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The influence of inbreeding depression on the stability of a small insular population : modeling inbreeding-stress… Runyan, Simone Elizabeth 2003

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THE INFLUENCE OF INBREEDING DEPRESSION ON THE STABILITY OF A S M A L L INSULAR POPULATION: M O D E L I N G INBREEDING-STRESS INTERACTIONS by SIMONE E L I Z A B E T H R U N Y A N B.Sc , The University of Victoria, 1997 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Forest Science) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A December 2003 © Simone Elizabeth Runyan, 2003 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of f~~or eS~V ^ ^ o ' x The University of British Columbia Vancouver, Canada Date Oec. % ^ . 2 . Q Q 3 <s C g . 5 Abstract I investigated the potential for inbreeding to increase variability in the population trajectory of a small insular population. Increased stochastic variability in population size increases extinction risk, especially in small populations, and is therefore of interest in conservation biology. Inbreeding depression may vary with severity of environmental stress such weather extremes. I explored three main inbreeding-stress interaction scenarios: 'no interaction', a 'linear interaction', and a 'threshold interaction' in which the survival rates of inbred individuals were greatly reduced above a stress threshold. I varied inbreeding depression in annual survival rate from 0% to 80%. I also explored the influence of immigration rate on population variability and extinction rate. I investigated these scenarios using an individual-based population model that traced a pedigree and determined an inbreeding coefficient for each individual. The model was parameterized using data from an island population of Song Sparrows (Melospiza melodia), monitored closely since 1975. Given constant immigration, little increase in variability in the population trajectory occurred as inbreeding depression increased from 0% to 80% with 'no interaction' between stress and inbreeding depression, and only slightly more with a 'linear interaction'. However, imposing a 'threshold interaction' between stress and inbreeding increased extinction rate markedly and led to regular crashes averaging 83% ± 1 1 % (mean ± SD) of population size at a mean interval of 11 ± 5 years. The immigration rate required to reduce the extinction rate to less than 5% increased with inbreeding depression, and with the level of the inbreeding-stress interaction. With inbreeding depression of 50%, an immigration rate of 1.25, or 5 immigrants in 4 years, was required given the 'threshold interaction'. By comparison, immigration rates of only 0.50 given a 'linear interaction', and 0.38 given 'no interaction', were required to reduce extinction rates to 5%. These results suggest that inbreeding-stress interactions have the potential to increase stochastic variability in a population trajectory and that higher levels of immigration may be required to reduce the risk of extinction if inbreeding depression is high or if inbreeding-stress interactions occur in nature. ii Table of Contents Abstract ii Table of Contents iii List of Tables '. vii List of Figures viii Acknowledgments x 1 Introduction 1 1.1.1 Deterministic versus Stochastic Causes of Extinction 2 1.1.2 Sources of Stochasticity 2 1.1.3 Environmental Stress 2 1.1.4 Synergism between Inbreeding and Stress 3 1.1.5 Objectives 4 1.1.6 My Approach 4 1.2 The Study System 5 1.2.1 Selection Against Inbred Individuals in a Population Bottleneck 6 1.3 Background 7 1.3.1 Inbreeding Depression 7 1.3.2' The Relationship between Fitness and Inbreeding May Not Be Linear 8 1.3.3 Stress Could Enhance the Degree of the Nonlinearity Between Fitness and Inbreeding 9 1.3.4 Experimental Observations of Synergism between Inbreeding and Stress 10 1.4 Possible Synergism Between Inbreeding and Stress on Mandarte 11 1.4.1 Susceptibility to Winter Stress through Compromised Metabolic Function 11 1.4.2 Susceptibility to Disease Stress through Compromised Immune Response 12 1.4.3 Susceptibility to Other Stresses 13 1.5 Purging of Deleterious Alleles 13 2 Methods 14 2.1 Field Methodology 14 2.2 Model Description 14 2.2.1 Parameters 15 2.2.2 Classes of Individuals 15 in 2.3 Model S equence 15 2.3.1 Immigration 15 2.3.2 Territoriality 16 2.3.3 Pairing Males and Females 16 2.3.4 Reproduction 17 2.3.5 Calculation of Inbreeding Coefficient, f 18 2.3.6 Over-Winter Survival 18 2.3.7 Summarize the Year 20 2.3.8 Begin Yearly Cycle Again 21 2.4 Collection of Model Statistics 21 2.4.1 Summarize the Model Run 21 2.4.2 Summarize Multiple Runs 22 2.4.3 Step Through a Range of Parameter Values 22 2.5 Model Options 22 2.5.1 Maintaining Age-Specific Survival Rates 22 2.5.2 Reproduction of Independent Young 23 2.5.3 Immigration 24 2.6 Model Runs 24 2.6.1 Survival Scenarios 24 2.6.2 Immi gration 25 2.6.3 Baseline Stochasticity and Default Settings 26 2.7 Model Validation 27 2.7.1 Inbreeding Coefficient Calculation 27 2.7.2 Survival Parameters 27 2.7.3 Reproduction 27 2.7.4 Initial Conditions 28 Results 2 8 3.1 Survival Scenarios I to V 29 3.1.1 Scenario I, No Interaction Between Stress and Inbreeding 30 3.1.2 Scenario II, Linear Interaction Between Inbreeding and Stress 31 3.1.3 Scenario III, Threshold Interaction, No Background Inbreeding Depression 32 3.1.4 Scenario IV and V, Threshold Interaction With Background Inbreeding Depression 34 iv 3.1.5 Average Inbreeding Levels Before and After Winter for Scenarios I to V 35 3.1.6 Testing for Cyclicity in the Trajectory 36 3.1.7 Detail of Scenario III at 50% Inbreeding Depression 37 3.2 Immigration 38 3.3 Baseline Stochasticity 40 4 Discussion 42 4.1 Main Conclusions 42 4.1.1 Inbreeding Depression with No Stress Interaction Did Not Increase Variability . 42 4.1.2 Inbreeding Depression with a Linear Stress Interaction Increased Variability Moderately 43 4.1.3 Inbreeding Depression with a Stress Threshold Greatly Increased Variability 43 4.1.4 Low Inbreeding Depression at Low Stress Reduced Extinction Rates 43 4.1.5 The Mandarte Trajectory Could Be Explained by an Inbreeding-Stress Threshold Interaction 44 4.1.6 Heterogeneity in Survival Rates May Reduce Extinction 45 4.1.7 Required Immigration Rate Increased With Inbreeding Depression 45 4.2 Conditionally Deleterious Alleles and the Potential for Purging 46 4.3 Implications of Model Structure to the Generality of the Results 46 4.4 Implications for the Management of Small Populations 47 4.5 Future Work 48 5 Conclusion 49 Summary 51 Literature Cited 75 Appendix 1. Wright's Inbreeding Coefficient 80 Appendix 2. Example of a Threshold Interaction: 80 Low Inbreeding Depression at Low Stress, but High Inbreeding Depression Beyond a Stress Threshold 80 Appendix 3. Parameterization 81 A. Immigration 81 B. Territoriality 81 C. Reproduction (Production of Independent Young) 81 D. Inbreeding Coefficient, f 82 E. Survival Rates 83 v Appendix 4 . Compact Disk 8 4 SurvivalCurves.xls 8 4 Model Code.doc 8 4 Code for Form 1 8 4 Appendix 5 . Alternative Survival Curve 8 4 Appendix 6 . Model Notes 8 5 Appendix 7 . Glossary 8 7 Catastrophe 8 7 Coefficient of Variation in Population Size, C V N 8 7 Deterministic variability • 8 7 Demographic stochasticity 8 7 Environmental stochasticity 8 7 Exogenous 8 8 Endogenous 8 8 Genetic stochasticity 8 8 Inbreeding depression 88 Jensen's inequality 8 8 Population trajectory 8 9 Stochastic variability 8 9 Stress threshold 8 9 Appendix 8 . Statistical Significance of Correlogram Results 8 9 vi List of Tables Table 1 . Model settings for over-winter survival parameters in Scenarios I to V 2 5 Table 2 . Summary of correlogram output corresponding to the maximum correlation coefficients for Scenarios I I I to V 3 7 Table 3 . Baseline variability in the population trajectory (number of adult females over time) due to six sources of stochasticity. The mean coefficient of variation is the average C V N o f 1 0 0 runs 4 1 Table 4 . Examples of inbreeding levels,/, produced by matings between related parents 8 0 Table 5 . Average rate of survival for each age and sex class from 1 9 7 5 to 2 0 0 2 8 3 v i i List of Figures Figure 1. Survival Curves for Scenarios I to V showing the manner in which survival rates decline as stress increases for individuals with a range of inbreeding levels (left), and inbreeding depression for an individual with/= 0.25, with arrows indicating the change in the curve as inbreeding depression averaged over all levels of stress increases (right) 52 Figure 2. Number of adult song sparrows on Mandarte Island at the spring census from 1975 to 2002 53 Figure 3. Number of adult females on Mandarte Island , adult survival rate in the previous winter and box plots showing distribution of inbreeding coefficients for adults in spring 54 Figure 4. Hypothesized decline in performance with inbreeding under reinforcing epistasis or a genotype-environment interaction 55 Figure 5. Model sequence showing the order of events for one annual cycle 56 Figure 6. The observed distribution of female immigrants arriving annually on Mandarte from 1975 to 2002 compared with a Poisson distribution with mean 0.82 57 Figure 7. The proportion of males that acquire territories declines slightly at high density and is well described by a fitted Type II functional response curve 58 Figure 8. Scatter plot of residual versus predicted values from a multiple linear regression using the variables maternal age, population density and maternal inbreeding coefficient to predict the number of independent young produced per female annually 59 Figure 9. Box plots showing the observed distribution of independent young produced on Mandarte Island from 1982 to 2001, corresponding to the predicted-reproduction category (predicted number of young rounded to the nearest integer) from linear regressions of maternal age, population density, maternal inbreeding coefficient (a) or maternal age and population density only (b) 60 Figure 10. Average survival rate for each age and sex class for the Mandarte Island song sparrows from 1975 to 2002 61 Figure 11. Observed density dependence in juvenile survival rate with regression line 62 Figure 12. Scenario I, No Interaction. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Right: Survival curves corresponding to each trajectory 63 Figure 13. Scenario II, Linear Interaction. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Right: Survival curves corresponding to each trajectory 64 Figure 14. Scenario III, Threshold Interaction. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Right: Survival curves corresponding to each trajectory 65 viii Figure 15. Scenario IV, Threshold Interaction with Constant Background. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Right: Survival curves corresponding to each trajectory 66 Figure 16. Scenario V, Threshold Interaction with Linear Interaction Background. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Right: Survival curves corresponding to each trajectory 67 Figure 17. Comparison of population mean (a), coefficient of variation (b), proportion of runs extinct (c), mean crash severity (d), and mean period length (e), for Scenario's I to V 68 Figure 18. Profiles of population size and variability over inbreeding depression of 0% to 98% for an individual with/ = 0.25, showing mean, median, minimum, maximum and quartiles 69 Figure 19. Comparison of average levels and variability of inbreeding,/ in the population between the five survival scenarios I to V 70 Figure 20. Model output from Scenario III, Threshold Interaction, at 50% inbreeding depression, showing number of adults females, females survival rate in previous winter, and box plots showing the distribution of inbreeding coefficients for adult females 71 Figure 21. Scatter plots showing the relation of the geometric population growth rate ( X* ) to over-winter stress (a), and to the inbreeding level of the population and population density at low stress (b and c) and high stress (d and e) 72 Figure 22. Extinction rate decreases as immigration rate increases from an average of zero to three immigrants per year for a range of levels of inbreeding depression 73 Figure 23. Decline in crash severity as immigration rate increases from zero to three for inbreeding depression of 50% for/= 0.25 74 ix Acknowledgments Special thanks to Peter Arcese who supervised my work and generously shared his ideas and encouragement throughout this project. His encyclopedic knowledge of ecological literature, assistance in the field and caring approach to supervision gave me an excellent experience as a graduate student. David Tait and Michael Whitlock provided stimulating discussions and feedback about the ideas discussed in this thesis, as well as excellent and thought provoking academic courses on modeling and population genetics respectively. I cannot repay Amy Marr for the assistance she gave me in conducting field work, statistical programming, understanding the history of the study, manipulating data and sharing her extensive knowledge of conservation genetics, as well as all of her encouraging and helpful comments. Jamie Smith brightened our field experience with live music and provided excellent editing advice. Lukas Keller and Jane Reid also shared their expertise in mist-netting. Katie O'Connor, Carolyn Saunders and Rob Landucci provided much help in the field. Our lab meetings were constructive and enjoyable, so I would like to thank Bruce Catton, Scott Wilson, Emily Gonzales, Justin Brashares and the other lab member whom I have already mentioned. Carl Walters, Steve Martel and Bob Lessard taught me about computer models and kindly provided programming assistance. Val LeMay and Sally Otto graciously provided advise on statistical problems. I am much indebted to all the workers who have maintained the Mandarte project over the years. I feel fortunate to have had access to well kept data from such an long term study. The Tsawout and Tseycum First Nations bands graciously allowed us to continue to work on Mandarte. Essential financial support for this research was provided