Evolutionary Models of Structured Populations: Integrating Methods

5 Sep 2018, 10:00 → 7 Sep 2018, 14:00 Europe/Berlin

Lecture Hall (MPI for Evolutionary Biology)

We aim at bringing together researchers working on different structures that influence the evolution of populations. Structures are here broadly conceived, they may be spatial structures (e.g. individuals living in a subdivided territory), interaction-based structures (e.g. food webs) as well as demographic structures (e.g. life cycles). The workshop is meant to be an occasion in which one gets to know techniques and ideas used by other scholars to deal with seemingly different structures that may extrapolate to the study of one’s own structure of interest.

The following speakers have confirmed their participation:

• Annette Baudisch (University of Southern Denmark)
• Oana Carja (University of Pennsylvania)
• Hal Caswell (University of Amsterdam)
• Nicole Creanza (Vanderbilt University)
• Florence Débarre (Centre Interdisciplinaire de Recherche en Biologie (CIRB))
• André M. de Roos (University of Amsterdam)
• Bartlomiej Waclaw (University of Edinburgh)

Scientific Organization: Stefano Giaimo & Laura Hindersin (MPI for Evolutionary Biology)

 

workshop-structured-populations.pdf

Wednesday, 5 September

10:00 → 12:00 Tutorial: Sensitivity Analysis: Hal Caswell

with separate registration only

10:00 Sensitivity Analysis

Speaker: Hal Caswell

14:00 → 14:05 Welcome

Speakers: Laura Hindersin, Stefano Giaimo

14:05 → 14:50 Keynote: Models of cultural evolution in structured populations

Many foundational models of social learning and cultural evolution are constructed within the framework of theoretical population genetics. With genetic evolution as a starting point, models of cultural evolution emphasize that cultural traits—learned behaviors such as beliefs, practices, and tools—can be transmitted between individuals and are subject to evolutionary forces such as
selection and drift. In contrast to the assumptions of these models, however, human (and animal) interactions are unlikely to be ideally represented by well-mixed populations. Humans and animals have complex contact networks, where interactions between some individuals are common and interactions between other individuals are rare or absent. Some of these differences in interaction might be due to the spatial distribution of individuals in a population; individuals located in geographic proximity are more likely to interact. Other differences might be driven by social structure, with interactions on a social network more likely to occur between genetically related individuals and between individuals sharing social contexts. Here, I discuss a set of new models that explore how spatial or network structure can affect the spread of a cultural trait compared to well-mixed populations. These models apply broadly to learned behaviors, from bird songs to human languages and cultural traditions. Understanding the spatial and environmental dynamics of cultural interactions could shed light on fundamental concepts in the evolution of behavior, such as social learning, cooperation, and cumulative culture.

Speaker: Nicole Creanza

14:50 → 15:10 The genetic consequences of social structure

In many species individuals live in socially structured populations, forming behaviorally cohesive groups with kinship structure, which can range from temporary pair-bounded aggregations to very complex societies, as observed in many mammals. Despite its ubiquity social structure is ignored by most population genetics models. The traditional genetic approach envisions and models populations of social species as a network of relatively small panmitic demes. There are genetic risks associated with small population sizes, including loss of genetic diversity and inbreeding depression. Thus many group-living species are believed to be at high risk of inbreeding-depression. This is indeed the case of many mammalian species, where ecologists often invoke inbreeding-avoidance strategies as an important mechanism to prevent inbreeding-depression effects. During the past years we have developed new software to simulate genetic and demographic data under some of the most common mating strategies found in primates. In this framework social groups consist of small age-structured units where a limited number of individuals monopolize reproduction and mate according to more or less complex strategies. Simulations under this model have shown that social groups are surprisingly efficient in maintaining genetic diversity in the form of outbreeding without selective mechanisms. This is highly counter intuitive and contradicts the belief that small groups are necessarily subject to strong genetic drift and at high risk of inbreeding depression. This new theoretical framework was already used to interpret the genetic diversity from a primate species and can also be used as a null model against which ecological theories can be tested.

