Evolutionarily stable levels of aposematic defence in prey populations.

Our understanding of aposematism (the conspicuous signalling of a defence for the deterrence of predators) has advanced notably since its first observation in the late nineteenth century. Indeed, it extends the scope of a well-established game-theoretical model of this very same process both from the analytical standpoint (by considering regimes of varying background mortality and colony size) and from the practical standpoint (by assessing its efficacy and limitations in predicting the evolution of prey traits in finite simulated populations). The nature of the manuscript at hand is more mathematical and its aim is two-fold: first, to determine the relationship between evolutionarily stable levels of defence and signal strength under various regimes of background mortality and colony size. Second, to compare these predictions with simulations of finite prey populations that are subject to random local mutation. We compare the roles of absolute resident fitness, mutant fitness and stochasticity in the evolution of prey traits and discuss the importance of population size in the above.

Eco-evolutionary dynamics in finite network-structured populations with migration.

We consider the effect of network structure on the evolution of a population. Models of this kind typically consider a population of fixed size and distribution. Here we consider eco-evolutionary dynamics where population size and distribution can change through birth, death and migration, all of which are separate processes. This allows complex interaction and migration behaviours that are dependent on competition. For migration, we assume that the response of individuals to competition is governed by tolerance to their group members, such that less tolerant individuals are more likely to move away due to competition. We look at the success of a mutant in the rare mutation limit for the complete, cycle and star networks. Unlike models with fixed population size and distribution, the distribution of the individuals per site is explicitly modelled by considering the dynamics of the population. This in turn determines the mutant appearance distribution for each network. Where a mutant appears impacts its success as it determines the competition it faces. For low and high migration rates the complete and cycle networks have similar mutant appearance distributions resulting in similar success levels for an invading mutant. A higher migration rate in the star network is detrimental for mutant success because migration results in a crowded central site where a mutant is more likely to appear.

A Markovian decision model of adaptive cancer treatment and quality of life.

This paper develops and analyzes a Markov chain model for the treatment of cancer. Cancer therapy is modeled as the patient's Markov Decision Problem, with the objective of maximizing the patient's discounted expected quality of life years. Patients make decisions on the duration of therapy based on the progression of the disease as well as their own preferences. We obtain a powerful analytic decision tool through which patients may select their preferred treatment strategy. We illustrate the tradeoffs patients in a numerical example and calculate the value lost to a cohort in suboptimal strategies. In a second model patients may make choices to include drug holidays. By delaying therapy, the patient temporarily forgoes the gains of therapy in order to delay its side effects. We obtain an analytic tool that allows numerical approximations of the optimal times of delay.

Aposematic signalling in prey-predator systems: determining evolutionary stability when prey populations consist of a single species.

Aposematism is the signalling of a defence for the deterrence of predators. We presently focus on aposematic organisms that exhibit chemical defences, which are usually signalled by some type of brightly coloured skin pigmentation (as is the case with poison frog species of the Dendrobatidae family), although our treatment is likely transferable to other forms of secondary defence. This setup is not only a natural one to consider but also opens up the possibility for rich mathematical modelling: the strength of aposematic traits (signalling and defence) can be unambiguously realised using variables that are continuously quantifiable, independent from one another and which together define a two-dimensional strategy space wherein the aposematic behaviour of any one organism can be represented by a single point. We presently develop an extensive mathematical model in which we explore the joint co-evolution of aposematic traits within the context of evolutionary stability. Even though empirical and model-based studies are conflicting regarding how aposematic traits are related to one another in nature, the majority of works allude to a positive correlation. We presently suggest that both positively and negatively correlated combinations of traits can achieve evolutionarily stable outcomes and further, that for a given level of signal strength there can be more than one optimal level of defence. Our findings are novel and pertinent to a sizeable body of physical evidence, which we discuss.

A mathematical model of kin selection in floral displays.

Plants can adjust their competitive traits for acquiring resources in response to the relatedness of their neighbours. Recently, it has been found that plants can alter their investment in traits of attracting pollinators based on kin-interaction. We build a mathematical model to study the optimal floral display to attract pollinators in a patch with kin structure. We show that when plants can attract pollinators to a whole patch through the magnet effect, the floral display should increase with the increasing relatedness of the plants in the patch. Our model also indicates that increasing investment into attracting pollinators is a form of altruism, reducing a plant's own seed production but increasing the contribution of other plants to its fitness. We also predict that seed production should increase with increasing relatedness in the patch. Our model provides the explicit conditions when resource allocation to attract pollinators in response to neighbour relatedness can be favoured by kin selection, and a possible mechanism for the plants to deal with the consequent loss of pollinator diversity and abundance.

