Adaptive Dynamics Approach Research Articles

Metamicrobiome diversity promotes the evolution of host-microbial mutualisms.

Ecological theory suggests that a host organism's internal spatial structure can promote the persistence of mutualistic microbes by allowing for the turnover of tissue occupied by non-beneficial or cheating microbes. This type of regulation, whereby a host preferentially rewards tissue occupied by beneficial members of its microbiome but sanctions tissue occupied by non-beneficial cheaters, is expected to generate a competition-extinction trade-off by allowing beneficial microbes to experience a lower extinction rate than competitively dominant cheaters. Using an adaptive dynamics approach, we demonstrate that although ecologically stable, microbial regulation via sanctioning is not stable in any evolutionary sense, as each individual host will be under pressure to reduce the costs incurred from cheater suppression in order to maximize its own fitness at the expense of the rest of the host population. However, increasing the diversity of non-beneficial cheaters in the host population metamicrobiome can lead to an increase in the relative fitness of hosts that actively sanction non-performing tissue, thus facilitating the evolutionary emergence and persistence of such strategies in host-microbial systems. These counter-intuitive results demonstrate how diversity at multiple levels of biological organization and spatiotemporal scales can interact to facilitate the establishment and maintenance of mutualistic relationships.

Adaptive Dynamics Simulation of Interference Phenomenon for Physical and Biological Systems.

Biological systems have been shown to have quantum-like behaviors by applying the adaptive dynamics view on their interaction networks. In particular, in the process of lactose-glucose metabolism, cells generate probabilistic interference patterns similarly to photons in the two-slit experiment. Such quantum-like interference patterns can be found in biological data, on all scales, from proteins to cognitive, ecological, and social systems. The adaptive dynamics approach covers both biological and physical phenomena, including the ones which are typically associated with quantum physics. We guess that the adaptive dynamics can be used for the clarification of quantum foundations, and the present paper is the first step in this direction. We suggest the use of an algorithm for the numerical simulation of the behavior of a billiard ball-like particle passing through two slits by explicitly considering the influence of the two-slit environment (experimental context). Our simulation successfully mimics the interference pattern obtained experimentally in quantum physics. The interference of photons or electrons by two slits is known as a typical quantum mechanical effect. We do not claim that the adaptive dynamics can reproduce the whole body of quantum mechanics, but we hope that this numerical simulation example will stimulate further extensive studies in this direction-the representation of quantum physical phenomena in an adaptive dynamical framework.

Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact.

Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Social dilemmas of sociality due to beneficial and costly contagion.

Levels of sociality in nature vary widely. Some species are solitary; others live in family groups; some form complex multi-family societies. Increased levels of social interaction can allow for the spread of useful innovations and beneficial information, but can also facilitate the spread of harmful contagions, such as infectious diseases. It is natural to assume that these contagion processes shape the evolution of complex social systems, but an explicit account of the dynamics of sociality under selection pressure imposed by contagion remains elusive. We consider a model for the evolution of sociality strategies in the presence of both a beneficial and costly contagion. We study the dynamics of this model at three timescales: using a susceptible-infectious-susceptible (SIS) model to describe contagion spread for given sociality strategies, a replicator equation to study the changing fractions of two different levels of sociality, and an adaptive dynamics approach to study the long-time evolution of the population level of sociality. For a wide range of assumptions about the benefits and costs of infection, we identify a social dilemma: the evolutionarily-stable sociality strategy (ESS) is distinct from the collective optimum-the level of sociality that would be best for all individuals. In particular, the ESS level of social interaction is greater (respectively less) than the social optimum when the good contagion spreads more (respectively less) readily than the bad contagion. Our results shed light on how contagion shapes the evolution of social interaction, but reveals that evolution may not necessarily lead populations to social structures that are good for any or all.

Competition among small individuals hinders adaptive radiation despite ecological opportunity.

