Immunity-driven evolution of virulence and diversity in respiratory diseases

The time-honored paradigm in the theory of virulence evolution assumes a positive relation between infectivity and harmfulness. However, the etiology of respiratory diseases yields a negative relation, with diseases of the lower respiratory tract being less infective and more harmful. We explore the evolutionary consequences in a simple model incorporating cross-immunity between disease strains that diminishes with their distance in the respiratory tract, assuming that docking rate follows the match between the local mix of cell surface types and the pathogen's surface and cross-immunity the similarity of the pathogens' surfaces. The assumed relation between fitness components causes virulent strains infecting the lower airways to evolve to milder more transmissible variants. Limited cross-immunity, generally, causes a readiness to diversify that increases with host population density. In respiratory diseases that diversity will be highest in the upper respiratory tract. More tentatively, emerging respiratory diseases are likely to start low and virulent, to evolve up, and become milder. Our results extend to a panoply of realistic generalizations of the disease's ecology to including additional epitope axes. These extensions allow us to apply our results quantitatively to elucidate the differences in diversification between rhino- and coronavirus caused common colds.

Evolution of Reproduction Periods in Seasonal Environments

AbstractMany species are subject to seasonal cycles in resource availability, affecting the timing of their reproduction. Using a stage-structured consumer-resource model in which juvenile development and maturation are resource dependent, we study how a species' reproductive schedule evolves, dependent on the seasonality of its resource. We find three qualitatively different reproduction modes. First, continuous income breeding (with adults reproducing throughout the year) evolves in the absence of significant seasonality. Second, seasonal income breeding (with adults reproducing unless they are starving) evolves when resource availability is sufficiently seasonal and juveniles are more efficient resource foragers. Third, seasonal capital breeding (with adults reproducing partly through the use of energy reserves) evolves when resource availability is sufficiently seasonal and adults are more efficient resource foragers. Such capital breeders start reproduction already while their offspring are still experiencing starvation. Changes in seasonality lead to continuous transitions between continuous and seasonal income breeding, but the change between income and capital breeding involves a hysteresis pattern, such that a population's evolutionarily stable reproduction pattern depends on its initial one. Taken together, our findings show how adaptation to seasonal environments can result in a rich array of outcomes, exhibiting seasonal or continuous reproduction with or without energy reserves.

Lotka-Volterra approximations for evolutionary trait-substitution processes

A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an appropriately chosen Lotka–Volterra model. To this end, the assumption is made that similar phenotypes in the community form clusters that are well-separated from each other, as is expected to be generally the case when evolution proceeds through small mutational steps. Each phenotypic cluster is represented by a single phenotype, which we call an approximate phenotype and assign the cluster’s total population density. We present our results in three steps. First, for a set of approximate phenotypes with arbitrary equilibrium population densities before the invasion, the Lotka–Volterra approximation is proved to apply if the changes of the population densities of these phenotypes are sufficiently small during the transient following the invasion. Second, quantitative conditions for such small changes of population densities are derived as a relationship between within-cluster differences and the leading eigenvalue of the community’s Jacobian matrix evaluated at the equilibrium population densities before the invasion. Third, to demonstrate the utility of our results, the ‘invasion implies substitution’ result for monomorphic populations is extended to arbitrarily polymorphic populations consisting of well-recognizable and -separated clusters.