A temporal model of territorial defence with antagonistic interactions.

Territorial behaviour is an important part of the lives of many animals. Once a territory has been acquired, an animal may spend its entire life on it, and may have to repeatedly defend it from conspecifics. Some species make great investments in the defence of a territory, and this defence can be costly, in terms of time, energy and risk of injury. Time costs in particular have rarely been explicitly factored into such models. In this paper we consider a model of territorial defence which includes both population dynamic and time delay elements, building upon recent advances in time constraint models. Populations may divide into two distinct types, where one type makes no effort to control territories. We shall call this type nomads, and the other type territorials. Here the territory owners must divide their time between patrolling and foraging, and this balance is their only strategic decision. We show how to find the evolutionarily stable patrolling strategy and the population composition of territorials and nomads, and consider some examples demonstrating key situations. We see that both time constraints and population density pressure are crucial to influencing behaviour. In particular we find cases with both territorial individuals and nomads where a mixed, either pure or both pure patrolling strategies are evolutionarily stable. In different conditions either nomads or territorials can be absent, and indeed for a significant range of parameter combinations the population can exhibit tristability, with three distinct ecologically stable population compositions: with both nomads and territorials, only nomads or only territorials.

Two-strategy games with time constraints on regular graphs.

Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.

Models and measures of animal aggregation and dispersal.

The dispersal of individuals within an animal population will depend upon local properties intrinsic to the environment that differentiate superior from inferior regions as well as properties of the population. Competing concerns can either draw conspecifics together in aggregation, such as collective defence against predators, or promote dispersal that minimizes local densities, for instance to reduce competition for food. In this paper we consider a range of models of non-independent movement. We include established models, such as the ideal free distribution, but also develop novel models, such as the wheel. We also develop several ways to combine different models to create a flexible model of addressing a variety of dispersal mechanisms. We further devise novel measures of movement coordination and show how to generate a population movement that achieves appropriate values of the measure specified. We find the value of these measures for each of the core models described, as well as discuss their use, and potential limitations, in discerning the underlying movement mechanisms. The movement framework that we develop is both of interest as a stand-alone process to explore movement, but also able to generate a variety of movement patterns that can be embedded into wider evolutionary models where movement is not the only consideration.

When optimal foragers meet in a game theoretical conflict: A model of kleptoparasitism.

Kleptoparasitism can be considered as a game theoretical problem and a foraging tactic at the same time, so the aim of this paper is to combine the basic ideas of two research lines: evolutionary game theory and optimal foraging theory. To unify these theories, firstly, we take into account the fact that kleptoparasitism between foragers has two consequences: the interaction takes time and affects the net energy intake of both contestants. This phenomenon is modeled by a matrix game under time constraints. Secondly, we also give freedom to each forager to avoid interactions, since in optimal foraging theory foragers can ignore each food type (we have two prey types: either a prey item in possession of another predator or a free prey individual is discovered). The main question of the present paper is whether the zero-one rule of optimal foraging theory (always or never select a prey type) is valid or not, in the case where foragers interact with each other? In our foraging game we consider predators who engage in contests (contestants) and those who never do (avoiders), and in general those who play a mixture of the two strategies. Here the classical zero-one rule does not hold. Firstly, the pure avoider phenotype is never an ESS. Secondly, the pure contestant can be a strict ESS, but we show this is not necessarily so. Thirdly, we give an example when there is mixed ESS.

Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time.

Fitness is often defined as the average payoff an animal obtains when it is engaged in several activities, each taking some time. We point out that the average can be calculated with respect to either the time distribution, or to the event distribution of these activities. We show that these two averages lead to the same fitness function. We illustrate this result through two examples from foraging theory, Holling II functional response and the diet choice model, and one game-theoretic example of Hamilton's rule applied to the time-constrained Prisoner's dilemma (PD). In particular, we show that in these models, fitness defined as expected gain per unit time equals fitness defined as expected gain divided by expected time. We also show how these fitnesses predict the optimal outcome for diet choice and the prevalence of cooperation in the repeated PD game.

Methods for approximating stochastic evolutionary dynamics on graphs.

Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.

A theory for investment across defences triggered at different stages of a predator-prey encounter.