Ontogenetic diet shifts, where individuals change their resource use during development, are the rule rather than the exception in the animal world. Here, we aim to understand how such changes in diet during development affect the conditions for an adaptive radiation in the presence of ecological opportunity. We use a size-structured consumer–resource model and the adaptive dynamics approach to study the ecological conditions for speciation. We assume that small individuals all feed on a shared resource. Large individuals, on the other hand, have access to multiple food sources on which they can specialize. We find that competition among small individuals can hinder an adaptive radiation to unfold, despite plenty of ecological opportunity for large individuals. When small individuals experience strong competition for food, they grow slowly and only a few individuals are recruited to the larger size classes. Hence, competition for food among large individuals is weak and there is therefore no disruptive selection. In addition, initial conditions determine if an adaptive radiation occurs or not. A consumer population initially dominated by small individuals will not radiate. On the other hand, a population initially dominated by large individuals may undergo adaptive radiation and diversify into multiple species.

Evolution of enzyme levels in metabolic pathways: A theoretical approach. Part 1

The central role of metabolism in cell functioning and adaptation has given rise to countless studies on the evolution of enzyme-coding genes and network topology. However, very few studies have addressed the question of how enzyme concentrations change in response to positive selective pressure on the flux, considered a proxy of fitness. In particular, the way cellular constraints, such as resource limitations and co-regulation, affect the adaptive landscape of a pathway under selection has never been analyzed theoretically. To fill this gap, we developed a model of the evolution of enzyme concentrations that combines metabolic control theory and an adaptive dynamics approach, and integrates possible dependencies between enzyme concentrations. We determined the evolutionary equilibria of enzyme concentrations and their range of neutral variation, and showed that they differ with the properties of the enzymes, the constraints applied to the system and the initial enzyme concentrations. Simulations of long-term evolution confirmed all analytical and numerical predictions, even though we relaxed the simplifying assumptions used in the analytical treatment.

Coevolution of the reckless prey and the patient predator

The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator–prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.

On the difficult evolutionary transition from the free-living lifestyle to obligate symbiosis.

Obligate symbiosis evolved from free-living individuals most likely via the intermediate stage of facultative symbiosis. However, why should facultative symbionts, who can live independently but also benefit from their partners if these are available, give up this best of both worlds? Using the adaptive dynamics approach, we analyse a simple model, focusing on one partner of the symbiosis, to gain more insight into the selective forces that make individuals forgo the ability to reproduce in the free-living state. Our results suggest that, similar to the parasitism-mutualism continuum, the free-living way of life and obligate symbiosis are two extremes of a continuum of the ability to reproduce independently of a partner. More importantly, facultative symbiosis should be the rule as for many parameter combinations completely giving up independent reproduction or adopting a pure free-living strategy is not so easy. We also show that if host encounter comes at a cost, individuals that put more effort into increasing the chances to meet with their partners are more likely to give up the ability to reproduce independently. Finally, our model does not specify the ecological interactions between hosts and symbionts but we discuss briefly how the ecological nature of an interaction can influence the transition from facultative to obligate symbiosis.

The Evolutionary Ecology of Metamorphosis.

Almost all animal species undergo metamorphosis, even though empirical data show that this life-history strategy evolved only a few times. Why is metamorphosis so widespread, and why has it evolved? Here we study the evolution of metamorphosis by using a fully size-structured population model in conjunction with the adaptive-dynamics approach. We assume that individuals compete for two food sources; one of these, the primary food source, is available to individuals of all sizes. The secondary food source is available only to large individuals. Without metamorphosis, unresolvable tensions arise for species faced with the opportunity of specializing on such a secondary food source. We show that metamorphosis can evolve as a way to resolve these tensions, such that small individuals specialize on the primary food source while large individuals specialize on the secondary food source. We find, however, that metamorphosis evolves only when the supply rate of the secondary food source exceeds a high threshold. Individuals postpone metamorphosis when the ecological conditions under which metamorphosis originally evolved deteriorate but will often not abandon this life-history strategy, even if it causes population extinction through evolutionary trapping. In summary, our results show that metamorphosis is not easy to evolve but that, once evolved, it is hard to lose. These findings can explain the widespread occurrence of metamorphosis in the animal kingdom despite its few evolutionary origins.

