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.     

Invasion fitness,inclusive fitness,and reproductive numbers in heterogeneous populations

How should fitness be measured to determine which phenotype or “strategy” is uninvadable when evolution occurs in a group‐structured population subject to local demographic and environmental heterogeneity? Several fitness measures, such as basic reproductive number, lifetime dispersal success of a local lineage, or inclusive fitness have been proposed to address this question, but the relationships between them and their generality remains unclear. Here, we ascertain uninvadability (all mutant strategies always go extinct) in terms of the asymptotic per capita number of mutant copies produced by a mutant lineage arising as a single copy in a resident population (“invasion fitness”). We show that from invasion fitness uninvadability is equivalently characterized by at least three conceptually distinct fitness measures: (i) lineage fitness, giving the average individual fitness of a randomly sampled mutant lineage member; (ii) inclusive fitness, giving a reproductive value weighted average of the direct fitness costs and relatedness weighted indirect fitness benefits accruing to a randomly sampled mutant lineage member; and (iii) basic reproductive number (and variations thereof) giving lifetime success of a lineage in a single group, and which is an invasion fitness proxy. Our analysis connects approaches that have been deemed different, generalizes the exact version of inclusive fitness to class‐structured populations, and provides a biological interpretation of natural selection on a mutant allele under arbitrary strength of selection.     

Rapid evolutionary escape by large populations from local fitness peaks is likely in nature

Fitness interactions between loci in the genome, or epistasis, can result in mutations that are individually deleterious but jointly beneficial. Such epistasis gives rise to multiple peaks on the genotypic fitness landscape. The problem of evolutionary escape from such local peaks has been a central problem of evolutionary genetics for at least 75 years. Much attention has focused on models of small populations, in which the sequential fixation of valley genotypes carrying individually deleterious mutations operates most quickly owing to genetic drift. However, valley genotypes can also be subject to mutation while transiently segregating, giving rise to copies of the high fitness escape genotype carrying the jointly beneficial mutations. In the absence of genetic recombination, these mutations may then fix simultaneously. The time for this process declines sharply with increasing population size, and it eventually comes to dominate evolutionary behavior. Here we develop an analytic expression for N(crit), the critical population size that defines the boundary between these regimes, which shows that both are likely to operate in nature. Frequent recombination may disrupt high-fitness escape genotypes produced in populations larger than N(crit) before they reach fixation, defining a third regime whose rate again slows with increasing population size. We develop a novel expression for this critical recombination rate, which shows that in large populations the simultaneous fixation of mutations that are beneficial only jointly is unlikely to be disrupted by genetic recombination if their map distance is on the order of the size of single genes. Thus, counterintuitively, mass selection alone offers a biologically realistic resolution to the problem of evolutionary escape from local fitness peaks in natural populations.     

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.     

The evolutionary dynamics of direct phenotypic overdominance: emergence possible, loss probable

An evolutionary dynamical system with explicit diploid genetics is used to investigate the likelihood of observing phenotypically overdominant heterozygotes versus heterozygous phenotypes that are intermediate between the homozygotes. In this model, body size evolves in a population with discrete demographic episodes and with competition limiting reproduction. A genotype-phenotype map for body size is used that can generate the two qualitative types of dominance interactions (overdominance versus intermediate dominance). It is written as a single-locus model with one focal locus and parameters summarizing the effects of alleles at other loci. Two types of evolutionarily stable strategy (ESS; continuously stable strategy, CSS) occur. The ESS is generated either (1) by the population ecology; or (2) by a local maximum of the genotype-phenotype map. Overdominant heterozygotes are expected to arise if the population evolves toward the second type of ESS, where nearly maximum body sizes are found. When other loci with partially dominant inheritance also evolve, the location of the maximum in the genotype-phenotype map repeatedly changes. It is unlikely that an evolving population will track these changes; ESSs of the second type now are at best quasi-stationary states of the evolutionary dynamics. Considering the restrictions on its probability, a pattern of phenotypic overdominance is expected to be rare.     

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.     

Hamilton's rule,inclusive fitness maximization,and the goal of individual behaviour in symmetric two‐player games

Hamilton's original work on inclusive fitness theory assumed additivity of costs and benefits. Recently, it has been argued that an exact version of Hamilton's rule for the spread of a pro‐social allele (rb > c) holds under nonadditive pay‐offs, so long as the cost and benefit terms are defined as partial regression coefficients rather than pay‐off parameters. This article examines whether one of the key components of Hamilton's original theory can be preserved when the rule is generalized to the nonadditive case in this way, namely that evolved organisms will behave as if trying to maximize their inclusive fitness in social encounters.     

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.     

Regression,least squares,and the general version of inclusive fitness

A general version of inclusive fitness based on regression is rederived with minimal mathematics and directly from the verbal interpretation of its terms that motivated the original formulation of the inclusive fitness concept. This verbal interpretation is here extended to provide the two relationships required to determine the two coefficients and b. These coefficients retain their definition as expected effects on the fitness of an individual, respectively of a change in allelic state of this individual, and of correlated change in allelic state of social partners. The known least‐squares formulation of the relationships determining b and c can be immediately deduced and shown to be equivalent to this new formulation. These results make clear that criticisms of the mathematical tools (in particular least‐squares regression) previously used to derive this version of inclusive fitness are misdirected.     

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.     

The fitness effect of mutations across environments: a survey in light of fitness landscape models

The fitness effects of mutations on a given genotype are rarely constant across environments to which this genotype is more or less adapted, that is, between more or less stressful conditions. This can have important implications, especially on the evolution of ecological specialization. Stress is thought to increase the variance of mutations' fitness effects, their average, or the number of expressed mutations. Although empirical evidence is available for these three mechanisms, their relative magnitude is poorly understood. In this paper, we propose a simple approach to discriminate between these mechanisms, using a survey of empirical measures of mutation effects in contrasted environments. This survey, across various species and environments, shows that stress mainly increases the variance of mutations' effects on fitness, with a much more limited impact on their average effect or on the number of expressed mutations. This pattern is consistent with a simple model in which fitness is a Gaussian function of phenotypes around an environmentally determined optimum. These results suggest that a simple, mathematically tractable landscape model may not be quantitatively as unrealistic as previously suggested. They also suggest that mutation parameter estimates may be strongly biased when measured in stressful environments.     

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.     

Modelling population dynamics in a unicellular social organism community using a minimal model and evolutionary game theory

Most unicellular organisms live in communities and express different phenotypes. Many efforts have been made to study the population dynamics of such complex communities of cells, coexisting as well-coordinated units. Minimal models based on ordinary differential equations are powerful tools that can help us understand complex phenomena. They represent an appropriate compromise between complexity and tractability; they allow a profound and comprehensive analysis, which is still easy to understand. Evolutionary game theory is another powerful tool that can help us understand the costs and benefits of the decision a particular cell of a unicellular social organism takes when faced with the challenges of the biotic and abiotic environment. This work is a binocular view at the population dynamics of such a community through the objectives of minimal modelling and evolutionary game theory. We test the behaviour of the community of a unicellular social organism at three levels of antibiotic stress. Even in the absence of the antibiotic, spikes in the fraction of resistant cells can be observed indicating the importance of bet hedging. At moderate level of antibiotic stress, we witness cyclic dynamics reminiscent of the renowned rock–paper–scissors game. At a very high level, the resistant type of strategy is the most favourable.