Jan Rychtář

Evolutionary games on graphs and the speed of the evolutionary process

In this paper, we investigate evolutionary games with the invasion process updating rules on three simple non-directed graphs: the star, the circle and the complete graph. Here, we present an analytical approach and derive the exact solutions of the stochastic evolutionary game dynamics. We present formulae for the fixation probability and also for the speed of the evolutionary process, namely for the mean time to absorption (either mutant fixation or extinction) and then the mean time to mutant fixation. Through numerical examples, we compare the different impact of the population size and the fitness of each type of individual on the above quantities on the three different structures. We do this comparison in two specific cases. Firstly, we consider the case where mutants have fixed fitness r and resident individuals have fitness 1. Then, we consider the case where the fitness is not constant but depends on games played among the individuals, and we introduce a ‘hawk–dove’ game as an example.

Evolutionary dynamics on small-order graphs

Abstract We study the stochastic birth-death model for structured finite populations popularized by Lieberman et al. [E. Lieberman, C. Hauert and M. A. Nowak (2005), Evolutionary dynamics on graphs, Nature, Vol. 433, pp. 312–316]. We consider all possible connected undirected graphs of orders three through eight. For each graph, using the Monte Carlo Markov Chain simulations, we determine the fixation probability of a mutant introduced at every possible vertex. We show that the fixation probability depends on the vertex and on the graph. A randomly placed mutant has the highest chances of fixation in a star graph, closely followed by star-like graphs, and the fixation probability is lowest for regular and almost regular graphs. We also find that within a fixed graph, the fixation probability of a mutant has a negative correlation with the degree of the starting vertex.

Two results on evolutionary processes on general non-directed graphs

The paper [Broom & Rychtař (2008)][1] analytically investigated the probability for mutants to fixate in an otherwise uniform population on two types of heterogeneous graphs (lines and stars) by evolutionary dynamics. The main motivation for concentrating on those two types of graphs only was the

Evolutionary Dynamics on Graphs - the Effect of Graph Structure and Initial Placement on Mutant Spread

We study the stochastic birth-death process in a finite and structured population and analyze how the fixation probability of a mutant depends on its initial placement. In particular, we study how the fixation probability depends on the degree of the vertex where the mutant is introduced, and which vertices are its neighbours. We find that within a fixed graph, the fixation probability of a mutant has a negative correlation with the degree of the starting vertex. For a general mutant fitness r, we give approximations of relative fixation probabilities in terms of the fixation probabilities of neighbours which will be useful for considering graphs of relatively simple structure but many vertices, for instance of the small world network type, and compare our approximations to simulation results. Further, we explore which types of graphs are conducive to mutant fixation and which are not. We find a high positive correlation between a fixation probability of a randomly placed mutant and the variation of vertex degrees on that graph.

A game-theoretic model of kleptoparasitic behavior in polymorphic populations ☆

Kleptoparasitism, the stealing of food by one animal from another, is a widespread biological phenomenon. In this paper we build upon earlier models to investigate a population of conspecifics involved in foraging and, potentially, kleptoparasitism. We assume that the population is composed of four types of individuals, according to their strategic choices when faced with an opportunity to steal and to resist an attack. The fitness of each type of individual depends upon various natural parameters, for example food density, the handling time of a food item and the probability of mounting a successful attack against resistance, as well as the choices that they make. We find the evolutionarily stable strategies (ESSs) for all parameter combinations and show that there are six possible ESSs, four pure and two mixtures of two strategies, that can occur. We show that there is always at least one ESS, and sometimes two or three. We further investigate the influence of the different parameters on when each type of solution occurs.

Evolutionary Games on Star Graphs Under Various Updating Rules

Evolutionary game dynamics have been traditionally studied in well-mixed populations where each individual is equally likely to interact with every other individual. Recent studies have shown that the outcome of the evolutionary process might be significantly affected if the population has a non-homogeneous structure. In this paper we study analytically an evolutionary game between two strategies interacting on an extreme heterogeneous graph, the star graph. We find explicit expressions for the fixation probability of mutants, and the time to absorption (elimination or fixation of mutants) and fixation (absorption conditional on fixation occurring). We investigate the evolutionary process considering four important update rules. For each of the update rules, we find appropriate conditions under which one strategy is favoured over the other. The process is considered in four different scenarios: the fixed fitness case, the Hawk–Dove game, the Prisoner’s dilemma and a coordination game. It is shown that in contrast with homogeneous populations, the choice of the update rule might be crucial for the evolution of a non-homogeneous population.

Variance-based selection may explain general mating patterns in social insects

Female mating frequency is one of the key parameters of social insect evolution. Several hypotheses have been suggested to explain multiple mating and considerable empirical research has led to conflicting results. Building on several earlier analyses, we present a simple general model that links the number of queen matings to variance in colony performance and this variance to average colony fitness. The model predicts selection for multiple mating if the average colony succeeds in a focal task, and selection for single mating if the average colony fails, irrespective of the proximate mechanism that links genetic diversity to colony fitness. Empirical support comes from interspecific comparisons, e.g. between the bee genera Apis and Bombus, and from data on several ant species, but more comprehensive empirical tests are needed.

Kleptoparasitic Melees—Modelling Food Stealing Featuring Contests with Multiple Individuals

Kleptoparasitism is the stealing of food by one animal from another. This has been modelled in various ways before, but all previous models have only allowed contests between two individuals. We investigate a model of kleptoparasitism where individuals are allowed to fight in groups of more than two, as often occurs in real populations. We find the equilibrium distribution of the population amongst various behavioural states, conditional upon the strategies played and environmental parameters, and then find evolutionarily stable challenging strategies. We find that there is always at least one ESS, but sometimes there are two or more, and discuss the circumstances when particular ESSs occur, and when there are likely to be multiple ESSs.

Overexpression of the Auxin Binding PROTEIN1 Modulates PIN-Dependent Auxin Transport in Tobacco Cells

Background Auxin binding protein 1 (ABP1) is a putative auxin receptor and its function is indispensable for plant growth and development. ABP1 has been shown to be involved in auxin-dependent regulation of cell division and expansion, in plasma-membrane-related processes such as changes in transmembrane potential, and in the regulation of clathrin-dependent endocytosis. However, the ABP1-regulated downstream pathway remains elusive. Methodology/Principal Findings Using auxin transport assays and quantitative analysis of cellular morphology we show that ABP1 regulates auxin efflux from tobacco BY-2 cells. The overexpression of ABP1can counterbalance increased auxin efflux and auxin starvation phenotypes caused by the overexpression of PIN auxin efflux carrier. Relevant mechanism involves the ABP1-controlled vesicle trafficking processes, including positive regulation of endocytosis of PIN auxin efflux carriers, as indicated by fluorescence recovery after photobleaching (FRAP) and pharmacological manipulations. Conclusions/Significance The findings indicate the involvement of ABP1 in control of rate of auxin transport across plasma membrane emphasizing the role of ABP1 in regulation of PIN activity at the plasma membrane, and highlighting the relevance of ABP1 for the formation of developmentally important, PIN-dependent auxin gradients.

Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.