by an Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship A, and the Bert Hoffmeister Scholarship in Forest Wildlife from the Faculty of Forestry at UBC. The Department of Forest Science at the University of British Columbia provided facilities to complete this thesis. Norman Hodges provided many hours of help in fixing my computer problems, for which I am grateful. Ryan Gandy and John Salter made office work more enjoyable with jokes and discussions. I am also hugely grateful to Brad Fedy for accommodating my allergies and being a pleasant office mate. My family has been very supportive of my work as a graduate student and, as always, I owe them everything for all that they have given me over the years. Special thanks to my wonderful mother and grandmothers. Most of all, I could not have completed this thesis without the love, support and delicious cooking of Scott Priestman. x 1 Introduction In conservation biology, a long-term debate has smoldered over the relative importance of genetic factors versus demographic and environmental factors, with regard to the persistence of endangered species (Lande 1988; Caro and Laurenson 1994; Caughley 1994; Lande 1995; Hedrick et al. 1996; Boyce 2002). Managers of small populations attempt to maintain genetic variability and prevent inbreeding depression by creating wildlife corridors to allow movement between reserves or by moving captured animals between isolated habitat fragments (Schonewald-Cox et al. 1983; Hedrick 1995; Moritz 1999; Woess et al. 2002). However, some conservation biologists have argued that non-genetic factors such as habitat loss, weather, predators, competitors, disease and demographic stochasticity are more important causes of extinction than the internal genetic structure of a population (Lande 1988; Caro and Laurenson 1994). Evidence is now accumulating that genetic factors such as inbreeding depression do play an important role in population persistence, and that genetic factor may interact with environmental factors synergistically. (Inbreeding depression is the reduction in fitness of offspring of related parents relative to outbred individuals.) Saccheri et al. (1998) found that those populations of the Glanville fritillary butterfly (Melitaea cinxia) with low heterozygosity, an indication of inbreeding, had significantly higher rates of extinction. Furthermore, Westemeier et al. (1998) documented a rebound in fertility rates when greater prairie chickens (Tympanuchus cupido) from a large population were introduced into an apparently inbred population that had been dwindling for 22 years. Evidence that inbreeding makes individuals more susceptible to environmentally inflicted mortality is also mounting (Keller et al. 1994; Bijlsma et al. 2000; Keller et al. 2002; Joron and Brakefield 2003). For example, Coltman, et al. (1999) found that inbred Soay sheep (Ovis aries) carried higher levels of parasitic nematodes which lowered their over-winter survival rates. Inbreeding depression vanished when the sheep were treated to remove the parasites. Inbreeding apparently reduced resistance to parasites which increased susceptibility to winter stress. This thesis explores the potential for inbreeding and inbreeding-stress interactions to increase extinction risk by destabilizing the dynamics of a small insular population. I used an individual-based model, parameterized based on an island population of song sparrows (Melospiza melodia) that have been individually tracked since 1975 and have a social pedigree from 1981 to the present. 1 1.1.1 Deterministic versus Stochastic Causes of Extinction In general, a population can go extinct due to a deterministic decline in the mean population size, stochastic variability in the population trajectory, or both factors (Goodman 1987; Pimm et al. 1988). Stochasticity is variability that is difficult to predict in a complex system due to numerous interacting factors (Shaffer 1987). Stochastic events are often modeled by drawing outcomes at random from a specified distribution. Here I investigate extinction via stochastic variability rather than deterministic decline. Any factor, including inbreeding depression, that lowers reproduction or survival rates such that the population fails to replace itself can lead to deterministic extinction. This thesis does not debate the potential for inbreeding depression to increase.risk of extinction by lowering the average fitness of a population and driving it into decline, as has likely been observed in the greater prairie chicken (Westemeier et al. 1998). However, inbreeding depression may also cause increased risk of extinction by generating increased variability in a population trajectory. As stated by Meffe et al. (1994), "a population with highly variable size has a much higher likelihood of extinction than a population with the same mean size, but lower variability, simply because the variable population approaches low numbers, or zero, more often" (Meffe et al. 1994, p. 223). 1.1.2 Sources of Stochasticity Sources of stochastic variability include demographic stochasticity (due to the random fortunes of individuals), genetic stochasticity (due to factors such as genetic drift, mutation and inbreeding), environmental stochasticity (due to fluctuations in extrinsic stressors such as weather, food supply, predators, parasites, disease and competitors) and catastrophe (due to storms, floods, fires, droughts etc.). Catastrophes can be seen as the extreme end of a skewed or bimodal frequency distribution of environmental stochasticity (Caughley 1994). Here, I consider catastrophes to be environmental impacts which result in a population decline of 75% or more. Synergism between these sources of stochasticity could magnify their effects on a population (Shaffer 1987, p. 71). 1.1.3 Environmental Stress In this thesis, I equate environmental effects with the degree of stress that individuals experience. Stress may be broadly defined as any environmental change that has the potential 2 to reduce the fitness of an organism (Hoffmann and Parsons 1991, p. 2). In population models, environmental stochasticity typically modulates rates of survival and fecundity, though in these modeling explorations only survival rates are impacted. Therefore, I consider the degree of stress to represent the severity of an environmental impact (or the cumulative effect of a number of impacts), with the mildest stress having very little impact on survival rates, and the most severe stress reducing survival rates greatly. 1.1.4 Synergism between Inbreeding and Stress Wright (1922) first speculated that stress could synergistically exacerbate the effects of inbreeding depression. One possible mechanism is that higher homozygosity in inbred individuals leads to less environmental buffering (Langridge 1962; Pederson 1968; Waller et al. in press). For example, a protein that functions well at a wide range of temperatures would be dominant to one that functions at a narrow temperature range. Inbred individuals with the conditionally deleterious recessive allele would not be compromised within their optimal temperature range, but would experience inbreeding depression outside that range. Reinforcing epistasis, in which one locus prevents or alters the expression of other loci (Crow and Kimura 1970, p. 77-81), could magnify the impact of conditionally deleterious recessive alleles on survival (Salathe and Ebert 2003). Keller (1998) noted that if poor environmental conditions cause inbreeding depression to become more pronounced, then inbreeding depression may have a major impact on persistence times of populations. An inbreeding-stress synergism which enhanced the impacts of normal environmental stochasticity on inbred individuals could increase the risk of extinction. Such an interaction might even cause catastrophe-like impacts on the population trajectory in the face of only moderate stress. However, the magnitude of the impact of such a synergism would be determined by the frequency distribution of individual inbreeding coefficients in the population. Selection against inbred individuals by inbreeding depression during low stress years may reduce the portion of highly inbred individuals (Falconer 1989, p. 253), preventing catastrophic impacts due to inbreeding-stress synergism. 3 1.1.5 Objectives The goal of this thesis is to explore the potential for inbreeding to affect population stability. The four specific objectives of this modeling investigation are to: 1. investigate the effect of the strength of inbreeding depression on population stability, given that a population is not in decline over the long term. 2. explore the effect of linear and curvilinear inbreeding-stress interactions. 3. explore the sensitivity of the population extinction rate to changing rates of immigration with and without inbreeding-stress interactions. 4. evaluate the possibility that catastrophic crashes of the magnitude and period observed in the Mandarte Island song sparrow population (86-92%) (Arcese et al. 1992; Keller et al. 1994) could be caused by an inbreeding-stress interaction. While some models have explored the effect of different levels of inbreeding depression on population growth rates (Haig et al. 1993; Mills and Smouse 1994; Jager et al. 2001), I am not aware of any studies on the contribution of inbreeding to stochastic variability in the population trajectory. Furthermore, although evidence that stress influences the expressed degree of inbreeding depression is accumulating (Bijlsma et al. 1999; Dahlgaard and Hoffmann 2000; Keller et al. 2002), I know of no studies exploring the possible effect of such an interaction on population dynamics. In the following sections I describe my approach in more detail, describe the study system used for parameterization of the model, explain some of the theory supporting the hypotheses, and give a few examples of possible mechanisms for inbreeding-stress interactions. 1.1.6 My Approach I investigated this problem using an individual-based model structured according to known life history, behavior and genetics in an extensively-studied wild population. The model traced a population pedigree and assigned Wright's (1969) inbreeding coefficients,/ to each individual. Crow (1970) interprets / as the probability that two alleles at a locus will be identical by descent. Inbreeding reduced survival rates of inbred birds based on three main scenarios: no interaction between stress and inbreeding (Scenario I), linear interaction (Scenario II) and three threshold interactions (Figure 1). In the first threshold interaction between stress and inbreeding 4 there was no inbreeding depression at low stress (Figure lc, Scenario III). In the second threshold interaction, there was a constant "background" level of 25% inbreeding depression at low stress (Figure Id, Scenario IV). And in the third threshold interaction, the background inbreeding depression at low stress occurred as a result of a linear inbreeding-stress interaction (Figure le, Scenario V). I do not investigate inbreeding depression as a deterministic cause of extinction that lowers demographic rates (i.e. survival and fecundity). To a large extent, this has been done. Mills and Smouse (1994) found increased rates of extinction with increased inbreeding load for both survival and fecundity. Large population size and high growth rate protected against inbreeding depression. Instead, I explore the effect of the shape and severity of the inbreeding depression-stress relationship on population instability and extinction rate. Higher inbreeding depression was achieved by both lowering survival rate of inbred individuals and raising survival rates of outbred individuals such that average survival rates in each age and sex class remained at the average levels observed on Mandarte over time. 1.2 The Study System The song sparrows of Mandarte Island, B.C., near Vancouver Island, are a good population on which to base a modeling investigation of the impact of an inbreeding-stress interaction on population dynamics. The population has been individually marked since 1975 and the pedigree is unbroken since 1981. Individual marking has allowed the detection of low levels of immigration and the calculation of pedigree-based inbreeding coefficients. As an insular, resident, island population, Mandarte is also a good model of endangered species in isolated habitat fragments and reserves. Due to the low rates of immigration (approximately one per year), small population size, and lack of inbreeding avoidance (Keller and Arcese 1998) many birds on Mandarte are inbred. Inbreeding depression in annual survival rate is about 25% for 7=0.25 (Keller 1998). However, due uncertainty in paternity, the exact severity of inbreeding depression in this and other populations is not known. The extra-pair fertilization rate on Mandarte is estimated at 30% (O'Connor 2003), so inbreeding depression is likely underestimated (Marr et al. in press). It is therefore reasonable to explore a range of severity of inbreeding depression. 5 The Mandarte population averages around 42 pairs and is capped by density dependent mechanisms, but is periodically affected by crashes or bottlenecks (Arcese et al. 1992), which have reduced the population by 86%, 92% and 63% (Figure 2). The cause of the first crash is unknown. The second and most severe crash, in 1988/89, was concurrent with a cold winter. And the third crash may have been due to a combination of disease and predation by a Cooper's hawk (Accipiter cooperii). Extrinsic factors such as weather, disease and predation may have acted alone to cause crashes. However, these impacts seem likely to occur more frequently than the crashes, suggesting an additional cause. 