Paul Blackwell
School of Mathematics and Statistics, University of Sheffield, UK

Modelling and inference for continuous-time animal movement

Abstract: Realistic representation of wildlife movement presents substantial challenges in terms of both modelling and statistical inference. The models need to take account of variation in behaviour over time, usually through an unobserved behavioural state, and ideally need to be formulated in continuous time, to allow coherent interpretation, comparison and handling of irregular observations. A suitable class of models, which are ecologically and intuitively appealing, can be defined as continuous-time Markov processes on a space which is a product of a discrete behavioural space and a continuous space describing the animals’s position and, optionally, velocity. The complex relationship between location, behaviour and movement in these models makes inference difficult, and often a simplistic discrete-time approximation is used.

In this talk I will describe some examples of models of this form, and some recent progress in Bayesian inference for them using a Markov chain Monte Carlo algorithm. The statistical method uses a uniformization approach to represent the unobserved changes in behaviour as a thinned Poisson process, avoiding time-discretization error.

These ideas will be illustrated using data on individual fishers (Martes pennanti, courtesy of Scott LaPoint, Max Planck Institute for Ornithology, Konstanz), wild boar (Sus scrofa, courtesy of Mark Lambert, Animal and Plant Health Agency, York) and reindeer (Rangifer tarandus, courtesy of Anna Skarin, Swedish University of Agricultural Sciences, Uppsala). I will also look briefly at extensions to multiple animals, using higher-dimensional versions of these models.

This is mainly joint work with Mu Niu, University of Glasgow, with contributions from Hajar Alkhezi, Keith Harris, Théo Michelot and Alison Parton, all at the University of Sheffield.

Ed Codling
Department of Mathematical Sciences, University of Essex, UK

Modelling the efficiency of animal navigation strategies

Abstract: In this talk I will discuss a simple theoretical navigation problem that is relevant to animal movement at a range of scales (from large scale migration to short scale search and target-finding). I will first consider a biased and correlated random walk (BCRW) model for individual movement that balances direct navigation with forward persistence. I will show how an approximation for the navigational efficiency can be derived mathematically and will demonstrate the counter-intuitive result that giving higher weighting to indirect navigational cues (such as persistence) rather than direct cues may in many cases be the most efficient navigation strategy. I will subsequently show how simulations can be used to extend the model to consider collective group navigation such as in migrating flocks of birds or schools of fish. I will demonstrate how using indirect navigation cues such as persistence or copying the direction of movement of group neighbours (or some combination of both) can lead in many cases to more efficient collective navigation.

Dmitri Finkelshtein
Department of Mathematics, Swansea University, UK

Perturbation expansion around spatial mean-field limit

Abstract: We describe a general method how to derive the equation for the next term of expansion beyond the solution to the spatial mean-field equation, on the example of the spatial logistic model (a.k.a. Bolker-Pacala-Dieckmann-Law model). We will show how to use the equation appeared to the study of the extinction threshold in the case of Gaussian dispersion and competition kernels on the plane.

Mike Fowler
Department of Biosciences, Swansea University, UK

Untangling the stability and diversity of Diversity-Stability relationships in community ecology

Abstract: Community ecology benefits from a relatively long, rich theoretical history built around both verbal and mathematical reasoning, yet this explicit link to mathematical formulations has not prevented a proliferation of terms and their meanings related to both diversity and stability. Many of the phrases have been co-opted and redefined by others, including other important stakeholders including governments, NGOS and other policy makers.  I will focus on a few specific stability concepts (local asymptotic stability and the temporal variability of fluctuating processes) in a simple Lotka-Volterra system, demonstrating that changes in different types of diversity can generate different relationships with these stability measures. I will also discuss recent work that investigates how different measures of stability are related.

Yan Fyodorov
Department of Mathematics, King’s College London, UK

How many stable equilibria will a large complex system have?

Abstract: We aim to provide the quantitative answer to the classical question posed by Robert May (1972) “Will a Large Complex System be Stable?”. To this end we analyse a generic autonomous nonlinear system of N>>1 randomly coupled ODE’ describing degrees of freedom relaxing with the common relaxation rate. We show that with decreasing the rate  such systems experience an abrupt transition at some critical value of the rate from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial ‘absolute instability’ regime for where equilibria are exponentially abundant, but typically all of them are unstable. Finally, at even smaller relaxation rates stable equilibria become exponentially abundant, but their fraction to totality of all equilibria remains exponentially small. The revealed picture goes much beyond the May’s linear analysis and is expected to be of relevance in the applications of complex systems to ecology, population biology, neural network theory and other areas. The presentation will be based on joint works with Gerard Ben Arous and Boris Khoruzhenko.