We introduce a general theoretical description of a combination of defences acting sequentially at different stages in the predatory sequence in order to make predictions about how animal prey should best allocate investment across different defensive stages. We predict that defensive investment will often be concentrated at stages early in the interaction between a predator individual and the prey (especially if investment is concentrated in only one defence, then it will be in the first defence). Key to making this prediction is the assumption that there is a cost to a prey when it has a defence tested by an enemy, for example because this incurs costs of deployment or tested costs as a defence is exposed to the enemies; and the assumption that the investment functions are the same among defences. But if investment functions are different across defences (e.g. the investment efficiency in making resources into defences is higher in later defences than in earlier defences), then the contrary could happen. The framework we propose can be applied to other victim-exploiter systems, such as insect herbivores feeding on plant tissues. This leads us to propose a novel explanation for the observation that herbivory damage is often not well explained by variation in concentrations of toxic plant secondary metabolites. We compare our general theoretical structure with related examples in the literature, and conclude that coevolutionary approaches will be profitable in future work.

Generalized Social Dilemmas: The Evolution of Cooperation in Populations with Variable Group Size.

Evolutionary game theory is an important tool to model animal and human behaviour. A key class of games is the social dilemmas, where cooperation benefits the group but defection benefits the individual within any group. Previous works have considered which games qualify as social dilemmas, and different categories of dilemmas, but have generally concentrated on fixed sizes of interacting groups. In this paper, we develop a systematic investigation of social dilemmas on all group sizes. This allows for a richer definition of social dilemmas. For example, while increasing a group size to include another defector is always bad for all existing group members, extra cooperators can be good or bad, depending upon the particular dilemma and group size. We consider a number of commonly used social dilemmas in this context and in particular show the effect of variability in group sizes for the example of a population comprising negative binomially distributed group sizes. The most striking effect is that increasing the variability in group sizes for non-threshold public goods games is favourable for the evolution of cooperation. The situation for threshold public goods games and commons dilemmas is more complex.

Models of kleptoparasitism on networks: the effect of population structure on food stealing behaviour.

The behaviour of populations consisting of animals that interact with each other for their survival and reproduction is usually investigated assuming homogeneity amongst the animals. However, real populations are non-homogeneous. We focus on an established model of kleptoparasitism and investigate whether and how much population heterogeneities can affect the behaviour of kleptoparasitic populations. We consider a situation where animals can either discover food items by themselves or attempt to steal the food already discovered by other animals through aggressive interactions. Representing the likely interactions between animals by a network, we develop pairwise and individual-based models to describe heterogeneities in both the population structure and other individual characteristics, including searching and fighting abilities. For each of the models developed we derive analytic solutions at the steady state. The high accuracy of the solutions is shown in various examples of populations with different degrees of heterogeneity. We observe that highly heterogeneous structures can significantly affect the food intake rate and therefore the fitness of animals. In particular, the more highly connected animals engage in more conflicts, and have a reduced food consumption rate compared to poorly connected animals. Further, for equivalent average level of connectedness, the average consumption rate of a population with heterogeneous structure can be higher.

The effect of fight cost structure on fighting behaviour involving simultaneous decisions and variable investment levels.

In the "producer-scrounger" model, a producer discovers a resource and is in turn discovered by a second individual, the scrounger, who attempts to steal it. This resource can be food or a territory, and in some situations, potentially divisible. In a previous paper we considered a producer and scrounger competing for an indivisible resource, where each individual could choose the level of energy that they would invest in the contest. The higher the investment, the higher the probability of success, but also the higher the costs incurred in the contest. In that paper decisions were sequential with the scrounger choosing their strategy before the producer. In this paper we consider a version of the game where decisions are made simultaneously. For the same cost functions as before, we analyse this case in detail, and then make comparisons between the two cases. Finally we discuss some real examples with potentially variable and asymmetric energetic investments, including intraspecific contests amongst spiders and amongst parasitoid wasps. In the case of the spiders, detailed estimates of energetic expenditure are available which demonstrate the asymmetric values assumed in our models. For the wasps the value of the resource can affect the probabilities of success of the defender and attacker, and differential energetic investment can be inferred. In general for real populations energy usage varies markedly depending upon crucial parameters extrinsic to the individual such as resource value and intrinsic ones such as age, and is thus an important factor to consider when modelling.

Evolutionary dynamics and the evolution of multiplayer cooperation in a subdivided population.

