Mathematical investigations into the mechanisms driving the spatiotemporal dynamics of cyclic populations. Matthew J. Smith. Heriot-Watt University

Transcription

1 Mathematical investigations into the mechanisms driving the spatiotemporal dynamics of cyclic populations. Matthew J. Smith Submitted for the degree of Doctor of Philosophy Heriot-Watt University School of Mathematical and Computer Sciences October, 27 This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that the copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the prior written consent of the author or of the University (as may be appropriate). i

2 ABSTRACT The research in this thesis uses mathematical modelling to investigate the importance of various biological mechanisms in determining the spatial and temporal dynamics of populations that show multi-year cycles in abundance. The ecological system of key interest is the field vole (Microtus agrestis) populations of Kielder Forest (northern UK) which show 3-5 year cycles that are spatiotemporally organised into periodic travelling waves. The first chapter introduces cyclic populations generally and current hypotheses for the mechanisms generating their spatiotemporal dynamics. The second chapter demonstrates that delayed density dependent reproductive timing, a phenomenon recently discovered in the Kielder Forest system, can generate population cycles. The third chapter then extends this finding, showing that plausible microparasitic disease effects on fecundity could generate cycles and delayed density dependent reproductive timing. The fourth chapter reviews the mathematical and empirical studies underpinning our current understanding of periodic travelling waves in cyclic populations, and identifies important research questions for the future. The fifth chapter demonstrates how curved habitat boundaries influence the wave train, and the population cycles, generated by such boundaries. The sixth chapter shows that the relative values of the dispersal rates of the interacting species generating the cycles can also have profound effects on the spatiotemporal dynamics. The seventh chapter demonstrates that the effects of dispersal rates that change as a function of local population density (a common phenomenon in nature) is relatively minor compared to the effects of the overall ratio of the dispersal rates. Finally, the eighth chapter makes detailed recommendations for future studies based on the findings of the previous chapters. ii

3 DEDICATION To Mum and Dad, for giving me a passion for the natural world and to Helen, for keeping me motivated and for keeping it all in perspective. Thanks also to William for making life so much fun. iii

4 ACKNOWLEDGEMENTS This has been a very productive few years and it s only been made possible by the generous help of others. First of all it would be difficult to find a better combination of supervisors than Jonathan Sherratt (Heriot-Watt University) and Xavier Lambin (Aberdeen University). Jonathan has an exceptional ability to explain mathematical concepts and without it this thesis would have taken at least twice as long to complete. I have hugely benefited from his open-door policy, his good nature, patience, generosity, and his inspirational academic thinking. Xavier is also a brilliant educator. His encyclopaedic knowledge of the biology, thorough explanations, and his hold-no-punches descriptions of the issues and controversies meant that obtaining the relevant biological details was made easier and fun. I also benefited from Xavier s expert demonstrations and assistance in the field. Both of my supervisors read and commented on more drafts than anyone else and I thank them for all the time they invested. Andy White (Heriot-Watt) also acted as if he was my supervisor and I especially benefited from his knowledge, experience, enthusiasm and advice. Thanks to Kevin Painter for his constructive annual assessments of my progress. Thanks to Mike Begon and Sandra Telfer (Liverpool University) who collaborated closely on the first two papers in this thesis (Chapters 2 and 3). I learned a lot from their expert advice and critical assessments of the drafts. Richard Sibly (Reading University) collaborated with me on the paper in Chapter 1 and I benefited greatly from his background knowledge and skills, and his good humour. Thanks also to Nicola Armstrong for useful discussions, for collaborating on one paper (Chapter 5) and, along with Elizabeth Miller, Dylan Bowden, Chenming Bao and Jozef Kollar for making life in room CMG11 that much more pleasant. Thanks to all of the field crew I worked with in Kielder Forest, especially Gill Telford who showed me the ropes and helped me get to, from, and around Kielder. Also thanks to Hazel McGregor, Andrea Birk, Lucasz Lukomski and Kevin Bown for field assistance. I am also indebted to the Post Office for running a reliable transport service between Kielder and the outside world. iv

5 Acknowledgements are also required for those who assisted in the production of the academic papers. Chapter 2: we thank Torbjørn Ergon (Oslo University) for providing useful comments on an earlier manuscript. We also thank Marcel Holyoak (University of California) and two anonymous referees for comments on an earlier draft. Chapter 3: we thank Alex Cook and Dugald Duncan (Heriot-Watt University) for technical advice. We also thank Prof. Mike Boots (Sheffield University) and two anonymous referees for comments on an earlier draft. Chapter 6: we thank Xavier Lambin (Aberdeen), Gabriel Lord (Heriot-Watt), Simon Malham (Heriot-Watt), Jens Rademacher (CWI, Amsterdam), Björn Sandstede (Surrey University), two anonymous reviewers and the participants and organisers of the "Travelling Waves: Theory and Applications" workshop (26-3 March 27, Surrey University) for helpful advice and discussions. Chapter 1: we are grateful to R.E. Longton (Reading University), D. Standing, M. Aitkenhead, J. McDonald (all Aberdeen University) and particularly J.P. Grime (Sheffield University) for constructive comments on an earlier version of the manuscript. For this study, the author was supported in part by Carnegie Trust undergraduate scholarships and a Nuffield Foundation undergraduate scholarship. I thank Fergus Massey and Sue Hartley (Sussex University) for collaborating in the, as yet, unpublished vegetation quality work discussed in Chapter 8. I also thank Tom Cope (Royal Botanic Gardens, Kew) for advice and discussions on Deschampsia spp. and Juncus spp. Colin McLuckie also generously provided assistance in shipping 6 boxes of frozen grass to Sussex, thank you. I also wish to thank those seemingly endlessly involved in the administration associated with the studies in this thesis. Christine McBride, Claire Porter and Pat Hampton dealt flawlessly with numerous requests to purchase or rent things, or to obtain information on this, that or the other. Thanks also to Steve Mowbray for all the miscellaneous computer-related enquiries and requests. This studentship was funded by the NERC Environmental Mathematics and Statistics programme and I thank them for their financial support. I am confident that their perceived benefits of funding studies at the interface between the natural and mathematical sciences has more than paid off in terms of productivity. v

6 DECLARATION STATEMENT (Research Thesis Submission Form should be placed here) vi

7 TABLE OF CONTENTS ABSTRACT...II DEDICATION... III ACKNOWLEDGEMENTS...IV LIST OF TABLES...XI LIST OF FIGURES... XII CHAPTER 1 INTRODUCTION Overview Introduction to cyclic populations The Kielder Forest field vole system Paper 1: Delayed density dependent reproduction could cause rodent population cycles Paper 2: Microparasite effects on reproduction may cause population cycles in seasonal environments The spatiotemporal dynamics of cyclic populations Paper 4: The effects of obstacle size on periodic travelling waves in oscillatory reaction-diffusion equations Paper 5: The effects of unequal diffusion coefficients on periodic travelling wave properties in oscillatory reaction-diffusion equations Paper 6: The effects of density dependent dispersal on periodic travelling waves in cyclic populations Thesis structure CHAPTER 2 DELAYED DENSITY DEPENDENT SEASON LENGTH ALONE CAN LEAD TO RODENT POPULATION CYCLES Role played by co-authors Abstract Introduction Models Results Discussion vii

8 2.7 Appendix CHAPTER 3 DISEASE EFFECTS ON REPRODUCTION CAN CAUSE POPULATION CYCLES IN SEASONAL ENVIRONMENTS Role played by co-authors Abstract Introduction Methods Infection threshold Mathematical analysis of the non-seasonal models The effects of seasonal birth rates on long terms host-parasite dynamics Discussion Appendix CHAPTER 4 PERIODIC TRAVELLING WAVES IN CYCLIC POPULATIONS: FIELD STUDIES AND REACTION-DIFFUSION MODELS Role played by co-authors Abstract Introduction Field data on periodic travelling waves Mathematics of periodic travelling waves I: Wave families Generation of periodic travelling waves Mathematics of periodic travelling waves II: Wave stability Discussion CHAPTER 5 THE EFFECTS OF OBSTACLE SIZE ON PERIODIC TRAVELLING WAVES IN OSCILLATORY REACTION-DIFFUSION EQUATIONS Role played by co-authors Abstract Introduction Wave selection for large obstacle radius in the λ ω equations Matching Behaviour near y = Application to a predator-prey model Discussion viii

9 5.9 Appendix CHAPTER 6 THE EFFECTS OF UNEQUAL DIFFUSION COEFFICIENTS ON PERIODIC TRAVELLING WAVES IN OSCILLATORY REACTION- DIFFUSION SYSTEMS Role played by co-authors Abstract Introduction Details on numerical computations The effect of unequal diffusion coefficients on the wave speed at the Hopf bifurcation in the travelling wave family equations The effect of unequal diffusion coefficients on the wave family properties The effect of unequal diffusion coefficients on periodic travelling waves generated by Dirichlet boundary conditions The effect of unequal diffusion coefficients on periodic travelling waves generated by predators invading a prey population Discussion Appendix CHAPTER 7 THE EFFECTS OF DENSITY DEPENDENT DISPERSAL ON THE SPATIOTEMPORAL DYNAMICS OF CYCLIC POPULATIONS Role played by co-authors Abstract Introduction Methods Results Discussion Appendix CHAPTER 8 - GENERAL DISCUSSION Introduction Incorporate more realism into the delayed density dependent season length models (Chapter 2) Incorporate more realism into the disease model (Chapter 3) Investigate the role of vegetation quality in the generation of population cycles in Kielder Forest field voles ix

10 8.5 Incorporate vole dispersal into the models in Chapter Study the effects of seasonal forcing in oscillatory reaction-diffusion models Study the effects of using different modelling frameworks Re-analyse the data in light of the predictions of reaction-diffusion models Conclusion CHAPTER 9 FULL RESULTS OF THE SYSTEMATIC ANALYSIS OF DISEASE PARAMETER SPACE FOR FIVE DIFFERENT RODENT POPULATION PARAMETERS Kielder Forest field voles Manor Wood bank voles French common voles Fennoscandian field voles Hokkaido grey-sided voles CHAPTER 1 IDENTIFICATION OF TRADEOFFS UNDERLYING THE PRIMARY STRATEGIES OF PLANTS Role played by co-authors Abstract Introduction Methods Results Discussion Appendix REFERENCES x

11 LIST OF TABLES 1.1 The most extensively studied populations exhibiting population cycles List of parameters, their definitions, units and values used The most extensively studied populations exhibiting population cycles Significant differences between the genetic traits of the four primary strategies Statistical comparisons of the phenotypic characterstics xi

12 LIST OF FIGURES 1.1 Analysis of the Kielder Forest field vole dataset (temporal dynamics only) Example limit cycle dynamics predicted by a predator-prey model Analysis of the Kielder Forest field vole dataset (spatiotemporal dynamics) Schematic representation of models A-C Functional form of equation (2.4) plotted against population density Numerical results for model B Sensitivity analysis of the dominant period of population fluctuations to variation in parameter values Comparison of the population dynamics predicted by model B with the Kielder Forest data Predicted long term dynamics for total population density and individual component densities for different values of τ Date in the year at which the infection threshold is first exceeded, as a function of 1 /τ and the critical τ at which the multi-year cycles occur as a function of the length of the reproductive season Effects of variation in disease parameters on the period and amplitude of the multi-year cycles predicted by model (3.1) Dominant period and amplitude predicted by the model as 1 / τ and f are varied, for the Kielder Forest field vole parameters Correlation between effective start date of the reproductive season and the population density at varying times in the past, and the correlation between seroprevalence and the population density at varying times in the past Analysis of the Kielder Forest field vole dataset (spatiotemporal dynamics) Example limit cycle dynamics predicted by a predator-prey model Comparison of travelling wave families for predator prey equations (4.2) and example spatiotemporal dynamics Solutions of the predator-prey model (4.2), showing the generation of periodic travelling waves by three separate obstacles Numerical simulation of periodic travelling wave generation by a large obstacle in the shape of Kielder Water An illustration of periodic travelling waves generated by the invasion of a prey population by predators in one space dimension xii

13 4.7 Examples of the generation of unstable periodic travelling waves for the predator prey model (4.2) The effect of α on the range of wavelengths in the travelling wave family for equations (4.2) A space-time plot of the solutions of the λ ω equations (5.1) for a circular obstacle An illustration of the variation in periodic travelling wave amplitude and spatial wavelength with obstacle size for the λ ω equations (5.1) Comparison of the solutions of the λ ω equations (5.1) for linear and circular obstacles The dependence of A 1 on a and ω An illustration of the wavelength of periodic travelling waves generated by circular obstacles for the predator-prey model (5.24) Numerical simulations of periodic wave generation in two dimensions, illustrating the transition from stable periodic travelling waves to irregular spatiotemporal oscillations as obstacle radius is increased The effect of obstacle geometry on wave amplitude predicted by equations (5.1) Comparison of the effects of varying α on the wave family predicted by travelling wave equations (6.8) with reaction kinetics of xiii λ ω form or predator-prey form The effect of varying α on the wave speed at the Hopf bifurcation for travelling wave equations (6.8) with reaction kinetics of λ ω form or predator-prey form The effect of varying α on the value of κ at the turning point of equation (6.13) The effect of varying α on the shape of the wave family and on wave stability when the kinetics of the travelling wave equations (6.8) are of λ ω form The effect of varying α on the shape of the wave family and on wave stability when the kinetics of the travelling wave equations (6.8) are of predator-prey form The α value associated with folds in the stability boundaries for varying κ when the kinetics of the travelling wave equations (6.8) are of the predator-prey form The effect of α on the maximum and minimum values of the wave properties in the travelling wave family (solutions of equations (6.8)), and on the wave properties picked out by zero Dirichlet boundary conditions in simulations of equations (6.1), for different reaction kinetics

