Evolutionary dynamics of habitat use

I examine the evolution of alternate genotypes that use two habitats that differ in vegetative cover, focusing on the interplay between ecological dynamics of the community and changes in selective advantage. Facultative habitat choice can stabilize a predator population that would cycle if isolated in the more open habitat. This has important implications for the evolution of habitat use strategies. Local stability arising from facultative habitat use allows any number of behavioural genotypes to co-exist: selective use of the open habitat, selective use of the dense habitat, opportunistic use of both habitats in proportion to availability, and facultative switching between habitats to maximize energy gain. Co-existence occurs because the fitness landscape is flat at the ecological equilibrium imposed by the facultative genotype. In contrast, ecological instability favours the evolution of genotypes with behavioural flexibility to avoid being in the wrong place at the wrong time or selective exploitation of one of the habitats. Uncertain information about habitat quality erodes the adaptive advantage of otherwise optimal behaviours, favouring a bet-hedging behavioural strategy synonymous with partial habitat preferences. These results suggest that ecological dynamics could have a strong influence on behavioural heterogeneity within forager populations and that a mixed ESS for habitat use should predominate.     

Evolutionary games on networks and payoff invariance under replicator dynamics

The commonly used accumulated payoff scheme is not invariant with respect to shifts of payoff values when applied locally in degree-inhomogeneous population structures. We propose a suitably modified payoff scheme and we show both formally and by numerical simulation, that it leaves the replicator dynamics invariant with respect to affine transformations of the game payoff matrix. We then show empirically that, using the modified payoff scheme, an interesting amount of cooperation can be reached in three paradigmatic non-cooperative two-person games in populations that are structured according to graphs that have a marked degree inhomogeneity, similar to actual graphs found in society. The three games are the Prisoner’s Dilemma, the Hawks-Doves and the Stag-Hunt. This confirms previous important observations that, under certain conditions, cooperation may emerge in such network-structured populations, even though standard replicator dynamics for mixing populations prescribes equilibria in which cooperation is totally absent in the Prisoner’s Dilemma, and it is less widespread in the other two games.     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Introduction. Evolutionary dynamics of wild populations: the use of long-term pedigree data

Studies of populations in the wild can provide unique insights into the forces driving evolutionary dynamics. This themed issue of Proc. R. Soc. B focuses on new developments in long-term analyses of animal populations where pedigree information has been collected. These address fundamental questions in evolutionary biology concerning the genetic basis of phenotypic diversity, patterns of natural and sexual selection, the occurrence of inbreeding and inbreeding depression, and speciation. Contributions include the analysis of evolutionary responses to climate change, exploration of the genetic basis of senescence, the exploitation of advances in molecular genetic technology, and reviews of developments in quantitative genetic methodology. We discuss here common themes, specific problems and pointers for future research.     

Evolutionary rates for multivariate traits: the role of selection and genetic variation

A fundamental question in evolutionary biology is the relative importance of selection and genetic architecture in determining evolutionary rates. Adaptive evolution can be described by the multivariate breeders'' equation (), which predicts evolutionary change for a suite of phenotypic traits () as a product of directional selection acting on them (β) and the genetic variance–covariance matrix for those traits (G). Despite being empirically challenging to estimate, there are enough published estimates of G and β to allow for synthesis of general patterns across species. We use published estimates to test the hypotheses that there are systematic differences in the rate of evolution among trait types, and that these differences are, in part, due to genetic architecture. We find some evidence that sexually selected traits exhibit faster rates of evolution compared with life-history or morphological traits. This difference does not appear to be related to stronger selection on sexually selected traits. Using numerous proposed approaches to quantifying the shape, size and structure of G, we examine how these parameters relate to one another, and how they vary among taxonomic and trait groupings. Despite considerable variation, they do not explain the observed differences in evolutionary rates.     

Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay

We construct two models of discrete-time replicator dynamics with time delay. In the social-type model, players imitate opponents taking into account average payoffs of games played some units of time ago. In the biological-type model, new players are born from parents who played in the past. We consider two-player games with two strategies and a unique mixed evolutionarily stable strategy. We show that in the first type of dynamics, it is asymptotically stable for small time delays and becomes unstable for big ones when the population oscillates around its stationary state. In the second type of dynamics, however, evolutionarily stable strategy is asymptotically stable for any size of a time delay.     