1.2.1 Selection Against Inbred Individuals in a Population Bottleneck The crash of 92% in the winter of 1988/89 is of particular interest as the severity of the crash may have been exacerbated by a synergism between inbreeding and stress. During the crash of 1988/89, inbred birds were selected against (Keller et al. 1994). The low survival rates for both adults (Figure 3) and juveniles during the 1988/89 crash resulted in low numbers (three females, eight males and one immigrant female). Highly inbred birds did not survive the crash, and the average inbreeding level of the population dropped (Figure 3). Following the crash, inbreeding levels rose rapidly because most members of the population were descended from the four females alive after the crash. Although allelic diversity dropped over the crash, new alleles were rapidly reintroduced by immigrants (Keller et al. 2001). In 1990 (study year 16) one male and three female immigrants arrived. Expected heterozygosity based on eight microsatellite loci dipped briefly over the crash but then recovered and remained at previous levels even as the average inbreeding level in the population climbed over the next four years (Keller et al. 2001). These observations confirmed Wright's (1931) suggestion that low levels of immigration should restore genetic variation rapidly. Sparrows in the smaller islets near Mandarte did not experience the severe mortality observed on Mandarte during the winter of 1988/89 (Rogers et al. 1991; Smith et al. 1996; Keller et al. 2001). Since the islands are close enough to experience the same weather, this suggests that some factor particular to Mandarte, in addition to bad winter weather, caused the crash on Mandarte. Mandarte has fewer trees and is somewhat more geographically isolated than other islands leading to lower immigration rates. While the immigration between the smaller islets was common (Smith et al. 1996), only about one immigrant per year entered the Mandarte 6 population. Although there is a possibility that the sparrows on Mandarte were more exposed to the weather due to the orientation of the island and lack of shelter, it seems plausible that low immigration rates and inbreeding on Mandarte caused the sparrows to be more inbred and more susceptible to mortality due to environmental stress than birds on other islands. 1.3 Background 1.3.1 Inbreeding Depression Inbreeding depression is the reduction in fitness of an inbred individual, relative to an outbred one: 8= (Wo- Wi) = 1 - Wi Wo Wo ( 1 ) where 8 is the level of inbreeding depression, Wo is the average fitness of an outbred individual, and Wi is the average fitness of an inbred individual. Here 8 varies from 0 to 1 and is expressed as a percentage. Because Wi varies with the degree of inbreeding, a given 8 must be qualified by stating the associated inbreeding coefficient, / In this thesis, inbreeding depression refers to the reduction in fitness experienced by an individual with/= 0.25, unless stated otherwise. Wright's inbreeding coefficient, f, can be calculated easily from simple pedigrees (for example, Haiti 2000). Offspring of a full-sibling mating have an inbreeding level, /=0.25. For more examples of matings that produce offspring with varying levels of inbreeding, see Table 4 in Appendix 1. Inbreeding depression is thought to result from increased homozygosity in inbred individuals which either increases the expression of deleterious recessive alleles or reduces the occurrence of overdominant allele combinations (Wright 1977; Fowler and Whitlock 1999). Recessive and partially recessive deleterious alleles now appear to account for most inbreeding depression (Charlesworth and Charlesworth 1999). In a small population, the average inbreeding level increases due to genetic drift because the number of possible mates is limited relative to a larger or infinite population (FST: inbreeding of the sub-population, relative to the total population) (Soule and Mills 1998). Inbreeding within sub-populations can further increase the inbreeding level of individuals (FIT: average 7 inbreeding of the individual, relative to the average of the total population; Fis: average inbreeding of the individual, relative to the average of the sub-population) (Lacy 1997). In this thesis, inbreeding levels of individuals are calculated relative to an outside population that is assumed to be unrelated, and relative to the initial population. Following Morton et al. (1956), it is generally assumed that inbreeding reduces survival according to a negative exponential function (Figure 4, a): Wi = Wo.exp(-Bf) (2) where B estimates the genetic load due to inbreeding in terms of lethal equivalents (Morton et al. 1956). The genetic load is the fraction by which average fitness in a population is reduced by a genetic factor such as mutation, drift or segregation, as compared with an otherwise identical population in which the factor of interest is absent (Crow and Kimura 1970; Keller and Waller 2002). Because it is difficult to distinguish a few deleterious alleles of large effect from many deleterious alleles of small effect, the genetic load due to inbreeding may be expressed in terms of lethal equivalents (Keller and Waller 2002). A collection of deleterious alleles at different loci amount to one lethal equivalent if, when made homozygous, they together have the same effect as one lethal mutation (Morton et al. 1956). The genetic load due to inbreeding depression may be calculated from 8: B = - Ln(l - S) f ( 3 ) In the following sections I outline some theory that suggests a curvilinear interaction between stress and inbreeding depression. I briefly discuss some of the relevant literature, describe possible mechanisms for an inbreeding-stress synergism on Mandarte, and discuss the issue of purging of deleterious alleles. 1.3.2 The Relationship between Fitness and Inbreeding May Not Be Linear The true nature of the relationship between Wi and/ suggested by Equation 2, is not known. If inbreeding depression is mainly due to deleterious recessive alleles, as appears likely (Charlesworth and Charlesworth 1999), and if loci affecting a particular trait act independently, then the log of the average value of the trait should be linear when plotted against / (Keller and 8 Waller 2002). However, if there is epistasis (interaction) between loci, then the rate of decline in the log of fitness may exceed linear expectations. Although empirical evidence for epistasis in inbreeding depression has been weak and variable (Mukai 1969; Willis 1993; Dudash et al. 1997), the possibility of epistasis is enough to concern conservation biologists, who worry about worst case scenarios for small endangered populations (e.g. Mills and Smouse 1994). Experimentally observed inbreeding depression of components of fitness tends to show linearity with respect to / (Falconer 1989, p. 252). However, several generations of inbreeding are generally required to obtain highly inbred individuals. Selection may occur in the inbred lines, increasing the fitness average of inbred individuals and causing overestimation of fitness at high inbreeding. Furthermore, Salathe and Ebert (2003) recently provided a statistically significant example of a strong nonlinear decrease in the logarithm of fitness with inbreeding. In the case of an interaction between two loci, the frequency of double heterozygotes should decline with a second order polynomial of/ (Crow and Kimura 1970, p. 77-81). Wi = Wo.exp(-Bf - Cf) ( 4 ) Therefore, as/increases, the rate of inbreeding depression increases, provided that there is synergistic epistasis, which reinforces inbreeding depression, rather than antagonistic epistasis, which reduces the effect of inbreeding depression (see Figure 4b) (Peters 1999). The number of loci involved determine the order (i.e. the highest order power) of the polynomial. The degree to which the relationship deviates from linear depends on the strength of the interaction between loci, and the number of loci involved. Theoretically, the synergism can take almost any form, and the polynomial may be viewed as a Taylor expansion (Figure 4f) (Mills and Smouse 1994). Wi = Wo.exp(-Bf-Cf-Df3 -Ef4.... ) ( 5 ) 1.3.3 Stress Could Enhance the Degree of the Nonlinearity Between Fitness and Inbreeding Keller and Waller (2002) suggest the possibility that some of the recessive alleles expressed due to inbreeding may be conditionally deleterious, reducing fitness only under environmental stress. Although few studies have been conducted and results are not conclusive, an interaction 9 between environmental stress and mutations, in the form of conditionally deleterious alleles, could strengthen synergistic epistasis (Salathe and Ebert 2003). Salathe and Ebert (2003) attribute some of their success in finding significant epistasis to the fact that they assessed fitness in a competition experiment. Fitness was measured as the ability of clonal Daphnia magna of various levels of inbreeding to increase in frequency while in competition with a tester strain. In the competition experiment, most fitness components contribute to the performance of a clone, lessening the chance that important factors could be missed, thus increasing the ability to detect epistasis. Furthermore, the stress of competition might increase inbreeding depression. In another study, Koelewijn (1998) found inbreeding depression, and some evidence of epistasis under harsh field conditions, but no evidence of either under more moderate greenhouse conditions, suggesting that stress may synergistically enhance inbreeding depression and epistasis. If stress enhances epistasis, then one could hypothesize that the magnitude of the coefficients in Eqn. 5 depend on the degree of stress. If the coefficients of the lower degree terms are small, while the larger degree terms are many times greater, a steep threshold results (Figure 4g). By adjusting the coefficients which determine the strength of the interaction, the threshold may be set at any value of / This scenario might occur if one or a few loci with conditionally deleterious alleles have a very strong effect on a multitude of loci, as with a lethal or semi-lethal allele. Due to the awkward formulation of these polynomials, the threshold effect was incorporated in the model using a logistic equation which allowed easy control of the location and steepness of the threshold via only two parameters, with no increase in the number of terms in the equation (see section 2.3.6, p. 18). 1.3.4 Experimental Observations of Synergism between Inbreeding and Stress While there is limited empirical support for strong synergism between loci, there is mounting evidence that inbreeding depression may increase under environmental stress. In speculating about the high mortality of continuously inbred guinea pigs relative to outbred controls during a severe winter, Wright (1922, p. 22) wrote: "It therefore seems probable that the inbreds had reached a critical stage [i.e. level of inbreeding], in which a given change for the worse in environmental conditions actually produced a disproportionately great effect on the mortality [rate of inbreds relative to outbreds]." 10 Tests of this intuitively appealing idea have generated mixed results. Mills and Smouse (1994) reviewed the literature of the time and concluded that inbreeding depression is more severe under environmental stress and is more likely to be observed under intra-or interspecific competition as in the wild. Some studies have found no effect of stress (Radwan 2003; Kandl 2001), or have even observed decreased inbreeding depression under stress (Fowler and Whitlock 2002). However, evidence that stress increases inbreeding depression does appear to be mounting (Coltman et al. 1999; Bijlsma et al. 2000; Dahlgaard and Hoffmann 2000; Keller et al. 2002; Joron and Brakefield 2003). A recent study on heat shock proteins provides a possible mechanism for a threshold relationship between fitness and inbreeding (Kristensen et al. 2002) (see Appendix 2). As noted by Keller and Waller (2002), a meta-analysis that accounts for publication bias would be worthwhile (see also Palmer 2000). 1.4 Possible Synergism Between Inbreeding and Stress on Mandarte In the Mandarte Island song sparrow population, at least two mechanisms could generate population level effects through an inbreeding depression-stress interaction ( a) compromised metabolic function or ( b ) reduced immune response. The mechanism may be expressed at high/when inbred individuals are challenged by either weather or disease. 1.4.1 Susceptibility to Winter Stress through Compromised Metabolic Function Severe winter weather is known to stress song sparrows. In a physiology study investigating metabolic responses to fasting in the white-crowned sparrow (Zonotrichia leucophrys gambelii), Ketterson and King (1977) found that most individuals did not survive longer than one day and two nights without food at air temperatures near freezing, though the unusual sparrow might survive three nights and two days. In cold weather birds die either from starvation because they are unable to fuel the body's need for heat production, or from hypothermia because they are unable to metabolize their reserves fast enough to meet demand (Newton 1998). Rogers et al. (1991) note that song sparrows on Mandarte Island maintain low fat reserves compared to North American ground-feeding finches. Passerine birds store fat in response to either current local temperatures or through cues which allow them to anticipate levels required to sustain themselves over the winter (Rogers et al. 1991). Sparrow on Mandarte Island do not accumulate large fat reserves, probably because Mandarte generally 11 experiences moderate winter conditions and song sparrow populations are non-migratory. However low fat reserves may make coastal song sparrows more vulnerable to cold weather when it does occur. During these periods of extreme stress, a metabolism which functions efficiently is necessary. Thus, it is plausible that individuals with metabolisms compromised due to inbreeding may be more susceptible to mortality during periods of environmental stress. The length of time fasting can occur before death may be related to inbreeding coefficient. A cold period of a given length could therefore produce high mortality in an inbred population, but not in a presumably outbred population with higher immigration rates, as may have been observed during the winter of 1988/89 on Mandarte and the surrounding small islands. 