Luca Giuggioli
Department of Engineering Mathematics and School of Biological Sciences, University of Bristol, UK

Stigmergic territorial systems

Abstract: When animals rely on processes external to themselves, they may coordinate their activity in an indirect manner. If the space retains memory of the passage or activity of an individual, animals respond indirectly to one another because interaction is mediated by the environment. This phenomenon, called stigmergy was coined by Pierre-Paul Grassé in the ’50s to explain nest building in termites. Besides the collective coordination in complex tasks, stigmergic processes may also serve the purpose of directing individuals towards or away from certain regions of space. Avoidance behaviour, specifically eschewing marks deposited by others, is the behaviour with which territorial patterns of various vertebrates form. Spatial segregation in minimally overlapping regions is accomplished by individuals leaving their own marks wherever they go, while retreating upon encountering foreign ones. I will present a model of scent-marking animals and quantify how different spatial patterns emerge as a function of the population density and the time for which the scent remains active after deposition.

Stephen Gourley
Department of Mathematics, University of Surrey, UK

Age-dependent toxicity in plant chemical defences and herbivore feeding behaviour

Abstract: This talk concerns the effects that woody plant chemical defenses may have on boreal hares that in winter feed almost entirely on twigs. Toxin concentration in twigs often varies with the age of twig segments. Youngest segments of the twigs are more defended by toxins than the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by hares is much higher than their intake of the biomass of the younger segments of twigs. This age-dependent toxicity of twig segments is modelled using age-structured equations that reduce to a system of delay differential equations. The model accounts for mortality of non-consumed younger twig segment biomass when older twig biomass is bitten off and consumed. I present basic mathematical properties of the model including upper and lower bounds on the solutions, necessary and sufficient conditions for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions for the global stability of that equilibrium. Numerical simulations demonstrate the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems. Thus, age dependence in plant chemical defenses has the capacity to generate hare – plant population cycles.

Andrew Morozov,
Department of Mathematics, University of Leicester, UK

Imperfect prey selectivity of a generalist predator promotes biodiversity and irregularity in food webs

Abstract: Ecological communities are often characterised by many species occupying the same trophic level and competing over a small number of vital resources. The mechanisms maintaining high biodiversity in such systems are still poorly understood. For example, a good understanding of the ‘paradox of plankton’ – the coexistence of many phytoplankton species competing over a few vital nutrient resources in an apparently homogeneous environment- is still lacking. In this talk, I revisit the role of prey selectivity by generalist predators in promoting biodiversity in food webs. Mathematically, I consider a generic tri-trophic food web, consisting of a single limiting nutrient resource, a large number of primary producers and a generalist predator. Firstly, I suggest a novel framework to describe the predator functional response combining active food selectivity for distinctly different functional prey groups with a proportion-based consumption of similar prey species. Model simulation reveals that intermediate levels of prey selectivity can explain high species richness, functional biodiversity, and variability among prey species observed in plankton communities. In contrast, perfect food selectivity or purely proportion-based food consumption (largely implemented in the current literature) would lead to a collapse of prey functional biodiversity.

Natalia Petrovskaya
School of Mathematics, University of Birmingham, UK

Evaluation of the total population size on coarse sampling grids: deterministic vs. probabilistic approach

Abstract: Many biological and ecological problems require accurate evaluation of the total population size. We discuss a sampling procedure used for evaluation of the population abundance from information collected on a grid of spatial sampling locations. It will be shown in our talk how insufficient information about the spatial population density obtained on a coarse sampling grid affects the accuracy of evaluation.  The insufficient information is collected because of inadequate spatial resolution of the population density on coarse grids and this is especially true when a heterogeneous spatial population density is sampled. It will be argued in the talk that the evaluation error is a random variable on coarse sampling grids because of the uncertainty in sampling spatial data and a probabilistic approach should be employed in the evaluation procedure. We also show that there exists a threshold number of sampling locations on a regular sampling grid where we can guarantee desired accuracy of evaluation. Information about the threshold number of sampling locations allows one to reconcile the probabilistic approach based on the assumption about randomness of sampling data with the deterministic approach based on the requirement that spatial data are collected only once as the sampling procedure cannot be repeated under the same conditions.