Speaker: Barbara Parreira

15:10 → 15:30 Invasion fitness for gene-culture co-evolution in family-structured populations

Human evolution depends crucially on the co-evolution between genetically determined behaviours and socially transmitted cultural information. Although vertical transmission of cultural information from parent to offspring is extremely common in hominins, its effects on cultural evolution are not well understood. We therefore investigated gene-culture co-evolution in a family-structured population of diploids. Specifically, we derived the invasion fitness of a mutant allele that influences a cultural variable (e.g., amount of knowledge or skill) to which carriers of the mutant are preferentially exposed in subsequent generations due to vertical transmission of culture. This allows for associations between genes and culture to last over multiple generations and thus sets the stage for cultural niche construction. We applied our invasion fitness to study how genetically determined phenotypes of individual and social learning co-evolve with the level of cultural adaptive information they generate. We find that due to kin selection effects, vertical transmission of information increases the level of adaptive information in the population. We also show that vertical transmission prevents evolutionary branching and the co-existence of highly-differentiated cultural morphs. Vertical transmission may therefore play an important qualitative role in gene–culture co-evolutionary dynamics. Importantly, our analysis of selection suggests that vertical transmission of culture can significantly increase levels of adaptive cultural information under the biologically plausible condition that information transmission between relatives is more efficient than between unrelated individuals.

Speaker: Charles Mullon

15:30 → 16:00 Coffee Break

16:00 → 16:45 Keynote: Imperfect transmission and the evolution of social behavior in spatially structured populations

The theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, i.e., for genetically transmitted traits, that mutation is very weak or absent. In this talk, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider spatially structured populations of fixed size N, in which two types of individuals,A and B, corresponding to two types of social behavior, are competing. Under the assumption of small phenotypic differences (weak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate this result with three common life-cycles (Wright-Fisher,Moran Birth-Death and Moran Death-Birth),and specific population structures. Qualitatively, we find that some life-cycles (Moran Birth-Death,Wright-Fisher, when social interactions affect fecundities) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. Imperfect strategy transmission also alters the balance between the benefits and costs of staying next to one's kin, leading to surprising results in subdivided populations, in that higher emigration probabilities can be favourable to the evolution of altruistic strategies.

Speaker: Florence Débarre

16:45 → 17:05 The replicator dynamics for multilevel selection in evolutionary games

We consider a model of evolutionary game theory in group-structured populations, extending Luo and Mattingly’s multilevel selection framework to account for frequency dependent selection. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, with particular emphasis placed on understanding the most frequent group compositions at steady-state. We observe, regardless of how weak within-group selection is relative to between-group selection, that the most prevalent group composition will contain more defectors than the type of group which maximizes average group payoff if, and only if, maximum average payoff is not achieved by full cooperator groups. In such cases, the dynamics at the two levels cannot be decoupled, and we even observe cases in which cooperation cannot be sustained even when between-group competition favors perfect coexistence of cooperators and defectors.

Speaker: Daniel Cooney

17:05 → 17:25 Stirring does not make populations well mixed: The effect of motion on fixation probability

In evolutionary dynamics, the notion of a “well-mixed” population is usually associated with all-to-all interactions at all times. This assumption simplifies the mathematics of evolutionary processes, and makes analytical solutions possible. At the same time the term “well-mixed” suggests that this situation can be achieved by physically stirring the population. Using simulations of populations in chaotic flows, we show that in most cases this is not true: conventional well-mixed theories do not predict fixation probabilities correctly, regardless of how fast or thorough the stirring is. We propose a new analytical description in the fast-flow limit. This approach is valid for processes with global and local selection, and accurately predicts the suppression of selection as competition becomes more local. It provides a modelling tool for biological or social systems with individuals in motion.

Speaker: Francisco Herrerías-Azcué

17:25 → 17:45 The cancellation effect at the group level

Group selection models combine selection pressure at the individual level with selection pressure at the group level (Boyd and Richerson, 2009; Luo, 2014; Simon, 2010; Simon et al., 2013; Sober and Wilson, 1998; Traulsen and Nowak, 2006; Wilson and Wilson, 2007). Cooperation can be costly for individuals, but beneficial for the group, and therefore, if groups are sufficiently much assorted, and cooperators find themselves in groups with disproportionately many other cooperators, cooperation can evolve. The existing literature on group selection generally assumes that competition between groups occurs in a well-mixed population of groups, where any given group competes with any other group equally intensely. Competition between groups however might very well occur locally; groups may compete more intensely with their neighbours than with far-away groups. We show that if competition is indeed local, then the evolution of cooperation can be hindered by the fact that groups with many cooperators will mostly compete against neighbouring groups that are similarly cooperative, and therefore harder to outcompete. At the individual level, a similar phenomenon is called the cancellation effect, and has been discovered by Wilson et al. (1992) and Taylor (1992a-b). We show that cancellation effects also occur at the group level, and that ignoring them makes empirical estimates of the benefit-to-cost ratios for which a given group structure could sustain cooperation too positive.

