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Models of adaptive evolution often have the property that change is guided by, but not fully determined by fit-ness. In a given situation many different mutant phenotypes may have a fitness advantage over the residents, and are thus potential invaders, implying that the mutational process plays an important role in deciding which par-ticular invasion will take place. By introducing an imaginary 'Darwinian demon' in charge of mutations, one can examine the maximal role that mutation could play in determining evolutionary change. Taking into account plei-otropic mutations and shifting fitness landscapes, it seems likely that a Darwinian demon could exert considerable influence and most likely would be able to produce any viable form of organism. This kind of perspective can be helpful in clarifying concepts of evolutionary stability.

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... In this original context, Law (1979) defined a 'Darwinian Demon' as an organism that 'can maximise all aspects of fitness simultaneously'. Two decades later and without reference to Law (1979), Leimar (2001) proposed that 'A Darwinian demon is a hypothetical being with the power to decide exactly which mutations appear, but without any influence over the ensuring consequences of natural selection' (italics added). According to Leimar (2001), this definition is coined as an analogue to 'Maxwell's demon', named after the physicist James C. Maxwell (1831Maxwell ( -1879 who employed a demon in a thought experiment to show that the second law of thermodynamics is a statistical certainty that governs equilibrium states (Maxwell 1908). ...

... Two decades later and without reference to Law (1979), Leimar (2001) proposed that 'A Darwinian demon is a hypothetical being with the power to decide exactly which mutations appear, but without any influence over the ensuring consequences of natural selection' (italics added). According to Leimar (2001), this definition is coined as an analogue to 'Maxwell's demon', named after the physicist James C. Maxwell (1831Maxwell ( -1879 who employed a demon in a thought experiment to show that the second law of thermodynamics is a statistical certainty that governs equilibrium states (Maxwell 1908). In Maxwell's thought experiment, a demon opens a door that allows fast molecules in a gas to enter one side of the container and slower molecules to enter the other side of the box. ...

... More recently, Krakauer (2011) introduced a third definition. Without reference to Law (1979) or Leimar (2001), he addressed the question as to how complex systems, such as multicellular animals, could have evolved by means of natural selection (Bonner 1988). With reference to Darwin's concept of natural selection (Darwin 1859) and 'Maxwell's demon device', Krakauer (2011) introduced a 'demonic selection principle' such that 'organisms cannot exceed the complexity of their environment'. ...

In evolutionary biology, the term ‘Darwinian fitness’ refers to the lifetime reproductive success of an individual within a population of conspecifics. The idea of a ‘Darwinian Demon’ emerged from this concept and is defined here as an organism that commences reproduction almost immediately after birth, has a maximum fitness, and lives forever. It has been argued that duckweeds (sub-family Lemnoideae, order Alismatales), a group containing five genera and 34 species of small aquatic monocotyledonous plants with a reduced body plan, can be interpreted as examples of ‘Darwinian Demons’. Here we focus on the species Spirodela polyrhiza (Great duckweed) and show that these miniaturised aquatic angiosperms display features that fit the definition of the hypothetical organism that we will call a ‘Darwin-Wallace Demon’ in recognition of the duel proponents of evolution by natural selection. A quantitative analysis (log-log bivariate plot of annual growth in dry biomass versus standing dry body mass of various green algae and land plants) revealed that duckweeds are thus far the most rapidly growing angiosperms in proportion to their body mass. In light of this finding, we discuss the disposable soma and metabolic optimising theories, summarise evidence for and against the proposition that the Lemnoideae (family Araceae) reflect an example of reductive evolution, and argue that, under real-world conditions (environmental constraints and other limitations), ‘Darwin-Wallace Demons’ cannot exist, although the concept remains useful in much the same way that the Hardy–Weinberg law does.

... 31. Strong convergence stable ✠᭜ (Leimar, 2001, in press). This is an extension of the convergence stability property to evolutionary games involving multidimensional traits in monomorphic or polymorphic population evolution. ...

... This is an extension of the convergence stability property to evolutionary games involving multidimensional traits in monomorphic or polymorphic population evolution. In the case of polymorphisms, no sufficient conditions are given (Leimar, 2001, p. 70). 32. ...

... 32. Absolute convergence stable ✠᭜ (Leimar, 2001, in press). This is the general extension of the convergence stability property to evolutionary games involving multidimensional traits in monomorphic or polymorphic population evolution to preclude any escape from the neighbourhood of the absolute convergence stable strategy (Leimar, 2001, p. 69). ...

Question: How are the three main stability concepts from evolutionary game theory – evolutionarily stable strategy (ESS), convergence stability, and neighbourhood invader strategy (NIS) – related to each other? Do they form a basis for the many other definitions proposed in the literature? Mathematical methods: Ecological and evolutionary dynamics of population sizes and heritable strategies respectively, and adaptive and NIS landscapes. Results: Only six of the eight combinations of ESS, convergence stability, and NIS are possible. An ESS that is NIS must also be convergence stable; and a non-ESS, non-NIS cannot be convergence stable. A simple example shows how a single model can easily generate solutions with all six combinations of stability properties and explains in part the proliferation of jargon, terminology, and apparent complexity that has appeared in the literature. A tabulation of most of the evolutionary stability acronyms, definitions, and terminologies is provided for comparison. Key conclusions: The tabulated list of definitions related to evolutionary stability are variants or combinations of the three main stability concepts.

... Such correlations will mainly occur for traits within a species. My aim here is to bring together various results from the literature (e.g. Leimar, 2001 Leimar, , 2005 Cressman et al., 2006; Dieckmann et al., 2006; Brown et al., 2007; Durinx et al., 2008) to address the question of how fitness interactions and genetic correlations can be taken Correspondence: O. Leimar, Department of Zoology, Stockholm University, SE-10691 Stockholm, Sweden. e-mail: olof.leimar@zoologi.su.se ...

... For instance, strong stability of an equilibrium (a singular point) does not guarantee that any gradualistic, adaptive path through trait space, consisting of a sequence of successful mutant invasions, must converge to or even remain near the point. On the contrary, unless fitness interactions are absent or have a very special form, an adaptive escape from any small neighbourhood of the point will be possible, although the sequence of mutations generating such invasions must have unlikely properties (Leimar, 2001). On the whole, there will usually be little reason to expect gradualistic evolution away from strongly convergence stable points. ...

... It is possible to formulate more robust criteria, at the expense of having fewer situations where these criteria are satisfied. For instance, one could require that a point should attract any nearby gradualistic, adaptive path, a property that may be called 'absolute convergence stability' (Leimar, 2001). In addition, there is the question of whether there is stabilizing selection at a singular point, preventing the appearance of polymorphisms (Christiansen, 1991). ...

Questions: Are there general stability conditions for the evolution of multidimensional traits, regardless of genetic correlations between traits? Can genetic correlations influence whether evolution converges to a stable trait vector? Mathematical methods: Adaptive dynamics theory and the weak selection limit of quantitative genetics. Key assumptions: Evolutionary change is represented as either (i) any gradualistic adaptive path in trait space, consisting of a sequence of small-effect mutant invasions, allowing for pleiotropic mutants, or (ii) a solution to the 'canonical equation' of adaptive dynamics with a gradually varying mutational covariance matrix. Assumption (ii) is a special case of (i). Conclusions: It is possible to formulate robust stability conditions for multidimensional traits, but most evolutionary equilibria will not satisfy these conditions. Under the liberal assumption (i), there will in general be no 'absolutely convergence stable' equilibria in multidimensional trait spaces (except for simplified models). Under the more restrictive assumption (ii), a much larger proportion of evolutionary equilibria is 'strongly convergence stable', i.e. are stable irrespective of genetic correlations.

... Thus an ESS for single species evolution has a better chance of being attained than an ESS coalition in a coevolution. Vincent et al. (1996) (also see Vincent and Cressman, 2000;Leimar, 2001; and references therein) give a definition of an ESS in multi-trait and/or multispecies evolutionary games. They give a criterion for an ESS to be classified as a local ESS or as a global ESS. ...

... As commonly done, we will call g i (v i , u, x) the invasion fitness of rare mutant individual adopting strategy v i in the environment (u, x) (see e.g., Leimar, 2001). This functional is also called fitness-generating function (Brown and Vincent, 1992). ...

... Motro (1994) also gives sufficient conditions for this stability. For other considerations of continuously stable strategy concept as applied to multidimensional trait spaces (see e.g., Lessard, 1990;Abrams, 1993;Dieckmann and Law, 1996;Marrow, 1996;Leimar, 2001). In what follows, we will largely explore the concept of ESNIS in a dimorphic population evolution in which individuals use onedimensional continuous strategies. ...

Dynamical attainability of an evolutionarily stable strategy (ESS) through the process of mutations and natural selection has mostly been addressed through the use of the continuously stable strategy (CSS) concept for species evolutionary games in which strategies are drawn from a continuum, and by the adaptive trait dynamics method. We address the issue of dynamical attainability of an ESS in coevolving species through the use of the concept of an ESNIS. It is shown that the definition of an ESNIS coalition for coevolving species is not in general equivalent to other definitions for CSS given in the literature. We show under some additional conditions that, in a dynamic system which involves the strategies of a dimorphic ESNIS coalition and at most two strategies that are not members of ESNIS coalition, the ESNIS coalition will emerge as the winner. In addition an ESNIS will be approached because of the invasion structure of strategies in its neighborhood. This proves that under the above conditions an ESNIS has a better chance of being attained than a strategy coalition which is a CSS. The theory developed is applied to a class of coevolutionary game models with Lotka–Volterra type interactions and we show that for such models, an ESS coalition will be dynamically attainable through mutations and natural selection if the ESS coalition is also an ESNIS coalition.

... As with much of classical population dynamics, AD typically focuses on demonstrating the stability of communities-even in the face of perpetual evolutionary arms races (Cortez & Patel, 2017;Cortez et al., 2020;Jian et al., 2016;Lehtinen & Geritz, 2019;Weitz et al., 2005). However, adaptive changes in interspecies interactions can occasionally lead to catastrophic displacements in equilibrium abundances, and even extinction, as was suggested theoretically (Boldin & Kisdi, 2016;Gyllenberg & Parvinen, 2001;Leimar, 2002;Marrow et al., 1996;Parvinen, 2005;Parvinen, 2010;Parvinen & Dieckmann, 2013) and empirically (Conover & Munch, 2002;Fiegna & Velicer, 2003;Howard et al., 2004;Muir & Howard, 1999;Olsen et al., 2004;Rankin & López-Sepulcre, 2005). This phenomenon is referred as 'evolutionary suicide'. ...

... This phenomenon is referred as 'evolutionary suicide'. Analytical studies of adaptive extinction have so far been limited to only single species mutating (Boldin & Kisdi, 2016;Gyllenberg & Parvinen, 2001;Parvinen, 2005;Parvinen & Dieckmann, 2013) or coevolving two-species communities (Leimar, 2002;Marrow et al., 1996;Parvinen, 2010). ...

In a complex community, species continuously adapt to each other. On rare occasions, the adaptation of a species can lead to the extinction of others, and even its own. “Adaptive dynamics” is the standard mathematical framework to describe evolutionary changes in community interactions, and in particular, predict adaptation driven extinction. Unfortunately, most authors implement the equations of adaptive dynamics through computer simulations, that require assuming a large number of questionable parameters and fitness functions. In this study we present analytical solutions to adaptive dynamics equations, thereby clarifying how outcomes depend on any computational input. We develop general formulas that predict equilibrium abundances over evolutionary time scales. Additionally, we predict which species will go extinct next, and when this will happen.

... Correlated mutations and different mutation rates affect the speed of evolution in the two traits. This can affect convergence and evolutionarily stability (Leimar 2001;2009). However, since we only find evolutionarily and convergence stable node-type equilibria, these assumptions do not affect the results. ...

... Correlated mutations and different mutation rates affect the speed of evolution in the two traits. This can affect convergence and evolutionarily stability (Leimar 2001;2009). However, since we only find evolutionarily and convergence stable node-type equilibria, these assumptions do not affect the results. ...

... and equals either a population's genetic variancev covariance matrix (quantitative genetics approach) or the variance-covariance matrix assumed for the distribution of evolutionary innovations (adaptive dynamics approach). Our geometric framework can accommodate this generalization simply by plotting the vectors instead v 7 g of g in diagrams like figures 3 and 4. Notice that, while this may affect which parts of a trade-off curve or which interior singular phenotypes are evolutionarily attracting (Leimar 2001), evolution along a trade-off curve is onedimensional and consequently does not involve a variancecovariance matrix. Evolutionary responses to frequency-dependent disruptive selection and thus establishment of population-level polymorphisms differ in models based alternatively on quantitative genetics or adaptive dynamics. ...

... Some models may incorporate more than a single trade-off, which may cause evolution to converge on interior attractors without ever being constrained by the focal trade-off. Analyses of multidimensional adaptive dynamics are then required; these are not the focus of this article and are addressed elsewhere (Dieckmann and Law 1996;Leimar 2001;Meszéna et al. 2001). ...

Life-history evolution is determined by the interplay between natural selection and adaptive constraints. The classical ap- proach to studying constrained life-history evolution—Richard Lev- ins's geometric comparison of fitness sets and adaptive functions— is applicable when selection pressures are frequency independent. Here we extend this widely used tool to frequency-dependent selec- tion. Such selection pressures vary with a population's phenotypic composition and are increasingly recognized as ubiquitous. Under frequency dependence, two independent properties have to be dis- tinguished: evolutionary stability (an evolutionarily stable strategy cannot be invaded once established) and convergence stability (only a convergence stable strategy can be attained through small, selec- tively advantageous steps). Combination of both properties results in four classes of possible evolutionary outcomes. We introduce a geometric mode of analysis that enables predicting, for any bivariate selection problem, evolutionary outcomes induced by trade-offs of given shape, shapes of trade-offs required for given evolutionary outcomes, the set of all evolutionary outcomes trade-offs can induce, and effects of ecological parameters on evolutionary outcomes in- dependent of trade-off shape.

... Strong convergence stability amounts to attractivity of the singular point whatever the mutational covariance matrix for the deterministic evolutionary dynamics that results when the mutational steps are not only rare but also very small and time is rescaled accordingly (Leimar, 2001(Leimar, , 2005). In the case of onedimensional trait spaces attractivity of the singular point is independent of the mutational variance and the condition for strong convergence stability reduces to the usual condition for Continuous Stability, Eshel, 1983;Taylor, 1989;Geritz et al, 1998. ...

... The determination of conditions for strong convergence for haplo-diploids is an open problem. The reason is that the canonical equation of adaptive dynamics (see Dieckmann and Law, 1996;Champagnat, 2003;and in particular Durinx et al., 2008), which was used by Leimar (2001Leimar ( , 2005) to derive his criteria, takes an unusual form for haplo-diploids (Metz and De Kovel, in prep). In diploids, if there are no parental effects, gene expression becomes additive for small enough mutational steps. ...

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.

... The mechanism proposed in this theory explains the origin of individual as well as population aging thereby foregoing the need to explore questionable methods of group selection to explain the latter. Furthermore, because aging is inseparably linked with reproductive fitness, emergence of a hypothetical "Darwinian demon" [135] is impossible. Paradoxically, because its origins are directly linked with the DP, aging could be considered programmed. ...

Formulating a novel concept about the origin of human aging has been constrained by the dominance of a “classic theory” that was proposed nearly 70 years ago. Despite concern over the validity of some of its assumptions, the theory remained basic to our understanding of aging’s relationship with natural selection (NS). However, the logic upon which it rests was tested and subsequently challenged. The present theory describes the single cause of human aging consistent with Darwin’s evolutionary requirement for selection of adaptive traits. It describes an emergent property of the developmental program (DP), that is expressed upon completion of ontogenesis. It involves redundant expression of regulatory processes from the last stage of the DP. That mechanism subsequently preserves a non-aging, stable interval of unchanging NS during which reproductive fitness is achieved. Thereafter, loss of DP regulatory redundancy due to reliability limits, stochastic mutation accumulation, reproductive and a specific type of DNA damage, initiates aging which causes an inexorable decline in strength of NS to begin. It starts approximately a decade later than proposed in the classic theory. Since reproduction and aging are inextricably linked by the same emergent property, selection of that regulatory mechanism makes both traits products of NS.

... Following the initial steps of biological invasions (Sakai et al., 2001;Dlugosch and Parker, 2008), evolutionary change (i.e., local adaptation) can allow invasive populations to reach new fitness peaks (Leimar, 2002), whereby pre-existing plasticity can either accelerate or slow down rates of genetic evolution (Ghalambor et al., 2007). Parts of this variation can be explained by the direction of environmentally-induced phenotypic alterations, i.e., whether the induced phenotype is closer to (adaptive plasticity) or further away from (non-adaptive plasticity) the favoured optimum (Ghalambor et al., 2007;López-Maury et al., 2008;van Gestel and Weissing, 2018). ...

Invasive alien species (IAS) have become a major threat to ecosystems worldwide. From an evolutionary ecological perspective, they allow teasing apart the relative contributions of plasticity and evolutionary divergence in driving rapid phenotypic diversification. When IAS spread across extensive geographic ranges, climatic variation may represent a source of strong natural selection through overwinter mortality and summer heat stress. This could favour local adaptation, i.e., evolutionary divergence of certain traits. IAS, however, are likely to show plasticity in survival-related traits, and environmental fluctuation in their new distribution range could favour the maintenance of this pre-existing phenotypic plasticity. By contrast, sexually selected traits are more likely to undergo evolutionary divergence when components of sexual selection differ geographically. Here, using data from a common-garden rearing experiment of Western mosquitofish (Gambusia affinis Baird and Girard, 1853) from five populations across the species' invasive range in China, we show that invasive mosquitofish have retained plasticity in key physiological (thermal tolerances), morphological and life-history traits even 100 years after their introduction to China, but exhibit heritable population differences in several sexually selected traits, including the shape of the male copulatory organ. Adaptive plasticity of traits linked to immediate survival in different thermal environments—while likely responsible for the species' extraordinary invasion success—could slow down genetic evolution. Several sexually selected traits could diverge geographically and show rapid evolutionary change, e.g., because climate alters selective landscapes arising from mate competition as an indirect consequence of variation in overwinter mortality.

... 3 -traits distributed as observed and independent from each other (62%); 4 -traits normally distributed and correlated as observed, approximately a hyperellipsoid (51%). Specifically, of all possible trait combinations -null model 1 assumes any combination of trait values can arise and escape natural selection with equal probability (Leimar, 2002; -only 9% are realized in contemporary mammal and bird ecological strategies and are therefore currently evolutionarily viable on Earth. ...

Global biodiversity loss threatens the continued provision of ecosystem function and ecosystem services, upon which we all rely. Biodiversity is multidimensional, encompassing taxonomic, phylogenetic and ecological diversity; yet taxonomic diversity has received the majority of research effort. In this thesis, I focus on the ecological diversity of the world’s mammals and birds, based on species traits, as ecological diversity strongly relates to species’ ecological roles and to the functions species perform. I show that mammals and birds are ecologically comparable and provide complementary and comparative macroecological perspectives. I find a global trade-off between the similarity of species roles (functional redundancy) and the breadth of roles across taxa (functional dispersion) (Chapter 2). I also demonstrate different contributions of mammals and birds to functional redundancy and functional dispersion, and unique geographic patterns of redundancy and dispersion by including both taxa. I then show that the ecological diversity of mammals and birds is structured by life-history speed (fast-slow) and body mass (small-large) in one dimension, and diet (invertivore-herbivore) and habitat breadth (generalist-specialist) in the other dimension (Chapter 3). Using a probabilistic extinction framework, I predict a greater decline in ecological diversity than expected at random over the next century, shifting the mammal and bird species pool towards small, fast-lived, highly fecund, insect-eating, generalists. I also quantify ecological distinctiveness for mammals and birds (Chapter 4), identifying conservation priority species with potentially irreplaceable ecological roles. I find that high ecological distinctiveness is associated with both highly threatened species, such as Amsterdam Albatross and Sumatran rhinoceros, and non-threatened hyper-generalists, such as Lesser Black-backed Gull and wild boar. Finally, using structural equation models, I determine a strong role of trophic interactions for global patterns of mammalian species richness, but a surprisingly weak role for functional diversity and phylogenetic diversity (Chapter 5). My thesis demonstrates that ecological diversity can offer novel and complementary insights and can inform the prioritization of conservation actions. Overall, I recommend maintaining the complex ecological diversity of the world’s mammals and birds as a fundamental goal for conservation.

... is often called the the phenotypic "selection gradient" in adaptive dynamics [53,99], inclusive fitness theory [96] and quantitative genetics [87]. The "gradient" terminology derives from the fact that S(z) measures the direction of selection on the phenotype with respect to fixation probability: a positive (negative) selection gradient implies that mutants with positive (negative) δ will have a higher (lower) chance of fixing than the resident type. ...

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

... We further find that the ecological strategy space currently occupied by mammals and birds is strongly restricted (9-62% occupation of null strategy spaces, all permutation tests P ≤ 0.001; Supplementary Table 2) when compared to four alternative null models: 29 1-traits uniformly distributed and independent from each other, approximately a hypercube (9% occupation); 2-traits normally distributed and independent from each other, approximately a hypersphere (37%); 3-traits distributed as observed and independent from each other (62%); 4-traits normally distributed and correlated as observed, approximately a hyperellipsoid (51%). Specifically, of all possible trait combinations-null model 1 assumes any combination of trait values can arise and escape natural selection with equal probability 29,43 -only 9% are realized in contemporary mammal and bird ecological strategies and are therefore currently evolutionarily viable on Earth. ...

Species, and their ecological strategies, are disappearing. Here we use species traits to quantify the current and projected future ecological strategy diversity for 15,484 land mammals and birds. We reveal an ecological strategy surface, structured by life-history (fast-slow) and body mass (small-large) as one major axis, and diet (invertivore-herbivore) and habitat breadth (generalist-specialist) as the other. We also find that of all possible trait combinations, only 9% are currently realized. Based on species’ extinction probabilities we predict this limited set of viable strategies will shrink further over the next 100 years, shifting the mammal and bird species pool towards small, fast-lived, highly fecund, insect-eating, generalists. In fact, our results show that this projected decline in ecological strategy diversity is much greater than if species were simply lost at random. Thus, halting the disproportionate loss of ecological strategies associated with highly threatened animals represents a key challenge for conservation.

... 3, these statements assume threshold cases (e.g., A or A C B negative semi-definite) do not arise. Based on Theorem 6 (a), Leimar (2009) defines the concept of strong convergence stability as a u that is convergence stable for all choices of C 1 .u/. 30 He goes on to show (see also Leimar 2001) that, in a more general canonical equation where C 1 .u/ need not be symmetric but only positive definite, u is convergence stable for all such choices (called absolute convergence stability) if and only if A C B is negative definite and symmetric. ...

Evolutionary game theory developed as a means to predict the expected distribution of individual behaviors in a biological system with a single species that evolves under natural selection. It has long since expanded beyond its biological roots and its initial emphasis on models based on symmetric games with a finite set of pure strategies where payoffs result from random one-time interactions between pairs of individuals (i.e., on matrix games). The theory has been extended in many directions (including nonrandom, multiplayer, or asymmetric interactions and games with continuous strategy (or trait) spaces) and has become increasingly important for analyzing human and/or social behavior as well. This chapter initially summarizes features of matrix games before showing how the theory changes when the two-player game has a continuum of traits or interactions become asymmetric. Its focus is on the connection between static game-theoretic solution concepts (e.g., ESS, CSS, NIS) and stable evolutionary outcomes for deterministic evolutionary game dynamics (e.g., the replicator equation, adaptive dynamics). © Springer International Publishing AG, part of Springer Nature 2018.

... Within the uni- verse of possible ecological strategies, the trait space actually occu- pied by a species pool is restricted by trade-offs among traits, as well as phylogenetic and ecological constraints. First, life-history trade-offs restrict trait spaces, for organisms cannot optimize their performance in all niche dimensions simultaneously (Leimar, 2001). Trade-offs between body form and physiological functions also limit the range of possible trait combinations. ...

Plant functional traits directly affect ecosystem functions. At the species level, trait combinations depend on trade-offs representing different ecological strategies, but at the community level trait combinations are expected to be decoupled from these trade-offs because different strategies can facilitate co-existence within communities. A key remaining question is to what extent community-level trait composition is globally filtered and how well it is related to global vs. local environmental drivers. Here, we perform a global, plot-level analysis of trait-environment relationships, using a database with more than 1.1 million vegetation plots and 26,632 plant species with trait information. Although we found a strong filtering of 17 functional traits, similar climate and soil conditions support communities differing greatly in mean trait values. The two main community trait axes which capture half of the global trait variation (plant stature and resource acquisitiveness) reflect the trade-offs at the species level but are weakly associated with climate and soil conditions at the global scale.Similarly, within-plot trait variation does not vary systematically with macro-environment. Our results indicate that, at fine spatial grain, macro-environmental drivers are much less important for functional trait composition than has been assumed from floristic analyses restricted to co-occurrence in large grid cells. Instead, trait combinations seem to be predominantly filtered by local-scale factors such as disturbance, fine-scale soil conditions, niche partitioning or biotic interactions.

... Correlated mutations and different mutation rates affect the speed of evolution in the two traits. This can affect convergence and evolutionarily stability (Leimar 2001(Leimar , 2009). ...

Despite empirical evidence for a positive relationship between dispersal and self‐fertilisation (selfing), theoretical work predicts that these traits should always be negatively correlated, and the Good Coloniser Syndrome of high dispersal and selfing (Cf. Baker's Law) should not evolve. Critically, previous work assumes that adult density is spatiotemporally homogeneous, so selfing results in identical offspring production for all patches, eliminating the benefit of dispersal for escaping from local resource competition. We investigate the joint evolution of dispersal and selfing in a demographically structured metapopulation model where local density is spatiotemporally heterogeneous due to extinction‐recolonisation dynamics. Selfing alleviates outcrossing failure due to low local density (an Allee Effect) while dispersal alleviates competition through dispersal of propagules from high‐ to low‐density patches. Because local density is spatiotemporally heterogenous in our model, selfing does not eliminate heterogeneity in competition, so dispersal remains beneficial even under full selfing. Hence the Good Coloniser Syndrome is evolutionarily stable under a broad range of conditions, and both negative and positive relationships between dispersal and selfing are possible, depending on the environment. Our model thus accommodates positive empirical relationships between dispersal and selfing not predicted by previous theoretical work and provides additional explanations for negative relationships.
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... Within the universe of possible ecological strategies, the trait space actually occupied by a species pool is restricted by trade-offs among traits, as well as phylogenetic and ecological constraints. First, life history trade-offs restrict trait spaces, for organisms cannot optimize their performance in all niche dimensions simultaneously (Leimar, 2001). Trade-offs between body form and physiological functions also limit the range of possible trait combinations. ...

Functional traits are commonly used in predictive models that link environmental drivers and community structure to ecosystem functioning. A prerequisite is to identify robust sets of continuous axes of trait variation, and to understand the ecological and evolutionary constraints that result in the functional trait space occupied by interacting species. Despite their diversity and role in ecosystem functioning, little is known of the constraints on the functional trait space of invertebrate biotas of entire biogeographic regions.
We examined the ecological strategies and constraints underlying the realized trait space of aquatic invertebrates, using data on 12 functional traits of 852 taxa collected in tank bromeliads from Mexico to Argentina. Principal Component Analysis was used to reduce trait dimensionality to significant axes of trait variation, and the proportion of potential trait space that is actually occupied by all taxa was compared to null model expectations. Permutational Analyses of Variance were used to test whether trait combinations were clade‐dependent.
The major axes of trait variation represented life‐history strategies optimizing resource use and antipredator adaptations. There was evidence for trophic, habitat, defence and life‐history niche axes. Bromeliad invertebrates only occupied 16%–23% of the potential space within these dimensions, due to greater concentrations than predicted under uniform or normal distributions. Thus, despite high taxonomic diversity, invertebrates only utilized a small number of successful ecological strategies.
Empty areas in trait space represented gaps between major phyla that arose from biological innovations, and trait combinations that are unviable in the bromeliad ecosystem. Only a few phylogenetically distant genera were neighbouring in trait space. Trait combinations aggregated taxa by family and then by order, suggesting that niche conservatism was a widespread mechanism in the diversification of ecological strategies.
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... [14]: Dieckmann and Law 1996). The criteria for stability can similarly be extended to multidimensional models (Leimar 2009), although the situation is complicated by the fact that in multidimensional trait-space mutation can influence the direction of evo-lutionary change in a way that is not possible in one dimension (Leimar 2002). Assessing stability can be likewise very challenging in models of structured populations (Day and Taylor 1998;Ajar 2003;Rousset 2004), but the underlying concepts are nevertheless similar to the simple cases studied here. ...

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

... This null model assumes that each of the six traits represents an independent axis of specialization (i.e. the traits define a six-dimensional manifold) and that the occurrence of extreme and central values is equally probable. This uni- form independent trait distribution represents a "Darwinian Demon" 35 scenario, where any combination of trait values can arise from mutation and escape from the natural selection process with equal probability. Simulated data are generated by randomly and independently sampling from independent uniform distribu- tions whose range limits are constrained to the 0.025 and 0.975 quantiles of the observed trait values. ...

... These traits are the probabilities s k that an individual assessing a site with k individuals will attach to that site. The principles behind adaptive dynamics of vectorvalued traits are no different from those of a single trait value, but the conditions for assessing convergence stability (Leimar 2001(Leimar , 2009) and evolutionary stability (Geritz et al. 2016) need to be extended to several dimensions. ...

Species that compete for access to or use of sites, such as parasitic mites attaching to honey bees or apple maggots laying eggs in fruits, can potentially increase their fitness by carefully selecting sites at which they face little or no competition. Here, we systematically investigate the evolution of site-selection strategies among animals competing for discrete sites. By developing and analyzing a mechanistic and population-dynamical model of site selection in which searching individuals encounter sites sequentially and can choose to accept or continue to search based on how many conspecifics are already there, we give a complete characterization of the different site-selection strategies that can evolve. We find that evolution of site-selection stabilizes population dynamics, promotes even distribution of individuals among sites, and occasionally causes evolutionary suicide. We also discuss the broader implications of our findings and propose how they can be reconciled with an earlier study (Nonaka et al. in J Theor Biol 317:96–104, 2013) that reported selection toward ever higher levels of aggregation among sites as a consequence of site-selection.

... Fortunately, the situation is less bleak than it may appear at fi rst sight. Some methods of evolutionary game theory appear rather robust and do not depend on genetic details (Leimar 2001 ). It can also be shown that in the limiting case of weak selection the conclusions derived from fi tness considerations are quite robust (Nagylaki et al . ...

Overview Evolutionary game theory may have done more to stimulate and refine research in animal behaviour than any other theoretical perspective. In this chapter, we will review some of the insights gained by applying game theory to animal behaviour. Our emphasis is on conceptual issues rather than on technical detail. We start by introducing some of the classical models, including the Hawk–Dove game and the Prisoner's Dilemma game. Then we discuss in detail the main ingredients of a game-theoretical approach: strategies, payoffs and ‘solution concepts’ such as evolutionary stability. It should become clear that first-generation models like the Hawk–Dove game, while of enormous conceptual importance, have severe limitations when applied to real-world scenarios. We close with a sketch of what we see as the most important gaps in our knowledge, and the most relevant current developments in evolutionary game theory. Introduction Social behaviour involves the interaction of several individuals. Therefore within most social contexts the best thing to do depends on what others are doing. In other words, within social contexts selection is typically frequency-dependent (Ayala & Campbell 1974, Heino et al. 1998). Game theory was originally formulated to predict behaviour when there is frequency dependence in economics, for example competition between firms (von Neumann & Morgenstern 1944, Luce & Raiffa 1957).

... Negative definiteness is a strong requirement. However, it appears that most published models that describe the evolutionary dynamics of a multivariate trait by means of the adaptive dynamics approximation fulfill this criterion (Leimar 2001;Vukics et al. 2003;Ackermann and Doebeli 2004;Beltman and Metz 2005;Ito and Shimada 2007;Ravigné et al. 2009;Doebeli and Ispolatov 2010;Svardal et al. 2011Svardal et al. , 2014. It is therefore of some relevance to know whether for this special but apparently regularly occurring case a similar dependency exists as for one-dimensional trait spaces. ...

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 [Formula: see text] in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to [Formula: see text]. 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 [Formula: see text], 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.

... Future studies will aim at investigating the possible differences in mating success between males and females with varying brain size to elucidate the link between brain size, cognition, attractiveness and mating success. But are the larger-brained males developing towards 'Darwinian demons' (Leimar, 2001) due to their superior cognitive ability and likely greater mating success? This is unlikely, because larger-brained guppies show a decreased fecundity (Kotrschal et al., 2013a ) and their smaller guts (Kotrschal et al., 2013a) may render them ill-adapted for low-food environments. ...

Brain size is an energetically costly trait to develop and maintain. Investments into other costly aspects of an organism's biology may therefore place important constraints on brain size evolution. Sexual traits are often costly and could therefore be traded-off against neural investment. However, brain size may itself be under sexual selection through mate choice on cognitive ability. Here we use guppy (Poecilia reticulata) lines selected for large and small brain size relative to body size to investigate the relationship between brain size, a large suite of male primary and secondary sexual traits, and body condition index. We found no evidence for trade-offs between brain size and sexual traits. Instead, larger-brained males had higher expression of several primary and pre-copulatory sexual traits - they had longer genitalia, were more colourful and developed longer tails than smaller-brained males. Larger-brained males were also in better body condition when housed in single-sex groups. There was no difference in post-copulatory sexual traits between males from the large- and small-brained lines. Our data do not support the hypothesis that investment into sexual traits is an important limiting factor to brain size evolution but instead suggest that brain size and several sexual traits are positively genetically correlated. This article is protected by copyright. All rights reserved.
This article is protected by copyright. All rights reserved.

... Obviously, such trajectories are highly improbable; real trajectories may converge even if the most extreme path does not. Convergence of the most extreme path has therefore been termed absolute convergence (Leimar, 2001). ...

Background: A recently developed geometric method makes it possible to explore how the shape of a trade-off determines the outcome of adaptive evolution in any complex model and without committing to a particular functional form of the trade-off function. Aim: Extend the method to the co-evolution of two species. (The two species may be distantly related such as a predator and its prey, or may be closely related like two strategies produced by evolutionary branching.) Results: Thresholds of the local convexity of the trade-off functions are obtained that guar-antee evolutionary and convergence stability when a given species pair is singular. In contrast to the single-species case, the condition for convergence is sufficient but not necessary. Criteria for evolutionary branching generalize from the single-species case. A cross-derivative of the invasion fitness determines whether evolutionary branching is possible; this quantity is independent of the trade-offs and if it is negative at a certain species pair, then the trade-offs can be chosen such that evolutionary branching occurs at this point. Worked example: A simple predator–prey model shows how these results can be used to identify trade-off functions such that evolution leads to an evolutionarily stable species pair or to evolutionary branching in either species.

... Combining the previous considerations shows that a good recipe for numerically finding possibly attracting candidate ESSes is running the canonical equation for a reasonable sample of initial conditions and mutational covariance matrices. Necessary and sufficient conditions for a guaranteed local convergence, independent of the mutational covariance matrix can be found in [21,22,23]. ...

The simplest behaviour one can hope for when studying a mathematical model of evolution by natural selection is when evolution
always maximises the value of some function of the trait under consideration, thus providing an absolute measure of fitness
for the model. We survey the role of such models, known as optimisation models in the literature, and give some general results
concerning the question of when a model turns out to be an optimisation model. The results presented vary from more abstract
results with a game-theoretical flavour to more detailed considerations of life history models. We also give a number of concrete
examples and discuss the role of optimisation models in the wider framework of adaptive dynamics.

... The latter three factors may all be functions of trait values. When equation (1) is generalized to multiple traits per species, C assumes a blockdiagonal structure, with the off-diagonal elements in each species' block describing additive genetic or mutational covariances among that species' traits (Lande, 1979;Falconer and Mackay, 1996;Leimar, 2001). ...

Question: What are the evolutionary consequences of extinctions in ecological communities?
Can evolution restore pre-extinction communities by replacing lost ecological strategies
with similar ones, or will communities change in fundamental ways and never be the same
again?
Mathematical approach: We develop and explore a new framework based on evolutionary
domains of attraction (EDAs), defined as sets of strategy combinations from which a particular
ESS community can be attained through gradual evolution. The latter dynamics may include
three types of evolutionary processes: continuous strategy adaptation in response to directional
selection, evolutionary branching in response to disruptive selection, and evolutionarily driven
extinction.
Key assumptions: We consider gradual frequency-dependent evolution in ecological
communities, with evolutionary dynamics being fully determined by the strategy composition
of a community’s resident species.
Results: The EDA approach distinguishes ESS communities that gradual evolution can
restore after extinctions from ESS communities for which this option does not exist or is
constrained. The EDA approach also offers a natural definition of ‘evolutionary keystone
species’ as species whose removal causes a community to shift from one EDA to another.
Our study highlights that environmentally driven extinctions can readily cause such shifts.
We explain why the evolutionary attainability of an ESS community through gradual
evolution from a single precursor species does not imply its evolutionary restorability
after extinctions. This shows that evolution driven by frequency-dependent selection may
lead to ‘Humpty-Dumpty’ effects and community closure on an evolutionary time scale.
By establishing EDAs for several example food webs, we discover that evolutionarily
driven extinctions may be crucially involved in the evolutionary restoration of ESS
communities.

... The central goal of adaptive dynamics theory is to identify and classify the stability properties of evolutionary singularities [16,32,36]. The classification is complete for one-dimensional traits [16,37,38]; the multi-dimensional case is found in the studies of Leimar [39,40]. In general, multiple evolutionary singularities may occur, some attractive, some repelling. ...

Adaptive dynamics theory has been devised to account for feedbacks between ecological and evolutionary processes. Doing so opens new dimensions to and raises new challenges about evolutionary rescue. Adaptive dynamics theory predicts that successive trait substitutions driven by eco-evolutionary feedbacks can gradually erode population size or growth rate, thus potentially raising the extinction risk. Even a single trait substitution can suffice to degrade population viability drastically at once and cause 'evolutionary suicide'. In a changing environment, a population may track a viable evolutionary attractor that leads to evolutionary suicide, a phenomenon called 'evolutionary trapping'. Evolutionary trapping and suicide are commonly observed in adaptive dynamics models in which the smooth variation of traits causes catastrophic changes in ecological state. In the face of trapping and suicide, evolutionary rescue requires that the population overcome evolutionary threats generated by the adaptive process itself. Evolutionary repellors play an important role in determining how variation in environmental conditions correlates with the occurrence of evolutionary trapping and suicide, and what evolutionary pathways rescue may follow. In contrast with standard predictions of evolutionary rescue theory, low genetic variation may attenuate the threat of evolutionary suicide and small population sizes may facilitate escape from evolutionary traps.

... The question of dynamical attainability of an ESS has received extensive attention (e.g. Geritz et al., 1998;Kisdi, 1999;Leimar 2001;Apaloo 2003;Vincent et al. 1996; and references therein). In particular, Apaloo (1997) and references therein, has addressed the question of dynamical attainability with the ESNIS concept and pointed out that dynamical attainability of the precise value of an ESS may not be possible when the ESS is not a NIS in a monomorphic population evolution in which no polymorphisms occur. ...

The evolutionary stability concepts continuously stable strategies (CSS) and evolutionarily stable neighborhood invader strategies (ESNIS) share two properties in common. First, they are both evolutionarily stable strategies (ESS). Secondly, given any strategy in the close neighborhood of the CSS or ESNIS, there are some strategies that are closer to the CSS or ESNIS that can invade it. An ESNIS is a CSS but the converse is not true in general. We examine evolutionary adaptive dynamics in the neighborhood of a CSS that is not an ESNIS. We show that if an evolutionary game possesses a CSS which is not an ESNIS, the succession of strategies mediated by natural selection become arbitrarily close to the CSS but the precise value of the CSS cannot be attained unless the CSS is the first strategy to invade into the environment and is henceforth never perturbed. Thus if evolution does not start with the CSS that is not an ESNIS, we will have a phenomenon of bounded evolutionary succession that does not come to an end. The analysis is applied to a class of monomorphic population evolutionary game models in which species ecological interaction is modeled by the Lotka-Volterra equations.

We examined the functional strategies and the trait space of 596 European taxa of freshwater macroinvertebrates characterized by 63 fuzzy coded traits belonging to 11 trait groups. Principal component analysis was used to reduce trait dimensionality, to explain ecological strategies, and to quantify the trait space occupied by taxa. Null models were used to compare observed occupancy with theoretical models, and randomization-based analyses were performed to test whether taxonomic relatedness, a proxy of phylogenetic signal, constrains the functional trait space of freshwater macroinvertebrates. We identified four major strategies along which functional traits of the taxa examined show trade-offs. In agreement with expectations and in contrast to existing evidence we found that life cycles and aquatic strategies are important in shaping functional structure of freshwater macroinvertebrates. Our results showed that the taxonomic groups examined fill remarkably different niches in the functional trait space. We found that the functional trait space of freshwater macroinvertebrates is reduced compared to the range of possibilities that would exist if traits varied independently. The observed decrease was between 23.44 and 44.61% depending on the formulation of the null expectations. We demonstrated also that taxonomic relatedness constrains the functional trait space of macroinvertebrates.

Modern evolutionary theory provides a theoretical framework for functional analyses of animal behaviour. In order to investigate the adaptive value of individual behaviour patterns, it is necessary to operationalize fitness and to characterize the evolutionary mechanisms that influence it. In terms of the most important fitness components - survival and reproductive success - four basic problems that each individual must successfully master can be derived: finding food, avoiding being eaten, reproducing successfully and raising offspring. Traits determining the reproductive component of fitness are identical with some of the most important life history traits. In this chapter, I will therefore outline the most important variables, mechanisms, processes and relationships among behaviour, evolution and life histories to provide a basis for the subsequent chapters, which will focus on these four basic problems.

Die moderne Evolutionstheorie liefert einen theoretischen Rahmen für funktionale Analysen einzelner Verhaltensweisen. Zur Untersuchung der Angepasstheit des Verhaltens ist es notwendig, Fitness zu operationalisieren und die evolutionären Mechanismen, die sie beeinflussen, näher zu charakterisieren. Aus den wichtigsten Fitnesskomponenten – Überleben und Fortpflanzungserfolg – lassen sich vier Grundprobleme ableiten, die jedes Individuum erfolgreich meistern muss: Fressen, Nicht-gefressen-Werden, Fortpflanzung und Jungenaufzucht. Merkmale, welche die Fortpflanzungskomponente der Fitness betreffen, sind identisch mit einigen der wichtigsten Merkmale der Life History (Lebensgeschichte). In diesem Kapitel skizziere ich die wichtigsten Variablen, Mechanismen, Prozesse und Zusammenhänge zwischen Verhalten, Evolution und Life Histories, um eine Grundlage für das Verständnis der nachfolgenden Kapitel zu schaffen, die sich inhaltlich an diesen vier Grundproblemen ausrichten.

We propose a novel mathematical model for a metapopulation in which dispersal occurs on two levels: juvenile dispersal from the natal site is mandatory but it may take place either locally within the natal patch or globally between patches. Within each patch, individuals live in sites. Each site can be inhabited by at most one individual at a time and it may be of high or low quality. A disperser immigrates into a high-quality site whenever it obtains one, but it immigrates into a low-quality site only with a certain probability that depends on the time within the dispersal season. The vector of these low-quality-site-acceptance probabilities is the site-selection strategy of an individual. We derive a proxy for the invasion fitness in this model and study the joint evolution of long-distance-dispersal propensity and site-seletion strategy. We focus on the way different ecological changes affect the evolutionary dynamics and study the interplay between global patch-to-patch dispersal and local site-selection. We show that ecological changes affect site-selection mainly via the severeness of competition for sites, which often leads to effects that may appear counterintuitive. Moreover, the metapopulation structure may result in extremely complex site-selection strategies and even in evolutionary cycles. The propensity for long-distance dispersal is mainly determined by the metapopulation-level ecological factors. It is, however, also strongly affected by the winter-survival of the site-holders within patches, which results in surprising non-monotonous effects in the evolution of site-selection due to interplay with long-distance dispersal. Altogether, our results give new additional support to the recent general conclusion that evolution of site-selection is often dominated by the indirect factors that take place via density-dependence, which means that evolutionary responses can rarely be predicted by intuition.

We propose a mathematical model that enables the evolutionary analysis of site-selection process of dispersing individuals that encounter sites of high or low quality. Since each site can be inhabited by at most one individual, all dispersers are not able to obtain a high-quality site. We study the evolutionary dynamics of the low-quality-site acceptance as a function of the time during the dispersal season using adaptive dynamics. We show that environmental changes affect the evolutionary dynamics in two ways: directly and indirectly via density-dependent factors. Direct evolutionary effects usually follow intuition, whereas indirect effects are often counter-intuitive and hence difficult to predict without mechanistic modeling. Therefore, the mechanistic derivation of the fitness function, with careful attention on density- and frequency dependence, is essential for predicting the consequences of environmental changes to site selection. For example, increasing fecundity in high-quality sites makes them more tempting for dispersers and hence the direct effect of this ecological change delays the acceptance of low-quality sites. However, increasing fecundity in high-quality sites also increases the population size, which makes the competition for sites more severe and thus, as an indirect effect, forces evolution to favor less picky individuals. Our results indicate that the indirect effects often dominate the intuitive effects, which emphasizes the need for mechanistic models of the immigration process.

Evolutionary game theory developed as a means to predict the expected distribution of individual behaviors in a biological system with a single species that evolves under natural selection. It has long since expanded beyond its biological roots and its initial emphasis on models based on symmetric games with a finite set of pure strategies where payoffs result from random one-time interactions between pairs of individuals (i.e., on matrix games). The theory has been extended in many directions (including nonrandom, multiplayer, or asymmetric interactions and games with continuous strategy (or trait) spaces) and has become increasingly important for analyzing human and/or social behavior as well. This chapter initially summarizes features of matrix games before showing how the theory changes when the two-player game has a continuum of traits or interactions become asymmetric. Its focus is on the connection between static game-theoretic solution concepts (e.g., ESS, CSS, NIS) and stable evolutionary outcomes for deterministic evolutionary game dynamics (e.g., the replicator equation, adaptive dynamics).

Jedes Tier durchläuft einen Lebenszyklus: es wird geboren, wächst, pflanzt sich fort und stirbt. Die Details eines Lebenszyklus unterscheiden sich dabei stark zwischen Arten sowie teilweise zwischen Individuen derselben Art. Diese Variabilität wird durch verschiedene Life history-Strategien beschrieben, mit denen Individuen versuchen, ihre Fitness zu optimieren.

Jedes Tier durchläuft einen Lebenszyklus: es wird geboren, wächst, pflanzt sich fort und stirbt. Die Details eines Lebenszyklus unterscheiden sich dabei stark zwischen Arten sowie teilweise zwischen Individuen derselben Art. Diese Variabilität wird durch verschiedene Life history-Strategien beschrieben mit denen Individuen versuchen, ihre Fitness zu optimieren. Innerartliche Variabilität in Merkmalen, die Life history-Strategien charakterisieren, reflektiert demnach individuelle Anpassungen. Weil manche Verhaltensweisen direkte Bezüge zu erfolgreichem Überleben und Fortpflanzen aufweisen, ist es daher im Rahmen einer ultimaten Analyse des Verhaltens (→ Kapitel 1.1) notwendig, bestimmte Verhaltensweisen und -muster im Kontext von Life history-Strategien zu betrachten. Außerdem erfordert es die Natur der Bestandteile von Strategien, dass sich Individuen entscheiden, was sie als nächstes tun. Life history-Strategien haben daher auch wichtige Konsequenzen für das Verhalten; oft in der Form, dass bestimmte Verhaltensweisen mehr oder weniger wahrscheinlich werden.
Ich möchte in diesem Kapitel die wichtigsten Life history-Merkmale näher beleuchten und dabei deren Verbindungen mit dem Verhalten der Tiere betonen. Dieser Ansatz ist notwendig, um zu verstehen, wie eng einzelne Verhaltensmerkmale im Lauf des Lebens eines Individuums mit anderen Aspekten der Physiologie, Anatomie und Ökologie eines Organismus verzahnt und mit diesen funktionell verknüpft sind.

A full text is currently unavailable.
Suggest contact the lead author: sdiaz@efn.uncor.edu
Kind regards,
andy g

Earth is home to a remarkable diversity of plant forms and life histories, yet comparatively few essential trait combinations have proved evolutionarily viable in today’s terrestrial biosphere. By analysing worldwide variation in six major traits critical to growth, survival and reproduction within the largest sample of vascular plant species ever compiled, we found that occupancy of six-dimensional trait space is strongly concentrated, indicating coordination and trade-offs. Three-quarters of trait variation is captured in a two-dimensional global spectrum of plant form and function. One major dimension within this plane reflects the size of whole plants and their parts; the other represents the leaf economics spectrum, which balances leaf construction costs against growth potential. The global plant trait spectrum provides a backdrop for elucidating constraints on evolution, for functionally qualifying species and ecosystems, and for improving models that predict future vegetation based on continuous variation in plant form and function.

Questions: Co-evolutionary models with one to multidimensional strategies can result in stable coalitions of many strategies. Are coalition strategies that are evolutionarily stable (ESS) and neighbourhood invader (NIS) also convergence stable? What is the implication for co-evolutionary models? Mathematical methods: Optimization of the fitness function and dynamical systems based on the selection gradient of the fitness function. Key assumptions: The fitness of a trait depends on its strategy value and on the environment consisting of strategy values of other traits and their population sizes. Co-evolutionary strategies in a close neighbourhood of a singular point of the canonical equation maintain stable population dynamic equilibria. Conclusions: In single-species evolutionary games with a multidimensional strategy set, a strategy that is both an ESS and NIS is also a strong convergence stable strategy and thus convergence stable. In co-evolutionary games, this implication is not guaranteed and there can never be a strategy that is strong NIS. Therefore, 'fast evolution', which can occur in single-species evolution when a singular point is both ESS and NIS, may not occur in co-evolutionary models.

Theoretical models suggest that resource competition can lead to the adaptive splitting of consumer populations into diverging lineages, that is, to adaptive diversification. In general, diversification is likely if consumers use only a narrow range of resources and thus have a small niche width. Here we use analytical and numerical methods to study the consequences for diversification if the niche width itself evolves. We found that the evolutionary outcome depends on the inherent costs or benefits of widening the niche. If widening the niche did not have costs in terms of overall resource uptake, then the consumer evolved a niche that was wide enough for disruptive selection on the niche position to vanish; adaptive diversification was no longer observed. However, if widening the niche was costly, then the niche widths remained relatively narrow, allowing for adaptive diversification in niche position. Adaptive diversification and speciation resulting from competition for a broadly distributed resource is thus likely if the niche width is fixed and relatively narrow or free to evolve but subject to costs. These results refine the conditions for adaptive diversification due to competition and formulate them in a way that might be more amenable for experimental investigations.

The evolution of social traits remains one of the most fascinating and feisty topics in evolutionary biology even after half a century of theoretical research. W. D. Hamilton shaped much of the field initially with his 1964 papers that laid out the foundation for understanding the effect of genetic relatedness on the evolution of social behavior. Early theoretical investigations revealed two critical assumptions required for Hamilton's rule to hold in dynamical models: weak selection and additive genetic interactions. However, only recently have analytical approaches from population genetics and evolutionary game theory developed sufficiently so that social evolution can be studied under the joint action of selection, mutation, and genetic drift. We review how these approaches suggest two timescales for evolution under weak mutation: (i) a short-term timescale where evolution occurs between a finite set of alleles, and (ii) a long-term timescale where a continuum of alleles are possible and populations evolve continuously from one monomorphic trait to another. We show how Hamilton's rule emerges from the short-term analysis under additivity and how non-additive genetic interactions can be accounted for more generally. This short-term approach reproduces, synthesizes, and generalizes many previous results including the one-third law from evolutionary game theory and risk dominance from economic game theory. Using the long-term approach, we illustrate how trait evolution can be described with a diffusion equation that is a stochastic analogue of the canonical equation of adaptive dynamics. Peaks in the stationary distribution of the diffusion capture classic notions of convergence stability from evolutionary game theory and generally depend on the additive genetic interactions inherent in Hamilton's rule. Surprisingly, the peaks of the long-term stationary distribution can predict the effects of simple kinds of non-additive interactions. Additionally, the peaks capture both weak and strong effects of social payoffs in a manner difficult to replicate with the short-term approach. Together, the results from the short and long-term approaches suggest both how Hamilton's insight may be robust in unexpected ways and how current analytical approaches can expand our understanding of social evolution far beyond Hamilton's original work.
Copyright © 2015. Published by Elsevier Inc.

Disruptive selection due to ecological causes can lead to different types of phenotypic polymorphism. For a broad range of ecological scenarios, we investigate the odds that disruptive selection leads to sexual dimorphism relative to polymorphisms that appear after evolutionary branching. These involve genetic polymorphism, such as sympatric species or Mendelian genes with strong dominance-recessivity. When models that allow for sexual dimorphism are compared with constrained models with equal phenotypes in males and females, a sexual dimorphism is expected to evolve instead of the evolutionary branching in the constrained model. This is an important general result on the odds of different types of ecological polymorphism. It implies that the possibility for sympatric speciation caused by ecological selection pressures can be removed by the evolution of ecological differences between the sexes. Evolutionary branching becomes more likely when: (1) there is a strong constraint on sex differentiation; (2) secondary branching events occur after sexual dimorphism has already evolved; (3) assortative mate choice occurs before trait divergence starts. The possibility of sexual selection driving sympatric speciation is not affected by our conclusions.

Cooperation is surprisingly common in life despite of its vulnerability to selfish cheating, i.e. defecting. Defectors do not contribute to common resources but take the advantage of cooperators' investments. Therefore, the emergence and maintenance of cooperation have been considered irrational phenomena. In this study, we focus on plastic, quantitative cooperation behaviour, especially on its evolution. We assume that individuals are capable to sense the population density in their neighbourhood and adjust their real-valued investments on public goods based on that information. The ecological setting is described with stochastic demographic events, e.g. birth and death, occurring at individual level. Individuals form small populations, which further constitute a structured metapopulation. For evolutionary investigations, we apply the adaptive dynamics framework. The cost of cooperative investment is incorporated into the model in two ways, by decreasing the birth rate or by increasing the death rate. In the first case, density-dependent cooperation evolves to be a decreasing function of population size as expected. In the latter case, however, the density-dependent cooperative investment can have a qualitatively different form as it may evolve to be highest in intermediate-sized populations. Indeed, we emphasize that some details in modelling may have a significant impact on the results obtained.

We investigate the joint evolution of public goods cooperation and dispersal in a metapopulation model with small local populations. Altruistic cooperation can evolve due to assortment and kin selection, and dispersal can evolve because of demographic stochasticity, catastrophes and kin selection. Metapopulation structures resulting in assortment have been shown to make selection for cooperation possible. But how does dispersal affect cooperation and vice versa, when both are allowed to evolve as continuous traits? We found four qualitatively different evolutionary outcomes. (1) Monomorphic evolution to full defection with positive dispersal. (2) Monomorphic evolution to an evolutionarily stable state with positive cooperation and dispersal. In this case, parameter changes selecting for increased cooperation typically also select for increased dispersal. (3) Evolutionary branching can result in the evolutionarily stable coexistence of defectors and cooperators. Although defectors could be expected to disperse more than cooperators, here we show that also the opposite case is possible: Defectors tend to disperse less than cooperators when the total amount of cooperation in the dimorphic population is low enough. (4) Selection for too low cooperation can cause the extinction of the evolving population. For moderate catastrophe rates dispersal needs to be initially very frequent for evolutionary suicide to occur. Although selection for less dispersal in principle could prevent such evolutionary suicide, in most cases this rescuing effect is not sufficient, because selection in the cooperation trait is typically much stronger. If the catastrophe rate is large enough, a part of the boundary of viability can be evolutionarily attracting with respect to both strategy components, in which case evolutionary suicide is expected from all initial conditions.

In this article we further develop the theory of adaptive dynamics of function-valued traits. Previous work has concentrated on models for which invasion fitness can be written as an integral in which the integrand for each argument value is a function of the strategy value at that argument value only. For this type of models of direct effect, singular strategies can be found using the calculus of variations, with singular strategies needing to satisfy Euler's equation with environmental feedback. In a broader, more mechanistically oriented class of models, the function-valued strategy affects a process described by differential equations, and fitness can be expressed as an integral in which the integrand for each argument value depends both on the strategy and on process variables at that argument value. In general, the calculus of variations cannot help analyzing this much broader class of models. Here we explain how to find singular strategies in this class of process-mediated models using optimal control theory. In particular, we show that singular strategies need to satisfy Pontryagin's maximum principle with environmental feedback. We demonstrate the utility of this approach by studying the evolution of strategies determining seasonal flowering schedules.

Continuous-trait game theory fills the niche of enabling analytically solvable models of the evolution of biologically realistically complex traits. Game theory provides a mathematical language for under- standing evolution by natural selection. Continuous-trait game the- ory starts with the notion of an evolutionarily stable strategy (ESS) and adds the concept of convergence stability (that the ESS is an evo- lutionary attractor). With these basic tools in hand, continuous-trait game theory can be easily extended to model evolution under con- ditions of disruptive selection and speciation, nonequilibrium pop- ulation dynamics, stochastic environments, coevolution, and more. Many models applying these tools to evolutionary ecology and co- evolution have been developed in the past two decades. Going for- ward we emphasize the communication of the conceptual simplicity and underlying unity of ideas inherent in continuous-trait game the- ory and the development of new applications to biological questions.

Adaptive dynamics (AD) is a recently developed framework geared towards making the transition from micro-evolution to long-term
evolution based on a time scale separation approximation. This assumption allows defining the fitness of a mutant as the rate
constant of initial exponential growth of the mutant population in the environment created by the resident community dynamics.
This definition makes that all resident types have fitness zero. If in addition it is assumed that mutational steps are small,
evolution can be visualized as an uphill walk in a fitness landscape that keeps changing as a result of the evolution it engenders.
The chapter summarises the main tools for analysing special eco-evolutionary models based on these simplifications. In addition
it describes a number of general predictions that directly derive from the AD perspective a such, without making any further
assumptions.
KeywordsAdaptive dynamics-Fitness landscapes-Evo-devo-Meso-evolution-Macro-evolution-Internal selection-Speciation

Fisher's runaway process is the standard explanation of the evolution of exaggerated female preferences. But mathematical formulations of Fisher's process (haploid and additive diploid) show it cannot cause stable exaggeration if female preference carries a cost. At equilibrium female fitness must be maximized. Our analysis shows that evolutionary stable exaggeration of female preference can be achieved if mutation pressure on the male character is biased, that is, mutation has a directional effect. At this equilibrium female fitness is not maximized. We discuss the reasons and evidence for believing that mutation pressure is typically biased. Our analysis highlights the previously unacknowledged importance of biased mutation for sexual selection.

Fisher's runaway process is the standard explanation of the evolution of exaggerated female preferences. But mathematical formulations of Fisher's process (haploid and additive diploid) show it cannot cause stable exaggeration if female preference carries a cost. At equilibrium female fitness must be maximized. Our analysis shows that evolutionary stable exaggeration of female preference can be achieved if mutation pressure on the male character is biased, that is, mutation has a directional effect. At this equilibrium female fitness is not maximized. We discuss the reasons and evidence for believing that mutation pressure is typically biased. Our analysis highlights the previously unacknowledged importance of biased mutation for sexual selection.

In a population that is fixed at an evolutionarily stable strategy (ESS), no mutant strategy can invade and spread. If, however, the strategy set is continuous, one can ask which mutations can be established in a population that is fixed not at an ESS but, rather, at a different, nearby strategy. This question gives rise to a possible distinction between the various ESSs with respect to their dynamic stability characteristics and is treated here for the case of asymmetric games. Two distinct types of ESSs can exist in such games: ESSs that are continuously stable (CSSs) and ESSs that are not. Any strategy in the neighborhood of a continuously stable ESS can always be invaded by mutants that are closer to the ESS. In contrast, any neighborhood of an ESS that is not a CSS contains a nonzero measure set of strategies that are not immune to any mutation that is further away from the ESS. Thus, in natural situations, one can expect more frequently to find populations at (or near) an ESS that is a CSS than at (or near) an ESS that is not continuously stable. The ideas are illustrated by two examples, the parental investment conflict and the dispersal conflict between males and females.

A model for the coevolution of body size of predators and their prey is described. Body sizes are assumed to affect the interactions between individuals, and the Lotka-Volterra population dynamics arising from these interactions provide the driving force for evolutionary change. The space of phenotypes of predator and prey contains a region, oval in shape, in which the predator and prey species coexist. Within this region, evolutionarily stable strategies (ESSS) and evolutionary saddles may be found, and coevolution may tend to an ESS, develop a Red Queen dynamic, or move to predator extinction. Ten qualitatively distinct kinds of phenotype space are described, depending mainly on the number of ESSS and evolutionary saddles. These varied outcomes are in part due to the range of ways in which density-dependent selection within the prey interacts with density- and frequency-dependent selection on the prey due to the predator. The results point to a `loser wins' principle, in which the evolution leads to a weakening of the interaction between predator and prey. The results also illustrate the deterioration of the environment associated with each evolutionary step of the species and the lack of a net improvement in their mean fitness.

We present models of adaptive change in continuous traits for the following situations: (1) adaptation of a single trait within a single population in which the fitness of a given individual depends on the population's mean trait value as well as its own trait value; (2) adaptation of two (or more) traits within a single population; (3) adaptation in two or more interacting species. We analyse a dynamic model of these adaptive scenarios in which the rate of change of the mean trait value is an increasing function of the fitness gradient (i.e. the rate of increase of individual fitness with the individual's trait value). Such models have been employed in evolutionary game theory and are often appropriate both for the evolution of quantitative genetic traits and for the behavioural adjustment of phenotypically plastic traits. The dynamics of the adaptation of several different ecologically important traits can result in characters that minimize individual fitness and can preclude evolution towards characters that maximize individual fitness. We discuss biological circumstances that are likely to produce such adaptive failures for situations involving foraging, predator avoidance, competition and coevolution. The results argue for greater attention to dynamical stability in models of the evolution of continuous traits.

We extend the ideas of evolutionary dynamics and stability to a very broad class of biological and other dynamical systems. We simultaneously develop the general mathematical theory and a discussion of some illustrative examples. After developing an appropriate formulation for the dynamics, we define the notion of an evolutionary stable attractor (ESA) and give some samples of ESAS with simple and complex dynamics. We discuss the relationship between our theory and that for ESSS in classical linear evolutionary game theory by considering some dynamical extensions. We then introduce and develop our main mathematical tool, the invasion exponent. This allows analytical and numerical analysis of relatively complex situations, such as the coevolution of multiple species with chaotic population dynamics. Using this, we introduce the notion of differential selective pressure which for generic systems is nonlinear and characterizes internal ESAS. We use this to analytically determine the ESAS in our previous examples. Then we introduce the phenotype dynamics which describe how a population with a distribution of phenotypes changes in time with or without mutations. We discuss the relation between the asymptotic states of this and the ESAS. Finally, we use our mathematical formulation to analyse a non-reproductive form of evolution in which various learning rules compete and evolve. We give a very tentative economic application which has interesting ESAS and phenotype dynamics.

Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals--the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator-prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator-prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species.

A stochastic process of long-term evolution due to mutation and selection is defined over an asexually reproducing population, with selection according to a population game with a one-dimensional continuity of pure strategies. Limiting the analysis to mutations of small effect, it is shown that long-term dynamic stability in such a process is equivalent to continuous stability in the relevant population game. In the case of a one-dimensional strategy set (but not necessarily if the strategy set is multi-dimensional), this result is virtually independent of the distribution of mutations.

Evolutionary games are introduced as models for repeated anonymous strategic interaction: actions (or behaviors) which are more "fit," given the current distribution of behaviors, tend over time to displace less fit behaviors. Cone fields characterize the continuous-time processes compatible with a given fitness (or payoff) function. For large classes of dynamics, it is shown that all stable steady states are Nash equilibria and that all Nash equilibria are steady states. The biologists' evolutionarily stable strategy condition is shown to be less closely related to the dynamic equilibria. Economic examples and a literature survey are also provided. Copyright 1991 by The Econometric Society.

We know very little about the genetic basis of adaptation. Indeed, we can make no theoretical predictions, however heuristic, about the distribution of phenotypic effects among factors fixed during adaptation nor about the expected "size" of the largest factor fixed. Study of this problem requires taking into account that populations gradually approach a phenotypic optimum during adaptation via the stepwise substitution of favorable mutations. Using Fisher's geometric model of adaptation, I analyze this approach to the optimum, and derive an approximate solution to the size distribution of factors fixed during adaptation. I further generalize these results to allow the input of any distribution of mutational effects. The distribution of factors fixed during adaptation assumes a pleasingly simple, exponential form. This result is remarkably insensitive to changes in the fitness function and in the distribution of mutational effects. An exponential trend among factors fixed appears to be a general property of adaptation toward a fixed optimum.

The population genetics of autosomal genes affecting the primary sex ratio is discussed. It is shown that a gene may be expected to increase in frequency if its increase will shift the sex ratio of the population nearer to 1 : 1. It is suggested that it should be possible to alter the primary sex ratio by selection.

The subject matter of evolutionary game theory is the analysis of conflict and cooperation in animals and plants. Originally, game theory was developed as a theory of human strategic behavior based on an idealized picture of rational decision making. Evolutionary game theory does not rely on rationality assumptions but on the idea that the Darwinian process of natural selection drives organisms toward the optimization of reproductive success. Most of evolutionary game theory focuses on those cases where stable equilibrium is reached. However, the dynamics of evolutionary processes in disequilibrium is also an active area of research. In principle, evolutionary game theory deals only with fully symmetric games. Asymmetric conflicts are embedded in symmetric games where each player has the same chance to be on each side of the conflict. The mathematical definition of evolutionary stability refers to symmetric games only. Because asymmetric conflicts can be embedded in symmetric games, this is no obstacle for the treatment of asymmetric conflicts.

Every form of behaviour is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realised how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centred not on the concept of rational players but on the population dynamics of behavioural programmes. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behaviour, and of the closely related interactions between species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions which can alter the basis of their success, i.e. to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions which punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms.

This paper concisely reviews the mathematical properties of the dominant Lyapunov exponent of a matrix sequence in the context of population biology. The concept of Lyapunov exponent provides a valuable tool for investigating processes of invasion in ecology or genetics, which are crucial in shaping community diversity, determining the spread of epidemics or the fixation of a new mutation. The appeal of the invasibility criterion based on the dominant Lyapunov exponent lies in the opportunity it offers to deal with population structure, complex life cycles, and complex population dynamics resulting from the model nonlinearities (oscillations, chaos), as well as random fluctuations arising from a stochastic environment. We put emphasis on the issues of the existence, numerical approximation, and regularity of the dominant Lyapunov exponent. Our presentation is aimed at showing that, despite our inability to compute the exponent analytically, which adds to its high intrinsic instability, important biological insights can nevertheless be achieved at the cost of fairly mild assumptions on the features of the models considered.

The two conditions for stability of an evolutionary equilibrium, the m-stability and the delta-stability conditions, are discussed. The m-stability condition is a condition for the convergence of the population toward the equilibrium, and the delta-stability condition corresponds to a local version of the classic evolutionarily stable strategy (ESS) condition. Together the two conditions provide the condition for a continuous stable strategy. The convergence stability condition corresponds to the requirement for convergence due to initial increase of rare alleles in a monomorphic population, and the local ESS stability condition corresponds to the stability of a monomorphic population at the evolutionary equilibrium against the increase of rare alleles. In this way, an evolutionary equilibrium that is convergence stable, but not local ESS stable, will tend to become polymorphic. The local ESS stability condition therefore contributes more to a description of the dynamics of variation at an evolutionary equilibrium than to the description of the stability of the evolutionary equilibrium. However, the characterization of a polymorphic evolutionary equilibrium cannot be reached by studying the initial increase of rare variant alleles, just as this method cannot describe all aspects of the convergence stability either. Combining these analyses provides a powerful tool in the initial exploration of evolutionary equilibriums of complicated systems, and convergence stable and local ESS unstable equilibriums point toward very interesting polymorphic evolutionary stable states. The analysis is illustrated on a model for intraspecific exploitative competition that may show monomorphic and polymorphic evolutionarily stable equilibriums.

We know very little about the genetic basis of adaptation. Indeed, we can make no theoretical predictions, however heuristic, about the distribution of phenotypic effects among factors fixed during adaptation nor about the expected 'size' of the largest factor fixed. Study of this problem requires taking into account that populations gradually approach a phenotypic optimum during adaptation via the stepwise substitution of favorable mutations. Using Fisher's geometric model of adaptation, I analyze this approach to the optimum, and derive an approximate solution to the size distribution of factors fixed during adaptation. I further generalize these results to allow the input of any distribution of mutational effects. The distribution of factors fixed during adaptation assumes a pleasingly simple, exponential form. This result is remarkably insensitive to changes in the fitness function and in the distribution of mutational effects. An exponential trend among factors fixed appears to be a general property of adaptation toward a fixed optimum.

We present a general framework for modelling adaptive trait dynamics in which we integrate various concepts and techniques from modern ESS-theory. The concept of evolutionarily singular strategies is introduced as a generalization of the ESS-concept. We give a full classification of the singular strategies in terms of ESS-stability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategy''s neighbourhood. Of particular interest is a type of singular strategy that is an evolutionary attractor from a great distance, but once in its neighbourhood a population becomes dimorphic and undergoes disruptive selection leading to evolutionary branching. Modelling the adaptive growth and branching of the evolutionary tree can thus be considered as a major application of the framework. A haploid version of Levene''s soft selection model is developed as a specific example to demonstrate evolutionary dynamics and branching in monomorphic and polymorphic populations.

Several definitions of evolutionary stability (evolutionarily stable strategy—ESS, continuously stable strategy—CSS, evolutionary genetic stability—EGS, evolutionarily stable state—ES state) are presented in a unifying framework.

A general notion of evolutionary stability is formulated in models in which the possible behaviours are parameterized by a continuous variable, and selection is assumed to be weak. Two local stability conditions are formulated, m-stability and (δ-stability, the former being first-order and the latter second-order in the mutant behavioural deviation. The conditions are interpreted in two standard formulations of a one-locus genetic model: a covariance approach and a structured population approach. A weak selection theorem is proved which says that m-stability can be calculated using the neutral covariances. These in turn can be calculated as relatedness coefficients; hence an inclusive fitness formulation is capable of checking m-stability. But (δ-stability, being second-order, is more difficult to handle.

A strategy in a population game is evolutionarily stable if, when adopted by large enough a majority in the population, it becomes advantageous against any mutant strategy. It is said to be continuously stable if, when the majority slightly deviates from it, some reduction of this deviation becomes individually advantageous. This definition is meaningful if a continuum of (pure) strategies is available to each individual in the population. For that case, a necessary and a sufficient condition for an evolutionary stable strategy being a continuously stable strategy is analyzed.

We set out to explore a class of stochastic processes, called "adaptive dynamics", which
supposedly capture some of the essentials of long term biological evolution. These processes have
a strong deterministic component. This allows a classification of their qualitative features which in
many aspects is similar to classifications from the theory of deterministic dynamical systems. But
they also display a good number of clear-cut novel dynamical phenomena.
The sample functions of an adaptive dynamics are piece-wise constant functions from R+ to
the finite subsets of some "trait" space X Ì Rk. Those subsets we call "adaptive conditions". Both
the range and the jumps of a sample function are governed by a function s, called "fitness",
mapping the present adaptive condition and the trait value of a potential "mutant" to R. Sign(s) tells
which subsets of X qualify as adaptive conditions, which mutants can potentially "invade", leading
to a jump in the sample function, and which adaptive condition(s) can result from such an
invasion.
Fitnesses supposedly satisfy certain constraints derived from their population/community
dynamical origin, such as the fact that all mutants which are equal to some "resident", i.e., element
of the present adaptive condition, have zero fitness. Apart from that we suppose that s is as smooth
as can possibly be condoned by its community dynamical origin. Moreover we assume that a
mutant can differ but little from its resident "progenitor".
In sections 1 and 2 we describe the biological background of our mathematical framework. In
section 1 we deal with the position of our framework relative to present and past evolutionary
research. In section 2 we discuss the community dynamical origins of s, and the reasons for
making a number of specific simplifications relative to the full complexity seen in nature.
In sections 3 and 4 we consider some general, mathematical as well as biological, conclusions
that can be drawn from our framework in its simplest guise, that is, when we assume that X is 1-
dimensional, and that the cardinality of the adaptive conditions stays low. The main result is a
classification of the adaptively singular points. These points comprise both the adaptive point
attractors, as well as the points where the adaptive trajectory can branch, thus attaining its
characteristic tree-like shape.
In section 5 we discuss how adaptive dynamics relate through a limiting argument to stochastic
models in which individual organisms are represented as separate entities. It is only through such a
limiting procedure that any class of population or evolutionary models can eventually be justified.
Our basic assumptions are (i) clonal reproduction, i.e., the resident individuals reproduce faithfully
without any of the complications of sex or Mendelian genetics, except for the occasional
occurrence of a mutant, (ii) a large system size and an even rarer occurrence of mutations per birth
event, (iii) uniqueness and global attractiveness of any interior attractor of the community dynamics
in the limit of infinite system size.
In section 6 we try to delineate, by a tentative listing of "axioms", the largest possible class of
processes that can result from the kind of limiting considerations spelled out in section 5. And in
section 7 we heuristically derive some very general predictions about macro-evolutionary patterns,
based on those weak assumptions only.
In the final section 8 we discuss (i) how the results from the preceding sections may fit into a
more encompassing view of biological evolution, and (ii) some directions for further research.

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Beginners in life history theory or evolutionary ecology seemingly face a variety of almost unrelated approaches. Yet the biomathematical literature of the last 10-20 years reflects the implicit acceptance of a common evolutionary framework, the core idea being that there exists a unique general fitness measure that concisely summarizes the overall time course of potential invasions by initially rare mutant phenotypes. Using such an invasion criterion to characterize fitness implicitly presupposes a scenario in which, during periods o f clear evolutionary change, the rate of evolution is set primarily by the random occurrence (and initial establishment) of favourable mutations. Evolutionarily stable life history strategies (ESSs) may then be regarded as traps for the evolutionary random walk.

I want in this article to trace the history of an idea. It is beginning to become clear that a range of problems in evolution theory can most appropriately be attacked by a modification of the theory of games, a branch of mathematics first formulated by Von Neumann and Morgenstern in 1944 for the analysis of human conflicts. The problems are diverse and include not only the behaviour of animals in contest situations but also some problems in the evolution of genetic mechanisms and in the evolution of ecosystems. It is not, however, sufficient to take over the theory as it has been developed in sociology and apply it to evolution. In sociology, and in economics, it is supposed that each contestant works out by reasoning the best strategy to adopt, assuming that his opponents are equally guided by reason. This leads to the concept of a ‘minimax’ strategy, in which a contestant behaves in such a way as to minimise his losses on the assumption that his opponent behaves so as to maximise them. Clearly, this would not be a valid approach to animal conflicts. A new concept has to be introduced, the concept of an ‘evolutionary stable strategy’.

We analyze monomorphic equilibria of long-term evolution for one or two continuous traits, controlled by an arbitrary number of autosomal loci and subject to constant viability selection. It turns out that fitness maximization always obtains at long term equilibria, but in the case of two traits, linkage determines the precise nature of the fitness measure that is maximized. We then consider local convergence to long term equilibria, for two multilocus traits subject to either constant or frequency dependent selection. From a model of long-term dynamics near an equilibrium we derive a criterion of local long-term stability for 2-dimensional equilibria. It turns out that mutation can be a decisive factor for stability.

In this paper we develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community's trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamic of the mean evolutionary path. To first order the resulting equation coincides with a dynamic that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

This paper investigates the problem of how to conceive a robust theory of phenotypic adaptation in non-trivial models of evolutionary biology. A particular effort is made to develop a foundation of this theory in the context of n-locus population genetics. Therefore, the evolution of phenotypic traits is considered that are coded for by more than one gene. The potential for epistatic gene interactions is not a priori excluded. Furthermore, emphasis is laid on the intricacies of frequency-dependent selection. It is first discussed how strongly the scope for phenotypic adaptation is restricted by the complex nature of 'reproduction mechanics' in sexually reproducing diploid populations. This discussion shows that one can easily lose the traces of Darwinism in n-locus models of population genetics. In order to retrieve these traces, the outline of a new theory is given that I call 'streetcar theory of evolution'. This theory is based on the same models that geneticists have used in order to demonstrate substantial problems with the 'adaptationist programme'. However, these models are now analyzed differently by including thoughts about the evolutionary removal of genetic constraints. This requires consideration of a sufficiently wide range of potential mutant alleles and careful examination of what to consider as a stable state of the evolutionary process. A particular notion of stability is introduced in order to describe population states that are phenotypically stable against the effects of all mutant alleles that are to be expected in the long-run. Surprisingly, a long-term stable state can be characterized at the phenotypic level as a fitness maximum, a Nash equilibrium or an ESS. The paper presents these mathematical results and discusses - at unusual length for a mathematical journal - their fundamental role in our current understanding of evolution.

The joint evolution of female mating preferences and secondary sexual characters of males is modeled for polygamous species in which males provide only genetic material to the next generation and females have many potential mates to choose among. Despite stabilizing natural selection on males, various types of mating preferences may create a runaway process in which the outcome of phenotypic evolution depends critically on the genetic variation parameters and initial conditions of a population. Even in the absence of genetic instability, rapid evolution can result from an interaction of natural and sexual selection with random genetic drift along lines of equilibria. The models elucidate genetic mechanisms that can initiate or contribute to rapid speciation by sexual isolation and divergence of secondary sexual characters.

this paper we integrate various concepts and techniques from modern ESS-theory into a single mathematical framework for modeling the dynamics of long-term phenotypic evolution. We introduce the concept of `evolutionarily singular strategy' as a generalization of the ESS-concept. Our main result is a classification of the singular strategies in terms of ESS-stability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategy's neighborhood. These four properties are to a large extend independent of one another and can occur in many combinations. Each combination represents a qualitatively different evolutionary scenario. A type of singular strategy that stands out in particular is convergence stable but lacks ESS-stability. We show that from larger distances it acts as an evolutionary attractor, but once nearby the population undergoes disruptive selection and splits up into two subsequently phenotypically diverging subpopulations. We therefore consider modeling the adaptive growth and branching of the evolutionary tree as a major application of the classification. We first develop the framework for monomorphic resident populations, and generalize some of our results to polymorphic populations later. We formulate a haploid version of Levene's (1953) `soft selection' model as a specific example to demonstrate evolutionary branching in both monomorphic and polymorphic populations. A more formal approach of the framework including generalizations for multi-dimensional (that is, vector-valued) strategies was presented by Metz et al.

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