Evolutionary mechanisms of senescence

This paper reviews theories of the evolution of senescence. The population genetic basis for the decline with age in sensitivity of fitness to changes in survival and fecundity is discussed. It is shown that this creates a presure of selection that disproportionately favors performance early in life. The extent of this bias is greater when there is a high level of extrinsic mortality; this accounts for much the diversity in life-history patterns among different taxa. The implications of quantitative genetic theory for experimental tests of alternative population genetic models of senescence are discussed. In particular, the negative genetic correlations between traits predicted by the antagonistic pleiotropy model may be obscured by positive correlations that are inevitable in a multivariate system, or by the effects of variation due to deleterious mutations. The status of the genetic evidence relevant to these theories is discussed.     

Evolutionary potential in the wild: more than meets the eye

The genus Aquilegia consists of 60–70 perennial plant species widely distributed throughout the northern hemisphere. Its flowers have a delicate and ornamental appearance that makes them a favourite of gardeners. In this genus, adaptive radiations for both floral and vegetative traits have occurred. These adaptive radiations, and the key phylogenetic placement of Aquilegia between Arabidopsis and rice, make this genus a ‘model system’ for plant evolution ( Kramer 2009 ). In this issue, Castellanos et al. (2011) use a marker‐based method to infer heritability for floral and vegetative traits in two Aquilegia species. Layered on top of this are estimates of the strength of natural selection. This novel joint estimation of heritability and selection in the wild showed that vegetative traits, compared to floral traits, have the highest evolutionarily potential. Evolutionary potential is the most important quantity to measure in wild populations. It combines inheritance and strength of selection and predicts the potential for populations to adapt to changing environments. The combination of molecular techniques with species in natural environments makes this work a model for molecular ecological investigations.     

Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

The replicator dynamics with n players and population structure

The well-known replicator dynamics is usually applied to 2-player games and random matching. Here we allow for games with n players, and for population structures other than random matching. This more general application leads to a version of the replicator dynamics of which the standard 2-player, well-mixed version is a special case, and which allows us to explore the dynamic implications of population structure. The replicator dynamics also allows for a reformulation of the central theorem in Van Veelen (2009), which claims that inclusive fitness gives the correct prediction for games with generalized equal gains from switching (or, in other words, when fitness effects are additive). If we furthermore also assume that relatedness is constant during selection - which is a reasonable assumption in a setting with kin recognition - then inclusive fitness even becomes a parameter that determines the speed as well as the direction of selection. For games with unequal gains from switching, inclusive fitness can give the wrong prediction. With equal gains however, not only the sign, but also even the value of inclusive fitness becomes meaningful.     

On the stability of a model of testosterone dynamics

We prove the global asymptotic stability of a well-known delayed negative-feedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.Supported in part by AFOSR Grant F49620-01-1-0063, NIH Grant P20 GM64375, and Dimacs.E.D. Sontag: Supported in part by AFOSR Grant F49620-01-1-0063 and NIH Grant R01 GM46383.Acknowledgement We would like to thank Augusto Ponce for useful suggestions.     

Prediction of the stability of coiled coils using molecular dynamics simulations

We have performed 40–80 ns-long molecular dynamics (MD) simulations of the GCN4 leucine zipper and synthetic coiled coils using the GROMOS96 (43a2) and OPLS-AA force fields, with the aim of predicting coiled coil stability. Starting with an initial configuration of two peptides placed in an ideal coiled coil configuration, we find that changing the amino acid sequence modestly or decreasing peptide length can lead to a decrease in the final α-helicity of coiled coils, although for peptides as long or longer than 16 residues, the values of helicity do not decrease to the low values seen in the experimental results of Lumb et al. (Biochemistry. 1994, 33, 7361–7367) or of Su et al. (Biochemistry. 1994, 33, 15501–15510), presumably because the simulations are not long enough. We find, however, that helicity correlates positively with the number of close hydrophobic interactions between the two peptides, showing that stable coiled coils in the simulations are tightly packed. The minimum interhelical distances are 0.50–0.66 nm for charged groups, indicating that favorable charge interactions are also important for the stability of the coiled coil.     