1.4.2 Susceptibility to Disease Stress through Compromised Immune Response Inbred individuals may also have compromised immune systems which could raise their susceptibility to disease. Inbreeding is known to affect immune response in the Mandarte song sparrows (Reid et al. 2003). Cell-mediated immune response (CMI) declined roughly linearly with inbreeding upon exposure to a non-specific mitogen that is known to induce an accumulation of leukocytes upon subcutaneous injection (Reid et al. 2003). Mounting an immune response may be energetically demanding and may depend on an individual's nutritional or hormonal state, suggesting that inbreeding may limit immune response indirectly through effects on metabolic or hormonal pathways (Reid et al. 2003). Alternatively, inbreeding may limit immune response directly, through loss of heterozygosity at the major histocompatibility complex (MHC) loci. The MHC is a group of genes that code for proteins on the surface of cells that help the immune system to recognize foreign substances. Heterozygosity at MHC loci enhances immune function (Carrington et al. 1999). MHC proteins are found in mammals and birds and have very high allelic diversity. Individuals that are heterozygous at the MHC loci may be able to present a greater variety of antigenic peptides to initiate CMI relative to homozygotes. Heterozygotes are therefore able to mount a more productive immune response against a diversity of pathogens (Carrington et al. 1999). The MHC loci suggest the possibility of strong interaction between an exogenic factor and inbreeding level, in which the lack of heterozygosity at a few loci causes the breakdown of the entire organism in the face of the disease stress. 12 1,4.3 Susceptibility to Other Stresses Reduced individual quality through inbreeding depression could be extended to explain reduced fitness of inbred individuals under almost any stress. For example, a Cooper's hawk may find it easier to catch inbred sparrows. The inbred sparrows may have slower reflexes and defend sub-optimal territories with less bush-covered foraging areas, increasing exposure to predation. 1.5 Purging of Deleterious Alleles In these modeling explorations, the inbreeding depression-stress relationship remained constant over time. Inbred individuals were selected against, which reduced the average level of inbreeding in the population. However, deleterious alleles were not purged by my model. Over time purging, or the removal of deleterious alleles from a population's gene pool, may gradually reduce the amount of inbreeding depression experienced on average by an individual with a given inbreeding level (Byers and Waller 1999; Keller and Waller 2002). Purging is likely much less effective if inbreeding depression is caused by weak selection against many mildly deleterious alleles rather than strong selection against a few highly deleterious alleles (Charlesworth and Charlesworth 1987, p. 243). In Drosophila, at least half the inbreeding load is caused by mildly deleterious alleles (Hedrick and Kalinowski 2000). Linkage between loci may also reduce the efficacy of purging (Keller and Waller 2002). Here it is assumed that any purging is balanced by immigration and mutation which re-introduce deleterious alleles at a rate such that inbreeding depression remains constant relative to inbreeding level. If inbreeding depression does not destabilize the population in this extreme case, then population instability resulting from inbreeding-stress interactions should be a low-priority concern for managers and researchers. Constructing a more detailed genetic model which allows purging, genetic drift and mutation would also not be warranted to explain large fluctuations in population size due to inbreeding depression-stress synergism. 13 2 Methods 2.1 Field Methodology Song sparrows have been studied continuously on Mandarte Island since 1975. Mandarte is a small (6 ha), narrow (125 m by 700 m) island running NW-SE in Haro Straight, ca. 9 km east of southern Vancouver Island, British Columbia. Bushes grow along the longitudinal axis of the island covering about 1 ha (Tompa 1962). The bush is surrounded by grassy meadows on either side which fall away to rocky intertidal along the NE shore and 15 m cliffs along the SW shore. The nearest island to Mandarte is Halibut Island which is 1.3 km south east. Sidney Island parallels the south west side of Mandarte at a distance of about 2 km. All sparrows are banded with a unique combination of one metal band and three colored plastic leg bands for field identification. The low shrub vegetation allows banded individuals to be resighted readily, lending high confidence that all territorial birds and offspring are recorded. Territories are monitored and nests found as early as possible so that chicks may be banded at around six days after hatching. Juvenile survival to fledging at 10 to 11 days and independence from parents at 24 to 30 days is also monitored. Immigrants are mist-netted and banded. The population size is taken as the number of territorial females at the spring census (April 30) each year. Because all of the shrub area is defended by territory holders, the terms "population size" and "population density" may be discussed interchangeably. For detailed field methodology, see Smith 1988, Hochachka et al. 1989, Smith and Arcese 1989 and Arcese et al. 1992. 2.2 Model Description The model was programmed in Visual Basic 6.0. Each simulation was run for 260 years. The model was initialized with a population of 11 unrelated birds ( /= 0) in year zero. Immigrants were also assumed to be unrelated to the Mandarte population and each other ( / = 0). Inbreeding coefficients, / , in the model are therefore relative to the initial population and to an outside "infinite" population. The first 30 years of output of each run were excluded from analyses to eliminate the effect of the starting conditions. Analyses used the 200 simulation years, from year 30 to 229, which I call the 'analysis window'. In order to test for cyclicity or autocorrelation with lag up to 30 years, it was necessary to have a 30 year buffer after the core 200 year observation period, so the model was run for 260 simulation years. 14 2.2.1 Parameters Details of parameter calculation and values appear in Appendix 3. Most rates and relationships used in the model were based on data gathered up to 2001. 2.2.2 Classes of Individuals The model recognizes five classes of individuals: floater, single, breeder, female and juvenile. The first three classes describe adult male status: floaters with no territory, single territory holders and breeders. Floaters and singles were assumed not to breed. Al l adult females pair and breed, so there is only one class of adult female birds. The model also recognizes juveniles (birds between independence and first breeding year) and tracks information on age, inbreeding level and immigrant status for individual birds. 2.3 Model Sequence The model is based on the annual cycle of the song sparrow (Figure 5). It is assumed that population size (the number of adult females) is censused yearly in the spring after immigrants arrive, but before breeding begins. Although this is an individual-based model with survival and reproduction probabilities based upon individual qualities such as age and inbreeding, it may be conceptualized simplistically in terms of a difference equation: Nt+, = Sa.N, + r»NfSj + N, - NE ( 6 ) where Nl+1 is the population size next year, T V , is the population size this year, Sa is the adult survival rate, Sj is the juvenile survival rate, r is the number of young produced per female, Nj is the number of immigrants, and NE is the number of emigrants. Emigration is implicitly incorporated into morality rates in the model. Model code is included in Appendix 4, the compact disk (ModelCode.doc). 2.3.1 Immigration The model determines the annual number of female immigrants by drawing an integer value from a Poisson distribution with a specified mean. The default model setting is an average of 1.0 females per year. Actual female immigration averaged 0.82 per year (about 4 immigrants in 15 5 years) and followed a Poisson distribution (Figure 6, Appendix 3A). Setting the mean immigration rate to 1 per year allows easy comparison with a constant rate of one female immigrant per year, which allows variability in immigration rate to be turned off. Male immigrants rarely breed in their first year on Mandarte (Appendix 3A). By setting the female immigrate rate above the observed level, I accounted for the effects of male immigrants. 2.3.2 Territoriality The number of territory holders is based on the total number of adult males in the population according to the following relationship, fitted using Mandarte data (Figure 7, Appendix 3B): T=0.98NM/(1 + 0.0027 NM) (7) where T is the number of territories, and NM is the number of males at the spring census before breeding. T is rounded to the nearest whole number. About 83% of males acquire territories, but the form of a Type II functional response curve in Eqn. 7 (Begon et al. 1990, p. 313) provided a slightly improved fit to the Mandarte data. This curve allows a marginal increase in the portion of males with territories at intermediate density (NM =50) and a decrease at high density. At high population densities the frequency of aggressive interactions between floaters and the resident territory holders increases (Tompa 1962; Arcese 1987; Smith and Arcese 1989) suggesting that a higher portion of subordinate birds are unable to acquire a territory. Territory holders that have survived from the previous year are assumed to retain their status. Empty territories are assigned to surviving floaters, giving priority to the oldest birds, and then to recruits. (If male immigrants are included in the model, they have lowest priority for territories.) Males that are not assigned a territory become floaters and are excluded from the breeding population. 2.3.3 Pairing Males and Females The model pairs individual females with male territory holders for breeding. All females are assumed to attempt breeding. Surviving pairs remain together. Older males have priority for females and become polygamous if the number of females exceed the number of territory holders. 16 2.3.4 Reproduction Reproduction, or the production of independent young, is related to maternal age (Smith et al. 1984), inbreeding level and density (Arcese et al. 1992). Nesting and fledging stages are not modeled. Inbreeding depression in reproduction averaging 31% can be turned on and off, but was left on for results presented in figures in this thesis. I based reproduction in the model on the observed numbers of independent young produced annually by females on Mandarte. I ran a multiple linear regression to predict reproduction using the variables maternal age, population density and maternal inbreeding coefficient (Appendix 3C). The resulting regression coefficients were used to predict the number of independent young produced by females in the model by rounding the predicted number to the nearest whole number (Figure 8). To add realistic variability in reproduction, I drew the number of independent young produced by a female in the model from the empirical distribution of residuals (Figure 8). I reconverted the residual values to whole numbers of independent young by adding the residuals to the corresponding predicted values. I then stratified the observed numbers of independent young produced per female into six 'predicted-reproduction categories' (labeled zero to five) with membership determined by rounding the corresponding predicted value to the nearest whole number (Figure 8). The number of young assigned to a female in the model was then drawn at random from the observed values within one of the six predicted-reproduction categories (Figure 9a). The predicted-reproduction category was determined by rounding the predicted number of young for the female in the model to the nearest whole number. Inbreeding depression in reproduction averaged 31% over all age categories in the Mandarte data, though the 95% confidence intervals stretched from zero to one. Inbreeding depression in reproduction could also be set to 0% by basing reproduction on a multiple linear regression that used the variables maternal age and population density, but not maternal inbreeding coefficient (Figure 9b). The option of no inbreeding depression in reproduction resulted in only minor changes in the survival scenario output, which are mentioned in the Results section. 17 Paternity was assigned assuming no extra-pair fertilization (EPF). Although the EPF rate on Mandarte has been estimated at 30%, particular males do not appear to be favored so this does not change the variance in reproductive success (O'Connor 2003). Therefore, EPF's do not significantly alter the effective population size (O'Connor 2003). Furthermore, females do not choose mates less related to them than their social mate through EPF's (O'Connor 2003). EPF's were therefore assumed not to be of major importance in shaping the distribution of inbreeding coefficients in the model population. Because field observations do not include the sex of independent young, but only of recruits, the sex ratio of independent young was assigned according to the sex ratio observed in recruits (age one non-immigrants) as 42.8% female, 57.2% male. Differential survival of juvenile males and females was not modeled. This approach produced the observed sex ratio in recruits. 2.3.5 Calculation of Inbreeding Coefficient, f I used a recursive matrix method described by Henderson (1976) and summarized by Mrode (1996) to calculate the inbreeding coefficients, of each independent young (Appendix 3D). 