Speaker: Aslıhan Akdeniz

18:00 → 20:00 Dinner

Thursday, 6 September

09:00 → 09:45 Keynote: Looking for fitness in all the wrong places: fitness as a multidimensional operator

Co-Author: Charlotte de Vries
Classical population genetics has this concept called "fitness." It is a scalar measure and in that theory, it serves three functions. It encapsulates, in a single number, all of the demographic processes of survival and reproduction that determine the transmission of genes. It also implies a projection of population size and composition over time, based on those demographic processes. Finally, it provides conditions for the coexistence or exclusion of genotypes. Analyses of structured populations have searched for a similar scalar measure that will serve these three functions (lambda, r, R0, reproductive value). Success has been limited. We will try to improve the situation. We show a framework that incorporates genetic processes into stage-classified matrix population models, and permits direct calculation of all three of these functions. It becomes clear that fitness is an operator, not a number, and that the three classical functions of fitness can be derived from this operator. However, unlike the basic population genetic case, no single number satisfies all three conditions. Our framework can help to integrate ecological and evolutionary processes.

Speaker: Hal Caswell

09:45 → 10:05 Biological pest control using cannibalistic predators and with provision of additional food

Cannibalism is a conspecific lethal interaction, a typical phenomenon in many natural populations, which is used as a ``life-boat strategy" to avoid circumstances leading to extinction. It is observed in many experimental studies that the cannibalistic nature of natural enemies deters the outcome of biological pest control programs. One of the ways to deviate natural enemies from conspecific lethal interactions is to provide them with additional food. In this paper, using the theory of dynamical systems, we analyse the dynamics of a cannibalistic predator-prey system when predators are provided with additional food. A detailed mathematical analysis is carried out to study the permanence, stability and various bifurcations occurring in the system. The system analysis reveals several interesting phenomena. Depending on the choice of quality (characterised by the predator's handling time towards additional food, and prey) and quantity of additional food, the system can exhibit multiple coexisting equilibria, leading to the emergence of a homoclinic loop. Further, it is observed that by varying the quality and quantity of additional food, one can not only limit and control the pest but also eradicate the predators. In the context of biological control programs, the current theoretical study aids eco-managers in choosing the appropriate additional food that is to be supplied for enhancing the biocontrol efficiency of cannibalistic predators.

Speaker: Sree Rama Vara Prasad Bhuvanagiri

10:05 → 10:25 Signatures and detection of selective sweeps in subdivided populations with various migration rates

There exists a large number of measures and algorithms designed to detect the effects of positive selection from population genetic or genomic data. However, most of these methods are based on a panmictic population model, while real-life populations often consist of subpopulations (demes) with limited migration. In such cases, a few extra factors exist compared to the standard selection model: a deme experiencing a selective sweep could have the adaptive allele arising within itself (native sweep) or immigrating from another deme (imported sweep); for an imported sweep, the allele may or may not be beneficial in the recipient deme. An imported allele from an adapted deme can increase its frequency faster than by drift alone, if the migration rate is high enough, even if it is not beneficial in the recipient deme. In addition, sweeps can arise from a single beneficial mutation (hard sweep) or from standing variation and environmental change (soft sweep). Finally, sweeps at different stages (e.g., ongoing vs. finished) produce different genetic footprints and can complicate detection if the model only accounts for one time snapshot. To distinguish these situations, we simulated spatially subdivided populations with a forward-in-time algorithm and under different selection and migration regimes. Using a combination of summary statistics based on frequency spectrum, haplotypes, linkage disequilibrium and coalescent topology, we employed machine learning algorithms to classify patterns and to determine the genomic location and the migration history of adaptive alleles.

