A succinct mathematical account of the co-evolution of defence and signalling of that defence (also known as aposematism) has been a long standing problem of ecology. The completion of the single-species game-theoretic model of Broom et al (2006) and its extension to the two-species case will have impact in a range of disciplines: from mathematical ecology and experimental biology (see for example Cortesi and Cheney 2010, Lee et al. 2010 and Lee et al. 2011, in the bibliography section) to game theory and complex systems theory. The specific aims of the proposed project are to1. Carry out an extensive review of the relevant literature (included in the Bibliography section); 2. Complete the one-species case (by uncovering the complete list of ESSs and characterising these in terms of their stability); 3. Develop the general model for the treatment of prey populations consisting of two species; 4. Determine the ESSs for two-species case (and characterise these in terms of their stability) for the case in which there is convergence in appearance (also known as two-point solution); and 5. Produce a model for Batesian mimicry (with strong emphasis in specific models of this). Background Prey species invest in constitutive and induced defences against attackers in order to increase their fitness. For attacks that occur over short time-scales constitutive defences admit the dominant mech-anism to avoid predation and may be classified as primary or secondary (Broom et al. 2006). The aim of the former is to reduce the rate at which prey species are attacked whereas the aim of the latter is two-fold: to reduce the probability that a mounted attack results in death (K) and to increase the aversive information (I) that predators have on similar looking prey. Investment in secondary defences may not be easily evaluated by predators ( as is the case with chemical defences) and is often coupled with a warning display which signals a prey species' investment in these defences (Broom et al. 2006). For the remainder of this discussion we will consider those secondary defences that require signalling in order for them to be evaluated by predators (Broom et al. 2010, Broom et al. 2013, Ruxton et al. 2009). The aim of the project that we are proposing is to complete the work on aposematism byroom et al. (2006) for the one-species case and to develop the general model for the two-species case. The latter will open up the possibility for studying particular models of Batesian mimicry (a form of mimicry in which harmless prey resemble harmful prey in order to increase their fitness) that are of interest in biology (Broom and Rychtar 2013, Cortesi and Cheney 2010, Lee et al. 2010 and Lee et al. 2011). Literature Review The co-evolution of secondary defences and signals of those defences have received certain, but not significant theoretical attention ( compared with studies on the mechanisms of action of these defences - Broom et al. 2006). A game-theoretic model of this co-evolution was first devised by Leimar et al.(1986) and continues to constitute dominant work in aposematism (Broom et al. 2006). Despite its novelty, this model is unable to account for the possibility of predators increasing their attack proba-bility towards prey species due to the presence of an ever-increasing inhibitory gradient. Additionally, many of the results rely on certain restrictive functional forms, which make it difficult to assess the generality of the model (Broom et al. 2006). In contrast with Leimar et al. (1986), which considers a set of (initially) naive predators (Broom et al. 2006), the (also game-theoretic) model of Broom et al. (2006), considers a set of predators modelled as a group in equilibrium for states of learning, hunger etc. and prey species (with average local relatedness a:) consisting of residents and mutants playing one of two distinct strategies. Of key importance to this model are evolutionarily stable strategies (ESSs) which are

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This project will explore the extent to which the predictive power of various forms of "Big Data" can be harnessed to overcome the impact of poor response to surveys - one of the major challenges facing social research today. Social surveys are a key tool used by the media, policy makers, and academics to understand more about public attitudes and behaviour. However, the value of surveys is put at risk by the fact that a large and growing number of those selected to take part in surveys do not respond. As non-respondents may be very different from respondents, nonresponse can introduce significant bias into the conclusions drawn from survey data. There is a pressing need therefore to understand more about the extent and sources of nonresponse bias. This requires having information about both respondents and non-respondents. In the absence of interview data being available for non-respondents, this information must be obtained from other, external, sources. The growth in "Big Data" i.e. routinely generated data arising from commercial transactions, online communication or public administration provides exciting new opportunities to supplement survey data with data from other sources. As opportunities for data linkage increase, there is a need for a detailed investigation into how such data can be used to understand and hopefully correct for nonresponse bias in general social surveys. This project will conduct such an investigation by adding pre-existing data from multiple sources to UK data from the European Social Survey (ESS), a methodologically rigorous survey of public attitudes. The project, drawing on the expertise of an inter-disciplinary team of survey researchers, statisticians and geographic information (GI) specialists, has three strands: First, the project will explore the opportunities that exist for matching data from three different sources to survey data. These include: small-area administrative data; commercial marketing data and geocoded information from the Ordnance Survey. Each data source will be evaluated in terms of: what information it can provide which may be matched to the survey records of respondents and non-respondents; the accuracy and completeness of this information; and the challenges that matching data presents in terms of the increased risk of individuals or households being identified from the combination of data held about them. Second, we will see how the matched data can provide information about potential biases that may be present in the survey data as a result of nonresponse. This will involve identifying any external variables associated both with the likelihood of nonresponse and the attitudes and behaviour the survey intendeds to measure. The project will consider how sources of nonresponse bias may vary geographically across the UK. Finally, we will assess whether using these external variables to create nonresponse weights to adjust for the possible over or under representation of certain types of respondent in the dataset has a significant effect on survey estimates and reduces bias in the data. This project has the potential to contribute significantly to our understanding not only of survey nonresponse bias but also the statistical tools available to remedy this bias, to improve survey data collection and generate more robust data to better understand public attitudes and behaviour. Lessons learnt will enhance general social surveys in the UK and internationally. This will have considerable benefits for the wide range of stakeholders involved in the funding, collection, and analysis of survey data and those who rely on the insights it provides. This includes academics, government agencies and other publically funded bodies, third sector organisations, policy makers and, ultimately, the general public.