Evolutionary dynamics of taxonomic structure

The distribution of species among genera and higher taxa has largely untapped potential to reveal among-clade variation in rates of origination and extinction. The probability distribution of the number of species within a genus is modelled with a stochastic, time-homogeneous birth-death model having two parameters: the rate of species extinction, μ, and the rate of genus origination, γ, each scaled as a multiple of the rate of within-genus speciation, λ. The distribution is more sensitive to γ than to μ, although μ affects the size of the largest genera. The species : genus ratio depends strongly on both γ and μ, and so is not a good diagnostic of evolutionary dynamics. The proportion of monotypic genera, however, depends mainly on γ, and so may provide an index of the genus origination rate. Application to living marine molluscs of New Zealand shows that bivalves have a higher relative rate of genus origination than gastropods. This is supported by the analysis of palaeontological data. This concordance suggests that analysis of living taxonomic distributions may allow inference of macroevolutionary dynamics even without a fossil record.     

Ecological stability,evolutionary stability and the ESS maximum principle

Summary Since the fitness of each individual organism in a biological community may be affected by the strategies of all other individuals in the community, the essential element of a game exists. This game is an evolutionary game where the individual organisms (players) inherit their strategies from continuous play of the game through time. Here, the strategies are assumed to be constants associated with certain adaptive parameters (such as sunlight conversion efficiency for plants or body length in animals) in a set of differential equations which describe the population dynamics of the community. By means of natural selection, these parameters will evolve to a set of strategy values that natural selection, by itself, can no longer modify, i.e. an evolutionarily stable strategy (ESS). For a given class of models, it is possible to predict the outcome of this evolutionary process by determining ESSs using an ESS maximum principle. However, heretofore, the proof of this principle has been based on a limited set of conditions. Herein, we generalize the proof by removing certain restrictions and use instead the concept of an ecological stable equilibrium (ESE). Individuals in a biological community will be at an ESE if fixing the strategies used by the individuals results in stable population densities subject to perturbations in those densities. We present both necessary and sufficient conditions for an ESE to exist and then use the ESE concept to provide a very simple proof of the ESS maximum principle (which is a necessary condition for an ESS). A simple example is used to illustrate the difference between a strategy that maximizes fitness and one that satisfies the ESS maximum principle. In general they are different. We also look for ESEs in Lotka—Volterra competition and use the maximum principle to determine when an ESE will be an ESS. Finally, we examine the applicability of these ideas to matrix games.     

Computational fluid dynamics simulation to evaluate aortic coarctation gradient with contrast-enhanced CT

Coarctation of aorta (CoA) is a narrowing of the aorta leading to a pressure gradient (ΔP) across the coarctation, increased afterload and reduced peripheral perfusion pressures. Indication to invasive treatment is based on values of maximal (systolic) trans-coarctation ΔP. A computational fluid dynamic (CFD) approach is herein presented for the non-invasive haemodynamic assessment of ΔP across CoA. Patient-specific CFD simulations were created from contrast-enhanced computed tomography (CT) and appropriate flow boundary conditions. Computed ΔP was validated with invasive intravascular trans-CoA pressure measurements. Haemodynamic indices, including pressure loss coefficient (PLc), time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI), were also quantified. CFD-estimated ΔP values were comparable to the invasive ones. Moreover, the aorta proximal to CoA was exposed to altered TAWSS and OSI suggesting hypertension. PLc was found as a further geometric marker of CoA severity. Finally, CFD-estimated ΔP confirmed a significant reduction after percutaneous balloon dilatation and stenting of the CoA in one patient (e.g. from ΔP～52 mmHg to ΔP～3 mmHg). The validation of the ΔP computations with catheterisation measurements suggests that CFD simulation, based on CT-derived anatomical data, is a useful tool to readily quantify CoA severity.     

Structure,dynamics and thermodynamic stability of high-pressure ices and clathrate hydrates

We present the recent development of simulation studies on structure and dynamics of high-pressure ices and filled ices. After surveying the representative structures of ices, focus is placed on some properties of one of the ice polymorphs, plastic ice, which has been theoretically predicted but not yet been found experimentally. Its intervention between ice VII and liquid water enables to account for large discrepancies among various experimental melting curves of ice VII. We also examine the dynamical properties of the filled ice in which hydrogen molecules are contained in void space of the low-pressure cubic ice. In contrast to the plastic ice, which has a bearing on ice VII, it exhibits a gradual change to each rotator phase of guest hydrogen and/or host water molecules.