Speaker: Yichen Zheng

10:25 → 11:00 Coffee Break

11:00 → 11:45 Keynote: Eco-evolutionary dynamics of life cycle complexity

The majority of species is characterised by a complex life cycle, in which individuals grow substantially during life or pass through several distinct life history stages and occupy different ecological niches during their life. In turn, these characteristics of the individual life history shape the ecological context in which evolution takes place through the impact of the population on its environment. To gain understanding about this interplay between the ecology and evolution of complex life cycles, it is hence necessary to take into account how the individual life history influences population and community dynamics and in turn how the population and community setting influences selection and adaptation of this life history.
Physiologically structured population models (PSPMs) is a class of models that consistently translates the individual life history to the population level, including the nonlinear interactions that individuals are exposed to. In recent years a general methodology has been developed for equilibrium and evolutionary analysis of PSPMs. The evolution of life history traits in PSPMs is carried out using the framework of Adaptive Dynamics by evaluating whether mutant types can invade a resident population under the ecological conditions that are determined by this resident. I will give a brief introduction to the general methodology and illustrate how it allows for evolutionary analysis starting from the key ingredients of an individual’s life history: its development, reproduction and mortality and its interactions, both intra- and interspecific, with its environment. I will subsequently use the PSPM methodology to study the evolution of metamorphosis using a fully size-structured population model in conjunction with the adaptive-dynamics approach. Even though almost all animal species undergo metamorphosis and empirical data show that this life-history strategy evolved only a few times, it is unclear why metamorphosis is so widespread and which factors played significant roles in its evolution. I will show that metamorphosis is not easy to evolve, but, once evolved, it is an evolutionary trap.

Speaker: André M. de Roos

11:45 → 12:05 Does developmental plasticity influence speciation?

Environmental cues affect phenotypic traits at the given life stage when they occur, but can also canalise later development down particular paths. The evolutionary consequences of developmental plasticity or canalisation are increasingly clear within species, but we lack fundamental data on these consequences for the macroevolutionary emergence of new species. Empirical evidence of evolutionary divergence among free-living organisms comes from the fossil record, which, while documenting the journey of life over vast tracts of time, does so via an incomplete static snapshot of death. Many organisms with life cycles as diverse as zooplankton, bivalves, trilobites and trees nevertheless contain records of their ontogenetic development that is extractable via modern imaging technology. Drawing data from experiments and developing feature extraction software for x-ray computed tomography, I'll use an integral projection model to motivate studies of how planktonic foraminiferal growth evolved in deep time. Empirical results show how somatic growth in individuals can evolve even when adult size remains constant, while the IPM predicts that greater lability during ontogeny will increase population mean fitness. The constraints imposed by logarithmic growth spirals hinderas selection leading to ever more variable developmental trajectories, but must at some point become relaxed to allow the emergence of new morphological forms. I'd welcome the opportunity at the workshop to explore alternative modelling treatments for use with large data sets to contextualise empirical evidence of how variation among individuals generates variation among species.

Speaker: Thomas Ezard

12:05 → 12:25 Evolution of life cycles in early multicellularity

Reproduction is a defining feature of living systems. To reproduce, aggregates of biological units (e.g., multicellular organisms or colonial bacteria) must fragment into smaller parts. Fragmentation modes in nature range from binary fission in bacteria to collective-level fragmentation and the production of unicellular propagules in multicellular organisms. Despite this apparent ubiquity, the adaptive significance of fragmentation modes has received little attention. Here, we develop a model in which groups arise from the division of single cells that do not separate but stay together until the moment of group fragmentation. We allow for all possible fragmentation patterns and calculate the population growth rate of each associated life cycle. Fragmentation modes that maximise growth rate comprise a restrictive set of patterns that include production of unicellular propagules and division into two similar size groups. Life cycles marked by single-cell bottlenecks maximise population growth rate under a wide range of conditions. This surprising result offers a new evolutionary explanation for the widespread occurrence of this mode of reproduction. All in all, our model provides a framework for exploring the adaptive significance of fragmentation modes and their associated life cycles.