2.3.6 Over- Winter Survival Winter survival probability is based on an individual's sex, age, inbreeding coefficient and the severity of the winter. All mortality is assumed to occur in winter. Age and sex-specific survival rates were taken from the Mandarte data (Figure 10, Appendix 3E). Females survive at slightly reduced rates compared to males. Demographic stochasticity is modeled by drawing a random number from a uniform distribution between 0 and 1. The individual survives if the number is less than the individual's specified probability of survival. Survival is determined independently for each individual. Juvenile survival probability is density dependent (Figure 11) and is calculated as follows: Sj = -0.0048 Np- + 0.6041 (8) where NF is the number of adult females alive in spring. Recall that independent young are produced in the sex ratio observed in recruits. Therefore, the juvenile survival rate does not incorporate the effects of sex. 18 Winter stress influenced the survival rate as follows: 50 = Sa(l + Sw.s) (9) where So is the survival rate of an outbred individual, Sa is the age and sex-specific survival rate (including the juvenile category, Sj), s is the degree of winter stress drawn annually from a uniform distribution between -0.5 and 0.5, and Sw scales the amount of environmental stochasticity. Sw was set low at 0.25 so that other effects could be detected above environmental stochasticity. In the mildest winter, when 5 = 0.5, survival rates are multiplied by 1.125, increasing them by 12.5%. In the harshest winter, when 5 = -0.5, survival rates are multiplied by 0.875, decreasing them by 12.5%. Survival rates of inbred birds are reduced according to survival Scenarios I to V (Figure 1). First the rate of constant background inbreeding depression, 5, for/=0.25 is transformed from a percentage (in the model interface) into inbreeding load: B = - ln(l - 5) f (10) where B is the inbreeding load in the annual survival rate. The rate of 'background' inbreeding depression is the inbreeding depression that occurs even at low stress. Following Morton et al. (1956), I assume that inbreeding reduces survival according to a negative exponential: Scenario I, no interaction (Figure la): 51 = So*exp(-Bf) (11) Scenario II, linear interaction between stress and inbreeding level, B = 0 (Figure lb): Si = So.exp(-(B+L s)J) (12) Scenario III to V, truncation interaction between stress and inbreeding level (Figure lc to d): Si = So.exp(-(B+ Ls)f)*LF (13) where Si is the survival rate of an inbred individual, So is the survival rate of an outbred bird, B is the inbreeding load, s is degree of winter stress, and L is the degree to which winter stress increases the inbreeding load linearly. (The behavior of these survival curves may be explored interactively by opening the file SurvivalCurves.xls, Logistic Threshold, in Appendix 4, the 19 compact disk.) LF is the logistic function, scaled so as to maintain a constant y-intercept ( LF = LFX/ LFO )'• LFX= 1- Exp(T, f(f/F,)F" (s/S,)s"-m 1 + ExpfTs ((f/Fs)F" (s/Ss) S" - Tj)) (14 ) where T$ is the slope of the threshold (the rate that survival declines beyond a certain level of stress), / is the inbreeding coefficient and s is the degree of winter stress. The threshold survival curves were designed to maintain the sudden drop in survival rates, and to keep the spread of the inflection points fory=0.03125 to 7=0.5 within about 0.35 on the stress axis, except for very high inbreeding depression when the inflection points become less spread out. To satisfy this objective, I used the following values for the scalars: Fs = 0.07, Ss =1.5 and 77 = 0.03. FP was a variable function of SP: FP = 0.7Sp0'15. By varying Sp, the threshold location moved up and down the stress axis. The logistic function is scaled by its own value at /= 0 and s = 0: L F 0 = 1 - ExpfTs (0-m 1 + Exp(Ts (0 - T,)) ( 1 5 ) An alternative, more compact formulation of the survival curves is given in Appendix 5. However, the logistic curves were used because they were easier to control. 2.3.7 Summarize the Year Detailed statistics are kept on the demographic outcomes of each year in two output files, the "survival summary" and the "year summary". The survival summary, contains a row for each bird alive in each model year. The survival summary lists the year, a unique identification number for each bird, sex, inbreeding coefficient, age, number of young produced for females, male category (floater, single, breeder), whether the individual survived, immigrant status, and the identification numbers of the sire and dam. The year summary contains a row for each model year with statistics describing the total number of adult males and females alive in spring, the number of territories, the number of males in each breeding category (floater, single or breeder), the number of immigrants that arrived in spring, the total independent young produced, the number of independent young that survived the winter, the degree of winter stress, the average level of inbreeding before and after winter and the average winter mortality rate. 20 2.3.8 Begin Yearly Cycle Again After survival is determined for each individual, the yearly cycle begins again with new immigrants entering the population. If the number of adult males or females, including the new immigrants which arrive in spring, is zero then the population is assumed to have gone extinct. 2.4 Collection of Model Statistics 2.4.1 Summarize the Model Run Summary statistics were calculated and recorded on the trajectory within the analysis window (year 30 to 229) for each run of the model. Information was recorded on time to extinction, mean, variance, maximum, minimum (excluding zero), 25th, 50th (median) and 75th percentiles of population size, the average period and severity of large scale fluctuations, the average inbreeding coefficient in the population before and after winter, average survival rates of age-one males with / = 0,f = 0.125,/= 0.25 and the survival rates of females of each age class (accounting for density dependence in juvenile survival). The run was recorded as having gone extinct if, by year 229, the number of males or females in spring after immigration reached zero. A correlogram was calculated for each run to determine whether the trajectory exhibited cyclicity with periods of up to 30 year (Begon et al. 1990, p. 530-533). Inbreeding depression experienced by age-one males across all levels of stress in the 200-year analysis window was calculated using the survival rates of/= 0,/= 0.125,/= 0.25 age-one males. This age class was used because age-one males are the most numerous adult age class, which reduced variability in the estimates. The average level of inbreeding depression of age-one males was used as a metric of comparison for the five survival scenarios. In order to measure the period and severity of large scale fluctuations, the population trajectory (number of adult females over time) was smoothed with a 3-year running average. This smoothed trajectory was compared with the population mean (calculated from the unsmoothed trajectory). Each time the smoothed trajectory crossed the population mean from below the mean to above the mean, the beginning of a 'cycle' was indicated and the corresponding model year was recorded. Then, using the unsmoothed trajectory, the population maximum, minimum and corresponding years were recorded for each 'cycle'. The severity of the fluctuation or 21 "crash severity" was calculated as (maximum - minimum)/maximum, and the period was taken as the number of years between consecutive minima. Only complete cycles were included in statistics. Several smoothing intervals were tried, but the 3-year running average appeared optimal for removing small scale variation, but not obliterating large scale fluctuations. 2.4.2 Summarize Multiple Runs 100 runs were performed on each variable setting to reduce variability in statistics. Mean, variance, maximum and minimum values were calculated over 100 runs for each statistic. Information was recorded on time to extinction, mean, variance and quartiles of population size, the average period and severity of large scale fluctuations, the average inbreeding coefficient in the population before and after winter, average inbreeding depression experienced by age-one males/ = 0.25, and the survival rates of females of each age class. 2.4.3 Step Through a Range of Parameter Values To explore the effect of varying parameter values, the desired parameter range was specified, as well as size of the step increment. The model did 100 runs with the lowest parameter value, calculated summary statistics, and added the step increment to the parameter value for the next set of runs. Step increments were added until the upper parameter value was reached. Mean values from each set of runs were summarized, including statistics on the population size (mean, maximum, minimum and quartiles), average variance, severity and period of major fluctuations, extinction rate, survival rates for each female, inbreeding depression for/= 0.25 (based on age-one males), average rate of reproduction per female, and average number young produced annually. The average coefficient of variation in population size over time, C V N was calculated by taking the square root of the average variance in population size and dividing by the average population mean size (see Appendix 6, Model Note 2 for comments regarding the calculation of C V N ) . 2.5 Model Options 2.5.1 Maintaining Age-Specific Survival Rates The Survival Intercept Adjustment option in the model allows average survival rates for each age sex class to be maintained at Mandarte rates, even though the strength of inbreeding 22 depression is being increased. When the strength of inbreeding depression was increased, survival rates of outbred birds, So for each sex and age class, were adjusted so as to maintain Mandarte survival rates for each age and sex class. This was accomplished by iteratively adjusting So for each female age class until rates for all ages classes were simultaneously within 0.01 of the desired rate. Rates were calculated as the average survival rate of the age class over the 200 year analysis window from year 30 to 229 inclusive. Model runs for the adjustment of So were not included in summary statistics. Usually less than 10 iterations were required to adjust So. The maximum number of iterations allowed was 200 at which point the correction acquired was accepted. 200 iterations were only required for very high inbreeding depression of 90 to 98%. Iterative adjustment of So was done for female birds only. To minimize calculations, the So for males was adjusted by the same increment. Because the juvenile recruitment rate is density dependent (Eqn. 8, p. 18), the intercept, or expected recruitment rate at zero density, was maintained at 0.6041, but the actual recruitment rate was allowed to vary with average population density. This was accomplished by assuming that the slope of the density dependent relationship was maintained: Sjio = Sjave + 0.0048 *NFave (16) where Sj,o is the observed juvenile survival intercept, Sjave is the observed average survival rate for the run, and Nfave is the average number of females for the run. The adjustment increment for the intercept is then: A=E-0 (17) where A is the adjustment, E is the expected value (0.6041) and O is the observed value, Sjio-Intercept adjustment for adults followed a similar procedure, without density dependence. In general, survival intercepts were adjusted when increasing inbreeding depression in the population, but not when varying the immigration rate. 2.5.2 Reproduction of Independent Young In the model, inbreeding depression in reproduction can be turned off by selecting the option of basing reproduction on the regression equation that excluded maternal inbreeding coefficient. Variability in production of independent young can be turned off by rounding the predicted number of independent young for each female to a whole number, rather than drawing a 23 number from the residual distribution. The default setting allowed inbreeding depression in reproduction, and drew actual numbers of young from the residual (Mandarte) distribution. 2.5.3 Immigration Immigration can be based on a Poisson distribution with a specified mean, set at a constant yearly integer value, drawn from a uniform distribution with a specified range, or based on the Mandarte distribution of male and females. The sex ratio of immigrants can also be specified. The default setting was an average of one female immigrant per year, based on a Poisson distribution. 2.6 Model Runs Model runs explored three main topics: the effect of the survival curves in Scenarios I to V, the effect of changing rates of immigration under several inbreeding levels in Scenarios I to V, and the baseline contribution of various sources of stochasticity in the model. Each combination of model settings was run 100 times for 260 model years to accommodate the 200 year analysis window. 2.6.1 Survival Scenarios To explore the effect of inbreeding depression under each scenario, the overall rate of inbreeding depression for an individual with/=0.25 was varied from 0% to 98% in increments about 2% (the actual step size varying slightly with each scenario). The model option settings for this investigation were as follows: Immigration was set to an average of one female per year, drawn from a Poisson distribution. Variability in reproduction was set to include both inbreeding depression and variability due to drawing the actual number of young produced from a residual distribution (Figure 9). Stress was set at ± 12.5%. And So, the survival rate of outbred individuals, was re-adjusted for each step using the Survival Intercept Adjustment option to keep age class survival rates constant. The survival parameter values were specified for each survival scenario (Table 1). (The parameters are defined in context in section 2.3.6, p. 18.) In Scenario I, No Interaction, 8, the level of constant inbreeding depression, was varied from 0 to 0.98 (Figure la). In Scenario II, Linear Interaction, the strength of the linear interaction between inbreeding depression and 24 stress, L , was varied so that inbreeding depression for /= 0.25 varied from 0% to 98% (Figure lb). In Scenario III, Threshold Interaction, both 8and L were set to zero, so that there was no background inbreeding depression at low stress (Figure lc). Inbreeding depression occurred entirely as a result of a steep threshold relationship produced by the logistic function. In Scenario IV, Threshold Interaction with Constant Background, L = 0, and background inbreeding depression was set to a constant rate of 8 = 0.25 (Figure Id). And in Scenario V , Threshold Interaction with Linear Interaction Background, 8 = 0, but L = 2.57 which provided background inbreeding depression related to stress, at a rate averaging 25% for /=0.25 (Figure le). Table 1. Model settings for over-winter survival parameters in Scenarios I to V . Scenario I II III IV V Parameter Constant Linear Threshold Threshold Threshold that was Inbreeding Interaction Inflection Inflection Inflection varied Depression Strength Point Point Point * (8 ) (L) (Sp) (SP) (SP) Other 1 = 0 8=0 8=0 8=025 8=0 Parameters 7/ = infinite Tj = infinite 1 = 0 7 = 0 7 = 2.57** Inbreeding depression Value of Parameter That Was Varied for/= 0.25 0% 0 0 14.1 - -25% 0.25 2.5 6.9 14.1 14.1 50% 0.50 6.5 3.9 5.4 4.8 75% 0.75 15 2.4 3.0 2.4 98% 0.98 120 0.9 0.9 0.9 •S=l-e-*. ** The setting L = 2.57 produces average inbreeding depression over all stress of 25% f o r / - 0.25 with no threshold. 