Multiple Evolutionary Routes to Monogamy: Modeling the Coevolution of Mating Decisions and Parental Investment.

The relationships between mating decisions and parental investment are central to evolution, but to date few theoretical treatments of their coevolution have been developed. Here we adopt a demographically explicit, adaptive dynamics approach to analyze the coevolution of female mating decisions and parental investment of both sexes in a self-consistent way. Our models predict that where females cannot interfere with one another's mating decisions and where they do not differ in their survival and fecundity prospects, monogamy should be rare, favored only under harsh environmental conditions, in sparse populations. However, allowing for interference or asymmetries among females leads to selection for monogamy over a much broader range of environments and demographies. Interference by paired, resident females may prevent unmated rivals from joining existing monogamous pairs, thus barring the formation of polygynous groups. Asymmetries between established, primary females and subsequently joining secondary females may increase the relative costs of early polygynous reproduction, compared to delayed monogamy for the latter. The models thus highlight different routes by which monogamy may evolve. We further track how parental investment by the sexes coevolves with female mating decisions, highlighting how sexual conflict over parental investment is both cause and effect of mating behavior.

Alternative assembly processes from trait-mediated co-evolution in mutualistic communities

Ecological and evolutionary dynamics observed in mutualistic communities can be shaped by several mechanisms, including ecological interactions and their co-evolutionary consequences. Here we explore how intra and interspecific competition, together with mutualistic interactions, can affect community assembly through their effects on adaptive diversification and the emergence of biodiversity. To capture both ecological and evolutionary processes simultaneously, we used the adaptive dynamics approach based on a Lotka-Volterra framework and simulated the ecological dynamics of populations as well as the evolutionary dynamics of phenotypic traits. Depending on the initial trait values, two possible alternative evolutionary regimes emerged: traits evolve towards either optimal utilization of environmental resources or maximizing the benefits from mutualistic interactions. Diversification and overall biodiversity are mostly driven by frequency-dependent competition, while mutualism plays an important role in enhancing ecosystem productivity and evolutionary stability. Because different initial trait values in a community can lead to alternative evolutionary regimes, species loss and biological invasions could not only alter ecological dynamics but also push the system onto an alternative successional climax or evolutionary end point. It thus becomes essential to clarify the past evolutionary dynamics so as to draw conclusions on key community assembly processes.

Sink or swim? Vertical movement and nutrient storage in phytoplankton

A simulation model of vertically migrating phytoplankton is presented, using a Lagrangian, individual-based computational approach. Algal cells acquire and store nutrient at the bottom of the habitat, using stored nutrient to grow while in shallower waters. Stored nutrient also governs movement: cells sink when their nutrient quota falls below a threshold; otherwise they rise (or at least sink more slowly). Although the bottom of the habitat provides the growth-limiting nutrient, it also entails a risk of mortality. For a parameter set representing phosphorus-limited algae with a fixed nutrient storage capacity, neither continual sinking nor continual rising are optimal strategies. Instead, an adaptive dynamics approach suggests there is an optimal movement strategy in which cells rise when their storage capacity is partially filled, and otherwise sink. When the movement strategy is fixed in such a way and storage capacity is free to evolve, storage capacity approaches an optimal value several times higher than the minimal quota permitting population growth. Vertical movement and nutrient storage affect the vertical distribution of total nutrient. When cells always sink, total nutrient declines exponentially from the nutrient source at the bottom to a surface minimum. When cells always rise, there is a peak of total nutrient at the bottom, and another at the surface, with a minimum between. When cells move optimally, the vertical distribution of total nutrient can be close to uniform, or have a peak at mid-depth.

Transmission Fitness in Co-colonization and the Persistence of Bacterial Pathogens.

