Ellen Baake (Bielefeld): Stochastic and deterministic aspects of recombination

Abstract: Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we recapitulate a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If only recombination is present (i.e., without resampling), the expected type frequencies in the finite population (of arbitrary size) equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.


Matthias Birkner (Berlin): Likelihood-based inference for multiple merger coalescents

Abstract: Multiple merger coalescents are generalisations of Kingman's coalescent, which have been proposed as models for genealogies in species with highly variable offspring numbers. We extends methods of Griffiths & Tavaré, which allow to estimate the likelihood of sequence observations using a Monte Carlo approach, to this setting, and illustrate our method using simulated and real datasets. (Joint work with Jochen Blath and Matthias Steinrücken, TU Berlin)


Tom JM Van Dooren (Leiden): Evolutionary stability on phenotype landscapes

Abstract: Individual phenotypes are determined through interactions between genes and environment. These interactions are mediated by developmental processes, which can be described by genotype-phenotype maps or phenotype landscapes. In order to understand selective advantages of genetic effects, it is essential to include a description of development into evolutionary models. In comparison with models that focus on the ecological effects of phenotypes, evolutionary models which include a genotype-phenotype mapping, or a phenotype-phenotype mapping, can show far richer evolutionary behaviour. That is because developmental constraints are included.
Different types of evolutionary endpoints which appear in adaptive dynamics approximations of such models are discussed, and also their evolutionary stability properties such as invasibility, whether they are reachable from nearby, and whether they allow the emergence of genotypic polymorphism.
Three main types of endpoints can be discerned,
(1) where ecological selection becomes non-directional
(2) evolutionarily stable configurations (sensu Wagner and Schwenk 2000) and
(3) trade-offs.
I will try to illustrate how this modelling style can help resolve discussions on the nature of evolutionary constraints, and point to some effects of standing genetic variation in evolving populations.


Nicolas Champagnat (Berlin): The limit of rare mutations in finite logistically regulated populations: including genetic drift in adaptive dynamics models

Abstract: The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE, called `canonical equation of adaptive dynamics'. However, in order to include the effect of genetic drift in this description, we have to apply a limit of weak selection to a finite stochastically fluctuating discrete population subject to logistic branching competition. We start with the study of the particular case of two competing subpopulations (resident and mutant) and seek explicit first-order formulae for the probability of fixation of the mutant, also interpreted as the mutant's fitness, in the vicinity of neutrality. In particular, the first-order term is a linear combination of products of functions of the initial mutant frequency times functions of the initial total population size, called invasibility coefficients (fertility, defence, aggressiveness, isolation, survival), which measures the robustness (stability wrt selective strengths) of the resident type. Then we apply a limit of rare mutations to a population subject to mutation, birth and competition where the number of coexisting types may fluctuate, while keeping the population size finite. This leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. Finally, we apply a limit of weak selection (small mutation steps) to this jump process, which leads to a diffusion process of evolution, called `canonical diffusion of adaptive dynamics', in which genetic drift is combined with directional selection driven by the fitness gradient.


Stefan Geritz (Helsinki): Trait substitution sequences as a model for discrete evolution

Abstract: Adaptive dynamics inks population dynamics to long-term evolution driven by mutation and selection, and therefore incorporates processes on two very different time scales: a fast ecological time scale and a slow evolutionary time scale. On the ecological time scale adaptive dynamics addresses such questions as which mutant strategies can invade a population of given resident strategies, and what would be the outcome of such an invasion in terms of which strategies remain and which are eliminated from the population once a new population dynamical attractor is reached. On the evolutionary time scale, adaptive dynamics is concerned with the long-term consequences of many such invasion-elimination events. As the set of strategies actually present in a population changes with each invasion-elimination event, successive invasion-elimination events generate a sequence of sets of strategies: a so-called trait substitution sequence. What can be said about the long-term behaviour of trait substitution sequences? This will be the focus of my talk.


Christine Jacob (Jouy-en-Josas): Semi-semi-Markov processes: a generalized renewal approach for individual based branching processes

Abstract: We consider the evolution of an individual-based branching population with semi-Markovian transitions (population with infinitely many types), and we generalize to this process the formalization of a classical individual semi-Markov model: we assume that for each individual of the branching population, the chain representing his jump states and jump times is a (time and population-dependent) Markov renewal process. Defining the jumps of the population by the set of jumps of the individuals, we calculate the kernel of the population process from the individual kernels, and then deduce the population probability law and some simulation algorithms. This formalization may also be viewed as the generalization of the classical superposition of i.i.d. renewal processes, to the superposition of marked individual (time and population-dependent) renewal processes resulting from a branching population. Such a model could be applied for example to an evolution tree where each node represents a species which may be characterized both by qualitative or/and quantitative traits, and where each individual transition is either the creation of a new species or the extinction of the species, and may depend of the whole population state through competition/adaptation/cooperative effects.


