Fig 4 - uploaded by Tom J M Van Dooren
Content may be subject to copyright.
Life cycle graphs that go with the examples in Section 3.2. The corresponding fitness proxies are given in Equation 12.
Source publication
We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject...
Contexts in source publication
Context 2
... the life cycle in Figure 4(a) with Q as in Equation (12a). If M = { ˜ f 13˜s13˜ 13˜s 32˜s32˜ 32˜s 21 , ˜ f 14˜s14˜ 14˜s 43˜s43˜ 43˜s 32˜s32˜ 32˜s 21 } and x = (s 21 , s 32 , s 11 ), then ...
Context 3
... principle: ψ(x) = q(α)(x) Example: Consider the life cycle in Figure 4(b) with Q as in Equation (12b). If M = { ˜ f 13˜s13˜ 13˜s 32˜s32˜ 32˜s 21 }, x = (s 11 , s 22 ) and R = (R s,21 , R s,32 , R s,33 , R f,13 ), then a(R) = f 13 R f,13 s 32 R ...
Context 5
... the life cycle in Figure 4(a) with Q as in Equation (12a). If M = { ˜ f 13˜s13˜ 13˜s 32˜s32˜ 32˜s 21 , ˜ f 14˜s14˜ 14˜s 43˜s43˜ 43˜s 32˜s32˜ 32˜s 21 }, x = (s 21 , s 32 , s 43 , f 13 , f 14 ) and R = (R s,21 , R s,32 , R s,11 ), then ...
Context 6
... the life cycle in Figure 4(c) with Q as in Equation (12c). If M = { ˜ f 13˜s13˜ 13˜s 32˜s32˜ 32˜s 21 , ˜ f 14˜s14˜ 14˜s 43˜s43˜ 43˜s 32˜s32˜ 32˜s 21 } and R = (R f,13 , R f,14 ) with R f,13 = R = R f,14 , then ...
Context 7
... simple life cycles it is often easy to check whether the terms in Q can be re- arranged such that Proposition 10 is applicable. However, in more complicated ones this can be tedious. In this subsection we present a comprehensive list of conditions such that Proposition 10 is applicable. We leave it as a challenge to the reader either to come up with cases not covered by our list or to prove that this list is complete. For each element in our list of optimisation principles we give an example. These examples are based on the life cycles shown in Figure 4 which we describe here briefly. Figure 4(a) shows the age-structured life cycle of an iteroparous plant species with seed bank. The fitness proxy Q is given ...
Context 8
... simple life cycles it is often easy to check whether the terms in Q can be re- arranged such that Proposition 10 is applicable. However, in more complicated ones this can be tedious. In this subsection we present a comprehensive list of conditions such that Proposition 10 is applicable. We leave it as a challenge to the reader either to come up with cases not covered by our list or to prove that this list is complete. For each element in our list of optimisation principles we give an example. These examples are based on the life cycles shown in Figure 4 which we describe here briefly. Figure 4(a) shows the age-structured life cycle of an iteroparous plant species with seed bank. The fitness proxy Q is given ...
Citations
... We plotted different shapes of the association trade-off along with fitness contours on the two-dimensional plane of the stage-specific associations X J and X A (Figure 5), which graphically illustrate how the partner would evolve the stage-specific association X i depending on the relative benefits of stage-specific mutualism B Pi and the juvenileadult stage distribution H i . The results showed that inter-stage partner sharing occurs (i.e., X J converge to the intermediate optima) when the association trade-off is weak while juvenile-specialized or adultspecialized association occurs (i.e., X J diverges to either zero or one depending on the initial trait value) when the association trade-off is strong ( Figure 5; see Rueffler et al., 2013 for similar examples). Further, the optimal X J is relatively high for weak association tradeoffs or X J is likely to converge to one (i.e., a divergent threshold is relatively small) for strong association trade-offs when the juvenile host is more abundant (e.g., compare Figures 5A,C) or when the juvenile-specific mutualism is more beneficial (e.g., compare Figures 5A,D). ...
