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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.

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.

We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and the process is viewed in the timescale of mutations. Using averaging techniques due to Kurtz (in Lecture Notes in Control and Inform. Sci., vol. 177, pp. 186-209, 1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (in Trends in Ecology and Evolution 7(6), 198-202, 1992), first worked out by Dieckman and Law (in Journal of Mathematical Biology 34(5-6), 579-612, 1996) and rigorously proved by Champagnat (in Theoretical Population Biology 69, 297-321, 2006): rigging the model such that "invasion implies substitution", we obtain in the limit a process that jumps from one population equilibrium to another when mutations occur and invade the population.

Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We exted 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 froma 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 M1 topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differenial equation in allele or in phenotype space of a type referred to in the adaptive dynamics literatue as 'canonical equation'.

Using a fitness-returns argument we derive an expression for the selection gradient for the age dependent allocation strategy in a common class of state variable based life-history models. By setting the selection gradient equal to zero as part of the calculation of the ESS-es for such models, we get a marginal value argument and through this recover the local version of Pontryagin's maximum principle. This fills in a minor gap in a recent paper by Parvinen, Heino and Dieckmann (2012; DOI10.1007/s00285-012-0549-2), who treated the calculation of the selection gradient and of the ESS-es as separate issues. As bonuses we (i) provide an extension of the framework of these authors that can handle also the more complicated evolutionary dynamics of the life histories that we consider, and (ii) derive also the full Pontryagin's maximum principle from a fitness-returns argument.

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 behavior 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 process, 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 distribution 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 archetypal 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 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.

Adaptive dynamics (AD) is a mathematical framework for dealing with eco-evolutionary problems, primarily based on the following simplifying assumptions: clonal reproduction, rare mutations, small mutational effects, smoothness of the demographic parameters in the traits, and well-behaved community attractors. However, often the results from AD models turn out to apply also under far less restrictive conditions. The main AD tools are its so-called canonical equation (CE) that captures how the trait value(s) currently present in the population should develop over evolutionary time, and graphical techniques for analyzing evolutionary progress for one-dimensional trait spaces like .pairwise invasibility plots. (PIPs) and .trait evolution plots. (TEPs). The equilibria of the CE, customarily referred to as evolutionarily singular strategies, or ess-es, comprise in addition to the evolutionary equilibria, or ESSes, also points in trait space where the population comes under a selective pressure to diversify. Such points mathematically capture the ecological conditions conducive to adaptive (Darwinian) speciation.

Part of the art of theory building is to construct effective basic concepts, with a large reach and yet powerful as tools for getting at conclusions. The most basic concept of population biology is that of individual. An appropriately reengineered form of this concept has become the basis for the theories of structured populations and adaptive dynamics. By appropriately delimiting individuals, followed by defining their states as well as their environment, it become possible to construct the general population equations that were introduced and studied by Odo Diekmann and his collaborators. In this essay I argue for taking the properties that led to these successes as the defining characteristics of the concept of individual, delegating the properties classically invoked by philosophers to the secondary role of possible empirical indicators for the presence of those characteristics. The essay starts with putting in place as rule for effective concept engineering that one should go for relations that can be used as basis for deductive structure building rather than for perceived ontological essence. By analysing how we want to use it in the mathematical arguments I then build up a concept of individual, first for use in population dynamical considerations and then for use in evolutionary ones. These two concepts do not coincide, and neither do they on all occasions agree with common intuition-based usage.

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 in- teraction 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 descrining the IBM can be approximated by the delay equation featured in (Diekmann et al., loc. cit.).

http://www.isi2011.ie/content/access-congress-proceedings.html; International audience; Population dynamics with age structure are very important in demography and ecology since the demographic parameters of most species change over their life (think of maturation and senescence) and many phenomena (\eg evolving life histories or kinship based social interactions such as cooperative breeding) require the introduction of age for their proper description. We consider here a long time scale diffusion limit for the dynamics of a large purely age structured population with closely matched birth an death rates. To this end we work out the technical details necessary to apply the general results in Méléard and Tran to this particular case. This way we derive, starting from a logistic age-structured birth and death process, a Feller diffusion with drift and diffusion coefficients that are averages over the age distribution.

The present chapter is complementary to the one by Warren Ewens. It contains almost no equations or genetics (or rather, both are implicitly there, but stay well hidden), fitnesses are not assumed to be constant, but keep changing (in genetical terms, selection is always seen as frequency dependent), the evolutionary path is considered as being shaped by the repeated substitution of novel mutations (as opposed to gene frequency change), and adaptive landscapes depict the fitness of potential mutant types over a space spanned by traits (instead of the mean fitness of a population over a space of gene frequencies). The reason for this difference in emphasis is that my first training is not as a mathematician or population geneticist but as a naturalist, i.e., field oriented taxonomist-ecologist, which shapes the questions that have my interest. As a consequence, within evolutionary biology I am interested primarily in meso-evolution, defined here as evolutionary changes in the values of traits of representative individuals and concomitant patterns of taxonomic diversification. This in contrast to micro-evolution, a term reserved for the changes in gene frequencies on a population dynamical time scale (the topic of Warren Ewens' chapter), and macro-evolution, a term that then can be reserved for large scale changes like anatomical innovations, where one cannot even speak in terms of a fixed set of traits. Thus meso-evolution acts on a time scale above the micro-evolutionary scale of gene substitutions but below the scale on which the intricacies of the developmental process start to have a large influence...

For structured populations in equilibrium with everybody born equal ln(R_0) is a useful fitness proxy for 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 e.g. 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 characterised by the trait vector Y in the (constant) environment engendered by a resident characterised by X. If the trait space is connected, global univadability 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 ESSes under haplo-diploid genetics (I), of Evolutionarily Steady genetic Dimorphisms with a priori proportionality of macro- and micro-gametic outputs (an assumption that is generally made but the fulfillment 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.

traduction en français; Taduction française de l'aticle paru dans la revue Evolutionary Ecology (DOI 10.1007/s10682-010-9387-0 ); In diesem Artikel wird die Auswirkung der periodischen Abwesenheit der Wirtspflanze auf die Entwicklung der Übertragungsraten von Pathogenen unter Verwendung von Maximierungstechniken von R0 untersucht. Die physiologische Folge einer erhöhten Übertragungsrate kann entweder eine erhöhte Virulenz sein (Kompromiss zwischen Übertragung und Virulenz) oder ein verringertes Überleben zwischen den Jahreszeiten (Kompromiss zwischen Übertragung und Überleben). Die Ergebnisse zeigen, dass der Typ des Kompromisses die Richtung der Auswahl bestimmt. Mit einem Kompromiss zwischen Übertragung und Virulenz wählen längere Abwesenheitszeiten vom Host die höheren Übertragungsraten aus. Mit einem Kompromiss zwischen Übertragung und Überleben zwischen den Jahreszeiten werden aber schwächere Übertragungsraten ausgewählt. Die Tatsache, dass für den Kompromiss zwischen Übertragung und Virulenz die beiden Kompromissparameter während der Anwesenheit der Wirtpflanze auftreten, während für den Kompromiss zwischen Übertragung und Überleben einer während der Anwesenheit der Wirtpflanze (Übertragung) und das andere (Überleben) während der Abwesenheit der Wirtpflanze ist die Hauptursache für diesen Unterschied in der Richtung der Selektion. Außerdem scheint die Abwesenheitszeit der Wirtpflanze der bestimmende Faktor für die Übertragungsrate des Erregers zu sein. Ein Vergleich von pflanzenpathologischen Systemen mit kontrastierenden biologischen Merkmalen zeigt, dass Pflanzenpathogene in der Luft bei längerer Abwesenheit vom Wirt anders reagieren als bodengetragene Pflanzenpathogene.; Cet article explore l'effet de l'absence périodique de l'hôte sur l'évolution des taux de transmission des agents pathogènes en utilisant des techniques de maximisation de R0. La conséquence physiologique d'un taux de transmission accru peut être soit une virulence accrue (compromis entre transmission et virulence), soit une réduction de la survie entre les saisons (compromis entre transmission et survie). Les résultats révèlent que le type de compromis détermine la direction de la sélection, avec des périodes plus longues d'absence de l'hôte qui sélectionnent des taux de transmission plus élevés avec un compromis entre la transmission et la virulence, mais des taux de transmission plus faibles avec un compromis entre la transmission et la survie entre les saisons. Le fait que, pour le compromis entre transmission et virulence, les deux paramètres de compromis interviennent pendant la présence de l'hôte, alors que pour le compromis transmission-survie, l'un opère pendant la présence de l'hôte (transmission) et l'autre (survie) pendant la période d'absence de l'hôte, est la principale cause de cette différence dans le sens de la sélection. De plus, la période d'absence de l'hôte semble être le facteur déterminant du taux de transmission du pathogène. La comparaison de systèmes phytopathologiques qui présentent des caractéristiques biologiques contrastées suggère que les agents pathogènes des plantes présents dans l'air réagissent différemment à des périodes d'absence plus longues de l'hôte que les agents pathogènes des plantes transmis par le sol.

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 R(0) 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 R*. 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 R(m) 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 R(m) can be calculated and expressed in terms of the key determinants of epidemiological dynamics.

Goal: Elucidating the role of the eco-evolutionary feedback loop in determining evolutionarily stable life histories, with particular reference to the methodological status of the optimisation procedures of classical evolutionary ecology. Conclusions: Even in cases where there happens to exist an optimisation principle, the evolutionary outcomes can be largely determined by other aspects of the population dynamical embedding. The existence of an optimisation principle is technically helpful, biologically very restrictive, and has in general no further biological relevance.

Goal: Elucidating the role of the eco-evolutionary feedback loop in determining evolutionarily stable life histories, with particular reference to the methodological status of the optimisation procedures of classical evolutionary ecology. Conclusion: A pure optimisation approach holds water only when the eco-evolutionary feedbacks are of a particularly simple kind.

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Diekmann et al. 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law 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.

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.

We review the state-of-the-art concerning a mathematical framework for general physiologically structured population models. When individual development is affected by the population density, such models lead to quasilinear equations. We show how to associate a dynamical system (defined on an infinite dimensional state space) to the model and how to determine the steady states. Concerning the principle of linearized stability, we offer a conjecture as well as some preliminary steps towards a proof.

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.

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, 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. The negative across-breed relationship is probably the consequence of short lifespans in large breeds. Artificial selection for extremely high growth rates in large breeds appears to have led to developmental diseases that seriously diminish longevity.