Amaury Lambert (Paris): Marginals, coalescence and sampling for 'general' branching processes

Abstract: A splitting tree is a real tree made of (all existence epochs of) individuals who have i.i.d. lifespans with general distribution, during which they give birth at constant rate, independently, to copies of themselves. Set X(t) the number of individuals alive at time t. Then the process (X(t); t>0) is a non-Markovian continuous-time branching process, known as the (binary) homogeneous Crump-Mode-Jagers (CMJ) process. Proving that the contour process of splitting trees is a Lévy process, we display new properties of CMJ processes (one-dimensional marginals, conditionings, limit theorems) and splitting trees (coalescence times, sampling partitions).


Torbjörn Lundh (Gothenburg): Sympatric speciation under reinforcement

I will describe an ongoing project with Johan Henriksson at Karolinska Institutet and Bernt Wennberg here at Mathematical Sciences. We are addressing the question of sympatric speciation in combination with a reinforcement dynamics. Our set up is aimed to be as simple as possible. We start with a population which has a phenotypic trait that is express as a real number, x. The population has an initial distribution centered around the origin with respect of this trait. Initially we assume that an offspring will get its parents' mean as its own parameter value. We then introduce a fitness function which has low fitness around the origin, and two local maximas at 1 and -1. This might lead to a split of the population into three groups with parameter values clustered around -1,0 and 1. This is the introductory sympatric speciation process. We then study how reinforcement in different forms might drive this three peaked population distribution towards a more efficient (with respect to the total fitness of the whole population) two peaked distribution centered at -1 and 1. The main question in this study is how sympatric speciation and reinforcement interact, but we will also address more specific questions such as dimensionality of the reinforced expressions.


John McNamara (Bristol): The importance of individual differences in conflict and the evolution of cooperation

Abstract: Game theoretical models often ignore differences between individuals. Using a series of examples I will demonstrate that such differences are not innocuous noise, but can fundamentally change the nature of a game. Difference can completely reverse the direction of evolution in a simple prisoner’s dilemma game. Differences can necessitate negotiation between parents over care of their young, and can interact with lifespan to determine how cooperative parent are with each other. Finally, differences in personality promote the need to be socially sensitive; and once individuals are socially sensitive, this can lead to the maintenance of differences.


J. A. J. Metz (Leiden): Attempts at a contribution to the post-modern synthesis: explorations at the interface between meso- and macro-evolution.

Abstract: Adaptive dynamics deals with meso-evolutionary time scales, in between the micro-evolutionary time scales of the population dynamics of gene substitutions and macro-evolutionary time scales, where the expression of genotypic memory through the developmental map drastically modifies mutational opportunities, and trait spaces change dimension due to key innovations. This talk will look into some morphological and developmental arguments that bear on the larger scale geometry of fitness landscapes. From this perspective the low dimensional fitness landscapes studied in adaptive dynamics can be seen as the surfaces of ridges in a much higher dimensional landscape of potential morphologies, with the abyss around the ridges created by the lack of a proper development, or functioning, of the corresponding morphologies. The location of the ridges and abysses is grossly the same for large sets of possible environmental conditions. Biological parlance expresses this constancy by referring to the corresponding selective processes as internal. The high dimension combined with the ridgyness conspire in a number of ways: 1. Developmental systems leading to mutation distributions that are in some way aligned with the ridges evolve much faster than systems where such is not the case. 2. The stabilizing selection in the off-ridge directions may be expected to have a great robustness of the developmental system as evolutionary side effect. Yet, the high dimension of genotype space makes that this robustness can never lead to a lack of suitable mutational variation, and thereby to the conservation of features. The fact that evolution largely proceeds through the quantitative variation in the size and shape of homologous parts is due only to the stabilizing internal selection that arises as a consequence of the organization of multicellular organisms. 3. Allopatric speciation supposedly occurs by separated populations wandering around in the high fitness maze, so that after a while their mixed offspring, having intermediate properties, ends up in the abyss. Random genotype to phenotype maps turn out not to be conducive to speciation. It is unclear how evolved genotype to phenotype maps might do better in this respect. 4. Large mutational steps far more often than not make an individual land in the abyss, and only the much rarer very small steps keep it on the top. This provides a justification for the assumption standard made in adaptive dynamics. However, the limit leading to the canonical equation bears reconsideration in this light.


