Summary 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 ESSstability, 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

/pdf/evolutionarily-singular-strategies-and-the-adaptive-growth-9tna03ynki.pdf

Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree

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.

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

Evolutionary branching occurs when frequency-dependent selection splits a phenotypically monomorphic population into two distinct phenotypic clusters. A prerequisite for evolutionary branching is that directional selection drives the population towards a fitness minimum in phenotype space. This paper demonstrates that selection regimes leading to evolutionary branching readily arise from a wide variety of different ecological interactions within and between species. We use classical ecological models for symmetric and asymmetric competition, for mutualism, and for predator-prey interactions to describe evolving populations with continuously varying characters. For these models, we investigate the ecological and evolutionary conditions that allow for evolutionary branching and establish that branching is a generic and robust phenomenon. Evolutionary branching becomes a model for sympatric speciation when population genetics and mating mechanisms are incorporated into ecological models. In sexual populations with random mating, the continual production of intermediate phenotypes from two incipient branches prevents evolutionary branching. In contrast, when mating is assortative for the ecological characters under study, evolutionary branching is possible in sexual populations and can lead to speciation. Therefore, we also study the evolution of assortative mating as a quantitative character. We show that evolution under branching conditions selects for assortativeness and thus allows sexual populations to escape from fitness minima. We conclude that evolutionary branching offers a general basis for understanding adaptive speciation and radiation under a wide range of different ecological conditions.

Evolutionary Branching and Sympatric Speciation Caused by Different Types of Ecological Interactions [Revised 18 September 2000]

Evolutionary branching occurs when frequency-depen- dent selection splits a phenotypically monomorphic population into two distinct phenotypic clusters. A prerequisite for evolutionary branching is that directional selection drives the population toward a fitness minimum in phenotype space. This article demonstrates that selection regimes leading to evolutionary branching readily arise from a wide variety of different ecological interactions within and between species. We use classical ecological models for symmetric and asymmetric competition, for mutualism, and for predator-prey interactions to describe evolving populations with continuously vary- ing characters. For these models, we investigate the ecological and evolutionary conditions that allow for evolutionary branching and establish that branching is a generic and robust phenomenon. Evo- lutionary branching becomes a model for sympatric speciation when population genetics and mating mechanisms are incorporated into ecological models. In sexual populations with random mating, the continual production of intermediate phenotypes from two incipient branches prevents evolutionary branching. In contrast, when mating is assortative for the ecological characters under study, evolutionary branching is possible in sexual populations and can lead to speciation. Therefore, we also study the evolution of assortative mating as a quantitative character. We show that evolution under branching con- ditions selects for assortativeness and thus allows sexual populations to escape from fitness minima. We conclude that evolutionary branching offers a general basis for understanding adaptive speciation and radiation under a wide range of different ecological conditions.

https://user.iiasa.ac.at/~dieckman/reprints/DoebeliDieckmann2000.pdf

Evolutionary Branching and Sympatric Speciation Caused by Different Types of Ecological Interactions.

https://hal.inria.fr/inria-00164784/document

Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models.

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.

Long-term evolution of multilocus traits.

A theoretical model of pollination that analyses the amount of geitonogamous pollen transfer among a group of plants is presented. We study how pollen transfer is affected by different round trips described by a pollinator, taking into account the fraction geitonogamy and the fraction pollen export in self-incompatible plant species. Deterministic and stochastic models of pollen transfer are developed, from which we find that when pollinators describe a uniform sequence (i.e. visit the same number of flowers in each plant), individuals receive maximum outcross pollen or minimum self pollen. Thus, from the point of view of group fitness, the optimal number of flowers visited in each plant depends on the number of flowers on the plant, the length of the foraging bout, and the number of plants in the population.

Cristina Di Pasquale

Papers

Long-term evolution of multilocus traits.

Dynamics of pollination: A model of insect-mediated pollen transfer in self-incompatible plants