Canonical Equation Of Adaptive Dynamics Research Articles

Evolutionary dynamics from deterministic microscopic ecological processes.

The central goal of a dynamical theory of evolution is to abstract the mean evolutionary trajectory in the trait space by considering ecological processes at the level of the individual. In this work we develop such a theory for a class of deterministic individual-based models describing individual births and deaths, which captures the essential features of standard stochastic individual-based models and becomes identical to the latter under maximal competition. The key motivation is to derive the canonical equation of adaptive dynamics from this microscopic ecological model, which can be regarded as a paradigm to study evolution in a simple way and give it an intuitive geometric interpretation. Another goal is to study evolution and sympatric speciation under maximal competition. We show that these models, in the deterministic limit of adaptive dynamics, lead to the same equations that describe the unraveling of the mean evolutionary trajectory as those obtained from the standard stochastic models. We further study conditions under which these models lead to evolutionary branching and find them to be similar to those obtained from the standard stochastic models. We find that, although deterministic models result in a strong competition that leads to a speedup in the temporal dynamics of a population cloud in the phenotypic space as well as an increase in the rate of generation of biodiversity, they do not seem to result in an absolute increase in biodiversity as far as the total number of species is concerned. Hence, they essentially capture all the features of the standard stochastic model. Interestingly, the notion of a fitness function does not explicitly enter in our derivation of the canonical equation, thereby advocating a mechanistic view of evolution based on fundamental birth-death events where fitness is a derived quantity rather than a fundamental ingredient. We illustrate our work with the help of several examples and qualitatively compare the rate of unraveling of evolutionary trajectory and generation of biodiversity for the deterministic and standard individual-based models by showing the motion of population clouds in the trait space.

A birth-death model of ageing: from individual-based dynamics to evolutive differential inclusions.

Ageing's sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual's fitness. Thanks to the recently described (Smurf) 2-phase model of ageing (Tricoire and Rera in PLoS ONE 10(11):e0141920, 2015) we propose a fresh angle for modeling the evolution of ageing. Indeed, by coupling a dramatic loss of fertility with a high-risk of impending death-amongst other multiple so-called hallmarks of ageing-the Smurf phenotype allowed us to consider ageing as a couple of sharp transitions. The birth-death model (later called bd-model) we describe here is a simple life-history trait model where each asexual and haploid individual is described by its fertility period [Formula: see text] and survival period [Formula: see text]. We show that, thanks to the Lansing effect, the effect through which the "progeny of old parents do not live as long as those of young parents", [Formula: see text] and [Formula: see text] converge during evolution to configurations [Formula: see text] in finite time. To do so, we built an individual-based stochastic model which describes the age and trait distribution dynamics of such a finite population. Then we rigorously derive the adaptive dynamics models, which describe the trait dynamics at the evolutionary time-scale. We extend the Trait Substitution Sequence with age structure to take into account the Lansing effect. Finally, we study the limiting behaviour of this jump process when mutations are small. We show that the limiting behaviour is described by a differential inclusion whose solutions [Formula: see text] reach the diagonal [Formula: see text] in finite time and then remain on it. This differential inclusion is a natural way to extend the canonical equation of adaptive dynamics in order to take into account the lack of regularity of the invasion fitness function on the diagonal [Formula: see text].

The Price Equation, Gradient Dynamics, and Continuous Trait Game Theory.

A recent article convincingly nominated the Price equation as the fundamental theorem of evolution and used it as a foundation to derive several other theorems. A major section of evolutionary theory that was not addressed is that of game theory and gradient dynamics of continuous traits with frequency-dependent fitness. Deriving fundamental results in these fields under the unifying framework of the Price equation illuminates similarities and differences between approaches and allows a simple, unified view of game-theoretical and dynamic concepts. Using Taylor polynomials and the Price equation, I derive a dynamic measure of evolutionary change, a condition for singular points, the convergence stability criterion, and an alternative interpretation of evolutionary stability. Furthermore, by applying the Price equation to a multivariable Taylor polynomial, the direct fitness approach to kin selection emerges. Finally, I compare these results to the mean gradient equation of quantitative genetics and the canonical equation of adaptive dynamics.

