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

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program off 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 presented.) in trait space squared harbouring protected dimorphism. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to (Formula presented.). However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evoltionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to (Formula presented.), just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibiliy that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es

The Evolution of Simple Life-Histories: Step Towards Classification

We present a classification of the evolutionary dynamics for a class of simple life-history models. The model class considered is characterised by discrete time population dynamics, density-dependent population growth, by the assumption that individuals can occur in two states, and that two evolving traits are coupled by a trade-off. Individual models differ in the choice of traits that are presumed to evolve and in the way population regulation is incorporated. The classification is based on a fitness measure that is sign equivalent to invasion fitness but algebraically simpler. We classify models according to curvature properties of the fitness landscape and whether the evolutionary dynamics can be analysed by means of an optimisation criterion. The first classification allows us to infer whether trait combinations that are characterised by a zero fitness gradient are susceptible to invasion by similar trait combinations. The second classification distinguishes models where evolutionary change is frequency-independent from models that give rise to frequency dependence. Given certain symmetry assumptions we can extend the classification in the latter case by splitting selection into a density-dependent and a frequency-dependent component. We apply our approach to several simple life-history models and demonstrate how our classification facilitates an analytical analysis. We conclude by discussing some general patterns that emerge from our analysis and by hinting at several possible extensions

The Evolution of Resource Specialization through Frequency-Dependent and Frequency-Independent Mechanisms

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

Adaptive Walks on Changing Landscapes: Levins' Approach Extended

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

Necessary and sufficient conditions for R0 to be a sum of contributions of fertility loops

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341.1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio R0 in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graph-reduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases R0 allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single individual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of R0 that may sometimes be used to good effect in cases where no simple formula for R0 exists

The components of directional and disruptive selection in heterogeneous group-structured populations.

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

Evolution of Assortative Mating in a Population Expressing Dominance

In this article, we study the influence of dominance on the evolution of assortative mating. We perform a population-genetic analysis of a two-locus two-allele model. We consider a quantitative trait that is under a mixture of frequency-independent stabilizing selection and density- and frequency-dependent selection caused by intraspecific competition for a continuum of resources. The trait is determined by a single (ecological) locus and expresses intermediate dominance. The second (modifier) locus determines the degree of assortative mating, which is expressed in females only. Assortative mating is based on similarities in the quantitative trait (‘magic trait’ model). Analytical conditions for the invasion of assortment modifiers are derived in the limit of weak selection and weak assortment. For the full model, extensive numerical iterations are performed to study the global dynamics. This allows us to gain a better understanding of the interaction of the different selective forces. Remarkably, depending on the size of modifier effects, dominance can have different effects on the evolution of assortment. We show that dominance hinders the evolution of assortment if modifier effects are small, but promotes it if modifier effects are large. These findings differ from those in previous work based on adaptive dynamics

Evolutionary Epidemiology of Drug-Resistance in Space

The spread of drug-resistant parasites erodes the efficacy of therapeutic treatments against many infectious diseases and is a major threat of the 21st century. The evolution of drug-resistance depends, among other things, on how the treatments are administered at the population level. “Resistance management” consists of finding optimal treatment strategies that both reduce the consequence of an infection at the individual host level, and limit the spread of drug-resistance in the pathogen population. Several studies have focused on the effect of mixing different treatments, or of alternating them in time. Here, we analyze another strategy, where the use of the drug varies spatially: there are places where no one receives any treatment. We find that such a spatial heterogeneity can totally prevent the rise of drug-resistance, provided that the size of treated patches is below a critical threshold. The range of parasite dispersal, the relative costs and benefits of being drug-resistant compared to being drug-sensitive, and the duration of an infection with drug-resistant parasites are the main factors determining the value of this threshold. Our analysis thus provides some general guidance regarding the optimal spatial use of drugs to prevent or limit the evolution of drug-resistance

Diversity and Relatedness Enhance Survival in Colour Polymorphic Grasshoppers

Evolutionary theory predicts that different resource utilization and behaviour by alternative phenotypes may reduce competition and enhance productivity and individual performance in polymorphic, as compared with monomorphic, groups of individuals. However, firm evidence that members of more heterogeneous groups benefit from enhanced survival has been scarce or lacking. Furthermore, benefits associated with phenotypic diversity may be counterbalanced by costs mediated by reduced relatedness, since closely related individuals typically are more similar. Pygmy grasshoppers (Tetrix subulata) are characterized by extensive polymorphism in colour pattern, morphology, behaviour and physiology. We studied experimental groups founded by different numbers of mothers and found that survival was higher in low than in high density, that survival peaked at intermediate colour morph diversity in high density, and that survival was independent of diversity in low density where competition was less intense. We further demonstrate that survival was enhanced by relatedness, as expected if antagonistic and competitive interactions are discriminately directed towards non-siblings. We therefore also performed behavioural observations and staged encounters which confirmed that individuals recognized and responded differently to siblings than to non-siblings. We conclude that negative effects associated with competition are less manifest in diverse groups, that there is conflicting selection for and against genetic diversity occurring simultaneously, and that diversity and relatedness may facilitate the productivity and ecological success of groups of interacting individuals

Contrasting Responses to Harvesting and Environmental Drivers of Fast and Slow Life History Species

According to their main life history traits, organisms can be arranged in a continuum from fast (species with small body size, short lifespan and high fecundity) to slow (species with opposite characteristics). Life history determines the responses of organisms to natural and anthropogenic factors, as slow species are expected to be more sensitive than fast species to perturbations. Owing to their contrasting traits, cephalopods and elasmobranchs are typical examples of fast and slow strategies, respectively. We investigated the responses of these two contrasting strategies to fishing exploitation and environmental conditions (temperature, productivity and depth) using generalized additive models. Our results confirmed the foreseen contrasting responses of cephalopods and elasmobranchs to natural (environment) and anthropogenic (harvesting) influences. Even though a priori foreseen, we did expect neither the clear-cut differential responses between groups nor the homogeneous sensitivity to the same factors within the two taxonomic groups. Apart from depth, which affected both groups equally, cephalopods and elasmobranchs were exclusively affected by environmental conditions and fishing exploitation, respectively. Owing to its short, annual cycle, cephalopods do not have overlapping generations and consequently lack the buffering effects conferred by different age classes observed in multi-aged species such as elasmobranchs. We suggest that cephalopods are sensitive to short-term perturbations, such as seasonal environmental changes, because they lack this buffering effect but they are in turn not influenced by continuous, long-term moderate disturbances such as fishing because of its high population growth and turnover. The contrary would apply to elasmobranchs, whose multi-aged population structure would buffer the seasonal environmental effects, but they would display strong responses to uninterrupted harvesting due to its low population resilience. Besides providing empirical evidence to the theoretically predicted contrasting responses of cephalopods and elasmobranchs to disturbances, our results are useful for the sustainable exploitation of these resourcesVersión del editor4,411