2.6.2 Immigration For each of the five survival scenarios, immigration was varied from a mean of 0 to 3 immigrants per year under 0%, 25%, 50% 75% and 98% inbreeding depression. So, the survival rate of outbred individuals, was only adjusted once, at the level of inbreeding depression of interest before varying the immigration rates in steps of 0.1. The number of immigrants for each year were drawn from a Poisson distribution with the specified mean. 25 2.6.3 Baseline Stochasticity and Default Settings There are several sources of stochasticity in the model which can be turned on and off. In order to assess the degree of variability contributed by each factor, the coefficient of variation, C V N , was calculated based on the average of 100 runs with and without each factor, as well as with a selection of combinations of the factors. The sources of stochasticity were due to demographic stochasticity in survival, immigration, reproduction, stress and inbreeding depression of 25%. Some demographic stochasticity, due to the individual-based nature of survival, could not be turned off. This provided a baseline for comparing additional sources of stochasticity. The default setting for immigration is an average of one age-one female per year, with the actual number drawn from a Poisson distribution with mean = 1. To remove this sources of stochasticity, immigration was set to a constant rate of one age-one female per model year. Reproduction contains two stochastic elements. The first is variability introduced by inbreeding depression in reproductive success. This was removed by selecting the regression equation based only on maternal age and female density (not maternal inbreeding), to determine the number of independent young per female. The second source of variability is due to selecting the actual number of independent young produced by each female from the residual distribution of the Mandarte data (Figure 9). This feature was turned off by using the predicted number of young rounded to the nearest whole number, rather than drawing from the distribution. Stress, or winter severity, affects survival rates by ± 12.5% (see Eqn. 9, p. 19). This percentage was set to zero. Inbreeding depression in survival rate of 25% was introduced with and without a linear interaction between stress and inbreeding depression (see Eqn.s 10 and 11, p. 19). These runs were repeated as part of the analyses for Scenario I and II, but were included in the analysis of baseline stochasticity for comparison. 26 2.7 Model Validation Care was taken to ensure the internal consistency of the model. Data for a single model run were generated in a format similar to the format used for the Mandarte song sparrow data (see Section 2.3.7, p. 20). This facilitated confirmation that parameters were being interpreted correctly. The same statistical approach used to calculate the parameters from the Mandarte data was used on the model output, and the results compared. The error checking functions in Visual Basic 6.0 allowed changes in arrays and variables to be easily viewed as the model code is stepped through, line by line. Written explanation accompanied each section of code. 2.7.1 Inbreeding Coefficient Calculation Model pedigrees were imported into SAS (PROC INBREED) to test the model pedigree algorithm. SAS calculated the same inbreeding coefficients as the model to nine decimal places. 2. 7.2 Survival Parameters When inbreeding depression in survival was set to 0%, survival rates remained at input levels (Appendix 3E, Table 5, p. 83). The correct function of the density dependence of the recruitment rate was verified by plotting the recruitment rate against female density. Under inbreeding depression, if the Survival Intercept Adjustment function was on, the survival rates of outbred individuals were increased to maintain constant survival rates within each age class. Output data verified that this mechanism functioned as intended. 2.7.3 Reproduction Multiple linear regression analysis like that which was used to parameterize reproduction using Mandarte data (section 2.3.4, p. 17) was run on the model output for a single model run. When the predicted number of independent young per female was recorded without rounding, and no variance was added by drawing from the residual distribution, the regression coefficients obtained on the model output were the same as those used to parameterize the model to at least 5 decimal places of accuracy. Rounding of the predicted number of independent young per female to the nearest whole number reduced the intercept from 4.63 to 4.27. This intercept is the estimated number of young produced annually by an age one, outbred female at zero 27 density. Other coefficients were similarly reduced in magnitude. However, rounding and adding variance by drawing the number of independents from the residual distribution did not appear to make the coefficients any lower than rounding alone. 2.7.4 Initial Conditions For these analyses, the model was initiated with a population of 11 unrelated birds: 6 females, 5 males, age 1 to 6. With immigration rates of one per year, no difference was apparent between trajectories or output statistics after year 20 when the model was initiated with a population of 60 unrelated individuals. The composition of the initial population did not influence F S T (the average level of inbreeding in the population, relative to an outside, unrelated population) by model year 20.1 therefore assume that the effects of initial conditions are avoided by beginning the analysis window at year 30. 3 Results I explored three main topics: the effect of the survival curves in Scenarios I to V , the effect of changing rates of immigration under Scenarios I to V , and the baseline stochasticity in the model. The primary output statistics were the mean population size in terms of adult females, the coefficient of variation of population size over time, C V N , the percentage of runs that went extinct, the magnitude of large fluctuations in population size or "crash severity", and the period length of large fluctuations or "crash period". I use the word "crash" to indicate major (>70%) downward fluctuations in the trajectory (see section 2.4.1, p. 21 for an explanation of major fluctuations). I also use the word "spike" to indicate a major (>70%) upward fluctuation. Inbreeding depression experienced by age-one males, averaged across all levels of stress, was used as a metric of comparison for inbreeding depression occurring under the different scenarios, because age-one males were the most numerous adult age category, which minimized variability in the estimate. Each output statistic is the average of 100 model runs for a particular parameter combination. 28 3.1 Survival Scenarios I to V In the survival scenario analyses, the model stepped from 0% to 98% inbreeding depression for each of the five survival scenarios (Figure 1). In each step, inbreeding depression was increased, and the survival rates for outbred birds were increased so as to maintain constant survival rates for each age-sex class. The following discussion of the model output for Scenarios I to V refers to five sets of output: Sample trajectories and corresponding survival curves are given for Scenarios I to V at inbreeding depression levels of 0%, 25%, 50%, 75% and 98%o (Figure 12 to 16). Descriptive statistics (mean population size, coefficient of variation, percent of runs extinct, crash magnitude and crash period) for these trajectories are compared for the five survival scenarios (Figure 17). And population profiles (mean, minimum, maximum, median, and quartiles) for the five scenarios under inbreeding depression ranging from 0% to 98%o (Figure 18). Average inbreeding levels before and after winter are also presented (Figure 19). Lastly, correlogram results are presented. The model output for inbreeding depression above 80% is presented to assist the reader in understanding the model mechanisms. However, inbreeding depression in annual survival rate above 80% is not likely in the wild, so the corresponding output will not be included in the discussion section. In general, intercept adjustment causes the survival rates of outbred (f = 0) individuals to increase with the intensity of inbreeding depression (Figures 12 and 13, a to e, right). However, in the threshold interaction scenarios (III to V), the adjusted survival rate of outbred age-one individual was lower under 98% inbreeding depression, than under 75% inbreeding depression (Figures 14 to 16, d and e, right). This decline occurred because the adjustment depends on the portion of inbred individuals in the age class, as well as the severity of inbreeding depression. At very high inbreeding depression (>90%) in the threshold scenarios, most inbred juveniles do not survive to age one. Because the age-one age class contains few inbred individuals, less of an increase in the survival rates of outbred individuals is required to keep the age class survival rate at levels observed on Mandarte. To explore mechanisms that might determine the cause of some results, I ran Scenarios II and III with constant immigration, as well as the default Poisson immigration with mean = 1 female immigrant per year. Constant immigration did not change the output, except where indicated in the following discussion. I was also concerned that model runs which went extinct might 29 produce different statistics from runs that didn't go extinct. Therefore I excluded these runs from analyses and ran Scenarios II and III again. Excluding runs that went extinct did not change any results. Lastly, I ran Scenarios I, II and III with no inbreeding depression in reproduction. 3.1.1 Scenario I, No Interaction Between Stress and Inbreeding 3.1.1.1 Mean and Median In Scenario I, the population mean and median remained relatively constant as the strength of inbreeding depression increased from 0% to 98% (Figure 18a). In fact, the mean increased slightly from an average of 42.5 females at 0% inbreeding depression to an average of 46.9 females at 98% inbreeding depression (Figure 17a). This effect vanished if there was no inbreeding depression in reproduction. As the strength of inbreeding depression increased, inbred females survived at lower rates. The population was therefore composed of a higher proportion of outbred females, so reproductive success was higher on average at high levels of inbreeding depression. 3.1.1.2 Range, CVN, Extinction Rate and Crash Severity The range from the population minimum to the maximum increased slightly as inbreeding depression increased (Figure 18a). This increase can also be seen in the variability of the trajectory itself (Figure 12a versus e) and in the marginal increase in the coefficient of variation, C V N , at high inbreeding depression (Figure 17b). None of the 100 runs went extinct, even at 98% inbreeding depression (Figure 17c). The crash severity stayed low at 35% to 38% at all levels of inbreeding depression (Figure 17d). 3.1.1.3 Crash Period The crash period increased gradually from 10.0 to 12.8 years as inbreeding levels rose (Figure 17e). This increase apparently resulted from dips of the population trajectory above or below the mean line at high levels of inbreeding depression due to two bad winters occurring in a row (for example, Figure 12e, year 180). Overall, the population trajectory in Scenario I showed little change as inbreeding depression increases from 0% to 98%. 30 3.1.2 Scenario II, Linear Interaction Between Inbreeding and Stress 3.1.2.1 Mean In Scenario II, the population mean increased slightly from 42.6 females at 0% inbreeding depression to 45.4 females at 90% inbreeding depression (Figure 18b). As in Scenario I, this increase was due to increased reproductive success at higher levels of inbreeding depression resulting from the smaller proportion of inbred birds. However, as inbreeding depression increased from 90% to 98%, the population size decreased to 38.4 females. This decrease occurred because even though the intercept adjustment gave outbred birds a survival rate of 100%, inbreeding was so severe that birds with/> 0 died rapidly. The influx of immigrants and their offspring was simply not enough to maintain the population. 