Adaptive Dynamics in Allele Space: Evolution of Genetic Polymorphism by Small Mutations in a Heterogeneous Environment

We investigate how a plymorphism of distinctly different alleles can evolve in an initially monomorphic population under frequency-dependent selection if mutations have only a small phenotypic effect. We consider the case of a single additive locus with a continuum of potential allele types in a diploid outbreeding population. As a specific example, we use a version of Levene's (1953) soft selection model, where stabilizing selection acts on a continuous trait within each of two habitats. If the optimal phenotypes within the habitats are sufficiently different, then two distinctly different alleles evolve gradually from a single ancestral allele. In a wide range of parameter values, the two locally optimal phenotypes will be realized by one of the homozygotes and the heterozygote, rather than the two homozygotes. Unlike the haploid analogue of the model, there can be multiple polymorphic evolutionary attractors with different probabilities of convergence

Red Queen Evolution by Cycles of Evolutionary Branching and Extinction

We use the theory of adaptive dynamics to construct and analyse a generic example of cycling evolution with alternating levels of polymorphism. A monomorphic population evolves towards larger trait values until it reaches a so-called evolutionary branching point. Disruptive selection at the branching point splits the population into two strategies. In the dimorphic population the strategies undergo parallel coevolution towards smaller trait values. Finally one of the two strategies goes extinct, and the remaining single strategy evolves upwards again to the branching point. The reversal of the direction of evolution is brought about by the changing level of polymorphism. Extinction is deterministic, i.e., it occurs inevitably and always at the same trait values; which of the two strategies goes extinct is, however, random. The present model is discussed in relation to other mechanisms for evolutionary cycles involving branching and extinction

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

Polymorphic evolution sequence and evolutionary branching

We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution sequence. Then we introduce a scaling of the size of mutations and we study the polymorphic evolution sequence in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. To this end we finely analyze the asymptotic behavior of 3-dimensional competitive Lotka-Volterra systems

Evolutionary Dynamics of Seed Size and Seedling Competitive Ability

We present a model for the evolutionary dynamics of seed size when seedlings from large seeds are better competitors than seedling from small seeds and there is a trade-off between seed size and seed number. We first consider two limiting cases where seed size either has no effect on the competitive ability of seedlings, or where seedlings from larger seeds always win from seedlings from smaller seeds if together from the same germination site. In the first case there is a single evolutionary optimal seed size excluding all other, whereas in the second case there is an evolutionary stable seed polymorphism with a continuous variation of seed sizes where plants with small (but numerous seeds) survive by exploiting sites that by chance remain unoccupied by plants with larger (but less numerous) seeds. We investigate how these two cases connect to one another via intermediate levels of competitive asymmetry. We find that strong competitive asymmetry and high resource levels favor coexistence of plants with different seed sizes when seed and seedling survival is moderately low but large seeds have a substantial competitive advantage over small seeds. Assuming mutation-limited evolution and assuming that single mutations have only a small phenotypic effect, an initially monomorphic population with a single seed size will reach the final evolutionarily stable polymorphic state through a series of discrete evolutionary branching events. At each branching event, a given lineage already present in the population divides into two daughter lines, each with its own seed size. If precompetitive seed and seedling survival is high for small and large seeds alike, evolutionary branching may be followed by extinction of one or more lineages (including mass-extinction), and thus not necessarily gives rise to evolutionarily stable seed polymorphism. Various results presented here are model-independent and point the way to a more general evolutionarily bifurcation theory describing how the number and stability properties of evolutionary equilibria can change as a consequence of changes in model parameters

The Dynamics of Adaptation and Evolutionary Branching

We present a formal framework for modeling evolutionary dynamics with special emphasis on the generation of diversity through branching of the evolutionary tree. Fitness is defined as the long term growth rate which is influenced by the biotic environment leading to frequency-dependent selection. Evolution can be described as a dynamics in space with variable number of dimensions corresponding to the number of different types present. The dynamics within a subspace is governed by the local fitness gradient. Entering a higher dimensional subspace is possible only at a particular type of attractors where the population undergoes evolutionary branching

Evolutionary Optimization Models and Matrix Games in the Unified Perspective of Adaptive Dynamics

Matrix game theory and optimization models offer two radically different perspectives on the outcome of evolution. Optimization models consider frequency-independent selection and envisage evolution as a hill-climbing process on a constant fitness landscape, with the optimal strategy corresponding to the fitness maximum. By contrast, in evolutionary matrix games selection is frequency-dependent and leads to fitness equality among alternative strategies once an evolutionarily stable strategy has been established. In this review we demonstrate that both optimization models and matrix games represent special cases within the general framework of adaptive dynamics. Adaptive dynamics theory considers arbitrary nonlinear frequency and density dependence and envisages evolution as proceeding on an adaptive landscape that changes its shape according to which strategies are present in the population. In adaptive dynamics, evolutionarily stable strategies correspond to conditional fitness maxima: the ESS is characterized by the fact that it has the highest fitness if it is the established strategy. In this framework it can also be shown that dynamical attainability, evolutionary stability, and invading potential of strategies are pairwise independent properties. In optimization models, on the other hand, these properties become linked such that the optimal strategy is always attracting, evolutionarily stable and can invade any other strategy. In matrix games fitness is a linear function of the potentially invading strategy and can thus never exhibit an interior maximum: Instead, the fitness landscape is a plane that becomes horizontal once the ESS is established. Due to this degeneracy, invading potential is part of the ESS definition for matrix games and dynamical attainability is a dependent property. We conclude that adaptive dynamics provides a unifying framework for overcoming the traditional divide between evolutionary optimization models and matrix games

Adaptive Dynamics: A Geometrical Study of the Consequences of Nearly Faithful Reproduction

We set out to explore a class of stochastic processes, called "adaptive dynamics", which supposedly capture some of the essentials of the 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 function from R_+ to the finite subsets of some "trait" space X in R^k. 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) tell us 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 invasion. Fitness supposedly satisfy certain constraints derived from their population/community dynamical origin, such as the fact that all mutants which are equal to some "residents", i.e., element of the present adaptive condition, have zero fitness. Apart from that, we suppose that s is as smooth as can be possibly 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 origin 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 the 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