Environmental dimensionality determines species coexistence

According to the competitive-exclusion principle, the number n of regulating variables describing a given community dynamics is an upper bound on the number of species (or types or morphs) that can coexist at equilibrium. On occasion, it is possible to reformulate a model with a lower number of regulating variables than appeared in the initial specification. We call the smallest number of such variables the dimension of the environmental feedback, or environmental dimension for short. For studying which species can invade a community, it is enough to know the sign of each species' long-term growth rate, i.e., invasion fitness. Therefore, different indicators of population growth - so-called fitness proxies, such as the basic reproduction number-are sometimes preferred. However, as we show, different fitness proxies may have different dimensions. Fundamental characteristics such as the environmental dimension should not depend on such arbitrary choices. Here, we resolve this difficulty by introducing a refined definition of environmental dimension that focuses on neutral fitness contours. On this basis, we show that this definition of environmental dimension is not only unambiguous, i.e., independent of the choice of fitness proxy, but also constructive, i.e., applicable without needing to assess an infinite number of possible fitness proxies. We then investigate how to determine environmental dimensions by analysing the two components of the environmental feedback: the impact map describing how a community's resident species affect the regulating variables and the sensitivity map describing how population growth depends on the regulating variables. The dimension of the impact map is lower than n when the set of feasible environments is of lower dimension than n, and the dimension of the sensitivity map is lower than n when not all n regulating variables affect the sign of population growth independently. While the minimum of the dimensions of the impact and sensitivity maps provides an upper bound on the environmental dimension, the combined effect of the two maps can result in an even lower environmental dimension, which happens when the sensitivity map is insensitive to some aspects of the impact map's image. To facilitate the applications of the framework introduced here, we illustrate all key concepts with detailed worked examples. In view of these results, we claim that the environmental dimension is the ultimate generalization of the traditional and widely used notions of the "number of regulating variables" or the "number of limiting factors", and is thus the sharpest generally applicable upper bound on the number of species that can robustly coexist in a community.

Finite dimensional state representation of physiologically structured populations

In a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations' outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input-output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODE-reducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environment-dependent distributions over the birth states weighted by environment independent 'population outputs'. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs.

Twelve fundamental life histories evolving through allocation-dependent fecundity and survival

An organism's life history is closely interlinked with its allocation of energy between growth and reproduction at different life stages. Theoretical models have established that diminishing returns from reproductive investment promote strategies with simultaneous investment into growth and reproduction (indeterminate growth) over strategies with distinct phases of growth and reproduction (determinate growth). We extend this traditional, binary classification by showing that allocation‐dependent fecundity and mortality rates allow for a large diversity of optimal allocation schedules. By analyzing a model of organisms that allocate energy between growth and reproduction, we find twelve types of optimal allocation schedules, differing qualitatively in how reproductive allocation increases with body mass. These twelve optimal allocation schedules include types with different combinations of continuous and discontinuous increase in reproduction allocation, in which phases of continuous increase can be decelerating or accelerating. We furthermore investigate how this variation influences growth curves and the expected maximum life span and body size. Our study thus reveals new links between eco‐physiological constraints and life‐history evolution and underscores how allocation‐dependent fitness components may underlie biological diversity.

Epidemiological, evolutionary, and economic determinants of eradication tails

Despite modern medical interventions, infectious diseases continue to generate huge socio-economic losses. The benefits of eradicating a disease are therefore high. While successful with smallpox and rinderpest, many other eradication attempts have failed. Eradications require huge and costly efforts, which can be sustained only if sufficient progress can be achieved. While initial successes are usually obtained more easily, progress often becomes harder as a disease becomes rare in the eradication endgame. A long eradication tail of slowly decreasing incidence levels can frustrate eradication efforts, as it becomes unclear whether progress toward eradication is still being made and how much more needs to be invested to push the targeted disease beyond its extinction threshold. Realistic disease dynamics are complicated by evolutionary responses to interventions and by interactions among different temporal and spatial scales. Models accounting for these complexities are required for understanding the shapes of eradication tails. In particular, such models allow predicting how hard or costly eradication will be, and may even inform in which manner progress has to be assessed during the eradication endgame. Here we outline a general procedure by analyzing the eradication tails of generic SIS diseases, taking into account two major ingredients of realistic complexity: a group-structured host population in which host contacts within groups are more likely than host contacts between groups, and virulence evolution subject to a trade-off between host infectivity within groups and host mobility among groups. Disentangling the epidemiological, evolutionary, and economic determinants of eradication tails, we show how tails of different shapes arise depending on salient model parameters and on how the extinction threshold is approached. We find that disease evolution generally extends the eradication tail and show how the cost structure of eradication measures plays a key role in shaping eradication tails.

Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock-scissor-paper configurations. Our concept differs from that of Heino et al. (TREE 13:367-370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.'s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock-scissor-paper configurations. Since in 'realistic' (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program of the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set D in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to D. However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evolutionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to D, just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibility that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es.

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.

Daphnias: from the individual based model to the large population equation

The class of deterministic 'Daphnia' models treated by Diekmann et al. (J Math Biol 61:277-318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23:114-135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4:82-109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and through their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., loc. cit.).