Sergei Petrovskii
Department of Mathematics, University of Leicester, UK

Catching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization

Abstract: Synchronization of population dynamics in different habitats is a frequently observed phenomenon. A common mathematical tool to reveal synchronization is the (cross-) correlation coefficient between time courses of values of the population size of a given species where the population size is evaluated from spatial sampling data. The corresponding sampling net or grid is often coarse, i.e. it does not resolve all details of the spatial configuration, and the evaluation error – i.e. the difference between the true value of the population size and its estimated value – can be considerable. We show that this estimation error can make the value of the correlation coefficient very inaccurate or even irrelevant. We consider several population models to show that the value of the correlation coefficient calculated on a coarse sampling grid rarely exceeds 0.5, even if the true value is close to 1, so that the synchronization is effectively lost. We also observe `ghost synchronization’ when the correlation coefficient calculated on a coarse sampling grid is close to 1 but in reality the dynamics are not correlated. Finally, we discuss a simple test to check the sampling grid coarseness and hence to distinguish between the true and artefactual values of the correlation coefficient.

Louise Riotte-Lambert
Institute of Biodiversity Animal Health and Comparative Medicine, University of Glasgow, UK

Consequences of memory-based movement at the individual and population levels

Abstract: Home ranges (HRs) are a remarkably common form of animal space use, but we still lack an integrated view of the individual-level processes that can lead to their emergence, particularly when individuals are in competition for resources. Moreover, population-level consequences of home ranging are still unknown.

I use a spatially explicit mechanistic movement model to investigate how simple memory-based foraging rules may enable animals to establish HRs and to what extent this increases their foraging efficiency. I then introduce demographic dynamics whereby reproduction and death depend on foraging efficiency, and compare the population dynamics of foragers using memory or not.

I show that memory-based movement enables individuals to perform better than individuals using the most efficient strategy that does not rely on memory and drives them to spatially segregate through avoidance of resource patches used by others. This striking result questions the common assumption that low HR overlaps are indicators of territorial behavior. I then show at the population level that memory use leads to a higher population size equilibrium and to a stronger depletion of the environment. Overall my results demonstrate the selective advantage of memory-based movement and that it has numerous non-negligible effects across several ecological organization levels.

Ulrike Schlägel
Institute of Biochemistry and Biology, Universität of Potsdam, Germany

Modelling cognition-based animal movement with random walks

Abstract: During station-keeping movements, cognition and spatial memory can help to accomplish tasks efficiently and we expect many animal species to have such capacities. Identifying and quantifying cognitive processes in movement behaviour is notoriously difficult, especially in natural settings that we cannot or do not want to manipulate. Discrete-time random walks offer an intuitive way to build statistical models for observed animal movement paths and can be extended to reflect fairly complex movement processes, including cognition. However, movement analyses based on such models suffer from the problem that their results are dependent on the temporal resolutions assumed for the model and data. In this talk, I summarize results from an in-depth analysis of movement models’ (non-) robustness against changes in temporal resolution and further discuss these in light of a model that accounts for wolf (Canis lupus) movement decisions based on a spatially heterogeneous environment and time since last visiting a location as a temporal component of an individual’s past experiences.

Daniel Strömbom
Department of Biosciences, Swansea University, UK, and Department of Mathematics, Uppsala University, Sweden

Effects of asynchrony in models of collective motion

Abstract: Animal groups such a schools of fish, flocks of birds, and herds of sheep often move together in a well-organised fashion. This despite the fact that each individual only interacts directly with its nearest neighbours and often no leader, or other form of central control, can be identified. How does that work? A common way to investigate this question is to postulate local interaction rules at the individual level and construct a so called self-propelled particle (spp) model based on them. The model is then analysed to determine what type of group-level properties emerge from repeated local interactions of the postulated type. This way of interpreting and presenting the results are common, i.e. ‘local interactions X produce groups of type A and B’. However, from a technical point of view it is clear that the situation is more complicated, as every model contains several other underlying assumptions and choices whose effects remain unknown. In this talk I will focus on one particular such choice that is always made, but rarely well-motivated or explored. Namely, whether all particles update their headings and position at exactly the same time (synchronous update), or not (asynchronous update). Unlike in related fields, e.g. cellular automata and multi-agent robotics, the effects of this choice has not been properly explored in the context of spp-models. Here I will illustrate, by example, that investigating this further may be critical to the study of collective motion by means of spp-models. In particular, using the simplest spp-model known to produce the three standard groups (swarms, mills and dynamic groups) I will show that dynamic groups form when certain asynchronous update schemes are used, but not when the synchronous update is used. This finding combined with the fact that synchronous update is the default choice in the field suggests that currently unobserved important group level phenomena may emerge in the standard models of collective motion if only another update scheme is chosen. For example, perhaps the elusive multistability and transition behaviour observed in schools of golden shiner fish is one of them.