14 6.8 Simulation of the invasion of a prey population by predators using equations (6.1) with predator-prey reaction kinetics (6.9), for different values of α and κ The effects of varying α on the speed of the travelling waves emerging behind the invasion front in numerical simulations of equations (6.1) with reaction kinetics (6.9) The effect of reducing wave speed towards zero on wave amplitude and wavelength for three different values of α The limiting contour lines of amplitude and wavelength for the particular predator-prey scenario detailed in the text An example of a travelling wave family predicted by a predator-prey reactiondiffusion model The shapes of the relationship between population density and the diffusion coefficient, D u (u) Comparison of varying the gradient of density dependent dispersal with varying the ratio of constant diffusion coefficients, on the shape of the travelling wave family predicted by equations (7.1) Comparison of varying the gradient of density dependent dispersal with varying the ratio of constant diffusion coefficients, on the on the wavelengths of periodic travelling waves picked out by simulations of equations (7.1) Comparison of varying the gradient of density dependent dispersal with varying the ratio of constant diffusion coefficients, on the on the stability profiles of the travelling wave families predicted by equations (7.1) Unstable periodic travelling waves arising from a landscape obstacle in a simulation of equations (7.1) with predator-prey reaction kinetics (7.2) Time in the year at which infected population density first peaks as a function of population density 6 months previously Contrasting population trajectories of field voles for 3 sites in the Kielder Forest, and the silica contents of the leaf samples taken from these sites, and from an increased grazing experiment Photograph of the 9 vole enclosures shortly after they were built in Spring Ch9 Full results of the systematic analysis of disease parameter space for five different rodent population parameters Birth rates, death rates, and fitness of each strategy in each environment Survivorship versus age relationships for each strategy in each environment xiv

15 1.3 Birth rate versus age relationships for each strategy in each environment Genetic (a, and phenotypic (c f) traits of the four evolved plant strategies Log mean and standard errors (interval bars) of the phenotypic characters of each invader and resident strategy in each environment xv

16 CHAPTER 1 INTRODUCTION 1.1 Overview This thesis presents five novel research papers that, in general, address the question: what mechanisms could be important in determining the spatial and temporal dynamics of cyclic populations? To be clear, the terms cyclic, multi-year cycles and population cycles are generally used in this thesis to refer to populations that show repeated oscillations in abundance with a cycle period of over one year in length. The remainder of this chapter serves as a general introduction to the papers but will focus on the background behind the first two papers, which present and analyse plausible mechanisms for the generation of population cycles (Chapters 2 and 3). Chapter 4, currently submitted as a review paper, then serves to introduce the other three papers, on the general topic of periodic travelling waves in cyclic populations (Chapters 5, 6, and 7). Chapter 8 makes recommendations for future work, based on the findings of all of the papers. Chapter 9 presents additional results from the research in Chapter 3 that was too cumbersome to include in that Chapter. Lastly, Chapter 1 presents one further research paper on the subject of plant evolutionary dynamics, which the author co-wrote and submitted during the course of his PhD studies. This last chapter is a self-contained manuscript and will not be referred to further in the main body of this thesis. All of the studies presented here were collaborative investigations. The contributions made by each of the co-authors are detailed at the start of each chapter. The author played a major role in the completion of all of these studies, and his role is also outlined. 1

17 1.2 Introduction to cyclic populations The overall aim of most population ecology research is to increase the understanding of the spatial and/or temporal dynamics of populations. Of all taxa studied to date, those exhibiting multi-year population cycles have received the most attention by population ecologists. This is partly because of the fascination such populations hold with ecologists and the general public. However, the most practical reason is that cyclic population dynamics show clear temporal trends despite the effects of unpredictable factors (stochasticity or noise ). In most other wild populations it is more difficult to disentangle the effects of noise from the effects of potentially more predictable mechanisms driving the population dynamics Some common questions are addressed by those working on cyclic populations, with the most important being: (i) What mechanisms are most important in generating and influencing the spatial and temporal dynamics? (ii) Are the dynamics in different populations generated by the same, or different mechanisms? (iii) Are the important mechanisms determining the dynamics of cyclic populations the same as for non-cyclic populations? (iv) Are the important mechanisms determining the spatial dynamics different from those determining the temporal dynamics? (v) If different mechanisms are important for the spatial and temporal dynamics then how do these interact? Addressing these questions will require extensive data from a wide range of cyclic and non-cyclic populations, and sufficient evidence so far is lacking. What is known is that a wide variety of taxa exhibit multi-year cycles in at least some of their populations. Some populations have been more intensively studied than others, and some of the most well understood are listed in Table

18 Taxon Cycle period (y) Possible hypothesis for temporal dynamics Spatial dynamics (TW= travelling waves) Possible hypothesis for spatial dynamics Larch budmoth, Zeiraphera diniana Plant-moth-parasitoid interaction 1 TW 2,3 Gradients in habitat connectivity and site productivity 3,4 Southern pine beetle, Dendroctonus frontalis Predator-prey interaction 1,5 Isolated patchy outbreaks 6,7 Diffusion driven instability by predator-prey interaction 7 Red grouse, Lagopus lagopus scoticus Parasite-grouse interaction 1,8-, kin selection 1,11,12 Region wide synchrony in some years 13, TW 14. Seasonal forcing 13, habitat boundary 23, productivity gradient 4 Fennoscandian voles, Microtus Predator-prey interaction 1,15,16 Landscape-scale Nomadic generalist predators 17 spp. and Clethrionomys spp. synchrony 17, TW 18 Fennoscandian lemmings, Vegetation-grazer interaction 1 Wide scale synchrony 19 Climatic forcing 19 Lemmus spp. Kielder Forest field voles, Local environment 2, not TW 22 Landscape obstacle (lake) 23 Microtus agrestis predator-prey 21. Grey-sided voles Clethrionomys rufocanus Predator-prey interaction 25, 27 Regional synchrony 25,26,27 Seasonal forcing 25,26, mobile predators,27 Canadian lynx, Lynx canadensis Vegetation-grazer-predator interaction 1, 28 Large scale synchrony 29-33, TW 31, Climatic forcing 3, 31, widespread movement by lynx. 32,33 Spruce needleminer, Epinotia Host-parasitoid interaction 35 Regionally synchronous 34 uncertain tedella Autumnal moth, Epirrita autumnata Host-parasite 36 TW 37, 38, region wide synchrony 36,39 Masting by trees 4 (disputed 41 ), Sunspots 42 (disputed 38 ), Table 1.1: The most extensively studied populations exhibiting population cycles, their associated spatial dynamics, and example hypotheses as to the mechanisms generating these dynamics (not comprehensive). Note that not all populations of the taxa detailed are cyclic. Superscripts denote: 1. Turchin Bjørnstad et al Johnson et al Johnson et al Turchin et al Okland et al Turchin et al Hudson et al Lambin et al Redpath et al Moss et al. 1996, 12. Mougeot et al Cattadori et al Moss et al Korpimaki and Norrdahl Klemola et al Ims and Andreassen 2 18 Ranta and Kaitala Angerbjorn et al Ergon et al. 21a 21. Graham and Lambin Bierman et al. 26, Lambin et al. 1998, Mackinnon et al Sherratt et al Stenseth et al Stenseth et al Haydon et al Bjørnstad et al. 1999a 28.Krebs et al Ranta et al Selås Stenseth et al Schwartz et al Rueness et al Munster-Swendsen Munster-Swendsen and Berryman Tanhuanpää et al Tenow et al Nilssen et al Klemola et al Selås et al Klemola et al Selås et al. 24 3

19 What is apparent from Table 1.1 is that a variety of different mechanisms have been identified as plausible for the generation of the cycles in different populations. Many more studies have investigated the temporal dynamics of cyclic populations than the spatial dynamics, or how the spatial and temporal dynamics interact. Support for hypotheses about the spatial dynamics is primarily from observational field studies and theoretical investigations. This introduction will focus on the studies relating to the temporal dynamics but will present at the end a brief introduction to the spatial aspects which will be covered in more depth in Chapter 4. Three recent reviews, one solely on rodent population cycles, have emphasised that some similar factors have been deemed as important in generating population cycles in different systems (Berryman, 22a, Turchin, 23, Stenseth, 1999). One generality is that the cycles in most populations are driven by some mechanism whose strength is a function of past population densities, or of something that is a correlate of past population densities. In other words, time-lags in interaction strengths, due to feedback loops within a cyclic population or between the component populations, play an important role in the generation of cyclic population dynamics. Mechanisms whose strength is a function of past population densities are termed delayed-density dependent mechanisms, whereas those whose strength is a function of current population densities are called direct density dependent mechanisms (Turchin, 23). Many studies of cyclic populations are concerned with the identification and analysis of direct and delayed density dependent mechanisms. Population cycles could alternatively be caused by external forcing, such as cyclic variation in some important component of a species biotic or abiotic environment. Multi-year cycles are known to occur in some abiotic factors as a result of phenomena such as the El Nino- Southern Oscillation and sun-spot dynamics, and these have been shown to affect the dynamics of some populations (Koelle et al., 25, Selås, 26). Furthermore, the population cycles could also occur simply through a taxons direct response to the multi-year cycles of some other species population, for example the cyclic population dynamics of Tengmalm s owls in northern Sweden occurs in response to the multi-year cycles of the local vole community, their main food source (Hipkiss and Hornfeldt, 24). However, empirical evidence commonly points to some intrinsically lagged biological mechanism as being ultimately responsible for generating population cycles. These findings are supported by a wide range of theoretical studies that show that time lags can generate population cycles. 4

20 Another general feature of cyclic populations is that environmental factors can interact with the biological mechanisms generating the cycles, to modify the population dynamics. Stenseth (1999), for example, speculated that differences in the relative lengths of the reproductive and non-reproductive season may be an important part of the explanation for why some rodent populations cycle and others do not. Furthermore, it has been found that characteristics of the spatiotemporal dynamics of some cyclic populations vary with environmental variables (Stenseth et al., 22, Hanski et al., 1991, Haydon et al., 22, Hudson et al., 22, Shaw et al., 24). The interaction between endogenously generated population cycles and exogenous seasonal forcing has also received increased attention in recent years due to the availability and analysis of sufficiently detailed datasets, particularly of disease prevalence in human populations (reviewed in Altizer et al., 26). A wide range of theoretical studies have shown that the nonlinear dynamics of biological interactions, such as host-parasite, can resonate with regular environmental fluctuations, to generate multi-year cycles (reviewed in Altizer et al., 26). Another generality, and the most controversial, is that consumer-resource interactions, such as predator-prey, host-parasite and grazer-vegetation, probably generate most population cycles (Berryman, 22a, Turchin, 23). That trophic interactions can generate cycles is well supported by theoretical models. However, controversy remains over whether there is sufficient empirical evidence, for any cyclic population, that a particular interaction is necessary or sufficient to generate the cycles (Lambin et al., 22). Few systems are sufficiently well understood, and there are often insufficient resources available, to allow biologists to conduct the key experimental tests that demonstrate the necessity of certain factors for generating the population cycles. For example, Hudson et al. (1998) claimed to have prevented multi-year cycles in red grouse populations by continually treating the population against one species of flatworm parasite. However, these conclusions were disputed by Lambin et al. (1999) for two key reasons; i) their use of numbers of birds shot as estimates of abundance overexaggerated the changes in cycle amplitude and, ii) it appears as though population cycles were still present in the treated areas, just at lower amplitudes. One further generality, deliberately omitted in the reviews of Berryman (22a), Turchin, (23) and Stenseth (1999) is that factors determining the spatial and temporal dynamics of cyclic populations are likely to interact. Mathematical models have 5

21 primarily provided the basis for this generality as empirical studies of sufficient scale and resolution are rare. One of the great revolutions in mathematical modelling in ecology is the realisation that the incorporation of dispersal into previously non-spatial models can fundamentally modify the population dynamics at any particular site (Tilman and Kareiva, 1997). In chapters 4-7 in this thesis, the underlying paradigm is that travelling waves in population density may be generated by dispersal acting on cyclic populations. The incorporation of dispersal in these studies means that the spatiotemporal model predicts different dynamics for a fixed point in space to the dynamics predicted by the non-spatial model. The papers in this thesis use mathematical modelling to assess the potential importance of recent empirical findings from studies of the cyclic field vole populations in Kielder Forest, for our understanding of the spatial and temporal dynamics of the cyclic population dynamics in general. The research therefore provides more theoretical findings that will aid in answering the questions 1-5 listed above, and will provide more support for the generalities just discussed. The next section will introduce the Keider Forest field vole system in more detail and relate the findings from studies of this system to what is known about other cyclic populations. 1.3 The Kielder Forest field vole system Kielder Forest is a large commercial forest plantation situated on the border between Scotland and England. Field voles (Microtus agrestis) inhabit the grassland dominated plant communities that form in the forest-clear cuts. The field vole populations in the general area of Kielder Forest were studied from 1929 to 194 by Charles Elton and coworkers to investigate rodent population cycles (Chitty, 1996). Investigations were resumed in the 198s and have continued to the present day. Between 1984 and 1998 signs of field voles were monitored three times a year by the same researcher in sites in Kielder Forest (Lambin et al., 2). At each site 25 randomly chosen quadrats (25cm x 25cm) were searched for the presence or absence of fresh grass clippings; these are left behind after a field vole has been eating the vegetation (field voles are the only herbivorous rodent in Kielder Forest). The frequency of such signs are commonly used as indicators of vole population size due to the prohibitive financial and time commitments required to estimate the population size 6

22 more accurately using trapping methods (Krebs, 26). Lambin et al. (2) presented the analysis of the ( ) data set. They first calibrated the vole sign index (number of quadrats out of 25 with clippings) with more accurate (but less numerous) estimates of vole density that were made using vole traps. Trapping methods are commonly used to estimate rodent population density: essentially traps are set according to a particular sampling protocol and the number of trappings (and possibly recaptures) is used to estimate vole density using standard statistical methods (Krebs, 26). Lambin et al. (2) showed that the number of quadrats with vole signs was a reasonable estimate of actual vole density at the site, tested by examining the strength of correlation between the abundance of vole signs and the density of voles estimated by trapping for a smaller dataset. Using these correlations, they estimated vole density at each site from the vole sign data. This revealed that field vole densities, for most individual sites and when averaged across all sites, cycled with a 3-4 year period. Much earlier data ( ) from the same area was also suggestive of population cycles. The vole sign sampling regime described by Lambin et al. (2) has continued to the present day, although the number and location of sites tested has varied as a result of different research projects being conducted since Fig. 1.1(a) gives the spring (March to May) and autumn (September to November) density estimates from the vole sign surveys conducted between 1984 and 27 (the data was partly collected by the author). From just looking at this figure it is clear that the population densities appear to have shown multi-year oscillations in abundance with a cycle period of between 3 and 5 years. Over the period of measurement, the average population density has varied by about an order of magnitude, from highs of around 2 voles ha -1 to lows of around 2 voles ha -1 ; the magnitude of variation for individual sites tends to be even greater (not shown). Autocorrelation analysis of the dataset indicates that population densities at time t are significantly positively correlated with population densities at 3 and 3.5 years previously. Furthermore, partial autocorrelation analysis of the spring or autumn data sets only (which simplifies the analysis) reveals that second-order negative feedbacks (corresponding to delayed-density dependence, see Turchin (23), Chapter 7 for further details) are a possible mechanism for the generation of the population cycles. However, a recent study of the data set (up to 24 for a more limited number of sites) revealed temporal changes to the population dynamics, and in particular, highlighted that the cycles may now have disappeared and been replaced by annual cycles (Bierman 7

23 et al., 26). Bierman et al. (26) showed that over a similar period the length and severity of the winter significantly declined and hypothesised that such effects could have led to the observed changes in the temporal dynamics. The analysis in Fig. 1.1 uses the whole dataset and therefore temporal changes to the population dynamics may be masked by the strong presence of cyclicity in the earlier years. Changes in the cycle characteristics are apparent in Fig. 1.1(, which plots the rates of change of population density over the time periods between the data points in Fig. 1.1(a). One characteristic of population cycles (in the northern hemisphere) is that population crashes can continue through the summer months and population increases can continue through the winter months (Chitty, 1996). This is apparent in the earlier half of the dataset shown in Fig. 1.1(. In contrast, the later half of the dataset is primarily characterised by population increases over the summer and declines over the winter, as one would expect for a seasonally reproducing population. However, between as recently as 21 and 24 in Fig. 1.1(a) there appears to be a population cycle, and visual inspection of the long term dynamics of some individual sites also suggests recent multi-year changes in population density (not shown). Therefore, questions remain as to whether any of the Kielder Forest field vole populations are still cyclic, and these will only be answered by the continued collection and analysis of field data. The Kielder Forest field vole cycles are similar in many ways to those reported by studies of other cyclic microtine (voles and lemmings) populations around the world (Stenseth, 1999), including the temporal changes in cyclicity. For example, the population cycles of bank voles in Finnish Lapland are around 5 years in length and have exhibited declines in the cycle amplitude over the 3 year monitoring period (Lambin et al., 22). It is therefore natural to ask whether similar mechanisms may be generating the similar dynamics in these different rodent populations. In the 199s one popular explanation for the cycles was that they were generated by the interaction between specialist predators, such as weasels, and their vole prey. To understand why requires a brief explanation of the specialist-predator hypothesis. 8

24 log 1 density a) Average Log 1 field vole density for Kielder Forest Spring Autumn year rate of change autocorrelation Rate of change of average density over summer or winter Winter Summer year c) ACF Lag (years) partial autocorrelation d) PACF Spring Autumn Lag (years) Fig. 1.1: a) Average population density estimates for Kielder Forest field voles every spring (March to May) and autumn (September to November), from autumn 1984 to autumn 27. For each estimate the number of 25x25cm quadrats out of 25 with fresh vole clippings was measured at a variable number of sites (minimum 12 sites, maximum 178, data was also usually collected in summer, but this data was omitted for the analysis). The calibration curves given in Lambin et al. (2) were used to estimate vole density from these abundance indices (see the main text for an explanation). The rate of change of population density over the summer or winter period between sampling intervals. c) Autocorrelation analysis of the dataset in (a), showing a significant positive correlation between current population density and population density 3 and 3.5 years previously. d) Partial autocorrelation analysis of only the spring (black bars) or autumn (grey bars) data sets in (a). This highlights that a likely model to generate the annual population dynamics (spring or autumn in (a)) is a second order model with negative feedback, as indicated by the significant partial autocorrelation at 2 years for both seasons (see Turchin (23), Chapter 7, for a fuller explanation). 9

25 1.3.1 The specialist-predator hypothesis Many mathematical models of trophic interactions, such as predator-prey, host parasite and grazer-vegetation, predict population cycles in their interacting components. For example a commonly used predator-prey model that predicts cyclic population dynamics is the Rosenzweig-Macarthur (1963) model du dt uv = u( 1 u), (1.1a) u+κ dv dt σuv = µ v, (1.1 u+ κ where u and v are the densities of prey and predators, respectively, µ is the predator death rate, σ is the prey to predator conversion rate, and κ is the half-saturation constant in the rate of prey consumption by predators. This model is used in Chapters 4-7, and further details are given there. The important point here is that this model has a stable limit cycle, as illustrated in Fig. 1.2, for some combinations of parameter values. The general mechanism for cycle generation in this and other ordinary differential equation models of trophic interactions is that the interacting components must have a sufficiently large effect on each others growth rates, with a sufficient temporal lag in the response of one population to changes in the other. In model (1.1), the maximum rate of change of prey from low densities, after the predator population has crashed, is relatively high, but the maximum rate of increase of predator populations from low densities, when prey is abundant, is relatively low. This causes a temporal lag between the time at which prey populations reach high densities and the time at which predator populations reach sufficiently high densities to cause the prey population to crash. 1

26 population size prey predator predator population size time prey population size Fig. 1.2: a) Example limit cycle dynamics predicted by predator-prey model (1.1) with σ =.15, µ =. 5, and κ =. 4. Phase space diagram for the same model with the same parameters. The vertical line and the parabola (solid black lines) are nullclines; the prey density at which the rate of change of predators is zero, and the predator density at which the rate of change of the prey population is zero, respectively. These lines cross at an unstable equilibrium point in equation 1.1. The small arrows plot the vector of change in predator and prey density at points in phase space. The coloured dots denote initial predator and prey densities for simulations of equations (1.1) and the coloured lines trace the change in predator and prey densities from these initial values. Note that, regardless of the initial conditions, the different population trajectories eventually settle down to the same population cycle. The Rosenzweig-Macarthur (1963) model is used to represent the interaction between a specialist predator, one that eats only one or a few very similar species, and its prey. The predator growth rate is solely a function of the amount of prey consumed, and is limited by its maximum consumption rate. In contrast, the dynamics of a generalist predator would not be so tightly coupled to the dynamics of one prey species. One simple way of modelling a generalist predator-prey interaction is simply to assume that the population of generalist predators is constant ( dv / dt = ) and model the abundance of their prey as equation (1.1a); this equation alone cannot predict population cycles. A more realistic way to model generalist predation is to assume that the predators switch to feeding on a particular prey type when it rises above a particular abundance (Turchin, 23). For example, equations (1.1) could be modified to read 11

27 du dt = u(1 u) uv u+ 2 ϑu 2 ϖ 2 κ u +, (1.2a) dv dt σuv = µ v, (1.2 u+ κ where ϑ is the maximum feeding rate of generalist predators and ϖ is the half saturation rate constant of the generalist predators. It is straightforward to show that as the impact of the generalist predator ϑ in the prey population is increased from zero, the dynamics predicted by equations (1.2) become less cyclic until they no longer predict multi-year cycles at all. These theoretical findings were reported by Hanski et al. (1991), although with a slightly different form for equation (1.2, and led them to hypothesise that the effects of generalist predators will tend to stabilise specialistpredator-prey population cycles. A range of empirical and theoretical studies have investigated the prediction that cyclic dynamics are generated by specialist predator-prey interactions in real populations. Most of these relate to the microtine populations of Fennoscandia (Norway, Sweden, and Finland). For example, Korpimaki and Norrdahl (1998) and Klemola et al. (1997) removed predators from an area and found that this prevented the characteristic summer decline in population density. Furthermore, Norrdahl and Korpimaki (1995) found that the predation rate of voles (several different species) by the specialist predator population (stoats and least weasels) was delayed-density dependent, also predicted by theoretical models. Further support came from data on population cycles throughout Fennoscandia, in combination with more detailed mathematical models. The field data shows a decline in the cycle period of Fennoscandian rodents with latitude, with cycles no longer being detectable in the datasets below 6 degrees north (Hanski et al., 1991). As stated above, mathematical models of a similar form to (1.2) predict this phenomenon when parameterised for the Fennoscandian systems (Hanski et al., 1991). The essential mechanism for cycles is predicted to be the same; a specialist predatorrodent interaction. The models also predict that the latitudinal gradient in cyclicity observed in the field occurs because of the increasing importance of generalist predation with decreasing latitude, and this is supported by field data (Hanski et al., 1991). 12

28 The findings from the Fennoscandian studies make a strong case for the importance of specialist predators in the generation of the population cycles. Furthermore, Hanski et al. (1991) predicted that if the same mechanisms operate in similar rodent populations in more southerly latitudes, such as the UK or mainland Europe, then their dynamics would not be cyclic, since the relatively strong effect of generalist predators should stabilise the dynamics. However, the occurrence of population cycles in Kielder Forest (at around 55 degrees north) shows that this prediction may not always be correct, as reported by Lambin et al. (2). Furthermore, Lambin et al. (2) estimated the predation rates of the different guilds of predators in Kielder Forest, albeit with relatively large degrees of uncertainty, and confirmed that if these rates are representative then population cycles should not occur according to the hypothesis of Hanski et al. (1991). It therefore seems likely that the most popular explanation for population cycles in Fennoscandian voles, and for the latitudinal gradient in cyclicity, the specialist-predator hypothesis, may not apply to the Kielder Forest field vole populations Experimental refutation of the specialist-predator hypothesis in Kielder Forest, and new hypotheses Graham and Lambin (22) conducted similar experiments in Kielder Forest to the Fennoscandian studies of Korpimaki and Norrdahl (1998) and Klemola et al. (1997), to test whether weasels, the main specialist predator of field voles in Kielder Forest, may be responsible for generating the population cycles. They found that weasel removal could not prevent the cyclic decline in vole populations, and that there were no significant differences between the vole population trajectories in predator-removal and control sites. They concluded that weasels were neither necessary nor sufficient to generate the population cycles in their field vole prey. Subsequent empirical investigations have provided data to support plausible alternatives to the specialistpredator hypothesis. Recall that the most convincing generality that can be applied to cyclic populations is that certain mechanisms act in a delayed-density dependent manner. In other words, the size of the effect of the important factor must depend on past population densities, rather than current population densities. It is this lag that destabilises the population dynamics. In two different experiments, Ergon et al. (21a, tested whether this 13

29 memory of past population densities (his words) resides in the voles or the environment. In the first of these Ergon et al. (21 captured voles from two populations in Kielder Forest with very different cyclic population trajectories, one in the increase phase of the population cycle and the other in the peak phase, and reared them under identical laboratory conditions over two generations. In general, they found that while reproductive traits, such as reproductive timing, were widely different between the two field populations, they were not significantly different between the laboratory populations from the different sites. They concluded that the variation observed in the field must be due to some factor that is present in the field and not due to some difference inherent in the voles themselves. In Ergon et al. (21a) voles were transplanted between sites with differing population trajectories and their reproductive traits were monitored. In this case the transplanted voles adopted the life history characteristics of the voles present in the sites to which they were transplanted. This is further evidence that it is the local environment, rather than the individuals themselves, which determines the life history characteristics of the field voles. Taken together, these experiments indicate that whatever was generating the cycles resided in the environment. This eliminated a number of hypotheses for the cause of the population cycles. In the absence of the findings of Graham and Lambin (22) the local density of specialist predators would be the most likely hypothesis for the environmental factor that could be generating the different population trajectories. With this hypothesis already being brought into doubt, researchers looked to possible alternatives. The first two papers in this thesis provide mathematical analysis that point towards such an alternative hypothesis. 1.4 Paper 1: Delayed density dependent reproduction could cause rodent population cycles The studies of Ergon et al. (21a, demonstrated that the timing of reproduction by overwintered female voles varied during the course of the population cycle. Ergon (23) elaborated on these findings to show that the date in the year in which over wintered female voles started their reproduction was significantly related to the local population density during the spring of the previous year, but not to the current population density. This delayed density dependent effect could be indicative of a mechanism generating the population cycles. The first paper in this thesis (Chapter 2) demonstrates mathematically that population cycles could indeed result from delayed 14