Humans are often colonized by polymorphic bacteria such as Streptococcus pneumoniae, Bordetella pertussis, Staphylococcus Aureus, and Haemophilus influenzae. Two co-colonizing pathogen clones may interact with each other upon host entry and during within-host dynamics, ranging from competition to facilitation. Here we examine the significance of these exploitation strategies for bacterial spread and persistence in host populations. We model SIS epidemiological dynamics to capture the global behavior of such multi-strain systems, focusing on different parameters of single and dual colonization. We analyze the impact of heterogeneity in clearance and transmission rates of single and dual colonization and find the criteria under which these asymmetries enhance endemic persistence. We obtain a backward bifurcation near [Formula: see text] if the reproductive value of the parasite in dually infected hosts is sufficiently higher than that in singly infected ones. In such cases, the parasite is able to persist even in sub-threshold conditions, and reducing the basic reproduction number below 1 would be insufficient for elimination. The fitness superiority in co-colonized hosts can be attained by lowering net parasite clearance rate ([Formula: see text]), by increasing transmission rate ([Formula: see text]), or both, and coupling between these traits critically constrains opportunities of pathogen survival in the [Formula: see text] regime. Finally, using an adaptive dynamics approach, we verify that despite their importance for sub-threshold endemicity, traits expressed exclusively in coinfection should generally evolve independently of single infection traits. In particular, for [Formula: see text] a saturating parabolic or hyperbolic function of [Formula: see text], co-colonization traits evolve to an intermediate optimum (evolutionarily stable strategy, ESS), determined only by host lifespan and the trade-off parameters linking [Formula: see text] and [Formula: see text]. Our study invites more empirical attention to the dynamics and evolution of parasite life-history traits expressed exclusively in coinfection.

Coevolution of patch-type dependent emigration and patch-type dependent immigration

The three phases of dispersal - emigration, transfer and immigration - are affecting each other and the former and latter decisions may depend on patch types. Despite the inevitable fact of the complexity of the dispersal process, patch-type dependencies of dispersal decisions modelled as emigration and immigration are usually missing in theoretical dispersal models. Here, I investigate the coevolution of patch-type dependent emigration and patch-type dependent immigration in an extended Hamilton–May model. The dispersing population inhabits a landscape structured into many patches of two types and disperses during a continuous-time season. The trait under consideration is a four dimensional vector consisting of two values for emigration probability from the patches and two values for immigration probability into the patches of each type. Using the adaptive dynamics approach I show that four qualitatively different dispersal strategies may evolve in different parameter regions, including a counterintuitive strategy, where patches of one type are fully dispersed from (emigration probability is one) but individuals nevertheless always immigrate into them during the dispersal season (immigration probability is one). I present examples of evolutionary branching in a wide parameter range, when the patches with high local death rate during the dispersal season guarantee a high expected disperser output. I find that two dispersal strategies can coexist after evolutionary branching: a strategy with full immigration only into the patches with high expected disperser output coexists with a strategy that immigrates into any patch. Stochastic simulations agree with the numerical predictions. Since evolutionary branching is also found when immigration evolves alone, the present study is adding coevolutionary constraints on the emigration traits and hence finds that the coevolution of a higher dimensional trait sometimes hinders evolutionary diversification.

A Parent-Offspring Trade-Off Limits the Evolution of an Ontogenetic Niche Shift.

Many free-living animal species, including the majority of fish, insects, and amphibians, change their food and habitat during their life. Even though these ontogenetic changes in niche are common, it is not well understood which ecological conditions have favored the evolution of these shifts. Using an adaptive dynamics approach, we show that it is evolutionarily advantageous to switch to an alternative food source in the course of ontogeny when this results in a higher intake rate for the switching consumers. Individuals are, however, not able to specialize on this new food source when this negatively affects the performance early in life on the original food source. Selection on these early life stages is so strong that in species with a complete diet shift, evolution results in large juveniles and adults that are maladapted to the alternative food source while their offspring are specialized on the original food source when young. These outcomes suggest strong selection to decouple the different life stages, such that they can maximize their performance on different food sources independently from each other. Metamorphosis could be a way to decouple the different life stages and therefore evolve in species that feed on multiple food sources during their life.