Ontogenetic symmetry and asymmetry in energetics

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as "ontogenetic symmetry". Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

A rigorous model study of the adaptive dynamics of Mendelian diploids

Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka-Volterra-like dynamics and single locus genetics (as in Kimura's in Proc Natl Acad Sci USA 54: 731-736, 1965 infinite allele model). We prove under the usual smoothness assumptions, starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves on the mutational time scale (the 'long' time scale of the literature on the genetical foundations of ESS theory) as a jump process moving between homozygous states (the trait substitution sequence of the adaptive dynamics literature). Essential technical ingredients are a rigorous estimate for the probability of invasion in a dynamic diploid population, a rigorous, geometric singular perturbation theory based, invasion implies substitution theorem, and the use of the Skorohod M 1 topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differential equation in allele or in phenotype space of a type referred to in the adaptive dynamics literature as 'canonical equation'.

What life cycle graphs can tell about the evolution of life histories

We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject to density-dependent population regulation and two or more demographic parameters to be subject to evolutionary change. Our aim is to identify structural features of life cycles and modes of population regulation that correspond to specific evolutionary dynamics. Our derivations are based on a fitness proxy that is an algebraically simple function of loops within the life cycle. This allows us to phrase the results in terms of properties of such loops which are readily interpreted biologically. The following results could be obtained. First, we give sufficient conditions for the existence of optimisation principles in models with an arbitrary number of evolving traits. These models are then classified with respect to their appropriate optimisation principle. Second, under the assumption of just two evolving traits we identify structural features of the life cycle that determine whether equilibria of the monomorphic adaptive dynamics (evolutionarily singular points) correspond to fitness minima or maxima. Third, for one class of frequency-dependent models, where optimisation is not possible, we present sufficient conditions that allow classifying singular points in terms of the curvature of the trade-off curve. Throughout the article we illustrate the utility of our framework with a variety of examples.

Invasion and persistence of infectious agents in fragmented host populations

One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the migration of hosts. Such fragmented population structures are studied extensively with metapopulation models. Being able to define and calculate an indicator for the success of invasion and persistence of an infectious agent is essential for obtaining general qualitative insights into infection dynamics, for the comparison of prevention and control scenarios, and for quantitative insights into specific systems. For homogeneous populations, the basic reproduction ratio plays this role. For metapopulations, defining such an ‘invasion indicator’ is not straightforward. Some indicators have been defined for specific situations, e.g., the household reproduction number . However, these existing indicators often fail to account for host demography and especially host migration. Here we show how to calculate a more broadly applicable indicator for the invasion and persistence of infectious agents in a host metapopulation of equally connected patches, for a wide range of possible epidemiological models. A strong feature of our method is that it explicitly accounts for host demography and host migration. Using a simple compartmental system as an example, we illustrate how can be calculated and expressed in terms of the key determinants of epidemiological dynamics.

A simple fitness proxy for structured populations with continuous traits, with case studies on the evolution of haplo-diploids and genetic dimorphisms

For structured populations in equilibrium with everybody born equal, ln(R (0)) is a useful fitness proxy for evolutionarily steady strategy (ESS) and most adaptive dynamics calculations, with R (0) the average lifetime number of offspring in the clonal and haploid cases, and half the average lifetime number of offspring fathered or mothered for Mendelian diploids. When individuals have variable birth states, as is, for example, the case in spatial models, R (0) is itself an eigenvalue, which usually cannot be expressed explicitly in the trait vectors under consideration. In that case, Q(Y| X):=-det (I-L(Y| X)) can often be used as fitness proxy, with L the next-generation matrix for a potential mutant characterized by the trait vector Y in the (constant) environment engendered by a resident characterized by X. If the trait space is connected, global uninvadability can be determined from it. Moreover, it can be used in all the usual local calculations like the determination of evolutionarily singular trait vectors and their local invadability and attractivity. We conclude with three extended case studies demonstrating the usefulness of Q: the calculation of ESSs under haplo-diploid genetics (I), of evolutionarily steady genetic dimorphisms (ESDs) with a priori proportionality of macro- and micro-gametic outputs (an assumption that is generally made but the fulfilment of which is a priori highly exceptional) (II), and of ESDs without such proportionality (III). These case studies should also have some interest in their own right for the spelled out calculation recipes and their underlying modelling methodology.