From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step

We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population ($K\rightarrow\infty$) size, rare mutations ($u\rightarrow 0$), and small mutational effects ($\sigma\rightarrow 0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, e.g. by Champagnat and Meleard, we take the three limits simultaneously, i.e. $u=u_K$ and $\sigma=\sigma_K$, tend to zero with $K$, subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that requires the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the system with the corresponding deterministic one. To solve this problem we develop a stochastic Euler scheme based on coupling arguments that allows to control the time evolution of the system over time-scales that diverge with $K$.

The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin's maximum principle.

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin’s maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.

The Branching Bifurcation of Adaptive Dynamics

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits — phenotypes or strategies — of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present — resident — types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits — the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching — the joint effect of resident-mutant coexistence and evolutionary instability — have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra- as well as inter-specific ecological interactions are described in the vicinity of a stationary regime.

Stochastic eco-evolutionary model of a prey-predator community

We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the polymorphic evolutionary sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these different limits with an example of prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.

Social evolution and genetic interactions in the short and long term

The evolution of social traits remains one of the most fascinating and feisty topics in evolutionary biology even after half a century of theoretical research. W.D. Hamilton shaped much of the field initially with his 1964 papers that laid out the foundation for understanding the effect of genetic relatedness on the evolution of social behavior. Early theoretical investigations revealed two critical assumptions required for Hamilton’s rule to hold in dynamical models: weak selection and additive genetic interactions. However, only recently have analytical approaches from population genetics and evolutionary game theory developed sufficiently so that social evolution can be studied under the joint action of selection, mutation, and genetic drift. We review how these approaches suggest two timescales for evolution under weak mutation: (i) a short-term timescale where evolution occurs between a finite set of alleles, and (ii) a long-term timescale where a continuum of alleles are possible and populations evolve continuously from one monomorphic trait to another. We show how Hamilton’s rule emerges from the short-term analysis under additivity and how non-additive genetic interactions can be accounted for more generally. This short-term approach reproduces, synthesizes, and generalizes many previous results including the one-third law from evolutionary game theory and risk dominance from economic game theory. Using the long-term approach, we illustrate how trait evolution can be described with a diffusion equation that is a stochastic analogue of the canonical equation of adaptive dynamics. Peaks in the stationary distribution of the diffusion capture classic notions of convergence stability from evolutionary game theory and generally depend on the additive genetic interactions inherent in Hamilton’s rule. Surprisingly, the peaks of the long-term stationary distribution can predict the effects of simple kinds of non-additive interactions. Additionally, the peaks capture both weak and strong effects of social payoffs in a manner difficult to replicate with the short-term approach. Together, the results from the short and long-term approaches suggest both how Hamilton’s insight may be robust in unexpected ways and how current analytical approaches can expand our understanding of social evolution far beyond Hamilton’s original work.

The canonical equation of adaptive dynamics for Mendelian diploids and haplo-diploids

One of the powerful tools of adaptive dynamics is its so-called canonical equation (CE), a differential equation describing how the prevailing trait vector changes over evolutionary time. The derivation of the CE is based on two simplifying assumptions, separation of population dynamical and mutational time scales and small mutational steps. (It may appear that these two conditions rarely go together. However, for small step sizes the time-scale separation need not be very strict.) The CE was derived in 1996, with mathematical rigour being added in 2003. Both papers consider only well-mixed clonal populations with the simplest possible life histories. In 2008, the CE's reach was heuristically extended to locally well-mixed populations with general life histories. We, again heuristically, extend it further to Mendelian diploids and haplo-diploids. Away from strict time-scale separation the CE does an even better approximation job in the Mendelian than in the clonal case owing to gene substitutions occurring effectively in parallel, which obviates slowing down by clonal interference.

