The ‘Hawk-Dove’ Game and the Speed of the Evolutionary Process in Small Heterogeneous Populations

Interaction rates, vital rates, background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk-Dove example.

Relation between travel strategy and social organization of migrating birds with special consideration of formation flight in the northern bald ibis

A considerable proportion of the world's bird species undertake seasonal long-distance migrations. These journeys are energetically demanding. Two major behavioural means to reduce energy expenditure have been suggested: the use of thermal uplifts for a soaring-gliding migration style and travelling in echelon or V-shaped formation. Both strategies have immediate consequences for the social organization of the birds as they either cause large aggregations or require travelling in small and stable groups. Here, we first discuss those consequences, and second present an analysis of formation flight in a flock of northern bald ibis on their first southbound migration. We observe clear correlations between leading and trailing on the dyadic level but only a weak correlation on the individual level during independent flight and no convincing correlation during the human guided part of the migration. This pattern is suggestive of direct reciprocation as a means for establishing cooperation during formation flight. In general, we conclude that behavioural adaptations for dealing with physiological constraints on long-distance migrations either necessitate or ultimately foster formation of social groups with different characteristics. Patterns and social organization of birds travelling in groups have been elusive to study; however, new tracking technology-foremost lightweight GPS units-will provide more insights in the near future.This article is part of the themed issue 'Physiological determinants of social behaviour in animals'.

Evolutionary dynamics of N-person Hawk-Dove games

In the animal world, the competition between individuals belonging to different species for a resource often requires the cooperation of several individuals in groups. This paper proposes a generalization of the Hawk-Dove Game for an arbitrary number of agents: the N-person Hawk-Dove Game. In this model, doves exemplify the cooperative behavior without intraspecies conflict, while hawks represent the aggressive behavior. In the absence of hawks, doves share the resource equally and avoid conflict, but having hawks around lead to doves escaping without fighting. Conversely, hawks fight for the resource at the cost of getting injured. Nevertheless, if doves are present in sufficient number to expel the hawks, they can aggregate to protect the resource, and thus avoid being plundered by hawks. We derive and numerically solve an exact equation for the evolution of the system in both finite and infinite well-mixed populations, finding the conditions for stable coexistence between both species. Furthermore, by varying the different parameters, we found a scenario of bifurcations that leads the system from dominating hawks and coexistence to bi-stability, multiple interior equilibria and dominating doves.

Matching times of leading and following suggest cooperation through direct reciprocity during V-formation flight in ibis

One conspicuous feature of several larger bird species is their annual migration in V-shaped or echelon formation. When birds are flying in these formations, energy savings can be achieved by using the aerodynamic up-wash produced by the preceding bird. As the leading bird in a formation cannot profit from this up-wash, a social dilemma arises around the question of who is going to fly in front? To investigate how this dilemma is solved, we studied the flight behavior of a flock of juvenile Northern bald ibis (Geronticus eremita) during a human-guided autumn migration. We could show that the amount of time a bird is leading a formation is strongly correlated with the time it can itself profit from flying in the wake of another bird. On the dyadic level, birds match the time they spend in the wake of each other by frequent pairwise switches of the leading position. Taken together, these results suggest that bald ibis cooperate by directly taking turns in leading a formation. On the proximate level, we propose that it is mainly the high number of iterations and the immediacy of reciprocation opportunities that favor direct reciprocation. Finally, we found evidence that the animals' propensity to reciprocate in leading has a substantial influence on the size and cohesion of the flight formations.

The evolutionary and ecological consequences of animal social networks: emerging issues

The first generation of research on animal social networks was primarily aimed at introducing the concept of social networks to the fields of animal behaviour and behavioural ecology. More recently, a diverse body of evidence has shown that social fine structure matters on a broader scale than initially expected, affecting many key ecological and evolutionary processes. Here, we review this development. We discuss the effects of social network structure on evolutionary dynamics (genetic drift, fixation probabilities, and frequency-dependent selection) and social evolution (cooperation and between-individual behavioural differences). We discuss how social network structure can affect important coevolutionary processes (host-pathogen interactions and mutualisms) and population stability. We also discuss the potentially important, but poorly studied, role of social network structure on dispersal and invasion. Throughout, we highlight important areas for future research.

Heterogeneity in background fitness acts as a suppressor of selection

We introduce the concept of heterogeneity in background fitness to evolutionary dynamics in finite populations. Background fitness is specific to an individual but not linked to its strategy. It can be thought of as a property that is related to the physical or societal position of an individual, but is not dependent on the strategy that is adopted in the evolutionary process under consideration. In our model, an individual's total fitness is the sum of its background fitness and the fitness derived from using a specific strategy. This approach has important implications for the imitation of behavioural strategies: if we imitate others for their success, but can only adopt their behaviour and not their social and economic ties, we may imitate in vain. We study the effect of heterogeneity in background fitness on the fixation of a mutant strategy with constant fitness. We find that heterogeneity suppresses selection, but also decreases the time until a novel strategy either takes over the population or is lost again. We derive analytical solutions of the fixation probability in small populations. In the case of large total background fitness in a population with maximum inequality, we find a particularly simple approximation of the fixation probability. Numerical simulations suggest that this simple approximation also holds for larger population sizes.

Approximating evolutionary dynamics on networks using a Neighbourhood Configuration model

Evolutionary dynamics have been traditionally studied on homogeneously mixed and infinitely large populations. However, real populations are finite and characterised by complex interactions among individuals. Recent studies have shown that the outcome of the evolutionary process might be significantly affected by the population structure. Although an analytic investigation of the process is possible when the contact structure of the population has a simple form, this is usually infeasible on complex structures and the use of various assumptions and approximations is necessary. In this paper, we adopt an approximation method which has been recently used for the modelling of infectious disease transmission to model evolutionary game dynamics on complex networks. Comparisons of the predictions of the model constructed with the results of computer simulations reveal the effectiveness of the method and the improved accuracy that it provides when, for example, compared to well-known pair approximation methods. This modelling framework offers a flexible way to carry out a systematic analysis of evolutionary game dynamics on graphs and to establish the link between network topology and potential system behaviours. As an example, we investigate how the Hawk and Dove strategies in a Hawk-Dove game spread in a population represented by a random regular graph, a random graph and a scale-free network, and we examine the features of the graph which affect the evolution of the population in this particular game.

A general framework for analysing multiplayer games in networks using territorial interactions as a case study

Recently, models of evolution have begun to incorporate structured populations, including spatial structure, through the modelling of evolutionary processes on graphs (evolutionary graph theory). One limitation of this otherwise quite general framework is that interactions are restricted to pairwise ones, through the edges connecting pairs of individuals. Yet, many animal interactions can involve many players, and theoretical models also describe such multiplayer interactions. We shall discuss a more general modelling framework of interactions of structured populations with the focus on competition between territorial animals, where each animal or animal group has a "home range" which overlaps with a number of others, and interactions between various group sizes are possible. Depending upon the behaviour concerned we can embed the results of different evolutionary games within our structure, as occurs for pairwise games such as the Prisoner's Dilemma or the Hawk-Dove game on graphs. We discuss some examples together with some important differences between this approach and evolutionary graph theory.