Mutualism is common in nature and is crucial for population dynamics, community structure, and ecosystem functioning. Studies have recently pointed out that life-history stage structure (e.g., juveniles and adults) is a key factor to better understand the ecological consequences of mutualism (termed stage-structured mutualism). Despite the potential importance, little is known about what kinds of stage-structured mutualism can evolve and when it is likely to occur. Here, we theoretically investigated how a mutualistic partner species should allocate efforts of mutualistic associations for different life-history stages of its host species to maximize its fitness. We assessed the partner’s optimal strategy by using a one host–one partner model with the host’s juvenile-adult stage structure. The results showed that different forms of stage-structured mutualism can evolve, such as juvenile-specialized association, adult-specialized association, and inter-stage partner sharing (i.e., the partner associates with both the juvenile and adult stages of the host) depending on the shape of association trade-off, i.e., how much association with one stage is weakened when the partner strengthens its association with the other stage. In general, stage-specialized association (either juvenile-specialized or adult-specialized association) tends to evolve when being associated with that stage is relatively beneficial. Meanwhile, when the association trade-off is weak, inter-stage partner sharing can occur if the mutualistic benefits of juvenile-specific and adult-specific associations are sufficiently large. We also found that when the association trade-off is strong, alternative stable states occur in which either juvenile-specialized or adult-specialized associations evolve depending on the initial trait value. These results suggest that pairwise interspecific mutualism is more complicated than previously thought, implying that we may under-or overestimate the strength of mutualistic interactions when looking at only certain life-history stages. This study provides a conceptual basis for better understanding the mechanisms underlying ontogenetic shifts of mutualistic partners and more complex mutualistic networks mediated by the life-history stages of organisms and their stage-structured interactions.
... Less well-trodden is disruptive selection in an agestructured population, which in fact we have not seen anywhere expressed as equation (3.4), together with (electronic supplementary material, appendix B.2.3 for derivation; e.g. [93] for other approaches to disruptive selection in age-structured populations). The term h w (z*) depends on how age-specific fitness components change nonlinearly with trait (with age-specific effects weighted accordingly, as in equation (3.7)). ...
Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration.
This article is part of the theme issue ‘Half a century of evolutionary games: a synthesis of theory, application and future directions’.
... Life histories of Asterina gibbosa (Pennant, 1777), Asterina phylactica Emson & Crump, 1979, Parvulastra vivipara (Dartnall, 1969), Cryptasterina pentagona (Muller & Troschel, 1840), Patiria miniata (Brandt, 1835), Aquilonastra yairi O'Loughlin & Rowe, 2006, Aquionastra burtonii (Gray, 1840, Nepanthia belcheri (Perrier, 1875), and Ailastra heteractis (H.L. Clark, 1938) were analyzed using graphs to help in the development of equations that summarize the life cycle (Caswell, 1982(Caswell, , 2001Ebert, 1996Ebert, , 1999. The utility of life-cycle graphs to the analysis of life-history evolution is presented by Rueffler et al. (2013); and life-cycle graphs have been used in studies of diverse species, including killer whales (Brault and Caswell, 1993), sea cucumbers (Ebert, 2010), perennial herbs (Hubbell and Werner, 1979), desert shrubs (Ebert and Ebert, 2006), and mosses (Rydgren and Økland, 2002). ...
The starfish family Asterinidae shows a diversity of reproductive modes, and a number of species have sufficient life-history data that can be used for analysis, using life-cycle graphs. These include four species that reproduce by fission (Aquilonastra yairi, Nepanthia belcheri, Aquilonastra burtonii, and Ailsastra heteractis), a viviparous species (Parvulastra vivipara), two species with benthic egg masses (Asterina gibbosa and Asterina phylactica), one with planktonic larvae that do not feed (Cryptasterina pentagona), and one with larvae that feed in the plankton (Patiria miniata). Species are compared using adult and first-year survival and, for some species, the age at first reproduction, number of offspring (eggs or newly released juveniles), and individual growth parameters of the von Bertalanffy model. The sensitivity of population growth, fitness, to changes in these traits is shown by elasticity analysis, which aids in understanding possible consequences of environmental forces as well as possible directions of selection.
... But this has not been analyzed so far even though it is captured implicitly when second-order derivatives of invasion fitness are computed as has been done in several previous works investigating evolutionary branching in some specific models of class-structured populations (e.g. Massol et al., 2011;Rueffler et al., 2013;Massol and Débarre, 2015;Kisdi, 2016;Parvinen et al., 2018, in press). ...
