Darwinian fitness

The term Darwinian fitness refers to the capacity of a variant type to invade and displace the resident population in competition for available resources. Classical models of this dynamical process claim that competitive outcome is a deterministic event which is regulated by the population growth rate, called the Malthusian parameter. Recent analytic studies of the dynamics of competition in terms of diffusion processes show that growth rate predicts invasion success only in populations of infinite size. In populations of finite size, competitive outcome is a stochastic process--contingent on resource constraints--which is determined by the rate at which a population returns to its steady state condition after a random perturbation in the individual birth and death rates. This return rate, a measure of robustness or population stability, is analytically characterized by the demographic parameter, evolutionary entropy, a measure of the uncertainty in the age of the mother of a randomly chosen newborn. This article appeals to computational and numerical methods to contrast the predictive power of the Malthusian and the entropic principles. The computational analysis rejects the Malthusian model and is consistent with of the entropic principle. These studies thus provide support for the general claim that entropy is the appropriate measure of Darwinian fitness and constitutes an evolutionary parameter with broad predictive and explanatory powers.     

Darwinian fitness and the intensity of natural selection: studies in sensitivity analysis

Darwinian fitness, the capacity of a variant type to establish itself in competition with the resident population, is determined by evolutionary entropy, a measure of the uncertainty in age of the mother of a randomly chosen newborn. This article shows that the intensity of natural selection, as measured by the sensitivity of entropy with respect to changes in the age-specific fecundity and mortality variables, is a convex function of age, decreasing at early and increasing at later ages. We exploit this result to provide quantitative evolutionary explanations of the large variation in survivorship curves observed in natural populations. Previous studies to explain variation in survivorship curves have been based on the proposition that Darwinian fitness is determined by the Malthusian parameter. Hence the intensity of natural selection will be determined by the sensitivity of the Malthusian parameter with respect to changes in the age-specific fecundity and mortality variables. This measure of the selection gradient is known to be a decreasing function of age, with implications which are inconsistent with empirical observations of survivorship curves in human and animal populations. The analysis described in this paper point to the mitigated import of sensitivity studies based on the Malthusian parameter. Our analysis provides theoretical and empirical support for the ecological and evolutionary significance of sensitivity analysis based on entropy, which is the appropriate measure of Darwinian fitness.     

Mutation in evolutionary games can increase average fitness at equilibrium

We study game dynamical interactions between two strategies, A and B, and analyse whether the average fitness of the population at equilibrium can be increased by adding mutation from A to B. Classifying all two by two games with payoff matrix [(a,b),(c,d)], we show that mutation from A to B enhances the average fitness of the whole population (i) if both a and d are less than (b + c)/2 and (ii) if c is less than b. Furthermore, we study conditions for maximizing the productivity of strategy A, and we analyse the effect of mutations in both directions. Depending on the biological system, a mutation in an evolutionary game can be interpreted as a genetic alteration, a cellular differentiation, a change in gene expression, an accidental or deliberate modification in cultural transmission, or a learning error. In a cultural context, our results indicate that the equilibrium payoff of the population can be increased if players sometimes choose the strategy with lower payoff. In a genetic context, we have shown that for frequency-dependent selection mutation can enhance the average fitness of the population at equilibrium.     

Pattern formation and chaos in spatial ecological public goodsgames

Cooperators and defectors can coexist in ecological public goods games. When the game is played in two-dimensional continuous space, a reaction diffusion model produces highly irregular dynamics, in which cooperators and defectors survive in ever-changing configurations (Wakano et al., 2009. Spatial dynamics of ecological public goods. Proc. Natl. Acad. Sci. 106, 7910-7914). The dynamics is related to the formation of Turing patterns, but the origin of the irregular dynamics is not well understood. In this paper, we present a classification of the spatio-temporal dynamics based on the dispersion relation, which reveals that the spontaneous pattern formation can be attributed to the dynamical interplay between two linearly unstable modes: temporal instability arising from a Hopf-bifurcation and spatial instability arising from a Turing-bifurcation. Moreover, we provide a detailed analysis of the highly irregular dynamics through Fourier analysis, the break-down of symmetry, the maximum Lyapunov exponent, and the excitability of the reaction-term dynamics. All results clearly support that the observed irregular dynamics qualifies as spatio-temporal chaos. A particularly interesting type of chaotic dynamics, which we call intermittent bursts, clearly demonstrates the effects of the two unstable modes where (local) periods of stasis alternate with rapid changes that may induce local extinction.     