Yi-Shan Wang
School of Mathematics and Statistics, University of Sheffield, UK

Continuous-time resource selection analysis for moving animals

Abstract: The analysis of animal movement has enriched our understanding of a variety of ecological phenomena. Particularly, mathematical analysis of observed data can enable us to identify switches in movement decisions of animals. Data may not be recorded at the precise point where the animal makes a decision to move. Consequently, continuous-time models are necessary to account for the fact that switches in movement decisions may have occurred at any point in time. Here, we focus on inferring switches in foraging decisions by moving animals. Our model incorporates a Resource Selection Function (RSF) into a switching Ornstein-Uhlenbeck (OU) process. We use a Markov Chain Monte Carlo (MCMC) algorithm to parametrise our OU model from simulated data of animals moving through a patchy landscape. Here, food quality can change over time, depleting and renewing dependent upon the presence or absence of animals. Our algorithm successfully infers both the OU movement parameters and the renewal and depletion rates of resources (the RSF parameters). We also show that it can be used to infer seasonally-varying RSF parameters.


Joe Bailey
Department of Mathematical Sciences, University of Essex, UK

Multiple movement behaviours with wrapped normally distributed turning angles can lead to mis-classification as a single movement strategy with a wrapped Cauchy distribution.

Abstract: Modelling the way in which animals move through their landscape is a vital and active area of research in movement ecology.  Analysis of such movement data often results in various parameters which can be used to reconstruct and interpret the movement behaviour, such as step lengths, turning angles, bout lengths etc.  When considering the distribution of the turning angles, one would expect to find a near normal distribution centred around some preferred direction in non-random movement, corresponding to a high number of turns with small deviations from the preferred direction and few larger turns.  However, the analysis of data can return heavy tailed distributions, indicating a propensity for highly directed movement with occasional medium and large turns, a result which appears counter intuitive to sensible movement strategies.  This type of behaviour has been noted in harbour seals [1], cow elk [2] and panthers [3].  In this presentation, I demonstrate how the mixing of two normally distribution random processes can lead to misclassification as a single heavy-tailed distribution (in this case a wrapped Cauchy distribution) when selected using best-fit model methods.  This in turn leads to a method for separating and identifying different movement behaviours from movement data.

[1] McClintock, B. T., Russell, D. J. F., Matthiopoulos, J. and King, R. (2013), Combining individual animal movement and ancillary biotelemetry data to investigate population-level activity budgets. Ecology, 94: 838–849.

[2] Morales, J. M., Haydon, D. T., Frair, J., Holsinger, K. E. and Fryxell, J. M. (2004), Extracting more out of relocation data:  Building movement models as mixtures of random walks. Ecology, 85: 2436–2445.

[3] van de Kerk, M., Onorato, D. P., Criffield, M. A., Bolker, B. M., Augustine, B. C., McKinley, S. A. and Oli, M. K. (2015), Hidden semi-Markov models reveal multiphasic movement of the endangered Florida panther. Journal of Animal Ecology, 84: 576–585.

Atheeta Ching
Department of Mathematics, University College London, UK

Diagonal Stability in Lotka-Volterra Systems

Abstract: Long term dynamics are a crucial part of population ecology. The interactions between different species in a Lotka-Volterra model can be represented by a matrix.  If this matrix is diagonally stable, there is a known Lyapunov function showing that any existing interior steady state will be globally stable, i.e. we have the co-existence of all species as time goes to infinity. Our work provides insight by exploring the geometric interpretation behind diagonally stable matrices in dimensions 3×3 and 4×4, allowing us to find other equivalent conditions. Zeeman and Zeeman introduced split-Lyapunov stability (originally in competitive systems) which also implies global stability of the interior steady state and is much easier to compute in higher dimensions. Numerical work was carried out to compare this class of matrices to diagonally stable matrices, examining whether it can be a viable alternative.