The puzzle of partial migration: Adaptive dynamics and evolutionary game theory perspectives

We consider the phenomenon of partial migration which is exhibited by populations in which some individuals migrate between habitats during their lifetime, but others do not. First, using an adaptive dynamics approach, we show that partial migration can be explained on the basis of negative density dependence in the per capita fertilities alone, provided that this density dependence is attenuated for increasing abundances of the subtypes that make up the population. We present an exact formula for the optimal proportion of migrants which is expressed in terms of the vital rates of migrant and non-migrant subtypes only. We show that this allocation strategy is both an evolutionary stable strategy (ESS) as well as a convergence stable strategy (CSS). To establish the former, we generalize the classical notion of an ESS because it is based on invasion exponents obtained from linearization arguments, which fail to capture the stabilizing effects of the nonlinear density dependence. These results clarify precisely when the notion of a “weak ESS”, as proposed in Lundberg (2013) for a related model, is a genuine ESS. Secondly, we use an evolutionary game theory approach, and confirm, once again, that partial migration can be attributed to negative density dependence alone. In this context, the result holds even when density dependence is not attenuated. In this case, the optimal allocation strategy towards migrants is the same as the ESS stemming from the analysis based on the adaptive dynamics.The key feature of the population models considered here is that they are monotone dynamical systems, which enables a rather comprehensive mathematical analysis.

A Dual-Mesh Unstructured Adaptive Cartesian Computational Fluid Dynamics Approach for Hover Prediction

A Dual-Mesh Unstructured Adaptive Cartesian Computational Fluid Dynamics Approach for Hover Prediction

Moment equations in spatial evolutionary ecology

How should we model evolution in spatially structured populations? Here, I review an evolutionary ecology approach based on the technique of spatial moment equations. I first provide a mathematical underpinning to the derivation of equations for the densities of various spatial configurations in network-based models. I then show how this spatial ecological framework can be coupled with an adaptive dynamics approach to compute the invasion fitness of a rare mutant in a resident population at equilibrium. Under the additional assumption that mutations have small phenotypic effects, I show that the selection gradient can be expressed as a function of neutral measures of genetic and demographic structure. I discuss the connections between this approach and inclusive fitness theory, as well as the applicability and limits of this technique. My main message is that spatial moment equations can be used as a means to obtain compact qualitative arguments about the evolution of life-history traits for a variety of life cycles.

Siderophore production and the evolution of investment in a public good: An adaptive dynamics approach to kin selection

Like many other bacteria, Pseudomonas aeruginosa sequesters iron from the environment through the secretion, and subsequent uptake, of iron-binding molecules. As these molecules can be taken up by other bacteria in the population than those who secreted them, this is a form of cooperation through a public good. Traditionally, this problem has been studied by comparing the relative fitnesses of siderophore-producing and non-producing strains, but this gives no information about the fate of strains that do produce intermediate amounts of siderophores. Here, we investigate theoretically how the amount invested in this form of cooperation evolves. We use a mechanistic description of the laboratory protocols used in experimental evolution studies to describe the competition and cooperation of the bacteria. From this dynamical model we derive the fitness following the adaptive dynamics method. The results show how selection is driven by local siderophore production and local competition. Because siderophore production reduces the growth rate, local competition decreases with the degree of relatedness (which is a dynamical variable in our model).Our model is not restricted to the analysis of small phenotypic differences and allows for theoretical exploration of the effects of large phenotypic differences between cooperators and cheats. We predict that an intermediate ESS level of cooperation (molecule production) should exist. The adaptive dynamics approach allows us to assess evolutionary stability, which is often not possible in other kin-selection models. We found that selection can lead to an intermediate strategy which in our model is always evolutionarily stable, yet can allow invasion of strategies that are much more cooperative. Our model describes the evolution of a public good in the context of the ecology of the microorganism, which allows us to relate the extent of production of the public good to the details of the interactions.

How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable.