UNIVERSITY OF CALGARY. The Effects of Dispersal Pattern on Spatial Synchrony. Jessica Hopson A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

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1 UNIVERSITY OF CALGARY The Effects of Dispersal Pattern on Spatial Synchrony by Jessica Hopson A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN BIOLOGICAL SCIENCES CALGARY, ALBERTA June, 2017 Jessica Hopson 2017

2 Abstract Separate populations of the same species often exhibit correlated fluctuations in abundance, a phenomenon known as spatial synchrony. Dispersal can generate spatial synchrony and more frequent dispersal increases synchrony. However, dispersal patterns investigated are typically simplistic and ignore occasional long-distance dispersal, which is common in natural systems. I used protist microcosms and theoretical models to investigate the impact of dispersal pattern on spatial synchrony. I found that occasional long-distance dispersal significantly increased spatial synchrony. This is due to the nonlinear relationship of synchrony to dispersal rate in the microcosm system. Increased long-distance dispersal at the expense of short-distance dispersal increased synchrony of widely separated populations without affecting short-distance synchrony. However, the theoretical models show that occasional long-distance dispersal can reduce short-distance synchrony without increasing long-distance synchrony, decreasing overall synchrony, depending on the shape of the dispersal rate-synchrony relation. i

3 Acknowledgements This thesis would not have been possible without the support of many people. First I would like to thank my supervisor Dr. Jeremy Fox for providing the guidance and support over the last two years that made this thesis possible. Thank you to my committee members Dr. Lawrence Harder and Dr. John Post for helping to guide the development of my thesis. Special thanks to my lab assistant Morgan Cotroneo who was already well established in the Fox lab when I joined and helped to design, set-up, and run my experiment. Without you as a lab assistant the summer spent counting protists would have been much less enjoyable. Thanks to the rest of the Fox lab, Rob Morgante for experiencing all of the ups and downs of graduate school with me and happily eating my stress baking so I didn t eat it all by myself and Lilian Guan who despite officially leaving the lab was still around for support. Thanks to the many other graduate students who made my two years here in Calgary a great experience. Special thanks go to Nick Rosenberger and Susan Anderson who both read over my entire thesis and gave extremely helpful feedback. Lastly, I must thank my family for constantly supporting and encouraging me throughout my life and especially during this last adventure. I would not be the person I am today without my family. To my sister, thanks for reading and giving me feedback on a second thesis, you are the best. ii

4 Table of Contents Abstract... i Acknowledgements... ii Table of Contents... iii List of Tables... v List of Figures... vi CHAPTER ONE: INTRODUCTION... 1 CHAPTER TWO: EFFECTS OF LONG- DISTANCE DISPERSAL ON SPATIAL SYNCHRONY IN PROTIST MICROCOSMS Introduction Spatial Synchrony and Dispersal Methods Study System Experimental Design Statistical Methods Results Discussion CHAPTER THREE: THEORETICAL EFFECTS OF LONG- DISTANCE DISPERSAL ON SPATIAL SYNCHRONY Introduction Methods iii

5 3.2.1 Discrete- Time Models Continuous- Time Model Results Discrete- Time Models Continuous- Time Model Discussion CHAPTER FOUR: Conclusion References Appendix A : Excluded Population Appendix B : Decomposition of terms for Gillespie algorithm iv

6 List of Tables Table 2.1 Overall synchrony as measured by mean (± SE) cross correlation for dispersal treatments Table 3.1 Within-patch demographic processes Table 3.2 Between-patch demographic processes v

7 List of Figures Figure 2.1 Wavelet analysis Figure 2.2 Representative dynamics of Tetrahymena Figure 2.3 Mean (± SE) correlation of Tetrahymena based on spatial lag Figure 2.4 Heat map of variation in mean likelihood wavelet modulus ratio Figure 3.1 Dispersal networks Figure 3.2 Relationship of synchrony (mean correlation) to dispersal proportion for the hostparasitoid model (a-c) and delayed-ricker model (d-f) Figure 3.3 Synchrony of hosts in the host-parasitoid model for different small-world fractions. 37 Figure 3.4 Synchrony versus spatial lag of the host-parasitoid model Figure 3.5 Synchrony in the delayed Ricker model for different small-world fractions Figure 3.6. Synchrony versus spatial lag for the delayed Ricker model Figure 3.7 Relationship of synchrony (mean correlation ± SE) to dispersal rate for Rosenzweig- MacArthur model Figure 3.8 Effects of increasing the amount of small-world dispersal on synchrony (mean correlation) in the Rosenzweig-MacArthur Gillespie model Figure 3.9 Synchrony versus spatial lag for the Rosenzweig-MacArthur model Figure 3.10 Possible relations of synchrony to dispersal proportion Figure 3.11 Underlying dynamics of the (a) Rosenzweig-MacArthur, (b) delayed Ricker, and (c) host-parasitoid models without dispersal Figure A.1 Plot of Tetrahymena density during the experiment for the excluded metapopulation vi

8 CHAPTER ONE: INTRODUCTION Synchronized phenomena occur commonly: birds migrate south simultaneously, diseases show simultaneously epidemic outbreaks in multiple locations, neurons fire simultaneously, and synchronized swimmers perform a routine together (Grassly, Fraser, & Garnett, 2005; Gwinner, 2003; Meister et al., 1991). The mechanism of synchrony is sometimes apparent, such as synchronized swimmers choosing to move in unison, but it is often less obvious in ecological dynamics. Synchrony occurs in many facets of ecological populations. Within populations, individuals may synchronize their behaviour, such as the harmonized calls of tree frogs or the coordinated diving of penguins (Takahashi et al., 2004; Tuttle & Ryan, 1982). As seen in brood sex ratios in goshawks (Byholm, Brommer, & Saurola, 2002) or synchronized coral spawning (Babcock et al., 1986), demographic properties such as sex ratios, timing of reproduction, abundances and mortality may vary synchronously between populations. Synchrony between separate populations, known as spatial synchrony, often takes the form of positively correlated fluctuations in abundance. For example, the abundance of Canadian lynx cycles synchronously across all of Canada (Liebhold, Koenig, & Bjørnstad, 2004; Moran, 1953a). Spatial synchrony is widespread, occurring in species in all major taxonomic groups: fungal and viral pathogens, plants, insects, molluscs, fish, amphibians, birds, and mammals (see Liebhold, Koenig, and Bjørnstad 2004 for review). Spatial synchrony can occur over very long distances, up to thousands of kilometres for larger organisms (Koenig, 2001; Moran, 1953a). However, nearby populations are typically more synchronized than distant populations (Koenig, 2002; Sutcliffe, Thomas, & Moss, 1996) suggesting that the underlying mechanisms decay with distance. 1

9 Two main causes of synchrony have been proposed: correlated environmental fluctuations and dispersal of individuals (Moran, 1953b; Ranta, Kaitala, & Lundberg, 1998; Ripa, 2000). Environmental fluctuations, such as in annual rainfall or mean temperatures, often correlate over large distances that include many populations, and they become less correlated with increasing distance, which could generate the observed distance-decay of synchrony (Koenig, 2002). Dispersal between sites mixes populations, thereby coupling their dynamics and synchronizing them. Because dispersal is distance dependent, fewer individuals dispersing long distances could also create distance-decaying synchrony (Bjørnstad & Bolker, 2000; Turchin, 1998). Environmental fluctuations and dispersal increase synchrony independently and together (Benton, Lapsley, & Beckerman, 2001; Fox et al., 2011; Goldwyn & Hastings, 2008; Hugueny, 2006; Kendall et al., 2000; Ripa, 2000). The similarity of effects caused by these mechanisms complicates disentangling the relative contributions to synchrony in natural populations. The effects of both mechanisms are expected to decline with distance, as progressively distant locations have less correlated environments (Koenig, 2002) and dispersal is commonly distance dependent (Bullock & Clarke, 2000; Koenig, Van Vuren, & Hooge, 1996; Sutherland et al., 2000). Some studies have examined populations for which one mechanism is not present to characterize the effect of the other mechanism. For example, island populations of sheep, for which dispersal is not possible, exhibit synchrony due to environmental fluctuations (Grenfell & Finkensta, 1998). In contrast, the effect of dispersal has been isolated only in laboratory settings, due to the impossibility of controlling environmental fluctuations elsewhere. Laboratory studies on bacteria and protists show that dispersal can synchronize populations when the environment is 2

10 either constant or fluctuating independently (Fox et al., 2011; Vogwill, Fenton, & Brockhurst, 2009). As dispersal can be manipulated in both laboratory and natural systems with varying ease, its effects on synchrony have been studied extensively. Dispersal readily synchronizes populations with cyclic dynamics, dispersal couples populations together and entrains the intrinsic cycles (Fox et al., 2011). In contrast, few examples illustrate strong synchrony of noncyclic populations (e.g. Pollard, 1991; Ranta, Kaitala, Lindstrom, & Helle, 1997). Many aspects of dispersal can affect the level of synchrony in metapopulations. Particularly important is the overall dispersal rate: the rate at which individuals leave a population and move to another. The effect of dispersal rate on synchrony has been assessed in many theoretical and empirical studies, which have found that synchrony generally increases with dispersal (Fox et al., 2013; Lande, Engen, & Sæther, 1999; Matter, 2001; Ranta, Kaitala, & Lundberg, 1998). Population connectivity also influences synchrony. Connectivity refers to either the spatial pattern of dispersal or the ease of access between different populations, such as dispersal asymmetries. Dispersal asymmetries, whereby individuals move more easily from population A to B than from B to A, have been shown theoretically both to increase and decrease overall synchrony depending on the direction of the asymmetry and differences in population parameters (Wang, Haegeman, & Loreau, 2015; Ylikarjula et al., 2000). Understanding the causes of spatial synchrony is important for understanding the basic principles of population regulation, such as how population size is determined and which factors affect changes in population size. These concepts underlie all of population ecology and have important implications for species conservation and management. Spatial synchrony affects 3

11 extinction risk in metapopulations (Heino et al., 1997; Matter, 2001). At high dispersal rates the strong synchrony of metapopulations increases the extinction risk of the entire metapopulation, whereas at low dispersal limits recolonization, allowing long-term extirpation of local populations. Thus, intermediate dispersal rates promote metapopulation persistence, as dispersal is not high enough to synchronize populations, but is high enough to recolonize extirpated sites quickly (Yaari et al., 2012). Although intermediate dispersal rates are well known to maximize metapopulation persistence, the possible interactions of dispersal rate and dispersal pattern have received relatively little attention. Yaari et al. (2012) compared the consequences of two dispersal patterns for persistence time using models with varying numbers of patches: with a stepping stone pattern dispersers moved only to adjacent patches; whereas fully connected dispersal allowed movement equally among all patches. They generally found longer metapopulation persistence with stepping stone dispersal, although their two models differ in some other respects, including the number of patches. Despite their relative simplicity, these models show that dispersal pattern and connectivity can affect synchrony. A few other studies of more complex dispersal patterns also found effects on spatial synchrony. Both Holland and Hastings (2008) and Ranta, Fowler, and Kaitala (2008) found that redistribution of some dispersal from short to long distances decreased spatial synchrony. Long-distance dispersal strongly affects other facets of population ecology, such as range expansion and genetic diversity (Trakhtenbrot et al., 2005) and likely plays an important role in spatial synchrony, although this remains to be demonstrated. In this thesis I examine experimentally and theoretically the impact of dispersal patterns and rates on the spatial synchrony of metapopulations. Specifically, I ask whether occasional long-distance dispersal increases or decreases the overall synchrony of a metapopulation and 4

12 why these changes occur. In Chapter 2, I examine the effects of long-distance dispersal in an experimental protist microcosm. I compare the synchrony of metapopulations subjected to strictly short-distance synchrony to metapopulations subjected to mostly short-distance dispersal with occasional long-distance dispersal. In Chapter 3, I examine the theoretical underpinnings of my experimental results using simulations to determine the generality of my experimental results. Finally, in Chapter 4 I present a synthesis of my experimental and theoretical work and discuss future work that would build on research in this thesis. 5

13 CHAPTER TWO: EFFECTS OF LONG-DISTANCE DISPERSAL ON SPATIAL SYNCHRONY IN PROTIST MICROCOSMS 2.1 Introduction Spatial Synchrony and Dispersal Dispersal commonly synchronizes metapopulation dynamics (Abbott, 2011; Liebhold, Koenig, & Bjørnstad, 2004; Paradis et al., 1999; Sutcliffe, Thomas, & Moss, 1996). When populations experience similar density-dependent processes, partial mixing of populations by dispersal causes population sizes to become more similar and synchronizes their dynamics. Metapopulation synchrony varies positively with dispersal rate in many situations, both theoretically (Hanski & Woiwod, 1993; Lande, Engen, & Sæther, 1999) and experimentally (Fox et al., 2013; Vogwill, Fenton, & Brockhurst, 2009). The effects of several different spatial patterns of among-population dispersal on synchrony have been assessed theoretically or experimentally. One of the most common is global dispersal, whereby all dispersers having an equal chance of moving to any other population (Arumugam, Dutta, & Banerjee, 2015; Goldwyn & Hastings, 2011; Ripa, 2000). This pattern is easily modeled and manipulated experimentally; however, it is unrealistic for larger metapopulations in which spatial structure and population proximity influence where individuals disperse. To incorporate spatial structure, studies often consider nearest-neighbour dispersal, whereby dispersers move only to adjacent populations (Goldwyn & Hastings, 2011; Holyoak & Lawler, 1996). Although nearest-neighbour dispersal captures the spatial aspect of dispersal better than global dispersal, it still does not fully represent the shape of many natural dispersal kernels. Dispersal kernels describe the frequency distribution of dispersal distance from a point source and often have a fat tail, meaning that most individuals move short distances, but some 6

14 individuals move very far (Nathan et al., 2012). Occasional long-distance dispersal could strongly influence the dynamics of a metapopulation. It might increase the synchrony between widely-separated populations and thus increase the overall level of synchrony in the metapopulation. Conversely, dispersers traveling long distances may have no effect on longdistance synchrony, but decrease short-distance synchrony. Whether either of these intuitions hold has been examined rarely theoretically and not at all experimentally. Two theoretical studies that included some form of long-distance dispersal found reduced synchrony compared to models with no long-distance dispersal (Holland & Hastings, 2008; Ranta, Fowler, & Kaitala, 2008). However, these predictions have not been experimentally tested. In this chapter, I experimentally address the effect of occasional long-distance dispersal on the synchrony of a metapopulation, using a protist predator-prey system. I ask whether occasional long-distance dispersal affects overall synchrony, whether synchrony at different distances is affected, and how changes in synchrony at short and long distances affect overall synchrony. Synchrony ought to decay over time in relation to the strength of the synchronizing mechanisms, with weak mechanisms resulting in faster attenuation than strong mechanisms. Thus, I also examine how initially high synchrony decays over time under different dispersal patterns. Specifically, can occasional long-distance dispersal slow the decline from initially high synchrony compared to short-distance dispersal? One challenge in studying the effect of dispersal on spatial synchrony in natural conditions is the large-scale of this effect. Experimental manipulation at the scale of synchrony in nature, hundreds to thousands of kilometres (Liebhold, Koenig, and Bjørnstad 2004), is impractical or impossible. Microcosms are useful surrogates, as dispersal rates and patterns can 7