Martin Möhle (Düsseldorff): On the number of cuts needed to isolate the root of a random recursive tree

Abstract: A weak convergence result is presented for the number of cuts needed to isolate the root of a random recursive tree. This result has direct consequences for the number of collision events that take place in the Bolthausen-Sznitman coalescent until there is just a single block. The proof is based on a coupling related to a certain random walk.


Peter Pfaffelhuber (Munich): The evolution of genealogical trees

Abstract: The typical situation a population geneticist faces is to consider genes from a small sample from a large population. The genetic sequences of his/her sample are correlated by common ancestry. A way to describe this is by coalescent-trees which describe the genealogical sample-tree. We want to replace this statical picture by a dynamical one and study how genealogical trees evolve in time. This results in a tree-valued Markov process. From the tree-valued process we derive a Markov dynamics which describes the evolution of sample tree-lengths. This is joint work with Anita Winter and Andreas Greven.


Andrea Pugliese (Trento): Evolution of pathogen virulence in a variable host population

Abstract: Several recent papers (for instance, Gilchrist and Sasaki (2002) and André-Gandon (2006)) have introduced explicit (simplified) models of hosts' immune response to study the evolution of pathogens, without the need of assuming/ a priori/ trade-offs between pathogens' virulence and transmission rates (or other traits). In all cases examined, it turns out that an intermediate level of pathogen replication (or of hosts' rate of immune response) is selected for. However, for most parameter values, this `intermediate' level of pathogen virulence corresponds to rather high lethality rates compared to normal infections. An aspect not considered by the previous models is variability (because of age, stress, nutritional conditions) in hosts' immune response. Assuming that immune response rates in the host population follow some probability distribution, computer simulations show that a much smaller replication rate is selected among pathogen, than if immune response rates were fixed; moreover, the overall lethality rate is smaller, and concentrated in the 'weakest' individuals. Since the overall model is rather complex and difficult to analyse, a toy model that shares some of its features is considered.
References: J.-B. André and S. Gandon, Vaccination, within-host dynamics, and virulence evolution. Evolution, 60(1), 2006, pp. 13-23
M. A. Gilchrist and A. Sasaki, Modeling Host-Parasite Coevolution: A Nested Approach Based on Mechanistic Models, J. theor. Biol. (2002) 218, pp. 289-308


Vladimir Vatutin (Moscow): Critical branching processes in random environment die slowly.

Abstract: A branching process \ $Z(n),n=0,1,...,$ is considered which evolves in a random environment generated by a sequence of independent identically distributed generating functions $% f_{0}(s),f_{1}(s),....$ Let $$S_{0}=0,\qquad S_{k}=\log f_{0}^{\prime }(1)+...+\log f_{k-1}^{\prime }(1),\qquad k\geq 1,$$ be the associated random walk and $T=\min\{k:\, Z(k-1)>0,Z(k)=0\} $. Assuming that the random walk satisfies the Spitzer condition \begin{equation*} \frac{1}{n}\sum_{k=1}^{n}\mathbf{P}\left( S_{k}>0\right) \rightarrow \rho \in (0,1),\;n\rightarrow \infty , \end{equation*}% we show that, contrary to the ordinary critical Galton-Watson process where (given $T=n$) the population size $Z(nt)$ has one and the same order for all $t\in (0,1)$, the branching process under consideration possesses an interesting feature: the population size of such process (given $T=n$) becomes drastically small in the vicinities of \textbf{random} moments where $\left\{ S_{k},\,0\leq k\leq n\right\}$ attains its local minima and is exponential large outside such vicinities, for instance at \textbf{nonrandom} moments like $nt,\,t\in (0,1)$. Thus, approaching the moment of extinction, the branching process in random environment passes through a number of bottlenecks and periods of recovering.


Anton Wakolbinger (Frankfurt): How often does the ratchet click? Facts, heuristics, asymptotics

Abstract: In an asexually reproducing population where (slightly) deleterious mutations accumulate along the individual lineages, how many generations does it take till "the ratchet clicks", i.e. till the current best class disappears from the population? Building on old ideas of Haigh (1978) we will propose a new one-dimensional diffusion approximation and identify a parameter depending on population size, mutation rate and selection strength which for this approximation and in a certain parameter regime gives information about the rate of the ratchet. Simulations confirm that our prediction works reasonably also for the full (multitype) Wright-Fisher model. (Joint work with Alison Etheridge and Peter Pfaffelhuber)