3.2.1.2 Median Concurrent with the drop in mean population size beyond 90% inbreeding depression, the median fell below the mean (Figure 18b). This drop occurred because the population spent more time at low levels, spiking to densities over 105 occasionally (Figure 13e). The spikes skewed the mean upwards, but affected the median less. 3.2.1.3 Range, CVM, Crash Severity and Extinction Rate The increased variability in the population trajectory as inbreeding depression increased is apparent in the increase in the range between maximum and minimum values (Figure 18b), the increased C V N , and the gradually increasing crash severity (Figure 17b and d). At very high inbreeding depression of 97% the extinction rate reached a maximum of 3% (Figure 17c). When Scenario II was re-run with constant immigration instead of variable immigration from a Poisson distribution, no extinction occurred. Therefore, stochasticity in immigration played a role in the extinction rate at very high inbreeding depression in Scenario II. 3.2.1.4 Upward Spikes in the Trajectory The upward spikes seen in the trajectory at 98% inbreeding depression (Figure 13e) were not due to stochasticity in immigration. They occurred when immigration was constant, as well as when immigrant number was drawn from a Poisson distribution. The spikes were due to the increasing sensitivity to variability in winter stress with increasing L (the strength of the linear interaction between stress and inbreeding). The upward spikes occurred following a good 31 winter in which most of the adults and juveniles survived. In the mildest quarter of winters, recruitment tended to be much higher. When the population was rebounding from low to moderate size, production of independent young and recruitment were not depressed by density causing the population occasionally to more than double, only to crash severely the following year if winter stress was high. Extremely high spikes often followed two mild winters. 3.2.1.5 Intercept Adjustment Causes Variability When inbreeding depression was high, survival intercept adjustments had difficulty in increasing the survival rate of outbred juveniles enough to maintain an average survival rate of 0.395. The variability in the estimates for Scenario II at high inbreeding depression (Figure 17, II) are likely due to variability in the inconsistency of the survival intercept adjustments. 3.1.3 Scenario III, Threshold Interaction, No Background Inbreeding Depression 3.1.3.1 Mean In Scenario III, the population mean fell slightly at inbreeding depression above 10% as large crashes began to impact the population (Figure 17a, III). Although survival rates were adjusted to remain the same for each age class over the 200 year analysis window, the average number of young produced each year declined. This decline is attributed to Jensen's inequality (see Glossary, Appendix 7, p. 87) with the observation that total independent young produced annually is a concave function of population size. The high variance in population size caused the population to be frequently too high or too low to produce optimal numbers of independent young. This drop occurred despite the compensatory nature of the density dependence. A further large decline in the mean occurred at very high levels of inbreeding depression above 85%. As in Scenario II, this drop occurred due to a drop in the average survival rates of juveniles for which intercept adjustment was unable to compensate. The population was maintained primarily by immigrants and their offspring with / = 0, but even though these individuals had very high survival rates, they could not maintain population size. 3.1.3.2 Median and Period Length The median was greater than the mean population size from 10% to 35% inbreeding depression (Figure 18c). The population size was most often at 40 to 45 females, but occasionally severe crashes occurred (Figure 14b). These crashes skewed the mean downward below the median. 32 They also caused the crash period to increase to 13.4 years (Figure 17e, III) because small fluctuations occurred above the mean, and were therefore not part of the analyses of large scale fluctuation period and severity. Above 40% inbreeding depression, the median dropped below the mean as the population began to spend more time at low density than at high density. At high inbreeding depression, above 80%, the period length increased again (Figure 17e) because the population was frequently at low density, but the mean was skewed upwards by spikes in population density so small scale variability was lost from period length calculations. 3.1.3.3 Range, Crash Severity, and CVN As severe crashes began to impact the population at around 10% inbreeding depression (Figure 17d, III), the minimum population size (not including extinctions) dropped to near zero (Figure 18c). The maximum population size roses only gradually from 10% to 40% inbreeding depression. Crash severity increased rapidly above 10% inbreeding depression and averaged 77% at 25% inbreeding depression. The coefficient of variation, C V N , also rose rapidly, though not as rapidly as the crash severity because C V N incorporated small scale variability (Figure 17b). Crash severity peaked at an average of 83.0% at 44% inbreeding depression. At 44% inbreeding depression, the average maximum crash for the 100 model runs that occurred in the 200-year analysis window was 95.7% and the average minimum crash was 60.0% 3.1.3.4 Maximum Above 40% inbreeding depression, the maximum population size began to rise more steeply, reaching an average high of 110 females at 87% inbreeding depression. These upward spikes occurred even with constant immigration rates. They appear to be the result of very high survival rates in low stress winters which allowed almost all juveniles to survive. The maximum magnitude of the spikes increased with increased inbreeding depression because survival rates of outbred individuals were higher due to survival intercept adjustment. Maximum population size dropped above 90% inbreeding depression (Figure 18c) for the following reason: At inbreeding depression above 90%, stress thresholds occur at low stress (Figure 14e, right). (The stress threshold is the stress level above which survival rates drop for an individual with a given inbreeding coefficient.) For example, a stress threshold of 0.05 for/ = 0.03125 causes survival rates to drop for all individuals with inbreeding greater than or equal to/ = 0.03125 whenever stress exceeds 0.05 (95% of the time because stress is drawn from a 33 uniform distribution). Low stress thresholds greatly reduce the frequency of years when survival rates are high for individuals with/> 0. At high inbreeding depression, fewer years with high survival rates caused spikes to become less frequent (Figure 14e). The low population mean above 90% inbreeding depression reduced the number of breeders and the number of juveniles produced. Even when survival rates were high, and most juveniles survived, the magnitude of the spikes was reduced. 3.1.3.5 Extinction Rate Extinction rates rose rapidly to a maximum of 27% at 39% inbreeding depression (Figure 17c, III). Beyond 39% inbreeding depression, extinction rates began to fall even though coefficient of variation, C V N , and crash severity remained high (Figure 17b and d, III). This decline in extinction rates was due to increased survival rates of outbred individuals. The outbred individuals were able to carry the population through severe winters, preventing extinctions, even in the face of severe crashes. However, at very severe inbreeding depression, above 90%, extinction rates increased to over 50%. With constant immigration, extinction rates were less than 7% at any level of inbreeding depression, therefore fluctuation in immigration, as well as environmental stochasticity, rate played a large role in extinctions in Scenario III. 3.1.4 Scenario IV and V, Threshold Interaction With Background Inbreeding Depression The output for Scenarios IV and V were quite similar to Scenario III, but they were compressed between 25% and 98%, rather than 0% and 98% (compare Figure 18 d and e with c). This compression occurred due to the background inbreeding depression of 25% in Scenarios IV and V. Even when stress was low, the population experienced 25% inbreeding depression. In Scenario IV, the background inbreeding depression was constant, and unrelated to the level of stress (Figure 15b, left). However, in Scenario V, inbreeding depression occurred as a linear interaction with stress (L - 2.57), so that inbreeding depression increased with stress (Figure 16b, left). Due to the presence of background inbreeding depression, for a given overall level of inbreeding depression (e.g. 50% for an age-1 male), the stress thresholds are shifted to higher values on the stress axis in Scenarios IV and V, as compared to Scenario III (e.g. Figure 15c versus Figure 14c). This shift in stress thresholds caused the output for Scenarios IV and V to be compressed between 25% and 98% inbreeding depression, as compared to Scenario III. 34 3.1.4.1 Mean The mean, median and range for Scenarios I V and V followed the same pattern with increasing inbreeding depression as discussed in Scenario I I I (Figure 18d and e, Figure 17a). The coefficient of variation, C V N , crash severity and period length also followed the same pattern as for Scenario I I I . However, the maximum crash severity and coefficient of variation were slightly less for Scenarios I V and V than for Scenario I I I . 3.1.4.2 Extinction Rates The biggest difference between Scenarios I V and V and Scenario I I I is that extinction rates only reached a maximum of 17% in Scenarios I V and V , rather than 27% as in Scenario I I I . Apparently the background inbreeding depression removed inbred individuals so that the population was more outbred and less prone to extinction via stress thresholds that affect inbred individuals. 3.1.4.3 Comparison of Scenario IV and V Differences between the output for Scenario I V and V were relatively minor. The maximum coefficient of variation and crash severity was slightly higher for Scenario V than for Scenario I V . The extinction rate was also somewhat higher for Scenario V over most levels of inbreeding depression. These differences were likely due to greater stochasticity in survival rates resulting from the linear interaction between stress and inbreeding. 3.1.5 Average Inbreeding Levels Before and After Winter for Scenarios ItoV As the strength of inbreeding depression increased, the magnitude of the decline in the average level of inbreeding in the population increased (Figure 19a). Overall, the average of the inbreeding coefficients for the population declined as inbred individuals were more efficiently removed with increasing inbreeding depression (Figure 19b and c). The average level of inbreeding in fall declined at about the same rate for the five survival scenarios as inbreeding depression increased (Figure 19b). However, the average level of inbreeding depression in spring, after over-winter selection against inbred individuals, was lower under the threshold interaction (Scenarios I I I to V ) , than under no interaction (Scenario I ) or the linear interaction (Scenario I I ; Figure 19c). As inbreeding depression increased to 98%, virtually all individuals 35 with/> 0 were eliminated from the population over winter leading to a population average of zero inbreeding (Figure 19c). Small population size above 90% inbreeding depression under the threshold interaction (Scenarios III to V; Figure 17a), caused increased mating among relatives. The increase in inbred matings produced juveniles that were more inbred, leading to an increase in average population inbreeding coefficient in fall under the threshold interaction at inbreeding depression above 90% (Figure 19b). The threshold interactions caused large crashes in the population trajectory which lowered the population size briefly (e.g. Figure 14b, left). If these crashes were followed by a low-stress winter, inbred juveniles produced by mating among relatives in the small population were able to survive the winter leading occasionally to high average inbreeding levels of up to 0.15 in spring (Figure 19d). The threshold interactions greatly increased the variance in the average inbreeding level in spring because low stress winters allowed survival of most inbred individuals, but high stress winters eliminated all inbred individuals (Figure 19e). In contrast, the maximum levels of inbreeding that occurred under no interaction (Scenario I) or the linear interaction (Scenario II), were about 0.05 lower than the threshold interactions (Figure 19d). The variance in inbreeding levels in spring under no interaction or linear interaction was also much lower than under the threshold interactions (Figure 19e). Spring inbreeding levels appeared to be marginally more variable under the linear interaction than under no interaction. 3.1.6 Testing for Cyclicity in the Trajectory The correlograms did not indicate significant cyclicity. The model produced a correlogram for each run with lag from 0 to 30 years. A correlogram is a graph with "lag" on the x-axis and "correlation coefficient" on the y-axis. For example, to determine the correlation coefficient for an offset of 5 year, the 200 observations of population size from year 30 to 229 (the analysis window) were correlated with the 200 observations of population size from year 35 to 234 (Begon et al. 1990, p. 530-533). The 100 correlograms produced for each parameter combination were averaged. The largest correlations for the averaged correlograms were r = 0.05 for Scenarios III, IV and V, which did not reach the significance level of r = 0.138 (see Appendix 8). Scenarios I and II showed no tendency toward cyclicity. Table 2 summarizes the 36 period and level of inbreeding depression corresponding to these correlation coefficients. These results indicate a tendency toward regular periods in Scenarios III-V. Despite the fact that crashes of large magnitude occurred frequently, they lacked true periodicity. Table 2. Summary of correlogram output corresponding to the maximum correlation coefficients for Scenarios III to V. Scenario r-value Lag (years) Inbreeding Depression III 0.05 10 42.5% I V 0.05 9 69.5% V 0.05 8 64.1% 3.1.7 Detail of Scenario III at 50% Inbreeding Depression The inbreeding structure of adult females in the model population can be seen to change as the population size increased and then crashed for Scenario III, at 50% inbreeding depression (Figure 20). Similarities can been seen when comparing the inbreeding structure, population trajectory and survival rates in the model population and the Mandarte population (Figure 3). For example, mean inbreeding level increased during years when survival rates were high in both the Mandarte population and the model population (Figure 3, Figure 20). Furthermore, when survival rates dropped, inbred individuals were removed and the mean inbreeding level in the population declined in both populations (Figure 3, Figure 20). Following a crash, inbreeding levels in the model population rose rapidly to an about / = 0.04 to 0.05, close to Mandarte levels (Figure 20). The probability of a severe population decline at high stress due to a stress threshold depends on the inbreeding level in the population, as well as the population density (Figure 21). At low levels of over-winter stress (< 0.54) population growth rates ( X ) tend to be greater than one, indicating population growth (Figure 21a), whereas when stress is high (> 0.54) population growth rates are variable because the threshold interaction sometimes causes population declines, depending on the inbreeding structure and density of the population (Figure 21a). When stress is low, correlation is low between X and the average level of inbreeding depression in the population, but high between X and population density (Figure 21b and c). When stress is 37 high, X depends more on the inbreeding level in the population and less on the population density (Figure 21 d and e). 