30 density dependent reproductive timing. It finds that population cycles could be generated for a wide range of realistic parameter values, including those derived from the Kielder Forest data. These findings led us to look for possible mechanisms that could be causing delayed density dependent reproductive timing, which led to the next paper. 1.5 Paper 2: Microparasite effects on reproduction may cause population cycles in seasonal environments Empirical results from several different studies led us to investigate whether microparasites could be responsible for the population cycles in Kielder Forest. Firstly, two separate studies of the effects of cowpox on bank voles and wood mice both showed that juvenile voles that are infected with cowpox may delay their development into reproductive adults (Telfer et al., 25, Feore et al., 1997). The period of reproductive delay was longer in field populations (Telfer et al., 25) than laboratory populations (Feore et al., 1997). It was therefore natural to hypothesise that such a mechanism might cause the reproductive delay in the Kielder Forest field voles reported by Ergon (23). Secondly, field studies in Kielder Forest showed that the seroprevalence of cowpox and the signs of vole tuberculosis varied in a delayed-density dependent manner (Cavanagh et al., 24). This led to the hypothesis that the effects of diseases such as cowpox and vole tuberculosis, including a period of reproductive delay, may be delayed-density dependent. Lastly, a more recent study showed that rodent survival in Kielder Forest was significantly reduced by cowpox infection (Burthe et al., 27). Taken together, these results suggest that cowpox may have a significant effect on the population dynamics of Kielder Forest field voles by reducing survival and reproduction, and that these effects may be delayed-density dependent. This has all the hallmarks of a destabilising trophic interaction. Furthermore, recent evidence has highlighted that a range of pathogens that exist, and potentially co-exist, in different cyclic rodent populations. All of these pathogens could potentially have effects on the vole population dynamics. Taken together these results led us to investigate whether such effects could lead to population cycles (Chapter 3), using a mathematical model. Our analysis indicated that microparasitic diseases could induce population cycles, but only in the presence of seasonally varying birth rates. Such disease-induced multi-year cycles were predicted by the model for a wide range of realistic parameter values, with 15

31 large amplitude cycles being predicted for diseases for which the period of infection is brief but full recovery of reproductive function is relatively slow. In particular, the model predicted conditions under which diseases such as cowpox virus would induce cycles in the Kielder Forest field vole populations and predicted that if certain diseases do induce population cycles then delayed density dependence in disease seroprevalence and in the effective date of onset of the reproductive season are also predicted: two phenomena reported in the field (Cavanagh et al., 24, Ergon, 23). 1.6 The spatiotemporal dynamics of cyclic populations Table 1.1 highlights that there are differences in the observed spatial dynamics of cyclic populations, as well as differences in the temporal dynamics. Some populations appear to exhibit synchronous population oscillations throughout the habitat, others exhibit more patchy dynamics, and some, such as the Kielder Forest system, appear to exhibit periodic travelling waves in abundance. What causes differences in these spatiotemporal dynamics is only beginning to be understood. Indeed, there is a backlog of predictions from theoretical models that require testing with empirical data and field experiments (Steinberg and Kareiva, 1997). The strict mathematical definition of periodic travelling waves refers to waves that have a constant form and move at a constant speed (Sherratt, 1996a). In empirical studies, periodic travelling waves are identified as spatially periodic fluctuations in population density that move in a particular direction through time. However, no empirical studies to date have determined whether the form of travelling waves is constant, and several studies have indicated that the estimated wave characteristics (such as wave speed) appear to vary through time and space (Bierman et al., 26, Bjørnstad et al., 22, Johnson et al., 24). It will rarely be the case that sufficient field data is available on a particular system to allow the determination of whether or not wave properties significantly change in time and space. Furthermore, taking into account spatial habitat variation, environmental change, stochasticity and measurement error, it seems unlikely that any study would actually find travelling waves of a fixed shape and speed in the field. Therefore, when referring to field data, this thesis will refer to travelling waves as travelling wave phenomena that have been observed in cyclic populations, but that may not necessarily have a fixed shape or speed. 16

32 Spatiotemporal dynamics in the field are most commonly characterised by the way in which synchrony between sites changes with distance. The measure of distance is usually either the Euclidean distance between sites or the distance between the perpendicular projections of the site locations onto a straight line of a given angle. Fig. 1.3(a) illustrates how the synchrony between the temporal dynamics at paired sites in Kielder Forest significantly declines with the Euclidean distance between sites (P<.1). This shows that the sites in Kielder Forest are, at least, not all fluctuating in synchrony, but that closer sites tend have more synchronous dynamics. The Mantel correlation is a measure of how steeply cross-site synchrony varies with distance; with significantly negative values corresponding to decreasing synchrony. Fig. 1.3(a) shows a significant negative Mantel R (correlation) statistic of -.16 (P<.1). If there are travelling waves of rodent population density in the field then cross site synchrony will vary differently with projected distance from a given site, depending on the direction of projection. One would expect synchrony to decline sharply with distance in the direction of movement of the travelling wave, whereas sites aligned perpendicularly to the wave direction should remain relatively synchronous, regardless of their distance apart. Fig. 1.3( illustrates how the Mantel R statistic varies with distance for different projections: there is significantly declining cross-site synchrony with projected distance for projection angles between 47 and 155 degrees north. In this analysis, the most significant Mantel R (correlation) statistic occurs at a projection angle of 85 degrees north (Mantel R=.23; P<.1; illustrated in Fig. 1.3(c)). In contrast, the relationship is not significant for the perpendicular projection angle of 175 degrees north (Fig. 1.3(d); P=.13). This brief analysis therefore provides some evidence for the presence of a directional travelling wave of vole density in Kielder Forest. 17

33 cross correlation cross correlation a) Euclidean pair wise distance (km) c) Mantel R Projected pair wise distance on 85 o N (km) cross correlation N.S. P< Projection angle d) Projected pair wise distance on 175 o N (km) Fig. 1.3: a) The cross correlation between the population dynamics of paired sites in Kielder Forest tends to decline with Euclidean distance (P<.1, linear regression: cross correlation = -.29 distance+.5672). Mantel s R statistic varies with projected distance between sites in Kielder Forest. c) Same as in (a) but with site distance being that once site coordinates have been projected onto a line of 85 degrees north (P<.1, linear regression: cross correlation = distance+.5747). d) Same as in (c) but with the projection angle being 175 degrees north (P=.13, linear regression: cross correlation = -.68 distance ). The raw data set for this analysis was spring and autumn densities at each site in Kielder Forest (omitting the data from neighbouring Redesdale, Kershope and Wark forests, in contrast to Mackinnon et al. (21)). Sites were grouped into 1km squares and their Log 1 densities averaged. The centre of the 1km squares was used as the spatial coordinate. The data set used for analysis was the rate of change of densities between consecutive measurements (spring to autumn and autumn to spring) for each 1km square. Cross correlation coefficients were calculated where there was four or more data points. 18

34 Two far more comprehensive studies of the spatiotemporal dynamics of the Kielder Forest field vole populations arrived at very similar findings to those above: that the multi-year cycles were spatially organised into periodic travelling waves (Lambin et al., 1998, Mackinnon et al., 21). These two studies differed in the spatial and temporal scales of the data analysed, but arrived at very similar conclusions, calculating the direction of the waves as 66 and 78 degrees North, and estimating the wave speed as 19 and 14 km yr -1, respectively. In these analyses, and the preliminary analysis above, the whole dataset was treated together for analysis. This will obviously mask any temporal changes in the spatiotemporal dynamics. Bierman et al. (26) looked at the changes over time in the evidence for periodic travelling waves for the Kielder Forest dataset and found that strong evidence for travelling waves only exists up to around 199, during the period in which the population cycles were most prominent. According to Bierman et al. (26), therefore, there is currently no evidence of periodic travelling waves in the Kielder Forest system. However, it is interesting to note that periodic travelling waves were detected in the years in which the population cycles were most prominent. As will be described in more detail in Chapter 4, the incorporation of spatial dispersal into ordinary differential equations predicting population cycles can give rise to periodic travelling waves. Consider, for example the commonly used reaction-diffusion equations u t Dispersal Birth and Death u 678 = Du + f ( u, v) x 2 (1.3a) v t 2 v = Dv + g( u, v) x Dispersal Birth and Death (1.3 where t is time, x is space, u and v are the component population densities, D u, are the constant dispersal rates, and the functions f and g are the effects of birth and death on the rate of change of u and v, respectively. The forms of f and g (termed the reaction kinetics ) will clearly vary depending on the system modelled but, for D v 19

35 example, could be the right hand sides of equations (1.1a) and (1.1, respectively. Therefore, in the absence of dispersal, equations (1.3) are straightforward ordinary differential equations that, we assume, predict population cycles (for example, those shown in Fig. 1.2). In the absence of the birth and death rate processes, equations (1.3a) and (1.3 are uncoupled and describe the diffusion of a population of individuals throughout the environment. Periodic travelling waves can arise from the combination of spatial dispersal with oscillatory population dynamics. The generation of waves in equations (1.3) requires some mechanism that forces the system away from environmentally synchronous oscillations (which are also one solution to equations (1.3)). For example, in Chapters 5-7 we consider wave generation by landscape obstacles, on the edge of which the interacting component population densities are fixed at zero. For the travelling waves in Kielder Forest it seems plausible that periodic travelling waves could arise in a similar general way to that demonstrated in mathematical models. i.e. that some mechanism generates population cycles and that dispersal by the voles, and possibly their interactants, in combination with some environmental forcing, can result in the cycles being spatially organised into periodic travelling waves. For example, the large lake in the centre of Kielder Forest is a landscape obstacle in the middle of the vole habitat. At the edge of this obstacle the vole density can reasonably be assumed to be zero. The dominant direction of the travelling waves is in a direction that is roughly perpendicular to the widest dimension of the lake, as is predicted by twodimensional simulations of reaction-diffusion equations (Sherratt et al., 23). The overall aim of the three studies in this section of the thesis is to perform detailed mathematical studies into the importance of various previously unconsidered biological features, for the emergence and properties of periodic travelling waves in oscillatory reaction-diffusion equations. 1.7 Paper 4: The effects of obstacle size on periodic travelling waves in oscillatory reaction-diffusion equations To gain mathematical insight into the process of periodic travelling wave generation by landscape obstacles, Sherratt (23) studied a special class of reaction-diffusion equations (lambda-omega) that are more amenable to mathematical analysis. By 2

36 assuming semi-infinite one dimensional space, with population densities fixed at zero at one domain edge to simulate a landscape obstacle, Sherratt (23) was able to derive a formula for the periodic travelling wave speed and amplitude that would be selected by the boundary conditions for these equations. Clearly if obstacles are generating cycles in the field then they will have a two dimensional form that is a significant departure from the simplified scenario studied by Sherratt (23). Two-dimensional problems are considerably harder to analyse mathematically, unless they can be simplified in some way. Assuming circular obstacles and symmetrical initial conditions means that the problem is again one dimensional, with the space coordinate now being the distance from the centre of the obstacle. With this arrangement, the effects of obstacle size were explored numerically by Sherratt et al. (23), who found that the wavelength and amplitude of selected waves declined with the radius of a circular obstacle in the centre of a habitat. In Chapter 5 we extend the work of Sherratt et al. (23) by mathematically studying the effects of circular obstacle radius on periodic travelling wave selection. To do this we rely on the solution for periodic travelling waves generated by a flat boundary from Sherratt (23), which corresponds to waves generated by a circular obstacle with infinite radius. We study the effects of introducing a slight curvature to this flat boundary. This modification introduces a small parameter to the equations; namely the reciprocal of obstacle radius (assuming large radii). This allows us to derive a leading order approximation to the properties of the wave generated by large obstacles. Whilst in principle this approximation only applies to large circular obstacles, it provides new information about how we might expect periodic travelling waves to be influenced by obstacle shape in the field. Our solution reveals that if periodic travelling waves are selected by curved obstacles then their properties may vary very gradually with distance from the obstacle edge, since the approach to the selected wave properties occurs algebraically with distance, rather than exponentially as in the case of a flat boundary (Sherratt, 23). Furthermore, our numerical studies reveal that the size of the landscape obstacle alone may be the difference between periodic travelling waves being observed and spatiotemporal irregularities. 21

37 1.8 Paper 5: The effects of unequal diffusion coefficients on periodic travelling wave properties in oscillatory reaction-diffusion equations Prior to this thesis there were no plausible candidate models for what could be causing the population cycles in Kielder Forest. Chapter 3 presents a new plausible mechanism for cycle generation. However studies of its predictions in a spatial context have not yet been looked at. The lack of available models has hampered theoretical investigations into the interaction between details of the biological mechanisms and properties of periodic travelling waves. Such information could indicate, for example, whether particular biological interactions are generating the periodic travelling waves (and cycles) in the field. One important biological feature that could potentially influence the spatiotemporal predictions of reaction-diffusion models are the dispersal rates of the interacting populations. It is possible to estimate the dispersal rates, at least roughly, for a range of different biological interactions that could generate population cycles. For example, Brandt and Lambin (27) recently estimated that the dispersal rates of voles and weasels differed by over two orders of magnitude. Furthermore, in a vegetation-grazer interaction the plants are stationary. That such differences in dispersal rates could be important for the spatiotemporal dynamics has been shown before in other models (Bjørnstad et al., 22, Comins et al., 1992). However, to the author s knowledge, there has been no systematic study performed on this for periodic travelling waves generated by reaction-diffusion models. In light of this, we investigated the importance of the relative dispersal rates of the interacting components in oscillatory reaction-diffusion equations (Chapter 6). Our basic questions in this paper were, how important are the dispersal rates for periodic travelling waves and, can we predict anything abut these waves by simply having knowledge of the dispersal rates? We picked two commonly used models for studying periodic travelling waves in oscillatory biological systems, the lambda-omega equations and predator-prey equations, and studied what happens to the wave properties predicted by these equations when we varied the ratio of their diffusion coefficients; varying the diffusion coefficients with their ratio constant simply scales the periodic travelling wave solutions. 22