Adaptation in a stochastic multi-resources chemostat model

We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (Lotka–Volterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multi-resources chemostat model, where the competition between individuals results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states (the polymorphic evolution sequence of the adaptive dynamics literature). An essential technical ingredient is the study of the long time behavior of a chemostat multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we prove a rigorous characterization of the evolutionary branching points.

Adaptive Dynamics with Interaction Structure

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the canonical equation of adaptive dynamics with interaction structure, characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.

MUTANT INVASIONS AND ADAPTIVE DYNAMICS IN VARIABLE ENVIRONMENTS

The evolution of natural organisms is ultimately driven by the invasion and possible fixation of mutant alleles. The invasion process is highly stochastic, however, and the probability of success is generally low, even for advantageous alleles. Additionally, all organisms live in a stochastic environment, which may have a large influence on what alleles are favorable, but also contributes to the uncertainty of the invasion process. We calculate the invasion probability of a beneficial, mutant allele in a monomorphic, large population subject to stochastic environmental fluctuations, taking into account density- and frequency-dependent selection, stochastic population dynamics and temporal autocorrelation of the environment. We treat both discrete and continuous time population dynamics, and allow for overlapping generations in the continuous time case. The results can be generalized to diploid, sexually reproducing organisms embedded in communities of interacting species. We further use these results to derive an extended canonical equation of adaptive dynamics, predicting the rate of evolutionary change of a heritable trait on long evolutionary time scales.

Symmetric competition as a general model for single-species adaptive dynamics

Adaptive dynamics is a widely used framework for modeling long-term evolution of continuous phenotypes. It is based on invasion fitness functions, which determine selection gradients and the canonical equation of adaptive dynamics. Even though the derivation of the adaptive dynamics from a given invasion fitness function is general and model-independent, the derivation of the invasion fitness function itself requires specification of an underlying ecological model. Therefore, evolutionary insights gained from adaptive dynamics models are generally model-dependent. Logistic models for symmetric, frequency-dependent competition are widely used in this context. Such models have the property that the selection gradients derived from them are gradients of scalar functions, which reflects a certain gradient property of the corresponding invasion fitness function. We show that any adaptive dynamics model that is based on an invasion fitness functions with this gradient property can be transformed into a generalized symmetric competition model. This provides a precise delineation of the generality of results derived from competition models. Roughly speaking, to understand the adaptive dynamics of the class of models satisfying a certain gradient condition, one only needs a complete understanding of the adaptive dynamics of symmetric, frequency-dependent competition. We show how this result can be applied to number of basic issues in evolutionary theory.

Cooperation and evolutionary dynamics in the public goods game with institutional incentives

The one-shot public goods game is extended to include institutional incentives (i.e. reward and/or punishment) that are meant to promote cooperation. It is shown that the Nash equilibrium (NE) outcomes predict either partial or fully cooperative behavior in these extended multi-player games with a continuous strategy space. Furthermore, for some incentive schemes, multiple NE outcomes are shown to emerge. Stability of all these equilibria under standard evolutionary dynamics (i.e. the replicator equation and the canonical equation of adaptive dynamics) is characterized.

EVOLUTIONARY DYNAMICS CAN BE CHAOTIC: A FIRST EXAMPLE

We present in this paper the first example of chaotic evolutionary dynamics in biology. We consider a Lotka–Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive phenotypic trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories approach a strange attractor in the three-dimensional trait space. The study is performed through the bifurcation analysis of the so-called canonical equation of Adaptive Dynamics, the most appropriate modeling approach to long-term evolutionary dynamics.

CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces

Static continuously stable strategy (CSS) and neighborhood invader strategy (NIS) conditions are developed for two-species models of frequency-dependent behavioral evolution when individuals have traits in continuous strategy spaces. These are intuitive stability conditions that predict the eventual outcome of evolution from a dynamic perspective. It is shown how the CSS is related to convergence stability for the canonical equation of adaptive dynamics and the NIS to convergence to a monomorphism for the replicator equation of evolutionary game theory. The CSS and NIS are also shown to be special cases of neighborhood p * - superiority for p * equal to one half and zero, respectively. The theory is illustrated when each species has a one-dimensional trait space.