Evolutionary novelties: the making and breaking of pleiotropic constraints

Body plans are remarkably well conserved, but on (very) rare occasions important novelties evolve. Such novelties involve changes at the genotypic and phenotypic level affecting both developmental and adult traits. At all levels, duplications play an important role in the evolution of novelties. Mutations for duplications, including mutations for duplications of body parts, as well as mutations for other changes in the body plan, in particular homeotic ones, occur surprisingly frequently. Hence the limitation of mutations appears to be relatively unimportant for the conservation of body plans. However, mutations for duplications of body parts and homeotic changes rarely persist in populations. We argue that the root cause of the conservation of body plans is the strong interactivity during the patterning of the embryonic axes, including the interactivity between patterning and proliferation processes. Due to this interactivity, mutations cause many negative pleiotropic effects (malformations and cancers) that dramatically lower fitness. As an example, we have shown that in humans there is extreme selection against negative pleiotropic effects of the, surprisingly frequent, mutations affecting the number of cervical vertebrae. Moreover, we argue for the relevance of relaxed selection, which temporarily allows just-arisen novelties to persist, for the effective breaking of pleiotropic constraints. We illustrate this with two empirical examples.

Do large dogs die young?

In most animal taxa, longevity increases with body size across species, as predicted by the oxidative stress theory of aging. In contrast, in within-species comparisons of mammals and especially domestic dogs (e.g. Patronek et al., '97; Michell, '99; Egenvall et al., 2000; Speakman et al., 2003), longevity decreases with body size. We explore two datasets for dogs and find support for a negative relationship between size and longevity if we consider variation across breeds. Within breeds, however, the relationship is not negative and is slightly, but significantly, positive in the larger of the two datasets. The negative across-breed relationship is probably the consequence of short life spans in large breeds. Artificial selection for extremely high growth rates in large breeds appears to have led to developmental diseases that seriously diminish longevity.

The Interplay between Behavior and Morphology in the Evolutionary Dynamics of Resource Specialization

We analyze the consequences of diet choice behavior for the evolutionary dynamics of foraging traits by means of a mathematical model. The model is characterized by the following features. Consumers feed on two different substitutable resources that are distributed in a fine-grained manner. On encounter with a resource item, consumers decide whether to attack it so as to maximize their energy intake. Simultaneously, evolutionary change occurs in morphological traits involved in the foraging process. The assumption here is that evolution is constrained by a trade-off in the consumer's ability to forage on the alternative resources. The model predicts that flexible diet choice behavior can guide the direction of evolutionary change and mediate coexistence of different consumer types. Such polymorphisms can evolve from a monomorphic population at evolutionary branching points and also at points where a small genetic change in a trait can provoke a sharp instantaneous and nongenetic change in choice behavior. In the case of weak trade-offs, the evolutionary dynamics of a dimorphic consumer population can lead to alternative evolutionarily stable communities. The robustness of these predictions is checked with individual-based simulations and by relaxing the assumption of optimally foraging consumers.

Extreme selection in humans against homeotic transformations of cervical vertebrae

Why do all mammals, except for sloths and manatees, have exactly seven cervical vertebrae? In other vertebrates and other regions, the vertebral number varies considerably. We investigated whether natural selection constrains the number of cervical vertebrae in humans. To this end, we determined the incidence of cervical ribs and other homeotic vertebral changes in radiographs of deceased human fetuses and infants, and analyzed several existing datasets on the incidence in infants and adults. Our data show that homeotic transformations that change the number of cervical vertebrae are extremely common in humans, but are strongly selected against: almost all individuals die before reproduction. Selection is most probably indirect, caused by a strong coupling of such changes with major congenital abnormalities. Changes in the number of thoracic vertebrae appear to be subject to weaker selection, in good correspondence with the weaker evolutionary constraint on these numbers. Our analysis highlights the role of prenatal selection in the conservation of our common body plan.