We derive how directional and disruptive selection operate on scalar traits in heterogeneous group-structured populations for a general class of models. In particular, we assume that each group in the population can be in one of a finite number of states, where states can affect group size and/or other environmental variables, at a given time. Using up to second-order perturbation expansions of the invasion fitness of a mutant allele, we derive expressions for the directional and disruptive selection coefficients, which are sufficient to classify the singular strategies of adaptive dynamics. These expressions include first- and second-order perturbations of individual fitness (expected number of settled offspring produced by an individual, possibly including self through survival); the first-order perturbation of the stationary distribution of mutants (derived here explicitly for the first time); the first-order perturbation of relatedness; and reproductive values, pairwise and third-order relatedness evaluated under neutrality. We introduce the concept of the individual k-fitness (defined as the expected number of settled offspring for which k-1 randomly chosen neighbors are lineage members) and show its usefulness for calculating relatedness and its perturbation. We then show that the directional and disruptive selection coefficients can be expressed in terms individual k-fitnesses with k=1, 2, 3 only. This allows for both a significant reduction in the dimensions of the system of equations describing population dynamics that needs to be solved to evaluate explicitly these selection coefficients and a biologically meaningful interpretation of their components. As an application of our methodology, we analyze directional and disruptive selection in a lottery model with either hard or soft selection and show that many previous results about selection in group-structured populations can be reproduced as special cases of our model.
... (Metz and Leimar 2011). That is, ln(R 0 ) > 0 if Q > 0, and for continuous B and a path connected trait space, C is an ESC if Q(Y |C) < 0 for all Y / ∈ C, and only if Q(Y |C) ≤ 0 for all Y / ∈ C. By the arguments in Appendix A of Rueffler et al. (2013) these properties also generically suffice for checking for frequency independence in the sense laid out in Sect. 4. However, it is not possible to conclude for a specific single Y from Q(Y |C) < (=) 0 that ln(R 0 (Y |C)) < (=) 0. As final point we mention that the concept of invasion fitness extends considerably further than the case of locally well-mixed populations. ...
... However, this construction requires infinitely many basic operations and to calculate ψ recourse has to be taken to Monte Carlo methods. On the other hand, there are by now many cases in which optimisation principles have been found by some finitary construction coming from an inspired analysis of the eco-evolutionary model (for a systematic overview of one class of examples see Rueffler et al. 2013). Only such a priori constructed optimisation principles can act as computational tools, the inferred ones only help in categorising observations on model outcomes. ...
The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock-scissor-paper configurations. Our concept differs from that of Heino et al. (TREE 13:367-370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.'s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock-scissor-paper configurations. Since in 'realistic' (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.
... Evolutionary branching points are a generic feature of ecoevolutionary models that incorporate intraspecific interactions (e.g., Metz et al. 1992;Geritz et al. 1998;Kisdi and Geritz 1999;Day 2000;Doebeli and Dieckmann 2000;Mathias et al. 2001;Day et al. 2002;Doebeli 2002;Schreiber and Tobiason 2003;Leimar 2005;Rueffler et al. 2006c;Massol et al. 2011;Doebeli 2011;Rueffler et al. 2013;Svardal et al. 2014). The analysis in these models is usually based on the technical assumption of large population size, rare mutations of small effect, and asexual reproduction. ...
We present two theoretical approaches to investigate whether organismal complexity, defined as the number of quantitative traits determining fitness, and the potential for adaptive diversification are correlated. The first approach is independent of any specific ecological model and based on curvature properties of the fitness landscape as a function of the dimension of the trait space. This approach indeed suggests a positive correlation between complexity and diversity. An assumption made in this first approach is that the potential for any pair of traits to interact in their effect on fitness is independent of the dimension of the trait space. In the second approach, we circumvent making this assumption by analyzing the evolutionary dynamics in an explicit consumer-resource model in which the shape of the fitness landscape emerges from the underlying mechanistic ecological model. In this model, consumers are characterized by several quantitative traits and feed on a multi-dimensional resource distribution. The consumer's feeding efficiency on the resource is determined by the match between consumer phenotype and resource item. This analysis supports a positive correlation between the complexity of the evolving consumer species and its potential to diversify with the additional insight that also increasing resource complexity facilitates diversification.This article is protected by copyright. All rights reserved.