Speaker: Yuriy Pichugin

12:25 → 14:00 Lunch Break

14:00 → 14:20 Phase transitions in evolutionary dynamics

The evolutionary dynamics of a finite population where resident individuals are replaced by invader or mutant ones depend on its spatial structure. The population adopts the form of an undirected graph where the place occupied by each individual is represented by a node and it is bidirectionally linked to the places that can be occupied by its clonal offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any node. But there also are undirected graph structures acting as suppressors of selection. Here, we show that some undirected graphs exhibit phase transitions between both evolutionary states when the mutant fitness varies. Firstly, as was already observed by Hindersin and Traulsen, we show that most graphs of order 10 or less are amplifiers of selection or suppressors for weak selection that become amplifiers from a unique transition phase. However, in order 7, we give examples of amplifiers for weak selection that become suppressors from some critical value. The analysis of all graphs from order 8 to 10 reveals a complex and rich evolutionary dynamics.

Speaker: Alvaró Lozano Rojo

14:20 → 14:40 Strong amplifiers of selection

In evolutionary graph theory, population of size $n$ is represented as a (connected) graph with $n$ vertices whose edges represent who can replace whom. Initially, all $n$ vertices have fitness 1 except for one vertex (mutant) that has fitness $r>1$. This initial mutant may have arised either spontaneously (aka uniform initialization), or during reproduction (aka temperature initialization). The population then evolves according to so-called Moran process until, eventually, all vertices become the same type — the single mutant either goes extinct or reaches fixation. Graphs that ensure fixation probability of 1 in the limit of infinite population size are called strong amplifiers. Previously, only a few examples of strong amplifiers were known for uniform initialization, and no strong amplifiers were known for temperature initialization. We show that self-loops and weighted edges are two key features for strong amplification. On one hand, we show that without either self-loops or weighted edges, strong amplification is (i) impossible under temperature initialization, and (ii) impossible for bounded-degree graphs under uniform initialization. On the other hand, we show that with both self-loops and weighted edges, strong amplification is ubiquitous: Almost any graph with self-loops can be assigned weights that make it a strong amplifier for both temperature and uniform initialization. This is a joint work with Andreas Pavlogiannis, Krishnendu Chatterjee, and Martin A. Nowak.

Speaker: Josef Tkadlec

14:40 → 15:10 Coffee Break

15:10 → 15:55 Keynote: Darwinian evolution in spatial models of cancer

Mathematical modelling of cancer has a long history but it traditionally focused on replicating growth laws observed for different tumours, the role of angiogenesis, or predicting the outcome of chemotherapy. Recently, advances in genomics have made it possible to investigate Darwinian evolution in populations of cancer cells. This has opened up many
interesting questions. In particular, as the cancerous tumour grows, cells accumulate further mutations. Are these mutations neutral “passengers” (i.e. they do not change the net growth rate) or are some of them “driver mutations” that increase the growth rate? Is there evidence of selection
acting on certain traits of cancer cells? How genetically diverse a typical tumour is? How is evolution affected by the spatial structure of the tumour?

In this talk I will show how computer models can be used to shed light on these questions. I will discuss models of increasing complexity: well-mixed, 3d lattice-based, and 3d off-lattice models. I will show how different
processes: replication, death, migration, and mechanical interactions between cells in a tumour affect its structure and genetic composition. I will also discuss how different models compare to experimental data, and implications for cancer therapy.

Speaker: Bartlomiej Waclaw

15:55 → 16:15 Modeling cancer: growth and size distribution of metastases under treatment

The observation and ability to form prognosis for the amount and sizes of possible metastases of a tumor is of high interest for oncologists. Mathematical models describing the seeding and growth of metastases are possibly of clinical use in optimizing individual therapy algorithms.
I adapted the well-known von-Foerster equation used by Iwata, Kawasaki and Shigesada (2000) to describe metastases’ dynamics when metastases are seeded at a cell size of exactly one. With redefined boundary conditions I was capable to reformulate the model to describe the seeding of metastasis of any size, also taking into account secondary metastases. As the model is defined in a continuous way, it can also be used to model a crossing of a T1N0M0 cancer towards an uprising metastatic disease.
The model is illustrated numerically with the data given by Iwata et al. to examine the differences of both approaches. Further, some therapy dynamics are included into the model equations for being able to analyze clinical data.