How can we model selectively neutral density dependence in evolutionary games

The problem of density dependence appears in all approaches to the modelling of population dynamics. It is pertinent to classic models (i.e., Lotka-Volterra's), and also population genetics and game theoretical models related to the replicator dynamics. There is no density dependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the question arises: How is unlimited population growth suppressed in frequency-dependent models? Two categories of solutions can be found in the literature. In the first, replicator dynamics is independent of background fitness. In the second type of solution, a multiplicative suppression coefficient is used, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory and basic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reaches the maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitly consider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium value of population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectories is the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and then remains constant and dependent on the rate of turn-over of individuals.     

Why Darwin would have loved evolutionary game theory

Humans have marvelled at the fit of form and function, the way organisms'' traits seem remarkably suited to their lifestyles and ecologies. While natural selection provides the scientific basis for the fit of form and function, Darwin found certain adaptations vexing or particularly intriguing: sex ratios, sexual selection and altruism. The logic behind these adaptations resides in frequency-dependent selection where the value of a given heritable phenotype (i.e. strategy) to an individual depends upon the strategies of others. Game theory is a branch of mathematics that is uniquely suited to solving such puzzles. While game theoretic thinking enters into Darwin''s arguments and those of evolutionists through much of the twentieth century, the tools of evolutionary game theory were not available to Darwin or most evolutionists until the 1970s, and its full scope has only unfolded in the last three decades. As a consequence, game theory is applied and appreciated rather spottily. Game theory not only applies to matrix games and social games, it also applies to speciation, macroevolution and perhaps even to cancer. I assert that life and natural selection are a game, and that game theory is the appropriate logic for framing and understanding adaptations. Its scope can include behaviours within species, state-dependent strategies (such as male, female and so much more), speciation and coevolution, and expands beyond microevolution to macroevolution. Game theory clarifies aspects of ecological and evolutionary stability in ways useful to understanding eco-evolutionary dynamics, niche construction and ecosystem engineering. In short, I would like to think that Darwin would have found game theory uniquely useful for his theory of natural selection. Let us see why this is so.     

The one-third law of evolutionary dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a "one-third law" of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency 13. This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the "number"13 is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.     

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646-650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency , strategy α fixates in a β-population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.     

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes—the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.     

Ecology and evolution of symbiosis in metapopulations

We present a model for symbionts in plant host metapopulation. Symbionts are assumed not only to form a systemic infection throughout the host and pass into the host seeds, but also to reproduce and infect new plants by spores. Thus, we study a metapopulation of qualitatively identical patches coupled through seeds and spores dispersal. Symbionts that are only vertically inherited cannot persist in such a uniform environment if they lower the host's fitness. They have to be beneficial in order to coexist with the host if they are not perfectly transmitted to the seeds; but evolution selects for 100% fidelity of infection inheritance. In this model we want to see how mixed strategies (both vertical and horizontal infection) affect the coexistence of uninfected and infected plants at equilibrium; also, what would evolution do for the host, for the symbionts and for their association. We present a detailed classification of the possible equilibria with examples. The stability of the steady states is rigorously proved for the first time in a metapopulation set-up.     

Active linking in evolutionary games

In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at which they seek new interactions. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In a limiting case, the linking dynamics introduces a simple transformation of the payoff matrix. We outline conditions for evolutionary stability. As a specific example, we study the interaction between cooperators and defectors. We find a simple relationship that characterizes those linking dynamics which allow natural selection to favour cooperation over defection.     

An evolutionary analysis of the relationship between spite and altruism

We investigate the selective pressures on a social trait when evolution occurs in a population of constant size. We show that any social trait that is spiteful simultaneously qualifies as altruistic. In other words, any trait that reduces the fitness of less related individuals necessarily increases that of related ones. Our analysis demonstrates that the distinction between Hamiltonian spite and Wilsonian spite is not justified on the basis of fitness effects. We illustrate this general result with an explicit model for the evolution of a social act that reduces the recipient's survival (harming trait). This model shows that the evolution of harming is favoured if local demes are of small size and migration is low (philopatry). Further, deme size and migration rate determine whether harming evolves as a selfish strategy by increasing the fitness of the actor, or as a spiteful/altruistic strategy through its positive effect on the fitness of close kin.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Evolutionary jumping and breakthrough in tree masting evolution