38 All oscillatory reaction-diffusion equations have a family of periodic travelling wave solutions. Any wave generated in simulations is just one member of this family, that has been selected by the initial and boundary conditions (see Chapter 4 for a fuller description). We found that varying the ratio of the diffusion coefficients significantly alters this family of travelling waves. In addition, using numerical computation of the stability spectrum of travelling wave solutions we determined which waves were stable or unstable on infinite domains, and found that the ratio of the diffusion rates significantly influences wave stability. Finally, by simulating the equations under different wave selection mechanisms (landscape obstacle and predator invasion) we illustrated how the ratio of the diffusion rates affects the waves that can emerge. In answer to the questions posed above therefore, we would expect the ratio of the dispersal rates of the interacting components to have important effects on the travelling wave properties observed in the field. However, we found that these effects are also largely dependent on the properties of the interaction being modelled (in this case either lambda-omega equations or predator-prey equations), which implies that it is unlikely that one could accurately predict the periodic travelling wave properties without knowledge of the underlying interaction being modelled. 1.9 Paper 6: The effects of density dependent dispersal on periodic travelling waves in cyclic populations Density dependent dispersal, in which movement rates vary as a function of local population density, is a biological phenomenon that has received a considerable amount of empirical investigation. As a result, several recent reviews have concluded that density dependent dispersal is not uncommon in animal populations and varies in its details amongst different taxa (Bowler and Benton, 25, Denno and Peterson, 1995, Matthysen, 25). For example, in several cyclic rodent populations it has been shown that dispersal between sites can be negatively density dependent (Matthysen, 25), however, the importance of this phenomenon for the spatiotemporal dynamics of cyclic populations is unexplored. In this paper (Chapter 7) we use similar techniques to those used in Chapter 6 to investigate whether density dependent dispersal could have important effects on periodic travelling waves that may be observed in the field. In contrast to the effects of varying the ratio of the diffusion rates, we find that varying the strength of density dependent dispersal (positive or negative) only has relatively minor effects on the properties of the wave families and on the waves emerging in simulations. 23

39 1.1 Thesis structure The chapters described above fall naturally into two parts. The first builds on recent empirical findings, primarily from studies on rodents in Kielder Forest and Manor Wood (UK), and uses modelling to explore their potential implications for the temporal dynamics of rodent populations (Chapters 2 and 3). Our priority with these studies is to aid future research into the Kielder Forest system, and then to look for implications relevant to other populations. The second part to the thesis explores the implications of incorporating biological detail into general reaction-diffusion models of oscillatory biological systems (Chapters 5-7). From the outset, these studies do not have any specific system in mind. Rather they are primarily intended to reveal the importance of previously unconsidered biological detail that is likely to occur in many or most cyclic populations exhibiting periodic travelling waves. Following these chapters is discussion about natural future research directions following on from all of the studies presented in this thesis (Chapter 8). 24

40 CHAPTER 2 DELAYED DENSITY DEPENDENT SEASON LENGTH ALONE CAN LEAD TO RODENT POPULATION CYCLES (Published in American Naturalist, 26, 167(5), ) 2.1 Role played by co-authors Matthew Smith, Andrew White, Xavier Lambin, Jonathan Sherratt and Michael Begon collaborated on this study. It was conceived by Andy White, Mike Begon, Xavier Lambin and Matthew Smith. The models were developed by Matthew Smith and Andy White, who also worked together to do the stability analyses (except model B). Jonathan Sherratt performed the stability analysis for model B and wrote the Appendix. Matthew Smith did all of the numerical studies and wrote the paper, with comments provided by the co-authors. 2.2 Abstract Studies of cyclic microtine populations (voles and lemmings) have suggested a relationship between the previous years population density and the subsequent timing of the onset of reproduction by over-wintered breeding females. No studies have yet explored the importance of this relationship in the generation of population cycles. Here we mathematically examine the implications of variation in reproductive season length caused by delayed density dependent changes in its start date. We demonstrate that when reproductive season length is a function of past population densities it is possible to get realistic population cycles without invoking any changes in birth rates or survival. When parameterized for field voles (Microtus agrestis) in Kielder Forest (northern England) our most realistic model predicts population cycles of similar periodicity to the Kielder populations. Our study highlights the potential importance of density dependent reproductive timing in microtine population cycles, and calls for investigations into the mechanism(s) underlying this phenomenon. 2.3 Introduction Despite ever-enlarging datasets on cyclic microtine (vole and lemming) populations around the world, there is still great uncertainty over the causal mechanisms behind 25

41 their population cycles (Stenseth, 1999, Turchin, 23). Analysis of these datasets has led to the consensus that both direct and delayed density dependent mechanisms operate on the populations (Stenseth, 1999, Lambin et al., 22, Turchin, 23). That is, the populations are influenced by factors that are a function of the current population density and by factors that are a function of the population density in the past. Direct density dependent mechanisms tend to stabilize population dynamics, making them less prone to cycle, whereas delayed density dependent mechanisms do the opposite (May, 1981, Murray, 22). There are several schools of thought concerning the cause of the delayed density dependence. The most commonly considered factors are the effects of specialist predators, resource (usually food) shortage, and intrinsic (e.g., maternal) factors (Stenseth, 1999, Berryman, 22b, Turchin, 23, Korpimäki et al., 24). Studies of cyclic Fennoscandian vole populations have generally concluded that the density of specialist predators is the important delayed density dependent factor (Hanski et al., 21, Turchin, 23), but for most other cyclic rodent populations, less of a consensus exists. For example, studies of field vole (Microtus agrestis) populations in Kielder Forest, northern England, have suggested that specialist predators play no causal role in the population cycles (Graham and Lambin, 22). Perhaps the only emerging consensus from these studies is that delayed density dependence acting on the populations in the winter is a prerequisite for the population cycles (Hansen et al., 1999, Stenseth et al., 23, Hörnfeldt, 24, Bierman et al., 26). Previous studies have shown that environmental influences on the length of the breeding season, are important in determining whether microtine populations cycle or not (Hansen et al., 1999, Stenseth et al., 22, Saitoh et al., 23). However, the timing of the breeding season can also vary in relation to the different phases of microtine population cycles (Krebs and Myers, 1974, Krebs, 1996, Boonstra et al., 1998, Batzli, 1999, Hansen et al., 1999). Therefore, not only does breeding season length vary in response to external factors, but it can also vary in response to past population densities. For example, Wiger (1982) proposed that variation in the date of onset of breeding season in Clethrionomys glareolus may be explained by direct density dependent territoriality in the breeding female population. Using a detail rich population model Stenseth and Fagerström (1986) found that this mechanism could lead to population cycles. However, those population cycles bore little resemblance to those found in 26

42 natural populations. Here we will investigate the effect of past densities on reproductive timing, and how this determines the population dynamics. Previous theoretical assessments have ignored this effect, perhaps because field populations are sampled too infrequently to yield reliable data on the timing of seasonal life history events. Furthermore, many field studies have sampled at a fixed time point (or time points) in the year. This fails to adequately capture variation in the actual timing of life history events in the studied populations. Recent, more intensive studies of the Kielder Forest system have also shown that vole life history traits are affected by their population density a year in the past (Ergon et al., 21a,. When field voles at Kielder Forest were transferred to other sites they rapidly took on the breeding characteristics of animals native to those sites (Ergon et al., 21a). This shows that the memory of past densities resides in the environment, not in the voles themselves (Ergon et al., 21a,, and rules out maternal or genetic effects as a cause for the cycles. Shortage of food is an obvious possible external mechanism through which breeding may be suppressed. However, a recent study also revealed that microparasite loads in the Kielder Forest field voles follow the vole population dynamics in a delayed density dependent manner (Cavanagh et al., 24). Regardless of the fundamental cause of the cycles in Kielder Forest, studies have shown that the external memory of past population densities influences the date at which the reproductive season starts (Ergon, 23). Specifically, Ergon (23) found a significant positive relationship between the date that 5% of the voles had produced their first litter in one year and the population density at the start of the previous spring. In the light of this, we ask whether such seasonal effects might contribute to cyclicity. We used a series of mathematical models to investigate whether the delayed density dependent effect of population density on the timing of onset of the reproductive season can cause realistic microtine population cycles. This contrasts with the studies of Wiger (1982) and Stenseth and Fagerström (1986) who assumed that season length was directly density dependent. In addition, we use the simplest possible model to study this phenomenon (in contrast to Stenseth and Fagerström, 1986). We assume that changes in the start date of the reproductive season directly relate to the reproductive season length (the end date of the reproduction season is assumed fixed). We acknowledge that many other factors are likely to affect reproductive season length. However, here we aim to 27

43 explore the implications of delayed density dependent season length, using the new data and findings derived from the Kielder Forest field vole populations. Our theoretical study will provide crucial evidence as to whether the seasonal effects discussed above should be included in assessments of the mechanisms that lead to (rather than result from) population cycles. We contend that if the models do show appropriate population cycles, then continued neglect of these seasonal effects in studies of the causes of microtine cycles cannot be justified. Our results below support this contention for the Kielder Forest system at least. 2.4 Models We analyze three models that differ only in the time frame over which delayed density dependent season length operates (as illustrated in Fig. 2.1). We refer to these models as model A (long delay), model B (medium delay), and model C (short delay). Note that we use N T to refer to the vole population density (voles ha -1 ) as a discrete function of time (at the end of the breeding season in year T), and n (t) to refer to it as a continuous function of time. For all models we assume that each year is divided into a reproductive and a nonreproductive season. Within the seasons, we assume identical individuals and complete mixing. This gives dn( t) dt = rn( t) in the reproductive season and (2.1) dn( t) dt = bn( t) in the non-reproductive season, (2.2) where n (t) is the population density (voles ha -1 ) at time t (years), r is the per capita rate of population increase in the reproductive season, and b is the per capita mortality rate in the non-reproductive season. The absence of direct density dependence in equations (2.1) and (2.2) reinforces the fact that we seek to understand the effect of one factor only, that of a variable season length. 28

44 Fig. 2.1: Schematic representation of models A-C. discrete time T, which is the end of the reproductive season, N T is the population density at n t is the population density at continuous time t, L is the length of the reproductive season. Although reproductive (L) and non-reproductive ( 1 L) seasons have the same population dynamics for all three models (see equations (2.1) and (2.2)) the delay over which previous population density affects the start date of the reproductive season differs. The curved arrow indicates how the relevant population density in the past relates to the season length for the three models. The ~ indicates the transition between the reproductive and non-reproductive seasons. The arrowed functional relationship to the right of each diagram represents the general functional relationship that can be derived for each model. 29

45 In all three model formulations the length of the reproductive season is a function of the population density at some time in the past ( L T = f ( n( t)), see Fig. 2.1). In model A, L T is a function of the population density at the start of the previous year ( N T 1 ). In model B, L T is a function of the population density at the start of the previous reproductive season, occurring at some time ( ~ t say) between T 1 and T. In model C, L T is a function of the population density at the end of the previous reproductive season ( N ). The delayed density dependence in model B corresponds to the field data from Kielder Forest (Ergon, 23) discussed above. We chose to analyze all three models to see how plausible delays of different lengths affect our results. T Equations (2.1) and (2.2) have simple analytical solutions for n (t). These can be combined with the framework for models A, B, and C (Fig. 2.1) to derive mathematical expressions for the three models (see the Appendix, 2.7.1, for their derivation) where in model A b+ ( r+ f ( NT 1 ) N T+ 1 = NT e, (2.3a) in model B N = b+ ( r+ f ( n( ~ t )) T+ 1 NTe, (2.3 and in model C N N e b+ ( r+ f ( N T ) T+ 1= T. (2.3c) For equations (2.3a-c) this means that the population density next year is equal to the population density in the current year multiplied by the net per capita growth from the reproductive season and proportional survival through the non-reproductive season. There are insufficient data to produce an accurate functional relationship for the delayed density dependence of season length for the field voles in Kielder Forest. Therefore we use the negative sigmoid relationship between reproductive season length and population density 1+ β 1 f ( n( t)) = α 1 + ( κ α), (2.4) β n( t) 1 β τ + e 3

46 where κ is the maximum reproductive season length, reproductive season length (here κ α is the minimum 1 >κ > α ), andβ and τ scale the slope of the sigmoid function. In equation (2.4) n (t) is the population density at the appropriate time (see Fig. 2.1). Using equation (2.4) we can then explore the effects of different shaped functional relationships on the model dynamics. Fig. 2.2 illustrates equation (4) for various values of α, β, and τ. 1.8 a) c) f(n).6.4 f(n).6.4 f(n) n 2 4 n 2 4 n Fig. 2.2: Functional form of equation (2.4) plotted against population density, where κ =.83 (1 months) and a) β = 5, τ = 3, and α is.2,.4,.6, and.8 (top to bottom), β = 5, α =. 83, and τ is 1, 3, 5, and 7 (left to right), and c) τ = 3, α =.83, and β is 1, 1, 1, and 1 (left to right). The non-trivial steady state solution for all three models (equation (2.3)) is b τ f ( n * ) = * β[ κ( r+ b] + α( r+ n = ln r+ b. (2.5) b ( κ α)( r+ Here * n is the population density that gives repeated and identical annual cycles. It corresponds to the population density at time T in models A and C, and at time t ~ in model B (Fig. 2.1). For realistic population densities this requires that ( κ α)( r + < b< ( βκ + α)( r+ / β, and (2.6a) β [ κ ( r + b] + α( r+ > b ( κ α)( r+. (2.6 31

47 It is possible to derive stability criteria for the annual cycles for all three models (see Appendix, 2.7.2). This gives the criteria for the local stability of * n as S crit < S <, (2.7a) where, S = ( r+ n * f '( n * ) ( β[ κ( r+ b] + α( r+ )( b ( κ α)( r+ ) b ( κ α)( r+. (2.7 = ln ( α+ αβ )( r+ β[ κ( r+ b] + α( r+ * Here, f '( n ) is the first order derivative of f ( n( t)) with respect to n (t) at * n, and S is an index of stability. Given the conditions for a positive equilibrium population density in equation (2.5), S must be negative. S = 1 for model A and S = 2 for model crit C. For model B, S crit depends on r and b and varies between -1 and -3 (see Appendix, 2.7.2). Our stability analysis shows that the parameter τ has no effect on the stability of the equilibrium solution (equation (2.7); however it does affect the equilibrium population density (equation (2.5)). crit Parameterization To investigate the stability and dynamics of the models above we need estimates for six parameters ( r, b, α, β, κ, and τ ). Therefore we define representative parameter values, from Kielder Forest data, and explore model stability and dynamics for variation in each parameter while keeping the remaining parameters fixed at their mean values. We estimate the per capita rate of increase, r, by assuming that the increase phase of the vole density cycle represents maximum annual growth rate. Any subsequent change in the annual population growth rate is therefore a result of the delayed density dependent season length function. Data from Burthe (25) for various trapping sites in Kielder Forest gave 1 < r < Using data from Graham and Lambin (22) and Burthe (25) gave monthly survival probabilities between.6 and.9. For our analysis we assume an instantaneous rate b = 2. 7 (monthly survival.8), and r =

48 We assume κ =. 83 (=1/12) since the maximum reproductive season length for field voles in Kielder Forest is 1 months (MacKinnon, 1998, Ergon et al., 21, and τ = 3 (τ does not affect the stability of the models). The calculation of α and β and the determination of the parameter ranges for the sensitivity analysis for all six parameters, are detailed and discussed in the results below. 2.5 Results Stability analysis of all three models, using equation (2.7) with the parameters defined in the previous section, showed that the equilibrium population density (equation (2.5)) becomes unstable to small perturbations above critical α and β values. For certain realistic parameter combinations, the models predict regular or quasiperiodic multi-year cycles (cyclic dynamics with a dominant but slightly variable period). Numerical analysis of model B (Fig. 2.3) illustrates the effect of changing α and β values on the population dynamics. For low α values in the range analyzed (Fig. 2.3(a-c)), β does not affect the overall stability and stable population dynamics are predicted (the dynamics are stable in discrete time and show an annual periodicity in continuous time). For higher α values (Fig. 2.3(d-i)) there is a threshold value of β, above which model B produces multi-year cycles. Qualitatively similar dynamics are produced by models A and C. Therefore, all three models predict that delayed density dependent season length can result in multi-year population density cycles. We next consider whether the predicted cycles are similar to those found in the Kielder Forest field voles, and whether their dominant period is sensitive to changes in parameter values. Sensitivity analysis of the dominant period of the population dynamics to parameter variation is shown in Fig We used α =. 5 and β = 1 when not varying these parameters (see Fig. 2.3(f) for example dynamics for model B). The parameter values for the sensitivity analysis were picked using sensible ranges from the Kielder Forest field vole population data, but additionally restricting the ranges to those that did not violate the conditions for a positive population density in equation (2.5). The sensitivity analysis clearly illustrates that delayed density dependent season length can result in multi-year population fluctuations, over a range of parameter values. Where the models do predict such cycles (dominant period > 1) model A predicts periodicities of 6-1 years whereas models B and C predict periodicities of 3-4 years. The periodicity of the 33

49 Kielder Forest cycles is around 3-4 years (Fig. 2.5(, Lambin et al., 2). Thus, only models B and C predict periodicities similar to the observed cycles. Furthermore, model B predicts periodicities that are similar to the Kielder Forest cycles over the largest range of parameter variation. For certain parameter combinations the dynamics of model B qualitatively resemble those of the Kielder Forest field vole population cycles (Fig. 2.5). In particular, note how the model results and field observations display pronounced cycles every 3-4 years with smaller fluctuations on an annual time scale. 15 a) β=1 β=1 β=1 c) n 1 5 α= d) e) f) n 5 α=.5 15 g) h) i) 1 n 5 α= Years Years Years Fig. 2.3: Numerical results for model B between 8 and 1 years where initial population density is 5, r = 1. 4, b = 2. 7, τ = 3, and κ =. 83, for different values of α (.2,.5, and.8, top to bottom, respectively) and β (1, 1, and 1, left to right, respectively). The thin line shows the within-year dynamics using a differential equation solver. The thick line connects the population densities at the end of each reproductive season to illustrate whether dynamics are stable or show multi-year population cycles. 34

50 Dominant Period Dominant Period Dominant Period a) r 1 c) α e) κ Dominant Period Dominant Period Dominant Period b d) f) β τ Fig. 2.4: Sensitivity analysis of the dominant period of population fluctuations to variation in parameter values. Dominant period was determined using power spectrum analysis. The constant parameters (other than when an individual parameter was varied in the analysis) are r = 1. 4, b = 2. 7, α =. 5, β = 1, τ = 3, and κ =. 83. The lines correspond to model A (thick), model B (thin) model C (dashed). When the dominant period equals 1 the populations are at equilibrium and do not show multi-year population cycles. Minor fluctuations in the lines represent either small chaotic windows in parameter space that otherwise predicts stable oscillations, or regions of parameter space giving chaotic dynamics but where the dominant period is sensitive to minor parameter variation. 35

51 n 1 a) n Years Fig. 2.5: a) Population dynamics from model B between years 8 and 1 where initial population density is 5, r = 1. 4, b = 2. 7, α =. 8, β = 4779, τ = 3, andκ =. 83. The dots represent the population density when measured at the same point (one quarter of the year or the start of April) in each year. Clearly sampling at this fixed time in the year only would fail to reveal the variable onset of reproductive season length, an important clue towards the mechanism underlying the dynamics. These dynamics appear qualitatively similar to the Kielder Forest field vole data shown in. The data in are estimated average field vole densities between 1984 and 24 for spring (black), summer (grey), and autumn (white), for the sites in Kielder Forest. Vole densities were derived using sign indices at each site, calibrated with capture-recapture estimates of vole density (see Lambin et al., 2 for further details). 2.6 Discussion The models analyzed here show clearly that delayed density dependent season length alone could play an important role in the generation of rodent population cycles. The sensitivity analysis (Fig. 2.4) indicates that this effect is robust to changes in parameter values of the system. Indeed Stenseths (1999) comment: The relative fraction of exposure to winter and summer conditions may keep the key to the enigma of the rodent cycle, is strongly supported by this study. For the Kielder Forest field voles in particular, our analysis confirms that the significant positive correlation found between the population density at the start of the previous years breeding season and the date in 36

52 the year that the breeding season starts (Ergon, 23) could point towards a mechanism that can result in population cycles. In further support, Model B, which most closely matches the field data, predicted periodicities within the 3-4 year range observed in the data for the largest range in parameter values (Fig. 2.4). Analysis of the character of the population cycles, including the within-year dynamics, reveals further features common to model output and real data. In addition to a qualitative resemblance to the observed trajectories (Fig. 2.5) the dynamics also show the characteristic delay in recovery following a population decline (Boonstra et al., 1998). As we have deliberately chosen the most parsimonious model, we need not expect such characteristics to match the population data, and further studies, incorporating delayed density dependent season length into more realistic models, are needed to indicate the causal agents behind particular characteristics of the population cycle. Numerical analysis also indicates that sampling populations at a few fixed points in the year (common practice) may miss important details of the true character of the annual population dynamics. To illustrate this, consider Fig. 2.5(a), a typical numerical realization of Model B. In Fig. 2.5(a) the densities at the beginning of April in years 81, 87, and 95 are similar. However, in year 81 the population density continues to drop, reaching a trough 3 months later, in year 87 it is at its trough, and in year 95 the population density has been increasing from a trough for almost 2 months. Only capture-recapture studies with short sampling intervals provide accurate estimates of the timings of key life history events in the breeding season and provide clues as to the importance of density dependent seasonality in reproduction (Yoccoz et al., 1998). It is also striking that, if sampled at fixed time intervals, simulated trajectories include periods with ill-defined cyclicity. This is a feature shared by more complex predator prey vole population models (Hanski et al., 1993, Hanski and Henttonen, 1996). Our analyses also indicated that populations should tend to show cyclic dynamics when i) the potential variation in season length (α ) is high, and ii) population densities over which season length varies (β ) are large (e.g., Fig. 2.3). These model predictions could be tested with field data. Our study highlights several avenues for future research. What causes the start date of the reproductive season to be delayed density dependent in Kielder Forest remains to be determined, as does the frequency of this phenomenon in other populations. For example, a similar phenomenon has been observed in Smallmouth bass (Micropterous 37

53 dolomieui) populations in the United States of America (Wiegmann et al., 25). The shape of the functional relationship between past densities and when the breeding season starts also requires further investigation. It also remains to be shown whether season length in Kielder Forest (and elsewhere) is delayed density dependent. However, the present study suggests that the omission of this phenomenon from future attempts to fully comprehend the cycles may be unwise. 2.7 Appendix: Derivation and stability analysis of models A-C Derivation of the three discrete models rl Equations (2.1) and (2.2) have simple analytical solutions, ( n ( t) = n() e and b(1 L) n( t) = n() e, respectively, where n () is the population density at the start of the respective season). For models A-C the product of these solutions is N rl b(1 L) b+ ( r+ L T+ 1 = NT e e NT e, (2.8) which calculates the population change for exactly a year in the future from the initial population size at the start of the previous year. Models A-C differ only in that L is a function of population density at different times in the past (Fig. 2.1). These are incorporated separately to give the three equations for each model, as shown in equation (2.3) Stability analysis Here we summarize the derivation of the stability condition for the three models. In each case we express the conditions in terms of the constant S = ( r+ N f '( N ). * * Model C is the simplest case. The model is the first order difference equation N b+ ( r+ f ( N T ) T+ 1= NTe. The steady state N T * N is therefore stable if and only if d dn b+ ( r+ f ( N ) [ Ne ] * < 1 N= N, which simplifies to (2.9a) 38

54 > S > 2. (2.9 Model A is the second order difference equation, N N e b+ ( r+ f ( NT 1 ) T+ 1 = T written as a system of two first order difference equations where. This can be N Y T+ 1 T+ 1 = Y = Y T T e, b+ ( r+ f ( N T ). (2.1) These have a stability matrix at N T * N given by 1 J A =. (2.11) S 1 The standard conditions for the stability of example Edelstein-Keshet, 1988). This simplifies to J A are 2 1+ Det ( J A ) > Tr( J A ) > (see for > S > 1. (2.12) Model B is the most difficult of the three cases because equations (2.1) and (2.2) do not immediately reduce to a difference equation. This makes it necessary to solve the underlying differential equations. The annual cycles predicted by the model have a reproductive season length b /( r+ ; the population density is reproductive season and N S * rb /( r+ n e * n at the start of the = at the end. To consider the deviation from these annual cycles we define ψ i to be the length of the reproductive season in year i and γ i+ 1 to be the population density at its end. In the non-reproductive season in year i the population density is initially N = γ, and evolves according to equation (2.2). Straightforward integration shows that the density b( 1 at the start of the reproductive season is γ i i e ψ ). To continue the solution we solve equation (2.1) for the reproductive season. This gives the population density at the end i of this season as b+( r+ γ i i e ψ. The reproductive season length and the population 39

55 density are related by the feedback function f (equation (2.4)). Therefore the model reduces to two coupled difference equations where γ ψ i+ 1 i+ 1 = γ e i b+ ( r+ ψ = f ( γ e i, b(1 ψ ) i i ). (2.13) The stability of the steady state cycles, can be found from the stability matrix γ = N, ψ = b /( r, corresponding to annual i s i + J B 1 = S ( r+ N S ( r+ N S b. (2.14) S r+ b The standard stability conditions (as for model C above) imply that only if * n is stable if and 2 > 1+ Sb /( r+ S and 1+ Sb /( r+ S > 1 Sb /( r+. This further implies that ( 1 b /( r+ ) S > 1, S < and ( 2b /( r+ 1) S > 2. We therefore see that < S <, where, S crit 3 1/(1 b /( r+ ) < b /( r+ S = 4 crit. (2.15) 3 2 /(2b /( r+ 1) < b /( r+ < 1 4 Note that S is a non-monotonic function of the ratio b /( r+. As b /( r+ is crit increased from zero, b /( r+ =. S crit then 3 S crit decreases from -1 reaching -4 at 4 increases as b /( r+ is increased further, reaching -2 at b /( r+ = 1. 4

56 CHAPTER 3 DISEASE EFFECTS ON REPRODUCTION CAN CAUSE POPULATION CYCLES IN SEASONAL ENVIRONMENTS (in press in the Journal of Animal Ecology) 3.1 Role played by co-authors Matthew Smith, Andrew White, Jonathan Sherratt, Xavier Lambin, Sandra Telfer and Michael Begon collaborated in this study. The mathematical model was conceived by Matthew Smith, Andrew White and Jonathan Sherratt. Xavier Lambin, Sandra Telfer and Michael Begon provided extensive advice on the empirical relevance of the model. Matthew Smith carried out the numerical studies (with advice from the co-authors) and wrote the paper, with comments provided by the co-authors. Matthew Smith performed the mathematical analyses in the Appendix ( 3.9) with guidance provided by Jonathan Sherratt and Andrew White. 3.2 Abstract Recent studies of rodent populations have demonstrated that certain parasites can cause juveniles to delay maturation until the next reproductive season. Furthermore, a variety of parasites may share the same host, and evidence is beginning to accumulate showing non-independent effects of different infections. We investigated the consequences for host population dynamics of a disease-induced period of no reproduction, and a chronic reduction in fecundity following recovery from infection (such as may be induced by secondary infections) using a modified SIR-model. We also included a seasonally varying birth rate as recent studies have demonstrated that seasonally varying parameters can have important effects on long term host-parasite dynamics. We investigated the model predictions using parameters derived from five different cyclic rodent populations. Delayed and reduced fecundity following recovery from infection have no effect on the ability of the disease to regulate the host population in the model since they have no effect on the basic reproductive rate. However, these factors can influence the long term dynamics including whether or not they exhibit multi-year cycles. The model predicts disease-induced multi-year cycles for a wide range of realistic parameter values. Host populations that recover relatively slowly following a disease-induced population crash are more likely to show multi-year cycles. Diseases 41

57 for which the period of infection is brief, but full recovery of reproductive function is relatively slow, could generate large amplitude multi-year cycles of several years in length. Chronically reduced fecundity following recovery can also induce multi-year cycles, in support of previous theoretical studies. When parameterised for cowpox virus in the cyclic field vole populations (Microtus agrestis) of Kielder Forest (northern England), the model predicts that the disease must chronically reduce host fecundity by more than 7%, following recovery from infection, for it to induce multi-year cycles. When the model predicts quasi-periodic multi-year cycles it also predicts that seroprevalence and the effective date of onset of the reproductive season are delayed density dependent, two phenomena that have been recorded in the field. 3.3 Introduction Effects of disease on host population dynamics come about fundamentally via effects on host fecundity rates, survival rates, or both. Such effects have been demonstrated in a range of wild populations (summarised in Tompkins and Begon 1999 and Tompkins et al. 22) and a few studies have demonstrated the resulting impact on the long term dynamics of their host populations (see Albon et al. 22, Hudson, Dobson and Newborn 1998, and Redpath et al. 26 as examples). Further complexity may arise if disease effects do not operate independently from other factors that may also influence survivorship or fecundity, such as environmental variation (Jolles, Etienne and Olff, 26). It is also common for different parasites to co-exist within the same host (reviewed in Cox 21, Roberts et al. 22, and Pedersen and Fenton 27). These could also potentially have interacting effects on host survival and fecundity. For example, epidemiological studies of humans have demonstrated that multiple infections can have important consequences for human health (Bradley and Jackson, 24, Druilhe, Tall and Sokhna, 25, Adams et al., 26, Sachs and Hotez, 26). Recent studies have also shown that the multi-year dynamics of certain diseases may be strongly influenced by seasonal fluctuations in one or more biological mechanisms (recently reviewed by Altizer et al. 26 and see Stone et al. 27 and He and Earn 27 for recent examples). Understanding how changes in seasonality could alter hostparasite dynamics in wild populations is crucial if we are to predict the emergence of zoonoses or diseases that could threaten species survival (Daszak, Cunningham and Hyatt, 2, Harvell et al., 22). Our work here builds on previous theoretical studies that have demonstrated that, if diseases influence survival or fecundity rates strongly 42

58 enough, then this can resonate with seasonal forcing to dramatically influence both host and parasite dynamics (Keeling, Rohani and Grenfell, 21, Keeling and Grenfell, 22, Dushoff et al., 24, Greenman, Kamo and Boots, 24, Hosseini, Dhondt and Dobson, 24, Ireland, Norman and Greenman, 24, Koelle, Pascual and Yunus, 25, He et al., 27, Ireland, Mestel and Norman, 27). Here we investigate mathematically how the dynamics of seasonally reproducing host populations may be affected by diseases that induce a period of no reproduction, or chronically reduce fecundity, following recovery from infection. Previous theoretical studies have already illustrated that both macro and microparasitic infection induced reductions in host fecundity can destabilise the dynamics of their host populations sufficiently to induce multi-year cycles (Anderson and May, 1981, Dobson and Hudson, 1992, White, Bowers and Begon, 1996, Boots and Norman, 2). However, to our knowledge, the implications of an infection-induced period of no reproduction have not yet been investigated. How the various disease effects may interact with seasonal forcing is also poorly understood for wildlife populations. We construct a model with cyclic microtine populations in mind primarily because evidence for disease effects on fecundity in these is particularly strong. Our choice is also convenient because data on population growth rates, survivorship rates, and seasonality in reproduction are available for a variety of populations and species, and the long term host dynamics for several different populations are known. Moreover, the impacts of disease on host fecundity are known in some cases allowing us to investigate the model predictions for certain diseases. A wide variety of pathogens have been detected in microtine rodent populations (Findlay and Middleton, 1934, Chitty, 1954, Soveri et al., 2, Noyes et al., 22, Olsson et al., 22, Telfer et al., 22, Bown et al., 23, Fichet-Calvet et al., 23, Bown, Bennett and Begon, 24, Cavanagh et al., 24, Fichet-Calvet et al., 24, Telfer et al., 25, Bown et al., 26, Burthe et al., 26, Niklasson et al., 26, Smith et al., 26a, Stenseth et al., 26, Telfer et al., 27). The effects of most of these are unknown or are only beginning to be understood. However, increased mortality rates as a result of infection from various diseases have been known for a long time and may play a role in the dynamics of rodent population crashes (Findlay et al., 1934, Elton, Davis and Findlay, 1935, Soveri et al., 2, Singleton et al., 25). Recent studies of cowpox virus infection have demonstrated its effects on both host survivorship and reproductive timing. In the case of wild populations of bank voles (Clethrionomys glareolus Schreber) and wood mice (Apodemus sylvaticus L.) in UK deciduous 43

59 woodland (Manor Wood), Telfer et al. (25) demonstrated that juvenile females infected with cowpox were likely to delay reproduction until the following breeding season. Such phenotypic plasticity in age to first breeding is predicted to occur in species in which reproductive maturity has significant survival costs, and is thought to be a general characteristic of rodent populations (Lambin and Yoccoz, 21, Telfer et al., 25). Depending on the time of year analysed, cowpox infection was also shown to increase or decrease survival in the Manor Wood populations. Contrastingly, infection by the same pathogen resulted in dramatic reductions in survival probabilities in cyclic field vole (Microtus agrestis L.) populations in UK grassland habitat (Kielder Forest) (Burthe et al., 27). However, the significance of these effects for long term population dynamics has not been investigated before now. Long term studies in Kielder Forest have shown that the prevalence of both vole tuberculosis infection and cowpox antibodies (i.e. cowpox seroprevalence: denoting the proportion of animals with recent or past infections) are significantly related to past population densities, with similar lags (Cavanagh et al., 24). Moreover the probability of a vole initially becoming infected with cowpox also shows a similar lag (Burthe et al., 26). Recently, we showed that if the timing of the onset of seasonal reproduction was influenced by past population densities, as found in Kielder Forest field voles, then this could result in multi-year population cycles (Smith et al., 26. However, it is as yet unknown whether microparasitic infections, through their effects on vole reproductive timing, could cause the onset of the reproductive season to be delayeddensity dependent. In this study, we first address whether microparasitic diseases could regulate or induce multi-year cycles in seasonally reproducing populations by influencing reproductive timing and reproductive output. We then ask how widespread disease-induced multiyear cycles could be in rodent populations living in seasonal environments. To address this question we explore the model dynamics for parameters representative of a range of rodent populations and a wide variety of microparasitic diseases. Lastly, we investigate the model predictions when parameterised for Kielder Forest field vole populations, since data on the prevalence of cowpox virus are available for this system. In addition, there is mounting evidence that a guild of potentially interacting parasites infect cyclic field voles in Kielder Forest (Birtles et al., 21, Noyes et al., 22, Bown et al., 23, Bown et al., 24, Cavanagh et al., 24, Bown et al., 26, Smith et al., 26a, Telfer 44

60 et al., 27). Lastly, we investigate for the Kielder Forest field vole populations, whether the empirically observed phenomena of delayed density dependent season length and seroprevalence, are predicted as epiphenomena of disease-induced multi-year cycles. 3.4 Methods Parameter definitions and model structure The model we use is a modification of the classical host-parasite framework (Anderson et al., 1981) to incorporate no reproduction by the infected class, an immune class that also cannot reproduce, and seasonal reproduction. The host population density is divided into four classes of individual, those that are susceptible to microparasitic infection S, infected individuals that cannot reproduce, I, recovered and immune individuals that cannot yet reproduce, Y, and recovered and immune individuals that can reproduce, Z, but potentially at a lower maximum reproductive rate than for those in the S class. A seasonal component to the model is included through a time dependent birth rate. The change in the densities of the host classes over continuous time, t, is modelled with the four ordinary differential equations ds dt = A( t)( S+ fz)(1 qn) βsi bs, (3.1a) di dt = β SI ( b+ α+ γ ) I, (3.1 dy dt = γ I ( b+ τ ) Y, and (3.1c) dz =τ Y bz, (3.1d) dt where, 45

61 a Τ< t<τ+ L A ( t) =. (3.1e) Τ+ L< t<τ+ 1 Here L is the reproductive season length in units of a fraction of one year, T is time in integer years and N = S+ I + Y + Z is the total population density. We assume that disease free per capita death rate, b, is constant throughout the year but that the per capita birth rate is seasonal ( A (t) ) with no births possible in the non-reproductive season ( A = ) and a constant maximum per capita birth rate in the reproductive season ( A= a ). Using other birth rate functions, such as a sinusoidal function (Barlow and Kean, 1998, Greenman et al., 24, Ireland et al., 24, He et al., 27) only has a minor quantitative effect on our results (not shown). a must be greater than b for voles (and the disease) to persist in the model; a> b is therefore assumed throughout this study. The birth rate is assumed to be density dependent and is modified due to a susceptibility to crowding coefficient, q, which is related to the carrying capacity K = ( a / aq. We assume density dependent transmission at rate β, although assuming frequency dependent infection rates ( β SI / N ) does not qualitatively affect our findings (not shown). Infected individuals potentially have an increased mortality rate due to effects of the disease (α ), and recover at a constant rate γ. Recovered individuals initially enter an immune but non-reproductive class which they leave at rate τ and regain a proportion of their reproductive capacity f ( < f < 1). The total average reproductive delay following infection is therefore ( 1/ γ ) + (1/ τ ) years Parameterization The model parameters can be divided into two groups: those that are specific to the rodent populations and are independent of the disease, and those that describe the effects of the disease on the rodent populations. We chose five combinations of the rodent specific parameters from published data sets to represent estimates from a variety of rodent populations (Table 3.1, see Appendix, 3.9.1, for their derivation). Differences between our estimates of the rodent population parameter values (e.g. the maximum per capita growth rate, r= a b ) at least partly reflect our uncertainly in the true parameter values, although true differences are expected between the different 46

62 species considered. We examined a broad range of disease parameter values for each set of rodent parameters (Table 3.1). The two recovery rates, γ and τ, were varied to give an average recovery time from two years (γ or τ =. 5 yr-1: an extremely low value given that very few rodents would live this long) to one week (γ or τ = 52 yr-1). The transmission rate, β, was varied between high ( β =. 9 ha yr-1) and low ( β =. 5 ha yr-1) infectivity. Disease induced mortality was varied from a 5% mortality per month ( α = 8. 4 yr-1) to no mortality ( α = ). We explored the full range of reduced fecundity following infection, f, from f = to f = Infection threshold It is straightforward to show for equations (3.1) that the threshold density of susceptible individuals required for the infected population density to increase is S > ( b+ α + γ ) / β =. (3.2) S C We use S C to visualise when the rate of change of the infected population is positive or negative in population time series plots. Another common way of expressing this threshold it to formulate it as the basic reproductive rate R = β S /( b+ α+ γ ) which is the expected number of secondary infections produced on average per infected individual (Anderson et al., 1981). This has the simple interpretation that I can only increase if R > 1. Therefore, R = 1 at S = SC. 47