Speaker: Pirmin Schlicke

16:15 → 16:35 Characterising the evolutionary modes of cancer and normal tissue

The nature of evolution within normal and neoplastic tissue is a subject of debate. I will present a highly flexible computational model that allows evolutionary dynamics resulting from diverse spatial structures to be compared in a single, minimal framework. Combining stochastic simulations with mathematical analysis, I will explain how tissue architecture governs the potential for subclonal expansion, the prevalence of selective sweeps, and spatial patterns of genetic heterogeneity. I will describe the conditions under which genetic diversity is most predictive of tumour progression, and I will discuss applications in optimising treatment protocols and understanding cancer risk variation. These findings help explain the observed multiformity of cancer and normal tissue evolution and contribute to establishing a theoretical foundation for predictive oncology.

Speaker: Robert Noble

16:45 → 19:45 Boat Trip & Dinner

Friday, 7 September

09:00 → 09:45 Keynote

Speaker: Oana Carja

09:45 → 10:05 Vacancies in a growing habitat and the evolution of cooperation

We consider a stochastic evolutionary game on a one-dimensional lattice with vacancies created by fitness-dependent death processes. Interactions between neighbors are mimicked by a prisoner’s dilemma game and an individual with higher payoff lives longer. In each time step, we choose a random site of the habitat. If the site is occupied, its inhabitant dies with probability depending its game payoff. If the chosen site is empty, the site is colonized by an offspring of its neighbors (provided at least one of the neighboring sites is occupied). We study the growth of populations from a single seed and find that its dynamics critically depends on the environment parameter. As the harshness of environment changes, phase transitions from growing population phases to vanishing population phases occur for both types of seeds, a cooperator and a defector. The transition point for cooperators is smaller than the transition point for defectors, implying the existence of a parameter region wherein only cooperative populations can exist. We also find that cooperators dominate defectors even when the environmental factor is moderate for surviving of defectors. Our observation shows that vacancies, introduced by fitness dependent death processes, provide a natural way to develop cooperative communities in a growing population.

Speaker: Hyejin Park

10:05 → 10:30 Coffee Break

10:30 → 11:15 Keynote: Promising perspectives on age-structured mortality data

Nature shapes patterns of birth and death for all living beings, including humans. Why does nature permit human lifespan to double from about 40 to 80, with progress still ongoing? What are the limits? Will we be in good health or not when old? How much can we influence, what is given? Life hides its secrets well under complex and endless variation – simple models and concepts provide helpful perspectives and guidance.

Speaker: Annette Baudisch

11:15 → 11:35 Life cycle of cooperation

Evolutionary game theory provides a powerful and flexible tool to the investigation of the evolution of interacting individuals. In this study we focus on the evolution of “staying together” groups, in which new individuals in a population emerge only by splitting the previous group. These groups deserve a special attention, since many multicellular organisms use this mode of the organism development. Interactions between players in a group, provided by a two- player matrix game, affects both the group growth time as a whole and the probabilities of each player type to reproduce. In our model, we mainly focus on the population of containing two phenotypes either A or B and quantise the interactions between cells. The combination of the game played within the group and the fragmentation mode of producing new groups determine the population growth rate. Thus, the mode of fragmentation providing the largest growth rate λ is considered to be evolutionary optimal reproductive strategy.

Speaker: Yuanxiao Gao

11:35 → 11:55 Applying evolutionary demographic theory to evolution on graphs

Population structure can strongly affect evolutionary dynamics. A popular way to describe such structures are graphs. A quantity of great interest in the study of evolutionary graphs is the probability that a novel beneficial mutation spreads through the entire population. Here, we propose an alternative way to understand the forces driving fixation by viewing graphs as life cycles. Adapting methods from evolutionary demography, we quantify the invasion fitness and the effective population size for different graphs. Both invasion fitness and effective population size appear in an approximation of the fixation probability of an advantageous mutant. The method is very general and applies to weighted graphs with node dependent fitness. However, we focus on analytical results for undirected graphs with node independent fitness. The method also contributes to conceptually integrate evolutionary graph theory with theoretical genetics of structured populations.

Speaker: Stefano Giaimo

11:55 → 12:00 Farewell