Danis Kiziridis
Department of Mathematics, Swansea University, UK

Rules of direct competition for space: fungi as exemplary combat system

Abstract: We investigated quantitatively the main forces driving the dynamics of interference competition for space, with wood decay fungi as model species. Three modelling approaches helped the generation of experimental hypotheses: lattice model, partial, and ordinary differential equations. Experimentally, two species were growing and competing on agar plates. We found that: (1) both species had constant speeds of expansion into unoccupied space; (2) species interactions were local, and same-species colonies were fusing; (3) only one of the two species could replace the other, after exceeding a colony size relative to the weaker competitor; (4) the stronger competitor replaced faster the younger colony parts of the weaker competitor; (5) competitors distributed uniformly their competitive power to the enemy front. Comparisons between experiment and models under various scenarios show that the five rules are essential for capturing the dynamics in the finite and bounded arenas of competition. Overall, this work suggests that by tracking experimentally these main forces, we could better explain and predict the dynamics of community succession.

Kirsty Lees
School of Biology, University of Newcastle, UK

Assessing the suitability of hidden Markov models for identifying potential behavioural states of the European lobster

Abstract: Homarus spp. lobsters are some of the best researched benthic organisms, however, despite their commercial importance knowledge gaps still exist regarding their movement patterns. Increased mobility of Homarus spp. has been correlated with increased trap exposure (Wiig et al., 2013), or catchability (Bowlby et al., 2007). Varying movement rates between demographic groups could have important implications for fisheries management. A popular approach to categorising unobserved behaviours assumes different behavioural `states’ characterised by differences in step length and the directionality of movement; straight continuous paths are considered as `transiting’ behaviour and more tortuous movements are indicative of area-restricted searching or `foraging’ behaviour (Morales et al., 2004). Acoustic transmitters were fitted to 58 European lobsters (H. gammarus) caught off the Northumberland coast (n = 44, 2013 and n = 14, 2016). High-resolution spatially-explicit data were gathered on their movements using a VEMCO Positioning System (VPS) (Amirix Systems Inc., Halifax, Canada) consisting of twelve hydrophone receivers covering an area ca.1.5 km. Although movement rates can be highly variable among individuals there is currently no definitive evidence of dichotomous behavioural states in H. gammarus and the influence of sex, body size, or reproductive state on movement rates is unclear (Moland et al., 2011; Skerritt et al., 2015). Using the R package moveHMM (Michelot et al., 2016) hidden Markov models (HMMs) were used to investigate the relationships between movement patterns and environmental covariates and how they differ among individuals. This study assesses the suitability of HMMs to correctly identify behavioural states of H. gammarus given its shelter-seeking behaviour within highly heterogeneous benthic habitats.

Wiig, J. R., Moland, E., Haugen, T. O. & Olsen, E. M. Can. J. Fish. Aquat. Sci. 70, 1468–1476 (2013).
Bowlby, H. D., Hanson, J. M. & Hutchings, J. A. Mar. Ecol. Prog. Ser. 331, 207–218 (2007).
Morales, J. M., Haydon, D. T., Frair, J., Holsinger, K. E. & Fryxell, J. M. Ecology 85, 2436–2445 (2004).
Moland, E. et al. Can. J. Fish. Aquat. Sci. 68, 1197–1210 (2011).
Michelot, T., Langrock, R. & Patterson, T. A. Methods Ecol. Evol. (2016).

Aled Morris
Department of Mathematics, Swansea University, UK

Individual variability in dispersal and invasion speed

Abstract: The ability to predict the speed at which a species expands its range is a fundamental problem in ecology. Of particular interest is understanding the influence of individual heterogeneity in movement and demographic parameters on the range expansion speed. Using a system of reaction-diffusion partial differential equations based on a Lotka-Volterra competition model, we model the growth, dispersal and mutation of two competing phenotypes in a domain – a fast disperser with a low reproductive rate and a slow disperser with a fast reproductive rate. Previous research, using numerical simulations and an assumption that the spreading speed of the system is determined by the linearisation of the reaction-diffusion system about the extinction state into which the species is spreading, had suggested that the spreading speed of such a heterogeneous population will be faster than single-phenotype populations. Here we prove this suggestion under certain conditions on parameters. Using this result we furthermore show that the spreading speed is a non-increasing function of the mutation rate and determine the ratio at which the phenotypes occur at the leading edge – a result of key ecological interest.