Evolutionary Predictions Should Be Based on Individual‐Level Traits

Recent theoretical studies have analyzed the evolution of habitat specialization using either the logistic or the Ricker equation. These studies have implemented evolutionary change directly in population-level parameters such as habitat-specific intrinsic growth rates r or carrying capacities K. This approach is a shortcut to a more detailed analysis where evolutionary change is studied in underlying morphological, physiological, or behavioral traits at the level of the individual that contribute to r or K. Here we describe two pitfalls that can occur when such a shortcut is employed. First, population-level parameters that appear as independent variables in a population dynamical model might not be independent when derived from processes at the individual level. Second, patterns of covariation between individual-level traits are usually not conserved when mapped to the level of demographic parameters. Nonlinear mappings constrain the curvature of trade-offs that can sensibly be assumed at the population level. To illustrate these results, we derive a two-habitat version of the logistic and Ricker equations from individual-level processes and compare the evolutionary dynamics of habitat-specific carrying capacities with those of underlying individual-level traits contributing to the carrying capacities. Finally, we sketch how our viewpoint affects the results of earlier studies.

Surprising evolutionary predictions from enhanced ecological realism

A focus on the eco-evolutionary feedback continually operating between a population's evolution and its environment helps to appreciate the generality of ESS theory. Here we illustrate, through a sequence of four examples, how respecting such feedback in the evolutionary dynamics of quantitative traits may result in qualitatively unexpected outcomes. Reviewing existing insights and complementing these with new results, we show (1) that evolutionary matrix games are fundamentally degenerate and allow a natural unfolding, (2) that selection-driven extinction may not be rare in nature, (3) that evolutionary epidemiology should not rely on R0 maximization, and (4) why the occurrence of Hardy-Weinberg proportions generically requires an evolutionary explanation.

Competitive exclusion and limiting similarity: A unified theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its impact and sensitivity niches are defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either the similarity of the impact niches or the similarity of the sensitivity niches results in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niche overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a continuum of populations is structurally unstable.

The evolution of resource specialization through frequency-dependent and frequency-independent mechanisms

Levins's fitness set approach has shaped the intuition of many evolutionary ecologists about resource specialization: if the set of possible phenotypes is convex, a generalist is favored, while either of the two specialists is predicted for concave phenotype sets. An important aspect of Levins's approach is that it explicitly excludes frequency-dependent selection. Frequency dependence emerged in a series of models that studied the degree of character displacement of two consumers coexisting on two resources. Surprisingly, the evolutionary dynamics of a single consumer type under frequency dependence has not been studied in detail. We analyze a model of one evolving consumer feeding on two resources and show that, depending on the trait considered to be subject to evolutionary change, selection is either frequency independent or frequency dependent. This difference is explained by the effects different foraging traits have on the consumer-resource interactions. If selection is frequency dependent, then the population can become dimorphic through evolutionary branching at the trait value of the generalist. Those traits with frequency-independent selection, however, do indeed follow the predictions based on Levins's fitness set approach. This dichotomy in the evolutionary dynamics of traits involved in the same foraging process was not previously recognized.

Link between population dynamics and dynamics of Darwinian evolution

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of similar populations. Similarity implies that the relative dynamics of the populations is slow compared to, and decoupled from, their aggregated dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the singular points of the evolutionary state space. Here "evolutionary branching" may occur. The diversification of life forms thus is demonstrated to be a natural consequence of the Darwinian process.

Hox genes, digit identities and the theropod/bird transition

Vargas and Fallon (2005. J Exp Zool (Mol Dev Evol) 304B:86-90) propose that Hox gene expression patterns indicate that the most anterior digit in bird wings is homologous to digit 1 rather than to digit 2 in other amniotes. This interpretation is based on the presence of Hoxd13 expression in combination with the absence of Hoxd12 expression in the second digit condensation from which this digit develops (the first condensation is transiently present). This is a pattern that is similar to that in the developing digit 1 of the chicken foot and the mouse hand and foot. They have tested this new hypothesis by analysing Hoxd12 and Hoxd13 expression patterns in two polydactylous chicken mutants, Silkie and talpid2. They conclude that the data support the notion that the most anterior remaining digit of the bird wing is homologous to digit 1 in other amniotes either in a standard phylogenetic sense, or alternatively in a (limited) developmental sense in agreement with the Frameshift Hypothesis of Wagner and Gautier (1999, i.e., that the developmental pathway is homologous to the one that leads to a digit 1 identity in other amniotes, although it occurs in the second instead of the first digit condensation). We argue that the Hoxd12 and Hoxd13 expression patterns found for these and other limb mutants do not allow distinguishing between the hypothesis of Vargas and Fallon (2005. J Exp Zool (Mol Dev Evol) 304B:86-90) and the alternative one, i.e., the most anterior digit in bird wings is homologous to digit 2 in other amniotes, in a phylogenetic or developmental sense. Therefore, at the moment the data on limb mutants does not present a challenge to the hypothesis, based on other developmental data (Holmgren, 1955. Acta Zool 36:243-328; Hinchliffe, 1984. In: Hecht M, Ostrom JH, Viohl G, Wellnhofer P, editors. The beginnings of birds. Eichstätt: Freunde des Jura-Museum. p 141-147; Burke and Feduccia, 1997. Science 278:666-668; Kundrát et al., 2002. J Exp Zool (Mol Dev Evol) 294B:151-159; Larsson and Wagner, 2002. J Exp Zool (Mol Dev Evol) 294B:146-151; Feduccia and Nowicki, 2002. Naturwissenschaften 89:391-393), that the digits of bird wings are homologous to digits 2,3,4 in amniotes. We recommend further testing of the hypothesis by comparing Hoxd expression patterns in different taxa.

Speciation: more likely through a genetic or through a learned habitat preference?

A problem in understanding sympatric speciation is establishing how reproductive isolation can arise when there is disruptive selection on an ecological trait. One of the solutions that has been proposed is that a habitat preference evolves, and that mates are chosen within the preferred habitat. We present a model where the habitat preference can evolve either by means of a genetic mechanism or by means of learning. Employing an adaptive-dynamical analysis, we show that evolution proceeds either to a single population of specialists with a genetic preference for their optimal habitat, or to a population of generalists without a habitat preference. The generalist population subsequently experiences disruptive selection. Learning promotes speciation because it increases the intensity of disruptive selection. An individual-based version of the model shows that, when loci are completely unlinked and learning confers little cost, the presence of disruptive selection most probably leads to speciation via the simultaneous evolution of a learned habitat preference. For high costs of learning, speciation is most likely to occur via the evolution of a genetic habitat preference. However, the latter only happens when the effect of mutations is large, or when there is linkage between genes coding for the different traits.

What we have also learned: adaptive speciation is theoretically plausible

A recent Perspectives article by Gavrilets (2003) on the theory of speciation ignored advances in understanding processes of adaptive speciation, in which the splitting of lineages is an adaptation caused by frequency-dependent selection. Adaptive, or sympatric, speciation has been modeled since the 1960s, but the large amount of attention from both empirical and theoretical biologists that adaptive speciation has received in recent years goes far beyond what was described in Gavrilets' paper. Due to conceptual advances based on the theory of adaptive dynamics, adaptive speciation has emerged as a theoretically plausible evolutionary process that can occur in many different ecological settings.

Adaptive walks on changing landscapes: Levins' approach extended

The assumption that trade-offs exist is fundamental in evolutionary theory. Levins (Am. Nat. 96 (1962) 361-372) introduced a widely adopted graphical method for analyzing evolution towards an optimal combination of two quantitative traits, which are traded off. His approach explicitly excluded the possibility of density- and frequency-dependent selection. Here we extend Levins method towards models, which include these selection regimes and where therefore fitness landscapes change with population state. We employ the same kind of curves Levins used: trade-off curves and fitness contours. However, fitness contours are not fixed but a function of the resident traits and we only consider those that divide the trait space into potentially successful mutants and mutants which are not able to invade ('invasion boundaries'). The developed approach allows to make a priori predictions about evolutionary endpoints and about their bifurcations. This is illustrated by applying the approach to several examples from the recent literature.

Anti-cancer selection as a source of developmental and evolutionary constraints

Recently at least two papers have appeared that look at cancer from an evolutionary perspective. That cancer has a negative effect on fitness needs no argument. However, cancer origination is not an isolated process, but the potential for it is linked in diverse ways to other genetically determined developmental events, complicating the way selection acts on it, and through it on the evolution of development. The two papers take a totally different line. Kavanagh argues that anti-cancer selection has led to developmental constraints. Leroi et al. argue that cancer is a side-effect of recent evolutionary changes that usually will disappear over time through anti-cancer selection. Here we place the papers in a wider perspective, and in so doing discuss various alternative developmental links cancer may have together with their evolutionary implications.