63 Symbol Definition Units Parameter values for each site. Ki MW FS Ho Fra K Maximum population voles ha density. Used to calculate q. r Maximum per capita yr growth rate. Used to calculate a. b Per capita death rate yr L Fraction of year that is reproductive season years 7/12 7/12 6/12 6/12 8/12 a Maximum per capita yr -1 a = ( r+ / L birth rate in the reproductive season q Density dependence 1/ (voles q = ( a / ak coefficient ha -1 ) α Disease induced yr , [, 2.1, 4.2, 6.3, 8.4] mortality rate β Infection rate ha yr , [.5,.1,.2,.5,.9] 1 /γ Mean recovery time years 2 years 1 week [2 years, 1 year, 6 months, 1 month, 1 week] 1 /τ Mean time to recover reproductive function following recovery years 2 years 1 week [2 years, 1 year, 6 months, 1 month, 1 week] from infection f Reduced birth rate following recovery proportion 1, [,.25,.5,.75, 1] Table 3.1: List of parameters, their definitions, units and values used in this study. Site and species codes are Ki= field voles in UK grassland (Microtus agrestis in Kielder Forest), MW= bank voles in UK mixed woodland (Clethrionomys glareolus in Manor Wood), FS = field voles in Fennoscandian grassland (M. agrestis. in northern Finland), Ho = grey-sided voles in Japanese natural woodland (C. rufocanus Sundevall in Hokkaido), Fra = French common vole in agricultural habitat (M. arvalis Pallas in south western France). Sources for parameter values are detailed in the Appendix, The model dynamics were systematically investigated for all possible combinations of the parameter values in square brackets for each set of site specific parameters. 3.6 Mathematical analysis of the non-seasonal models In this section we briefly summarise the dynamics predicted by the separate reproductive and non-reproductive season components of equations (3.1). The importance of seasonally forced birth rates in determining the multi-year dynamics can 48

64 then be assessed. The Appendix, 3.9.2, gives details of the analyses performed in this section. It is straightforward to show that when the disease is absent, the susceptible population density declines monotonically to zero when A = (non reproductive season equations) and increases monotonically to S = K when A= a (reproductive season equations). It is also straightforward to show that the long term dynamics predicted by equations 1, when A = and disease is present, is for the exponential decay of all component population densities to zero. For the reproductive season equations ( can become endemic in the population if A= a ), the disease K > SC. In this case, for most values of the other parameters, an equilibrium is approached through a series of damped oscillations to S = SC. The parameters τ and f, have no effect on inequality (3.2) and therefore ultimately will have no effect on the regulation of the host population. However they do influence the equilibrium population densities of I, Y and Z, and the rate at which these equilibria are approached (Appendix, 3.9.2). This analysis therefore indicates that in the full seasonal model each year will consist of a period of exponential decay to the zero-steady state in the non-reproductive season and an approach to a positive steady state in the reproductive season. 3.7 The effects of seasonal birth rates on long term host-parasite dynamics In the absence of disease, the full seasonal model (equations (3.1) with S > and I = Y = Z = ) can be solved analytically (see the Appendix, 3.9.3, for details). The analytical solution indicates that if the reproductive season is less than a critical length ( L < b / a ) then the host population will decay to zero over time whereas if L > b / a then the long term dynamics will be annual cycles that are exactly repeated each year. If the model starts with a positive initial population density of infected individuals then it predicts three generally different types of long term multi-year dynamics. The first is when the disease dies out over time either because the susceptible host population is always less than S C, or because it is greater than S C for such a short time each year that it results in net annual losses to the infected population density (as also shown by Ireland et al. 24). In these scenarios, the long term dynamics of the susceptible 49

65 population density are the disease free long term dynamics. The second set of dynamics has the disease remaining endemic in the susceptible population, with annual cycles in the component population densities that are repeated exactly each year (e.g. Fig 3.1(af)). With the third class of dynamics the disease again remains endemic but the population density oscillations are not repeated exactly every year (e.g. Fig 3.1(g-x)). In this case we observe either regularly repeated multi-year cycles (e.g. Fig. 3.1(g-l,s-x)), or quasi-periodic multi-year cycles (Fig. 3.1(m-r), with a dominant period of 4 years). The above results show that providing the vole population can persist, the model always predicts oscillations in one (in the case of no-endemic disease) or all of the population components, due to the seasonally varied birth rate. To aid clarity below we define regular annual cycles as the cases where the annual oscillations are exactly repeated every year (as in Fig. 3.1(a-f)) and multi-year cycles as the cases where they are not (as in Fig. 3.1(g-x)). 5

66 1/τ=22 days 1/τ=36 days 1/τ=189 days 1/τ=564 days Voles ha 1 15 a) 1 5 g) m) s) N(T) Voles ha h) n) t) N(t) Voles ha 1 Voles ha 1 15 c) 1 5 Sc 15 d) 1 5 i) j) o) p) u) v) S I Voles ha 1 15 e) 1 5 k) q) w) Y Voles ha 1 15 f) 1 5 l) r) x) Z Years Years Years Years Fig. 3.1: Predicted long term dynamics for total population density ( N ) and individual component densities (susceptible, S, infected, I, recovered but not reproductive, Y, recovered and potentially reproductive, Z ) for different values of τ. The thin dotted vertical lines denote the annual transition from the reproductive season to the nonreproductive season. The top row plots N through time, when it is sampled once a year only (at time T ). The second row plots N as a function for continuous time t. Parameter values are the Kielder Forest field vole parameters (Table 3.1) with, α = 4. 3, β =.9, f =. 225, 1 / γ = 14 days and 1 / τ as detailed at the top of each row. S C is the critical density of S required for I to increase (as detailed in the main text). All simulations started with S = 49 voles ha -1, I = 1 voles ha -1 and Y = Z = voles ha

67 3.7.1 Why does the model predict a variety of multi-year dynamics? Fig. 3.1(a-x) illustrates a transition from regular annual cycles to multi-year cycles, caused by reducing the rate of recovery of reproduction following infection, τ (increasing the time taken to recover), while holding all other parameter values constant. The decision to vary τ is not crucial and similar transitions could be illustrated by varying any other disease parameter. The key difference between the model predicting regular annual cycles and multi-year cycles is a difference in the rate at which the susceptible population recovers following a population crash. If this rate allows the period of oscillation of the host-disease dynamics to match the period of seasonal forcing (i.e. one year), usually with one or two annual disease oscillations per year, then the model settles on regular annual cycles. For example, when the time taken to recover reproductive function is sufficiently fast in Fig. 3.1 ( 1/ τ 22 days) then the predicted long term dynamics are regular annual cycles (Fig. 3.1(a-f)) with the population densities of all four population components peaking once each year. With a slight increase in the time taken to recover reproductive function ( 1/ τ 36 days) the model predicts low amplitude two year multi-year cycles (Fig. 3.1(g-l)). In this case the rate of recovery of susceptible individuals is sufficiently slow for the annual population dynamics not to be repeated in consecutive years. Fig. 3.2(a) illustrates that as 1 / τ is increased from 22 days (Fig. 3.1(a-f)) to 36 days (Fig. 3.1(g-l)) there is a critical point at which the multi-year dynamics change from the susceptible population first exceeding S C at the same time every year to this event occurring relatively late one year and relatively early in the following year. Analysis of this transition reveals that the critical reproductive lag at which this transition occurs, τ C, is dependent on the initial population densities used in simulations. In particular, between 1 / τ = 23 days and 1 / τ = 66 days in Fig. 3.2(a) the model either predicts regular annual cycles or two year multi-year cycles, depending on initial conditions; mathematically, there is a hysteresis in the bifurcation diagram (not shown). However, the remainder of the bifurcation diagram is unaffected by initial conditions. As 1 / τ is increased further in Fig. 3.2(a) the difference between the dates of the early and late disease outbreaks diverge until there is a period doubling bifurcation, closely followed by further bifurcations and chaos, at around 1 / τ = 8 days in this example. Fig. 3.1(m-r) is an example from the chaotic region in Fig. 3.2(a) ( 1 / τ = 189 days). In 52

68 this scenario, the times in the year at which the different component populations peak are different in different years (Fig. 3.1(m-r)). Extensive numerical simulations suggest that the transition from regular annual cycles to multi-year cycles is always associated with the susceptible population density exceeding S C later in the reproductive season. Reducing τ and/or f (which increases the recovery lag) and varying other parameters (more details in next section) can all reduce the rate at which the susceptible population recovers from low densities. This in turn can induce multi-year cycles. The smallest value of τ ( 1/ τ 1. 6 years) in Fig. 3.1 (Fig. 3.1(s-x)) causes the population dynamics to settle on regular two-year cycles. In this case the infected population density peaks every two years exactly (Fig. 3.1(v)). This is because it takes two years for the susceptible population density to induce a sufficiently large outbreak in the infected population density to cause the susceptible population density to crash. In other scenarios (not illustrated) a disease outbreak may reduce the susceptible population to such an extent that it may take longer than one year for the susceptible population density to recover past S C, also resulting in multi-year cycles. In a few rare scenarios, increasing the time it takes for the susceptible population to recover from a population crash can result in a transition from multi-year cycles to regular annual cycles. Such transitions are associated with a reduction in the number of disease outbreaks per year. For example when the model predicts multi-year cycles with the disease breaking out on average twice a year, reducing the rate of recovery of the susceptible population following a population crash (say, by reducing τ ) can then cause the disease to only break out once a year and can induce regular annual cycles. 53

69 1 a) time in year when S C first exceeded Fig.1a f 1/τ=22 days Fig.1g l 1/τ=36 days Fig.1m r 1/τ=189 days Fig.1s x 1/τ=564 days.5 1/52 1/12 1/2 1 2 post recovery reproductive lag (1/τ years, log axis) 1 Fig. 3.2: a) Date in the year at which the infection threshold ( N ) is first exceeded, as a function of 1 / τ. Parameter values are as in Fig Dotted lines denote 1 / τ values used for the examples in Fig The critical τ at which the multi-year cycles occur ( C τ ) is at 1 / τ = 27 days in this example. τ C as a function of the length of the reproductive season, L. Regular annual cycles occur in the regions denoted NO M.Y. CYCLES. Other parameter values and initial conditions are as in Fig. 3.1(m-r). We wrote a numerical code to run the model simulations for the full range of L with these parameter values, and calculate τ C to three decimal places when it could be found (dots), or give details of the predicted dynamics when τ C could not be found. C 54

70 3.7.2 The importance of seasonal forcing in determining the multi-year dynamics In Fig. 3.2( we illustrate the importance of seasonal reproduction for the multi-year dynamics by varying the relative lengths of the reproductive and non reproductive season and plotting the critical τ ( τ C ) required to induce multi-year cycles (with all other parameters held constant and starting from identical initial conditions). Slightly above the critical reproductive season length necessary for the voles to exist ( L = b / a=. 35 in Fig. 3.2() is a region in which the hosts exist at densities that are too low to allow the disease to become endemic (.35< L <. 39 ), and slightly beyond this is a small region in which the disease persists at such low prevalence that it only has a minor impact on the host population (.39< L <. 41). Once the reproductive season length is sufficiently long there is a region (.41< L <. 57 ) in which the disease induces multi-year cycles for all values of τ. Note however that in this region the presence of the disease is still necessary to induce multi-year cycles. For higher reproductive season lengths there is a region (.57< L <. 94 ) in which τ C is a non-monotonic function of reproductive season length. Analysis of the numerical dynamics along the τ C line reveals that the bump (at L =. 78) corresponds to the transition between the disease breaking out once a year (lower L ) and twice a year (higher L ). Therefore, once the reproductive season is sufficiently long, the disease can have a significant impact on the stability of the multiyear host population dynamics. Finally, as the length of the non-reproductive season becomes very short ( L >. 94 ) there is a region in which multi-year cycles do not occur for any values of τ (Fig. 3.2(). This illustrates our general finding that the non-reproductive season must have a sufficiently large impact on the population dynamics for the model to predict multi-year cycles Systematic sampling of parameter space: how common could disease-induced cycles be? In this section we describe the regions of disease parameter space that predict diseaseregulation and multi-year cycles in the population dynamics. This will indicate the sorts of values of τ and f necessary to induce multi-year cycles. 55

71 Fig. 3.3 shows how the dominant period or amplitude of the multi-year cycles predicted by the model are affected by variation in four of the disease parameters for four sets of rodent parameter values (see Chapter 9 for the results of the full analysis). For illustration purposes we chose sufficiently high β values for each set of rodent parameters in Fig. 3.3, so that the disease can become endemic for most of the combinations of the other disease parameter values (detailed in Table 3.1). Varying β has less effect on the model predictions than varying the other parameter values provided that it is sufficiently high for the disease to become endemic in the population. For brevity we omit showing representative results for the grey-sided voles dataset (Table 3.1) as they were very similar to the results obtained for the field voles in Fennoscandia (See Chapter 9 for these results). Each sub-figure in Fig. 3.3 has a series of plots of the dominant period or amplitude of the multi-year cycles. We plot these as a function of 1 / τ and f for 25 sets (per subfigure) of the other parameters. The individual 1 / τ against f plots all share some basic features. Regular annual cycles are predicted when the rate of recovery of reproductive function is fast and f is large ( 1 / τ = 7 days, f = 1) and multi-year cycles are predicted when the rate of recovery of reproductive function is slow and f is small ( 1/ τ 1 year, f = ) (Fig. 3.3). Between these two corners of 1/ τ - f space there is a bifurcation structure similar to that illustrated in Fig. 3.2(a), with a region of low amplitude, period- 2 multi-year cycles close to the stability transition and behind it, a region of perioddoubling and chaos, associated with larger amplitude cycles. Generally, increasing the time taken to recover reproductive function results in multi-year population cycles being predicted at higher values of f. Numerical investigation showed that the dynamics predicted when the reproductive delay is shortest ( 1 / τ = 7 days) are qualitatively similar to those when recovery of reproductive function following infection is instantaneous ( τ = ) (not shown), in which case the model collapses down to a classical host-parasite framework (Anderson et al., 1981). Increasing α generally increases the size of the region predicting multi-year cycles in the individual against f plots. Note, however, that cycles can be predicted by the model even when infection causes no increase in the mortality rate ( α = ). The effect of variation in γ appears more complicated, with the size of parameter space predicting multi-year cycles 1 / τ 56

72 sometimes initially contracting, and then expanding, as 2 years (explained below). 1 / γ is increased from 7 days to 57