15 be controlled along with confounding environmental factors, experiments can be conducted over relatively large spatial scales, and replication is straightforward. 2.2 Methods Study System This experiment involved the ciliate protists Euplotes patella as predator and Tetrahymena pyriformis as prey, which have been used previously to study spatial synchrony experimentally (Fox et al. 2013; Vasseur and Fox 2009). These species have short generation times, c. 4 h for Tetrahymena and c. 24 h for Euplotes. This difference results in predator-prey cycles with a period of about days, which allowed me to observe multiple cycles during the experiment. I used culture methods established by Fox et al. (2011) with slight modifications. Culture vessels were 100-mL glass jars capped with foam stoppers. Each vessel contained 40 ml of nutrient medium and half a wheat seed to serve as a carbon source for the unknown bacteria species on which Tetrahymena fed. Nutrient medium was composed of protozoan pellets (Carolina Biological Supply Company) dissolved at a concentration of 0.15 g/l in spring water obtained from Big Hill Springs Provincial Park (51 15' N ' W). The experimental units were metapopulations of 15 culture jars arranged in a ring. All metapopulations were maintained in a dark incubator at 20 C. To initiate the experiment, I autoclaved nutrient medium and inoculated it with unknown bacteria species cultured from protist stock cultures. After 24 h, I inoculated the medium with Tetrahymena stock cultures to give expected densities of 8500 L -1. Tetrahymena grew for 4 days until reaching high densities and I then transferred 40 ml to each culture vessels with an expected Tetrahymena density of approximately 580 per ml. I added 1 ml of Euplotes stock 8

16 culture at a density of 35 ml -1 to each jar. Sampling started three days later and the experiment ran for 81 days Experimental Design To assess the effects of occasional long-distance dispersal, I subjected the 14 experimental metapopulations to one of two treatments. Both treatments involved the same overall dispersal rate: each dispersal event involved 10% of well-mixed medium from each population. This imposed a constant density-independent per-capita dispersal rate for both species. The treatments differed in the spatial scale of dispersal. For the short-distance dispersal treatment, 2 ml of medium was transferred to the populations on either side of the dispersing population. For the long-distance dispersal treatment, each 2 ml of medium had a 30% probability of being transferred to a randomly chosen non-source population instead of to the nearest-neighbour. I used a 10% dispersal rate because previous work had demonstrated that it is high enough to increase synchrony appreciably in this experimental system (Fox et al., 2013) while also being within the range of dispersal rates seen in fragmented landscapes (Bowne & Bowers, 2004). Previous theory indicated that a 30% long-distance dispersal probability should be sufficient to see an effect of dispersal pattern (Ranta, Fowler, & Kaitala, 2008). For every dispersal event, I removed all medium to be dispersed from a focal jar before adding any medium to prevent individuals from moving more than once during a single dispersal event. I identified the recipient populations for the long-distance treatment prior to dispersal. Each 2 ml subsample going left and right was assigned a probability drawn randomly from a uniform distribution with a range of 0-1. If this probability was 0.3 the subsample was assigned a long-distance transfer; otherwise it was moved to the adjacent population in the appropriate direction. This destination population for long-distance transfers was randomly 9

17 chosen from all non-source populations, with each population having an equal chance of being selected. The same long-distance dispersal pattern applied for all metapopulations in a given treatment in a given experimental day, with different patterns for each dispersal day. The algorithm for assigning dispersal patterns was implemented in R ( I blocked the sampling of metapopulations by time and counted each block twice a week: block 1 on Mondays and Thursdays and block 2 on Tuesdays and Fridays. This allowed for much more replication than an unblocked design due to the time required for sampling. Block 1 contained four long-distance dispersal and three short-distance dispersal metapopulations and block 2 contained three long-distance dispersal and four short-distance dispersal metapopulations. Dispersal occurred three times a week: on Monday, Wednesday, and Friday for both blocks. To obtain density estimates I sampled each metapopulation twice a week. To sample, I agitated culture vessels to mix the contents and distribute the protists homogenously, pipetted out 0.3 ml of medium, and counted the prey and predator individuals under a binocular microscope. If Tetrahymena was too abundant to be counted accurately, I first counted the Euplotes in the sample and then diluted, subsampled 0.33 ml, and counted the Tetrahymena. Once a week, I removed 4 ml of mixed medium and replaced it with 4 ml of sterile medium plus the volume needed to bring each culture vessel to a total volume of 40 ml. This prevented exhaustion of resources, replaced medium lost to sampling, and maintained similar culture conditions for all jars. Medium replacement occurred on Wednesdays before dispersal. Medium replacement is a small synchronous perturbation that could increase synchrony slightly, but it could not create differences between treatments. 10

18 2.2.3 Statistical Methods I calculated density per ml from the abundance data and then transformed the data using log(n/ml+1). This transformation has been used in similar previous studies (Fox et al., 2011, 2013) and is a normalizing transformation for count data that includes zeroes. I excluded one metapopulation in the long-distance dispersal treatment from all analyses, as Tetrahymena abundance declined from the initially high abundance and never recovered (Appendix A). To measure metapopulation synchrony, I first calculated the correlation of transformed abundances for all pairs of jars in a metapopulation for both the prey and predator species. I used the mean of all pair-wise correlations for a metapopulation as a measure of overall synchrony and used Welch s two-sample t-test to assess the effect of dispersal treatment on overall synchrony. To assess how synchrony changed with the distance between populations, I examined the correlations between populations separated by a specific number of intervening nearestneighbour dispersal steps (spatial separation). Adjacent populations have a spatial separation of 1, whereas populations on opposite sides of the metapopulation ring are at a spatial separation of 7, because 7 dispersal steps connect them. Each spatial separation in a metapopulation was represented by 15 pairwise correlations; the mean of which was used as the spatial synchrony at that separation for an individual metapopulation. These mean correlations at the 7 spatial lags constitute a multivariate response variable, as mean correlation at any given lag is not independent of mean correlation at the other lags. To test the effect of the two dispersal treatments, I used a Hotelling s T-squared permutation test to test for differences in the two multivariate responses using the Hotelling package in R (Curran, 2013). To aid the interpretation of the multivariate test, I also conducted Welch s two-sample t-test at each spatial lag. I verified that the assumptions of normality were met for all t-tests by examination of qqnorm plots. 11

19 Mean pair-wise correlations as a measure of synchrony averages over changes in synchrony over time. Correlation also measures only whether the time series change in the same direction at the same time, not whether cycles are synchronized. As dispersal promotes synchrony by synchronizing the already existing predator-prey cycles, synchronized cycles should have a frequency (1/cycle length) of the predator-prey cycle. Sampling error causes small fluctuations in density over brief periods, especially when organisms rare and near the detection limit. Correlations in these fluctuations are not of interest in determining the effect of dispersal on synchrony as they are due to sampling error. I identified how synchrony changed over time and assessed synchrony specifically at the frequency of the predator-prey cycle using wavelet analysis. Wavelet analysis allows detection of signals of different frequencies in a time series that vary over time. To do this, a mother wavelet is chosen and this wavelet can be compressed or expanded, so it cycles at different frequencies (Figure 2.1a-c). Comparing these transformed wavelets to the time series and shifting them through time allows assessment of how well they match, with high positive values of the wavelet transform for matching well, 0 for no match, and high negative values if cycles are out of phase with each other (Figure 2.1d-e). As wavelets of different frequencies are compared over time, the result is a time- and frequency-resolved measure describing the fluctuations of the original time series. 12

20 a b c Frequency d e 0 Low Value High Positive Value High Negative Value T1 T2 T3 Time Figure 2.1 Wavelet analysis. a-c: The sine wavelet compressed or expanded so it represents different frequencies. d: Wavelet b is matched to the time series and shifted along it through time. At T1 the wavelet is mismatched with the time series, resulting in a low value for this time and scale. At T2 the wavelet matches the time series well, giving a high, positive value. And at T3 the wavelet is out of phase with the time series, resulting in a high negative value. Application of this method for many wavelets at different frequencies along the entire time series provides values for different frequencies throughout time. These values indicate how much of the original fluctuations in the time series are described by the wavelet at that frequency and time. Figure adapted from Cazelles et al This process is conducted for each time series. The resulting wavelets summarize the fluctuations of the time series at different frequencies over time and can then be used to 13

21 determine whether the fluctuations at the different frequencies and times are synchronized. Specifically, the localized wavelet modulus ratio (LWMR) is the ratio of actual amplitudes of fluctuations in the metapopulation to the maximum possible amplitude if all populations were perfectly synchronized for each time and frequency. Thus, if at a particular frequency and time two populations are out of phase, the amplitudes will cancel each other out in the numerator and the amplitudes will be less than the maximum and LWMR will be less than one. LWMR is bounded between zero and one, with one indicating perfect synchronization. I focused on the frequency of the predator-prey cycle to assess how synchrony between populations changed over time specifically because of synchronization of the predator-prey cycle. I used only the prey data for wavelet analysis, because the predators spent much more time below the detection limit owing to their lower abundances. For the prey time series, x k,l, for each jar (k) in each metapopulation (l), the continuous wavelet transform W k,l is the convolution of x k,l with the conjugate of the scaled and translated Morlet wavelet φ (Torrence & Compo, 1998; Vasseur et al., 2014): W!,! n, s = s!!.!!!!! x!,! (τ! ) φ!!!!! where t i is the sampling day during the experiment and N is the total number of samples of the time series. Parameters n and s are respectively the time and scale localization of the Morlet wavelet: φ τ = π!!!e!!!! e!!!!. I set ω!, the wavenumber of the Morlet wavelet, to 6, as in previous studies (Keitt, 2008; Torrence & Compo, 1998; Vasseur et al., 2014). Following Vasseur et al. (2014), I generated an array of scales using s = !" where δ = 0.1 and j is a sequence of integers. I set the minimum j so that the smallest scale was 10.5, or three times the average time between 14

22 sampling. I set the maximum j so that the largest scale did not exceed half the experiment duration. This resulted in a wavelet transform for each jar. To estimate a quantitative measure of synchrony, I then calculated the localized modulus ratio (LWMR) for each metapopulation using the previously calculated wavelet transforms: ρ! n, s = Λ!,!(! W!,! n, s ) Λ!,! (! W!,! n, s ) Here is the complex modulus and Λ!,! = e!!!!!!! /!! dn is a Gaussian localization function in time n. I then plotted the mean of the LWMR for each treatment at the frequency of the predator-prey cycle over time to illustrate the synchrony. 2.3 Results All but one metapopulation exhibited predator-prey cycles with a period of approximately days (Figure 2.2). Metapopulations subject to 30% long-distance dispersal had significantly higher overall synchrony than those with strictly short-distance dispersal for prey (Welch s t-test, t=3.65 df=10.99, p=4) but not for predators (Welch s t-test, t=0.60 df=8.88, p=0.56) (Table 2.1). Long-distance dispersal caused consistently high synchrony for prey regardless of spatial lag, whereas short-distance dispersal resulted in prey synchrony declining with spatial lag (Figure 2.3). Long-distance dispersal resulted in significantly higher synchrony than shortdistance dispersal (Hotelling T 2 permutation test, p=95, number of permutations=10000) and long-distance dispersal resulted in significantly higher synchrony at all population separations except for lags 1 and 2 (Follow-up two sample t-tests, spatial separation 1 and 2: t < 2.04, df = 11, p-value > 0.06; all other separations: t>2.5, df=11, p<0.029). Predator synchrony exhibited a similar (Figure 2.3), but non-significant difference between long-distance dispersal and short- 15

23 distance dispersal (Hotelling T 2 permutation test, p=0.1, B=10000). For the prey, the LWMR showed a decay of synchrony over time for short-distance dispersal, but not for the long-distance dispersal (Figure 2.4). Tetrahymena Density Day Figure 2.2 Representative dynamics of Tetrahymena. Each line shows densities from one of the 15 populations comprising a single metapopulation. This metapopulation experienced longdistance dispersal and dynamics are representative of other metapopulations. Cycles have a period of around days and synchrony is highest at the beginning of the experiment. Table 2.1 Overall synchrony as measured by mean (± SE) cross correlation for dispersal treatments. Correlation Prey Predator Long-distance Dispersal 0.81± ±0.02 Short-distance Dispersal 0.70± ±

24 Treatment Short Distance Dispersal Long Distance Dispersal 0.8 Mean Correlation Prey Predator Spatial Lag Figure 2.3 Mean (± SE) correlation based on spatial lag. Synchrony decreases with distance only with long-distance dispersal (red) and not for short-distance dispersal (black). The prey species, Tetrahymena, is shown on top and the predator, Euplotes, is shown on the bottom. 17

25 Figure 2.4 Heat map of variation in mean likelihood wavelet modulus ratio. Time along the x-axis is the experimental day during the experiment and the y-axis is the period of the predatorprey cycle for long-distance dispersal treatments. Color represents the LWMR, which ranges from intermediate synchrony (blue) to high synchrony (red). Only Tetrahymena densities were used to calculate the LWMR. 18

26 2.4 Discussion Both dispersal patterns resulted in relatively high synchrony compared with natural systems (Ranta, Kaitala, & Lindstrom, 1999) and long-distance dispersal for the prey enhanced synchrony relative to short-distance dispersal. The latter result arose because long-distance dispersal maintained synchrony at all spatial lags throughout the experiment. In contrast, metapopulations subjected to short-distance dispersal showed initially high synchrony that decreased quickly as synchrony decayed at longer spatial lags. Predators showed a similar but non-significant trend, likely due to overall lower abundance and increased impact of sampling error. Small fluctuations around the detection limit in the predator had a much greater impact on the observed synchrony, as they spent significantly more time with population sizes near this detection limit. Noisy predator data has been observed previously in this system (Fox et al., 2011; Vasseur & Fox, 2009). The relationship of synchrony to spatial lag differed between treatments and is the underlying reason for the overall difference in synchrony. At short distances, a spatial lag of one or two, prey synchrony did not differ between the two treatments. Thus, the reduced proportion of nearest-neighbour dispersal in the long-distance dispersal treatment did not affect synchrony at short distances. However, incorporation of some long-distance dispersal increased synchrony at longer distances (spatial lags 3) compared to short-distance dispersal. Thus, occasional longdistance dispersal synchronized more widely separated populations and thus increased overall synchrony. Note that the relationship of synchrony to dispersal rate in this system is known to be non-linear, with synchrony plateauing at high dispersal rates (Fox et al., 2013). This nonlinearity was determined from pairs of jars and demonstrates that dispersal exceeding a threshold 19

27 rate does not further increase synchrony. The dispersal rate used in this experiment, 10% three times a week, is well within this region of high synchrony of the dispersal rate-synchrony curve, so that synchrony at short distances is not expected to change with small changes in the nearestneighbour dispersal rate. Synchrony also increases quickly with increasing dispersal at very low dispersal rates (Fox et al., 2013), so changing the dispersal rate of distant populations from no dispersal to very low dispersal would be expected to greatly increase long-distance synchrony. Because of these two aspects of the synchrony-dispersal rate relation, redistributing some short dispersal to longer dispersal is expected to have little impact on synchrony between close populations and greatly increased long-distance synchrony, which is what my experimental results showed. Although redistribution of dispersal caused the result predicted by the relationship of synchrony to dispersal rate described by Fox et al. (2013), synchrony in this experiment was much higher than expected. For example, Fox et al. s relationship and the effective dispersal rate suggests mean correlations of about 0.5 for adjacent populations and about 0.03 for long-distance synchrony. This discrepancy in synchrony between observations and prediction arose partially from differences in initial conditions. Fox et al. (2013) started their pairs of populations out of phase, whereas populations in my metapopulations started with similar abundances. Although this difference in initial conditions contributed to the greater synchrony in my metapopulations, it likely was not the only reason, especially for the greater spatial lags. The dispersal rate between two populations is not the only influence on their synchrony. Dispersal of a single individual connects two populations and influences their synchrony, thereby indirectly influencing the synchrony between other pairs of patches linked via dispersal to the focal pair. This is clearly evident for nearest-neighbour dispersal. For example, such dispersal between patches A and B 20

28 and between B and C should synchronize A and C indirectly, even though they are not connected by dispersal. Such phase-locking can create more extensive spatial synchrony of cyclic population dynamics than the scale at which dispersal operates (Fox et al., 2011; Jansen, 1999; Vasseur & Fox, 2009). Although the metapopulations subject to short-distance dispersal did not show complete phase-locking, it is evident that dispersal was affecting a wider range of populations than were directly connected. Indirect effects are less obvious for long-distance dispersal because populations connected by long-distance synchrony exchange dispersers only occasionally. But if longdistance dispersal increases synchrony of two widely-separated populations and they are synchronized with their nearest-neighbours, then the synchrony should increase within the metapopulation as a whole. Thus, long-distance dispersal should affect synchrony more at greater spatial lags than would be expected simply from the long-distance dispersal rate. For this reason, a metapopulation is much more connected than expected solely from the number of long-distance connections. Connectedness has been looked at in the physics literature using cyclic oscillators and is known as percolation (Newman & Watts, 1999). If populations are represented by connected nodes of a graph, percolation theory asks on average what fraction of populations must be randomly occupied for all occupied populations to be connected to each other? When some long-distance connections are introduced to a network, the proportion of nodes that must be occupied to achieve global connectivity is much lower than with strictly nearest-neighbour connectivity (Newman & Watts 1999). Thus, long-distance connections greatly increase the overall connectivity of a metapopulation. Accordingly, the long-distance dispersal connections in the experimental metapopulation likely affected connectivity more than expected from the long-distance dispersal rate connecting 21

29 two populations. More connected metapopulations should have a higher level of synchrony, since increased connectivity allows the synchronizing effect of dispersal to be propagated throughout the metapopulation more easily. The ability of dispersal to synchronize any metapopulation decreases with increasing metapopulation size (Yaari et al., 2012). For the same system Fox et al. (2011) showed that nearest-neighbour dispersal increased synchrony similarly for all spatial lags in a linear array of 6 populations (Fox et al., 2011). In contrast, in the much larger metapopulations that I examined with the same (short-distance) dispersal pattern, synchrony declined with distance. This shows that although short-distance dispersal is be able to phase-lock a small number of populations, increased metapopulation size makes it more difficult for strictly short-distance dispersal to phase-lock dynamics. These results contrast starkly with theory that predicted reduced synchrony with longdistance dispersal (Holland & Hastings, 2008; Ranta, Fowler, & Kaitala, 2008). This contrast may indicate that the effect of occasional long-distance dispersal depends on the system and dispersal rates. The most important difference between this experimental and previous theory is the inherent demographic stochasticity in natural systems that was not incorporated into the models. Ranta et al. (2008) found that even very limited nearest-neighbour dispersal caused extreme synchrony because of phase-locking. In the absence of demographic stochasticity, nearest-neighbour dispersal between phase-locked populations involves exchange of the same number of dispersers each time. In this situation, variable long-distance dispersal between specific populations, as in my experiment, imposes small, recurring disturbances, continually disrupting the ability of the system to phase-lock perfectly, because different numbers of dispersers were being exchanged. If demographic stochasticity were incorporated in 22

30 metapopulation models, phase-locking should be incomplete, as demographic stochasticity would continuously perturb dynamics (Simonis, 2012). Overall, I found that occasional long-distance increased the overall metapopulation synchrony by connecting distant populations and increasing their synchrony. The generality of these results is not clear. Metapopulation size importantly determines the effectiveness of dispersal in synchronizing populations. For metapopulations larger than those studied here, longdistance dispersal may have less effect on overall synchrony. Furthermore, I did not incorporate distance between populations explicitly, which could have led to higher levels of synchrony than if distance had been incorporated. That said, quantifying the level and pattern of long-distance dispersal is difficult (Bullock & Clarke, 2000). However, further study of the effect of dispersal pattern and long-distance dispersal is warranted, as it is evident that long-distance dispersal can have large impacts on metapopulation synchrony. 23

31 CHAPTER THREE: THEORETICAL EFFECTS OF LONG-DISTANCE DISPERSAL ON SPATIAL SYNCHRONY 3.1 Introduction Most organisms disperse at some time during their lives and dispersal has important consequences for populations and communities. Dispersal can influence population regulation, stability, extinction risk, competition, and coexistence (Levin et al., 2003). The consequences of dispersal depend on two features: where individuals move and the rate that they move (Bahn, Krohn, & O Connor, 2008; Levin et al., 2003; Vuilleumier, Bolker, & Lévêque Olivier, 2010). These factors are linked; the rate of movement between populations varies because individuals move to nearby locations more often than to distant locations (Nathan et al., 2012). Dispersal is also constrained by the ease of access to locations. Differences in connectivity arise due to dispersal corridors, such as protected lands, rivers, roadside verges facilitating dispersal (Johansson, Nilsson, & Nilsson, 1996; Säumel & Kowarik, 2010; Tikka, Högmander, & Koski, 2001; Vermeulen & Opdam, 1995), whereas rivers, roads, or mountain ranges impede dispersal (Frantz et al., 2010; Hayes & Sewlal, 2004; Long et al., 2010). Consequently, dispersal is commonly non-random. When spatial structure is explicit, dispersal can be modeled using dispersal networks. This approach represents populations network nodes and the connections between nodes are the dispersal paths. Networks can range from regular networks (Figure 3.1a) to random networks (Figure 3.1c). Regular networks have connections between a set number of neighbours. These connections can then by randomly rewired to connect random populations, leading to smallworld networks that have some regular connections and some random connections (Figure 3.1b). When all connections are randomly assigned, the network is considered random. (Figure 3.1c). For small-world and random networks, connections can be fixed over time, always connecting 24

32 the same populations, or they can be flexible with the connections between populations changing over time. a b c Figure 3.1 Dispersal networks. Arrangements of dispersal networks include (a) regular, (b) small-world, and (c) random. Adapted from Holland and Hastings Dispersal networks in natural populations are often not strictly regular, with close populations more connected by dispersal than distant populations (Nathan et al., 2012). Smallworld networks with flexible rewiring can be used to model this dispersal pattern, as most connections link nearby populations with the occasional rewiring of connections allowing intermittent long-distance dispersal. This dispersal pattern would be expected for a species that usually disperses short distances but can occasionally disperses much further without following fixed dispersal corridors, such as plants with wind-dispersed seeds (Bullock & Clarke, 2000; Levin et al., 2003). In both non-spatial models and those with only nearest-neighbour dispersal, dispersal can synchronize metapopulations and increased dispersal usually increases synchrony (Arumugam, Dutta, & Banerjee, 2015; Goldwyn & Hastings, 2011; Kendall et al., 2000; Ripa, 2000). Only Holland & Hastings (2008) and Ranta, Fowler, & Kaitala (2008) have considered the consequences of small-world dispersal for synchrony has in the contest of population ecology. Holland & Hastings (2008) examined the impact of small-world dispersal on the number of 25

33 synchronous clusters with a continuous-time deterministic model. They found that the number of small-world connections positively affected the number of clusters and thus negatively affected synchrony. However, they considered asymptotic dynamics, which may have limited relevance for natural systems, given their inherent stochasticity. They also used fixed connections in their small-world networks, which may have promoted clustering compared to a flexible network. Ranta et al. (2008) examined the impact of small-world networks on synchrony with three different discrete-time models that incorporated environmental stochasticity. They also found that increasing the proportion of small-world dispersal decreased synchrony. This pattern held for a range of dispersal rates. However, their metapopulations had phase-locked dynamics, cycling perfectly in phase, even though they comprised 100 patches. In this case small-world dispersal acts as a perturbation, disrupting this phase-locking and lowering synchrony. When short-distance dispersal does not phase-lock dynamics its effect may be different. The experimental results described in Chapter 2 revealed increased synchrony under occasional long-distance dispersal, contrary to the results of previous small-world models. This contrast suggests that incorporating demographic stochasticity in a small-world model would change the impact of dispersal, perhaps predicting metapopulation dynamics consistent with the experimental results. In particular, the impact of small-world dynamics can be assessed by redistributing some nearest-neighbour dispersal to long-distance dispersal in different models and assessing the effects of dispersal on both short and long-distance synchrony. In the experiment occasional long-distance dispersal increased long-distance synchrony without decreasing short-distance synchrony, but the generality of these results and sensitivity to specifics of the model are unknown. I thus explore the effects of small-world dispersal on short and long-distance synchrony and on overall synchrony for multiple models. Specifically, I 26

34 compare three models, two discrete-time models based on Ranta et al. (2008) and a continuoustime model that better represents the specifics of the experimental system. 3.2 Methods Discrete-Time Models I used a delayed Ricker function (Ricker, 1954) and a host-parasitoid model (Beddington, Free, & Lawton, 1975) as these were used by Ranta et al. (2008) and would allow me to compare their results to my results with demographic stochasticity incorporated. These two models also have very different underlying dynamics, which allows for greater generalizations of the effect of occasional long-distance dispersal. The first discrete-time model I used was the delayed Ricker function: N i,t = N i,t!1 exp r + b 1 N i,t!1 + b 2 N i,t!2 ( 1 ) where r (1.01) is the intrinsic growth rate, b 1 (-01) controls direct density dependence, and b 2 (-6) controls delayed density dependence. These parameter values were used by Ranta et al. (2008) and produce cycles with a period of 6 time steps. I initialized patches with a population size of 100. The second discrete-time model I used was a host-parasitoid model: H i,t = H i,t!1 exp r 1 H i,t!1 K c 1 P i,t!1 P i,t = c 2 H i,t!1 1 exp c 1 P i,t!1 ( 2 ) where r (1) is the intrinsic growth rate of the host, K (100) is the host equilibrium in the absence of parasitoids, c 1 (0.035) is the parasitoid search efficiency, and c 2 (1) is the parasitoid s conversion efficiency. These parameter values were used by Ranta et al. (2008) and produce dynamics that are stabilized by dispersal. I initialized patches with a host (H) population of

35 individuals and parasitoid (P) population of 10 individuals. Populations were arranged in a ring, so each population had two immediate neighbours, one on either side. To model dispersal, a total of N i,t d dispersers left a focal patch during each dispersal event, where d is the dispersing fraction and N i,t is the number of individuals in patch i at time t. For nearest-neighbour dispersal, all dispersers were distributed evenly between populations on either side of the focal patch. To incorporate the possibility of long-distance dispersal, a random uniform number, 0 u 1, was drawn for each population. If this number exceeded the probability of small-world dispersal, p, nearest-neighbour dispersal occurred for that population as described above; otherwise small-world dispersal occurred. For small-world dispersal, N i,t d(1- f) dispersers engaged in nearest-neighbour dispersal, whereas N i,t fd dispersed to a randomly chosen population, whereas f is the small-world dispersal fraction. Having two parameters controlling small-world dispersal allows control of both the frequency of long-distance dispersal, by varying the probability of small-world dispersal, and the number of long-distance dispersers, by varying the fraction dispersing via a small-world connection. To incorporate demographic stochasticity, I used the value returned by the population renewal functions as the mean of a Poisson distribution and drew the population size from that distribution. Each patch in the metapopulation underwent population renewal independently, followed by dispersal between the patches according to the above-mentioned rules, and then the metapopulation was censused. For both models, 15 populations were arranged in a ring to form each metapopulation. Each realization of the discrete-time models was run for 600 time steps and the last 200 time steps were used to calculate synchrony. For each combination of parameter values for a particular model I ran 100 realizations. 28