The classical models of evolution have been developed to incorporate structured populations using evolutionary graph theory and, more recently, a new framework has been developed to allow for more flexible population structures which potentially change through time and can accommodate multiplayer games with variable group sizes. In this paper we extend this work in three key ways. Firstly by developing a complete set of evolutionary dynamics so that the range of dynamic processes used in classical evolutionary graph theory can be applied. Secondly, by building upon previous models to allow for a general subpopulation structure, where all subpopulation members have a common movement distribution. Subpopulations can have varying levels of stability, represented by the proportion of interactions occurring between subpopulation members; in our representation of the population all subpopulation members are represented by a single vertex. In conjunction with this we extend the important concept of temperature (the temperature of a vertex is the sum of all the weights coming into that vertex; generally, the higher the temperature, the higher the rate of turnover of individuals at a vertex). Finally, we have used these new developments to consider the evolution of cooperation in a class of populations which possess this subpopulation structure using a multiplayer public goods game. We show that cooperation can evolve providing that subpopulations are sufficiently stable, with the smaller the subpopulations the easier it is for cooperation to evolve. We introduce a new concept of temperature, namely "subgroup temperature", which can be used to explain our results.

A Game-Theoretical Winner and Loser Model of Dominance Hierarchy Formation.

Many animals spend large parts of their lives in groups. Within such groups, they need to find efficient ways of dividing available resources between them. This is often achieved by means of a dominance hierarchy, which in its most extreme linear form allocates a strict priority order to the individuals. Once a hierarchy is formed, it is often stable over long periods, but the formation of hierarchies among individuals with little or no knowledge of each other can involve aggressive contests. The outcome of such contests can have significant effects on later contests, with previous winners more likely to win (winner effects) and previous losers more likely to lose (loser effects). This scenario has been modelled by a number of authors, in particular by Dugatkin. In his model, individuals engage in aggressive contests if the assessment of their fighting ability relative to their opponent is above a threshold [Formula: see text]. Here we present a model where each individual can choose its own value [Formula: see text]. This enables us to address questions such as how aggressive should individuals be in order to take up one of the first places in the hierarchy? We find that a unique strategy evolves, as opposed to a mixture of strategies. Thus, in any scenario there exists a unique best level of aggression, and individuals should not switch between strategies. We find that for optimal strategy choice, the hierarchy forms quickly, after which there are no mutually aggressive contests.

Modelling Dominance Hierarchies Under Winner and Loser Effects.

Animals that live in groups commonly form themselves into dominance hierarchies which are used to allocate important resources such as access to mating opportunities and food. In this paper, we develop a model of dominance hierarchy formation based upon the concept of winner and loser effects using a simulation-based model and consider the linearity of our hierarchy using existing and new statistical measures. Two models are analysed: when each individual in a group does not know the real ability of their opponents to win a fight and when they can estimate their opponents' ability every time they fight. This estimation may be accurate or fall within an error bound. For both models, we investigate if we can achieve hierarchy linearity, and if so, when it is established. We are particularly interested in the question of how many fights are necessary to establish a dominance hierarchy.

The effect of fight cost structure on fighting behaviour.

A common feature of animal populations is the stealing by animals of resources such as food from other animals. This has previously been the subject of a range of modelling approaches, one of which is the so called "producer-scrounger" model. In this model a producer finds a resource that takes some time to be consumed, and some time later a (generally) conspecific scrounger discovers the producer with its resource and potentially attempts to steal it. In this paper we consider a variant of this scenario where each individual can choose to invest an amount of energy into this contest, and the level of investment of each individual determines the probability of it winning the contest, but also the additional cost it has to bear. We analyse the model for a specific set of cost functions and maximum investment levels and show how the evolutionarily stable behaviour depends upon them. In particular we see that for high levels of maximum investment, the producer keeps the resource without a fight for concave cost functions, but for convex functions the scrounger obtains the resource (albeit at some cost).

A study of the dynamics of multi-player games on small networks using territorial interactions.

Recently, the study of structured populations using models of evolutionary processes on graphs has begun to incorporate a more general type of interaction between individuals, allowing multi-player games to be played among the population. In this paper, we develop a birth-death dynamics for use in such models and consider the evolution of populations for special cases of very small graphs where we can easily identify all of the population states and carry out exact analyses. To do so, we study two multi-player games, a Hawk-Dove game and a public goods game. Our focus is on finding the fixation probability of an individual from one type, cooperator or defector in the case of the public goods game, within a population of the other type. We compare this value for both games on several graphs under different parameter values and assumptions, and identify some interesting general features of our model. In particular there is a very close relationship between the fixation probability and the mean temperature, with high temperatures helping fitter individuals and punishing unfit ones and so enhancing selection, whereas low temperatures give a levelling effect which suppresses selection.