The evolution of dispersal under demographic stochasticity

Temporal and spatial variations of the environment are important factors favoring the evolution of dispersal. With few exceptions, these variations have been considered to be exclusively fluctuations of habitat quality. However, since the presence of conspecifics forms part of an individual's environment, demographic stochasticity may be a component of this variability as well, in particular when local populations are small. To study this effect, we analyzed the evolution of juvenile dispersal in a metapopulation model in which habitat quality is constant in space and time but occupancy fluctuates because of demographic stochasticity. Our analysis extends previous studies in that it includes competition of resources and competition for space. Also, juvenile dispersal is not given by a fixed probability but is made conditional on the presence of free territories in a patch, whereas individuals born in full patches will always disperse. Using a combination of analytical and numerical approaches, we show that demographic stochasticity in itself may provide enough variability to favor dispersal even from patches that are not fully occupied. However, there is no simple relationship between the evolution of dispersal and various indicators of demographic stochasticity. Selected dispersal depends on all aspects of the life-history profile, including kin selection.

On the concept of attractor for community-dynamical processes II: the case of structured populations

In Part I of this paper Jacobs and Metz (2003) extended the concept of the Conley-Ruelle, or chain, attractor in a way relevant to unstructured community ecological models. Their modified theory incorporated the facts that certain parts of the boundary of the state space correspond to the situation of at least one species being extinct and that an extinct species can not be rescued by noise. In this part we extend the theory to communities of physiologically structured populations. One difference between the structured and unstructured cases is that a structured population may be doomed to extinction and not rescuable by any biologically relevant noise before actual extinction has taken place. Another difference is that in the structured case we have to use different topologies to define continuity of orbits and to measure noise. Biologically meaningful noise is furthermore related to the linear structure of the community state space. The construction of extinction preserving chain attractors developed in this paper takes all these points into account.

On the concept of attractor for community-dynamical processes I: the case of unstructured populations

We introduce a notion of attractor adapted to dynamical processes as they are studied in community-ecological models and their computer simulations. This attractor concept is modeled after that of Ruelle as presented in [11] and [12]. It incorporates the fact that in an immigration-free community populations can go extinct at low values of their densities.

Steady-state analysis of structured population models

Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing on steady states we shall restrict to constant functions). The steady-state problem has two components: (i). the environmental condition should be such that the existing populations do neither grow nor decline; (ii). a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.

Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity

In this paper, we predict the outcome of dispersal evolution in metapopulations based on the following assumptions: (i) population dynamics within patches are density-regulated by realistic growth functions; (ii) demographic stochasticity resulting from finite population sizes within patches is accounted for; and (iii) the transition of individuals between patches is explicitly modelled by a disperser pool. We show, first, that evolutionarily stable dispersal rates do not necessarily increase with rates for the local extinction of populations due to external disturbances in habitable patches. Second, we describe how demographic stochasticity affects the evolution of dispersal rates: evolutionarily stable dispersal rates remain high even when disturbance-related rates of local extinction are low, and a variety of qualitatively different responses of adapted dispersal rates to varied levels of disturbance become possible. This paper shows, for the first time, that evolution of dispersal rates may give rise to monotonically increasing or decreasing responses, as well as to intermediate maxima or minima.

Conservation of the segmented germband stage: robustness or pleiotropy?

Gene expression patterns of the segment polarity genes in the extended and segmented germband stage are remarkably conserved among insects. To explain the conservation of these stages, two hypotheses have been proposed. One hypothesis states that the conservation reflects a high interactivity between modules, so that mutations would have several pleiotropic effects in other parts of the body, resulting in stabilizing selection against mutational variation. The other hypothesis states that the conservation is caused by robustness of the segment polarity network against mutational changes. When evaluating the empirical evidence for these hypotheses, we found strong support for pleiotropy and little evidence supporting robustness of the segment polarity network. This points to a key role for stabilizing selection in the conservation of these stages. Finally, we discuss the implications for robustness of organizers and long-term conservation in general.