... In fact, the same patterns of specialization can be theoretically explained without assuming that trade-offs exist (Whitlock 1996). Theoretical conditions under which weak trade-offs select for specialists and strong trade-offs select for generalists can also be found in Rueffler et al. (2006aRueffler et al. ( , 2013. ...
Although trade-off curves between fitness components are essential in theoretical studies of ecological specialization, few empirical studies have actually determined these curves experimentally. Using the snail-feeding carabid beetle Damaster blaptoides, which is endemic to the Japanese archipelago, we estimated the trade-off curve for feeding success with alternative foraging behaviors that are linked to varying morphology. First, we crossed a stout-bodied and a slender-bodied subspecies and produced their F1 and backcross hybrids, which exhibited intermediate body shapes. Then we compared the snail-feeding success of these beetles. Stout beetles could eat small snails by crushing shells, whereas slender beetles could eat large snails by inserting their heads into shells. Although hybrids with intermediate body shapes attempted to employ both strategies, they frequently failed at both. The relationship between feeding success rate and beetle body shape was represented by an inward bending curve, which implies a strong trade-off that can cause disruptive selection, leading to ecological specialization. We suggest that the intermediately shaped beetles were maladapted for snail-feeding and that disruptive selection may have played an important role in the morphological divergence of these beetles.
... Presently two manuscripts are floating around, by Roger Bowers [36] and by Claus Rueffler and co-workers [37], that relate life cycle structure to properties of the associated invasion fitness function. Both manuscripts deal with finite state individuals in constant environments, i.e., community dynamical equilibria. ...
... We start with the plethora of sufficient conditions for the existence of an optimisation principle derived by Rueffler et al. [37]. Two immediate trivial cases are Proposition 3.20. ...
... Moreover, it is not clear yet whether possibly any further special cases are still out in the wild. Hence we give only one relatively simple example and refer to [37] for the details. ...
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 cost of germline maintenance gives rise to a trade-off between lowering the deleterious mutation rate and investing in life history functions. Therefore, life history and the mutation rate coevolve, but this coevolution is not well understood. We develop a mathematical model to analyse the evolution of resource allocation traits, which simultaneously affect life history and the deleterious mutation rate. First, we show that the invasion fitness of such resource allocation traits can be approximated by the basic reproductive number of the least-loaded class; the expected lifetime production of offspring without deleterious mutations born to individuals without deleterious mutations. Second, we apply the model to investigate (i) the coevolution of reproductive effort and germline maintenance and (ii) the coevolution of age-at-maturity and germline maintenance. This analysis provides two resource allocation predictions when exposure to environmental mutagens is higher. First, selection favours higher allocation to germline maintenance, even if it comes at the expense of life history functions, and leads to a shift in allocation towards reproduction rather than survival. Second, life histories tend to be faster, characterized by individuals with shorter lifespans and smaller body sizes at maturity. Our results suggest that mutation accumulation via the cost of germline maintenance can be a major force shaping life-history traits.
We derive how directional and disruptive selection operate on scalar traits in a heterogeneous group-structured population for a general class of models. In particular, we assume that each group in the population can be in one of a finite number of states, where states can affect group size and/or other environmental variables, at a given time. Using up to second-order perturbation expansions of the invasion fitness of a mutant allele, we derive expressions for the directional and disruptive selection coefficients, which are sufficient to classify the singular strategies of adaptive dynamics. These expressions include first- and second-order perturbations of individual fitness (expected number of settled offspring produced by an individual, possibly including self through survival); the first-order perturbation of the stationary distribution of mutants (derived here explicitly for the first time); the first-order perturbation of pairwise relatedness; and reproductive values, pairwise and three-way relatedness, and stationary distribution of mutants, each evaluated under neutrality. We introduce the concept of individual k-fitness (defined as the expected number of settled offspring of an individual for which k-1 randomly chosen neighbors are lineage members) and show its usefulness for calculating relatedness and its perturbation. We then demonstrate that the directional and disruptive selection coefficients can be expressed in terms individual k-fitnesses with k=1,2,3 only. This representation has two important benefits. First, it allows for a significant reduction in the dimensions of the system of equations describing the mutant dynamics that needs to be solved to evaluate explicitly the two selection coefficients. Second, it leads to a biologically meaningful interpretation of their components. As an application of our methodology, we analyze directional and disruptive selection in a lottery model with either hard or soft selection and show that many previous results about selection in group-structured populations can be reproduced as special cases of our model.
