstances there is an infinite array of these aposematic ESSs, where the precise appearance is unimportant as long as it is highly visible and shared by all members of the population. Yet another class of solutions is possible where there is strong between-individual variation in appearance between conspicuous, poorly defended prey.     

Evolutionarily stable range limits set by interspecific competition

A combination of abiotic and biotic factors probably restricts the range of many species. Recent evolutionary models and tests of those models have asked how a gradual change in environmental conditions can set the range limit, with a prominent idea being that gene flow disrupts local adaptation. We investigate how biotic factors, explicitly competition for limited resources, result in evolutionarily stable range limits even in the absence of the disruptive effect of gene flow. We model two competing species occupying different segments of the resource spectrum. If one segment of the resource spectrum declines across space, a species that specializes on that segment can be driven to extinction, even though in the absence of competition it would evolve to exploit other abundant resources and so be saved. The result is that a species range limit is set in both evolutionary and ecological time, as the resources associated with its niche decline. Factors promoting this outcome include: (i) inherent gaps in the resource distribution, (ii) relatively high fitness of the species when in its own niche, and low fitness in the alternative niche, even when resource abundances are similar in each niche, (iii) strong interspecific competition, and (iv) asymmetric interspecific competition. We suggest that these features are likely to be common in multispecies communities, thereby setting evolutionarily stable range limits.     

Specific leaf area and leaf dry matter content as alternative predictors of plant strategies

A key element of most recently proposed plant strategy schemes is an axis of resource capture, usage and availability. In the search for a simple, robust plant trait (or traits) that will allow plants to be located on this axis, specific leaf area is one of the leading contenders. Using a large new unpublished database, we examine the variability of specific leaf area and other leaf traits, the relationships between them, and their ability to predict position on the resource use axis. Specific leaf area is found to suffer from a number of drawbacks; it is both very variable between replicates and much influenced by leaf thickness. Leaf dry-matter content (sometimes referred to as tissue density) is much less variable, largely independent of leaf thickness and a better predictor of location on an axis of resource capture, usage and availability. However, it is not clear how useful dry matter content will be outside northwest Europe, and in particular in dry climates with many succulents.     

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.