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This research would not have been possible without the generous EPSRC stipend; we are very grateful for the awarding of this studentship to Alan Scaramangas. 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 it’s 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. Publisher PDF

Evolutionary game theory has proved to be a powerful tool to probe the self-organisation of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not only on the payoffs received after each encounter but also on the population's size. Here, we study $2\times2$ games in well-mixed finite populations by analysing the fixation probabilities of single mutants as functions of population size. We proved that 9 out of the 24 possible games always lead to monotonically decreasing functions, similarly to fixed fitness scenarios. However, fixation functions showed increasing regions under 12 distinct anti-coordination, coordination, and dominance games. Perhaps counter-intuitively, this establishes that single-mutant strategies often benefit from being in larger populations. Fixation functions that increase from a global minimum to a positive asymptotic value are pervasive but may have been easily concealed by the weak selection limit. We obtained sufficient conditions to observe fixation increasing for small populations and three distinct ways this can occur. Finally, we describe fixation functions with increasing regions bounded by two extremes under intermediate population sizes. We associate their occurrence with transitions from having one global extreme to other shapes., Comment: 23 pages, 8 figures

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.

This paper will consider a group of individuals who each have a target number of contacts they would like to have with other group members. We are interested in how close this can be to being realized, considering the group's long-term outcome under reasonable dynamics on the number of contacts. We formulate this as a graph realization problem for undirected graphs, with the individuals as vertices and the number of desired contacts as the vertex degree. It is well known that not all degree sequences can be realized as undirected graphs, and the Havel-Hakimi algorithm characterizes those that can. When we ask how close the degree sequences can be to realization, we ask for graphs that minimize the total deviation between what is desired and possible. The sets of all such graphs and all such associated sequences are termed the minimal sets. Broom and Cannings have previously considered this problem in many papers, and it is hard to tackle for general target sequences. This paper revisited the minimal set in general, investigating two particular classes of sequence in particular. We consider the n-element arithmetic sequence (n-1, n-2, … 1, 0) for general n, including obtaining a formula that generates the size of the minimal set for a given arithmetic sequence, and the all or nothing sequences, where targets are either 0 or n-1, where a recurrence relation for such a formula is found. Further, we consider the question of the size of the minimal set of sequences in general. We consider a strategic version of the model where the individuals are involved in a multiplayer game, each trying to achieve their target, and show that optimal play can lead to the minimal set being left, thus answering an open question from earlier work.

In previous work we considered a model of a population where individuals have an optimum level of social interaction, governed by a graph representing social connections between the individuals, who formed or broke those links to achieve their target number of contacts. In the original work an improvement in the number of links was carried out by breaking or joining to a randomly selected individual. In the most recent work, however, these actions were often not random, but chosen strategically, and this led to significant complications. One of these was that in any state, multiple individuals might wish to change their number of links. In this paper we consider a systematic analysis of the structure of the simplest class of non-trivial cases, where in general only a single individual has reason to make a change, and prove some general results. We then consider in detail an example game, and introduce a method of analysis for our chosen class based upon cycles on a graph. We see that whilst we can gain significant insight into the general structure of the state space, the analysis for specific games remains difficult.

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 choose the number of treatment rounds they wish to administer based on the progression of the disease as well as their own preferences. We obtain a powerful analytic decision tool by which patients may select their preferred treatment strategy. In a second model patients may make choices on the timing of treatment rounds as well. By delaying a round of therapy the patient forgoes the gains of therapy for a time in order to delay its side effects. We obtain an analytic tool that allows numerical approximations of the optimal times of delay.

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.

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.