Many long-lived plants such as trees show masting or intermittent and synchronized reproduction. In a coupled chaos system describing the dynamics of individual-plant resource budgets, masting occurs when the resource depletion coefficient k (ratio of the reproductive expenditure to the excess resource reserve) is large. Here, we mathematically studied the condition for masting evolution. In an infinitely large population, we obtained a deterministic dynamical system, to which we applied the pairwise invasibility plot and convergence stability of evolutionary singularity analyses. We prove that plants reproducing at the same rate every year are not evolutionarily stable. The resource depletion coefficient k increases, and the system oscillates with a period of 2 years (high and low reproduction) if k<1. Alternatively, k may evolve further and jump to a value >1, resulting in the sudden start of intermittent reproduction. We confirm that a high survivorship of young plants (seedlings) in the light-limited understory favors masting evolution, as previously suggested by computer simulations and field observations. The stochasticity caused by the finiteness of population size also promotes masting evolution.     

Evolutionary dynamics of finite populations in games with polymorphic fitness equilibria

The hawk-dove (HD) game, as defined by Maynard Smith [1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge], allows for a polymorphic fitness equilibrium (PFE) to exist between its two pure strategies; this polymorphism is the attractor of the standard replicator dynamics [Taylor, P.D., Jonker, L., 1978. Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145-156; Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge] operating on an infinite population of pure-strategists. Here, we consider stochastic replicator dynamics, operating on a finite population of pure-strategists playing games similar to HD; in particular, we examine the transient behavior of the system, before it enters an absorbing state due to sampling error. Though stochastic replication prevents the population from fixing onto the PFE, selection always favors the under-represented strategy. Thus, we may naively expect that the mean population state (of the pre-absorption transient) will correspond to the PFE. The empirical results of Fogel et al. [1997. On the instability of evolutionary stable states. BioSystems 44, 135-152] show that the mean population state, in fact, deviates from the PFE with statistical significance. We provide theoretical results that explain their observations. We show that such deviation away from the PFE occurs when the selection pressures that surround the fitness-equilibrium point are asymmetric. Further, we analyze a Markov model to prove that a finite population will generate a distribution over population states that equilibrates selection-pressure asymmetry; the mean of this distribution is generally not the fitness-equilibrium state.     

Chaotic dynamics of the surface potential of human skeletal muscles determined in electromyography

A new method for differential evaluation of electromyographic data on straited muscles of human lower extremities was developed. This method is based on nonlinear dynamics and thermodynamics and can be used for identification of pathologies. The distance between two trajectories of the potential of two symmetric muscles was the main measured characteristic of coordinated muscle work. These data were used to determine the Lyapunov exponent and the time of forgetting initial conditions, which reflect the generally chaotic dynamics of muscle activity. Application of the theory of deterministic chaos to analysis of electromyographic patterns can improve the diagnosis of peripheral nervous system diseases and the efficacy of treatment control. Quantitation of nonlinear dynamic parameters of muscle activity, clear data representation, high prognostic information content of the Lyapunov exponent and Kolmogorov entropy are among the advantages of the new method.     

Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction

It has recently been demonstrated that ecological feedback mechanisms can facilitate the emergence and maintenance of cooperation in public goods interactions: the replicator dynamics of defectors and cooperators can result, for example, in the ecological coexistence of cooperators and defectors. Here we show that these results change dramatically if cooperation strategy is not fixed but instead is a continuously varying trait under natural selection. For low values of the factor with which the value of resources is multiplied before they are shared among all participants, evolution will always favour lower cooperation strategies until the population falls below an Allee threshold and goes extinct, thus evolutionary suicide occurs. For higher values of the factor, there exists a unique evolutionarily singular strategy, which is convergence stable. Because the fitness function is linear with respect to the strategy of the mutant, this singular strategy is neutral against mutant invasions. This neutrality disappears if a nonlinear functional response in receiving benefits is assumed. For strictly concave functional responses, singular strategies become uninvadable. Evolutionary branching, which could result in the evolutionary emergence of cooperators and defectors, can occur only with locally convex functional responses, but we illustrate that it can also result in coevolutionary extinction.