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

Evolution of discrete populations and the canonical diffusion of adaptive dynamics

The biological theory of adaptive dynamics proposes a description of the long-term evolution of a structured asexual population. It is based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE describing the evolution of the dominant type, called the 'canonical equation of adaptive dynamics'. Here, in order to include the effect of stochasticity (genetic drift), we consider self-regulated randomly fluctuating populations subject to mutation, so that the number of coexisting types may fluctuate. We apply a limit of rare mutations to these populations, while keeping the population size finite. This leads to a jump process, the so-called 'trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. Then we apply a limit of small mutation steps (weak selection) to this jump process, that leads to a diffusion process that we call the 'canonical diffusion of adaptive dynamics', in which genetic drift is combined with directional selection driven by the gradient of the fixation probability, also interpreted as an invasion fitness. Finally, we study in detail the particular case of multitype logistic branching populations and seek explicit formulae for the invasion fitness of a mutant deviating slightly from the resident type. In particular, second-order terms of the fixation probability are products of functions of the initial mutant frequency, times functions of the initial total population size, called the invasibility coefficients of the resident by increased fertility, defence, aggressiveness, isolation, or survival.

Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models

A distinctive signature of living systems is Darwinian evolution, that is, a propensity to generate as well as self-select individual diversity. To capture this essential feature of life while describing the dynamics of populations, mathematical models must be rooted in the microscopic, stochastic description of discrete individuals characterized by one or several adaptive traits and interacting with each other. The simplest models assume asexual reproduction and haploid genetics: an offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take a mutation step to new trait values; selection follows from ecological interactions among individuals. Here we present a rigorous construction of the microscopic population process that captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by the trait values of each individual, and interactions between individuals. A by-product of this formal construction is a general algorithm for efficient numerical simulation of the individual-level model. Once the microscopic process is in place, we derive different macroscopic models of adaptive evolution. These models differ in the renormalization they assume, i.e. in the limits taken, in specific orders, on population size, mutation rate, mutation step, while rescaling time accordingly. The macroscopic models also differ in their mathematical nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. These models include extensions of Kimura's equation (and of its approximation for small mutation effects) to frequency- and density-dependent selection. A novel class of macroscopic models obtains when assuming that individual birth and death occur on a short timescale compared with the timescale of typical population growth. On a timescale of very rare mutations, we establish rigorously the models of “trait substitution sequences” and their approximation known as the “canonical equation of adaptive dynamics”. We extend these models to account for mutation bias and random drift between multiple evolutionary attractors. The renormalization approach used in this study also opens promising avenues to study and predict patterns of life-history allometries, thereby bridging individual physiology, genetic variation, and ecological interactions in a common evolutionary framework.

The adaptive dynamics of function-valued traits

This study extends the framework of adaptive dynamics to function-valued traits. Such adaptive traits naturally arise in a great variety of settings: variable or heterogeneous environments, age-structured populations, phenotypic plasticity, patterns of growth and form, resource gradients, and in many other areas of evolutionary ecology. Adaptive dynamics theory allows analysing the long-term evolution of such traits under the density-dependent and frequency-dependent selection pressures resulting from feedback between evolving populations and their ecological environment. Starting from individual-based considerations, we derive equations describing the expected dynamics of a function-valued trait in asexually reproducing populations under mutation-limited evolution, thus generalizing the canonical equation of adaptive dynamics to function-valued traits. We explain in detail how to account for various kinds of evolutionary constraints on the adaptive dynamics of function-valued traits. To illustrate the utility of our approach, we present applications to two specific examples that address, respectively, the evolution of metabolic investment strategies along resource gradients, and the evolution of seasonal flowering schedules in temporally varying environments.