3.2 Immigration Extinction frequency depended on both the severity of inbreeding depression that a population experienced and the frequency of immigration (Figure 22). For these analyses, the desired inbreeding level and survival scenario was specified, immigration was set to average one per year, drawn from a Poisson distribution, and then the survival rates of outbred individuals were adjusted to maintain age class survival rates at Mandarte levels. After survival rates had been adjusted, the survival intercept adjustment function was turned off, and mean immigration rate was varied from 0 to 3 in steps of 0.1. These analyses were meant to explore the effect of changing immigration rate on extinction probability, given that the population was not declining deterministically, under an immigration rate of one per year. Extinction rates were the chance of extinction within 230 model years. In general, Scenarios III to V required the highest immigration levels to prevent extinctions, followed by Scenario II, and then Scenario I (e.g. Figure 22c). Because the results are parallel for 5%, 50% and 95% extinction rates, I will focus on patterns at 5% extinction. For Scenarios I and II, the level of immigration required to prevent extinctions increased linearly as inbreeding depression increased. Slightly higher levels of immigration were required to prevent extinctions in Scenario II than in Scenario I. For example, at 50% inbreeding depression, an immigration rate of 0.38 was needed under Scenario I before only 5% of runs went extinct, but a rate of 0.50 was required under Scenario II. With no inbreeding depression in survival and no immigration, 40% to 60% of runs crashed due to inbreeding depression in reproduction (Figure 22a). The lack of immigrants caused average inbreeding levels in the population to rise which lead to a gradual decline in reproduction until the population went extinct. At most levels of inbreeding depression, Scenarios III to V required about the same rate of immigration to keep the extinction rate below 5%. However, threshold declines do not impact 38 survival under Scenarios IV and V at 25% inbreeding depression, so they only require an immigration rate of about 0.26 and 0.29, similar to Scenario I and II respectively (Figure 22b). However, threshold declines do impact Scenario III at 25%> inbreeding depression, so it requires an immigration rate of 1.27 to reduce extinction rate to 5%. Immigration requirements remain the about same, at 1.25, for Scenario III when inbreeding depression is increased to 50% (Figure 22c). Similar immigration rates are required under Scenarios IV and V. At 75% inbreeding depression, the rate of immigration needed to reduce extinction rates to 5% falls below the levels required at 50% to about 1.08 (Figure 22d). At very high inbreeding depression of 98%>, immigration rates of about 1.38 are required to keep extinction rates at 5% (Figure 22e). This drop followed by an increase in immigration requirements for Scenarios III to V apparently reflects the same process that caused extinction rates to reach a maximum at about 40% inbreeding depression, and then decline with higher inbreeding depression (Figure 17c, III). At inbreeding depression of around 75%, survival rates of outbred individuals are high, and these individuals are able to carry the population through crashes, reducing extinction rates, even with somewhat fewer immigrants. Crash severity declines with increasing immigration rate (Figure 23). With an average of one immigrant per year, as occurs on Mandarte Island, under no interaction (Scenario I), crash severity averaged only 35.5%. Under the linear interaction (Scenario II), crash severity average 45.8%. But under the threshold interaction (Scenarios III to V), crash severity averaged 81.4%. With an average of three immigrants per year, as might occur on some of the smaller islands near Mandarte, crash severity under the threshold interaction was greatly reduced, especially with background inbreeding depression. Crash severity averaged 56.2%, 44.0% and 49.5% for Scenarios III to V respectively. Crash severity was lower at high immigration under constant background inbreeding depression (Scenario IV) than under the linear interaction background inbreeding depression (Scenario V), perhaps because the constant background inbreeding depression was more efficient in removing inbred birds producing a consistently healthier population 39 3.3 Baseline Stochasticity The evaluation of baseline sources of stochasticity is summarized in Table 3. The coefficient of variation, C V N , with only demographic stochasticity in survival rates was 12.70 (Table 3a). Inbreeding depression of 25% for / = 0.25 with no inbreeding depression-stress interaction produced little increase in stochasticity over the baseline demographic stochasticity (Table 3b). Immigration based on a Poisson distribution with mean = 1 produced a marginal increase in C V N of 0.24%, over constant immigration (Table 3c). Inbreeding depression in reproduction increased C V N by 0.42% over the demographic baseline (Table 3d), but drawing the number of independents from the residual distribution of observed Mandarte reproduction increased C V N by 0.99% (Table 3e). Stress was the largest single contributor to stochasticity, increasing C V N by 3.39% (Table 3f). When several factors acted together, they generally increased the C V N above the baseline variability such that the more factors acting, the greater the C V N - The exception to this rule appears to be when inbreeding depression was in both survival (delta = 0.25) and in reproduction. For example, in comparing Table 3m with Table 3n, even though four factors are involved in 'n', less stochasticity is evident than in'm'. The same can be seen in comparing'd' with 'g'. Possibly the inbreeding depression in survival rates acted to remove inbred females, thus reducing variability in the number of young produced each year. 40 -pn 3 X> Pi ~ ... i> >> CS > cd C o 73« 3 , M a. cd-P^.Q led-„ 3 CO' O O " t f t-- o vO o O O " t f ON CN c N C N m c o s o c N c o r o -tf OS O (N..- —' so ,ci "tf o oo "tf CN -tf r> . . . G -• 3 - 2 . • = / 3 c/i >'C *:5 6J) c '-5 - J "J .2 B .2,, in CN) o 5/ 3x , s - J 'cs o o -*-» CO j j j j j j ! V ! ~ ? v - " v . - s s ~s 7 N N N N f f . " T 'r< 'r> JS D. cd l-00 o 60 •3 cd o c o II o Q ^ £1 41 4 D i s c u s s i o n In this thesis, I explored the potential for inbreeding depression in annual survival rates to cause fluctuations in population size in a population which is not experiencing deterministic decline. I also investigated the destabilizing effect of inbreeding-stress interactions that increase the susceptibility of inbred individuals to stress. I used the inbreeding depression experienced by an age-one male, averaged across all levels of stress, as a metric of comparison for the different survival scenarios. I hypothesized that stress may increase inbreeding depression linearly, or, as a worst case scenario, a threshold interaction may occur in which inbreeding depression drastically reduces survival rates of inbred individuals beyond a stress threshold. Therefore, there were three main inbreeding-stress interaction scenarios: 'no interaction', 'linear interaction' and 'threshold interaction'. I also investigated the effect of adding 25% 'background' inbreeding depression under low stress to the threshold interaction. I included background inbreeding depression that was constant and that occurred via a linear interaction with stress, for a total of five inbreeding-stress interaction scenarios (Figure 1). Finally, I explored the effect of changing immigration rates on the probability of extinction under the five inbreeding-stress interaction scenarios. Because inbreeding depression in annual survival rate above 80% has not been demonstrated in the wild, I focus my discussion on inbreeding depression rates of 0% to 80%. I discuss the main conclusions, the likelihood of these survival scenarios, implications for the management of small populations, and possibilities for future investigation. 4.1 Main Conclusions 4.1.1 Inbreeding Depression with No Stress Interaction Did Not Increase Variability Under constant immigration averaging one female per year, if inbreeding depression was independent of stress, even severe inbreeding depression did not increase variation in numbers or extinction risk (Figure 17 b and c, I). Provided that low survival rates of inbred individuals were balanced by high survival rate of outbred individuals so that the survival rates within each age class remain the same, the behavior of the model populations changed little with inbreeding depression ranging from 0% to 80%. 42 4.1.2 Inbreeding Depression with a Linear Stress Interaction Increased Variability Moderately Under a linear interaction between inbreeding and stress, the coefficient of variation of population size over time increased by 42%, from 16.7% at 0% inbreeding depression to 23.7% at 50% inbreeding depression (Figure 17b, II). This increase occurred because the inbred portion of the population, experienced increased susceptibility to stress or environmental stochasticity. However, the linear interaction between inbreeding and stress only increased the probability of extinction by 1% when inbreeding depression reached 80% (Figure 17c, II). 4.1.3 Inbreeding Depression with a Stress Threshold Greatly Increased Variability Under a threshold interaction between inbreeding and stress, extinction frequency and variability in the population trajectory increased greatly compared to the linear interaction or no interaction. Under a threshold interaction with no background inbreeding depression at low stress, the average crash severity was maximized at 83% and the coefficient of variation of population size more than tripled to 60.5% at 44% inbreeding depression (Figure 17 b and c, III). Even with background inbreeding depression of 25% at low stress, the crash severity and the coefficient of variation were only slightly lower than with no background inbreeding depression (Figure 17b and d, IV and V versus III). Although the likelihood of the steepness of the threshold used may be debated, these results demonstrate the destabilizing potential of an inbreeding-stress threshold interaction and validate the possibility of further investigation in this area. 4.1.4 Low Inbreeding Depression at Low Stress Reduced Extinction Rates When background inbreeding depression of 25% at low stress was added to an inbreeding-stress threshold interaction, extinction rates were reduced from a maximum of 27% with no background inbreeding depression to a maximum of 17% with background inbreeding depression (Figure 17c, IV and V versus III). This drop in extinction rate occurred despite the fact that the background inbreeding depression reduced the coefficient of variation of the population size and the crash severity only slightly (Figure 17b and d, IV and V versus III). Although background inbreeding depression reduced the proportion of inbred individuals enough to lower the extinction rate, it did not prevent large crashes from occurring. Little 43 difference was apparent if the background inbreeding depression at low stress resulted from an inbreeding-stress interaction or from inbreeding independent of stress. 4.1.5 The Mandarte Trajectory Could Be Explained by an Inbreeding-Stress Threshold Interaction Crashes of the magnitude observed in the Mandarte Island song sparrow population (86% to 92%) do occur under the threshold interaction between inbreeding and stress with an immigration rate of one per year. The maximum average crash severity under the threshold interaction was 83%, but crashes ranged from 60% to 96% with an average interval of 12 years. Furthermore, the changes in inbreeding structure of the model population after a crash was reminiscent of the 1988/89 crash in the Mandarte population (Figure 3 and Figure 20). The average inbreeding level of the population dropped and highly inbred individuals were removed following a crash. The presence of background inbreeding depression which removed inbred individuals even at low stress reduced crash severity only slightly. Therefore, the operation of moderate inbreeding depression during non-crash years on Mandarte does not preclude large crashes due to an inbreeding-stress threshold interaction. When immigration was increased to three per year, as might occur on some of the smaller islands near Mandarte, the average crash severity was only 44%> with constant background inbreeding depression, and 56% without background inbreeding depression. The reduction of crash magnitude under higher immigration is consistent with the observation that song sparrow populations on the smaller islands, which experience higher immigration rates, did not decline as severely as the Mandarte population over the winter of 1988/89. Although the magnitude of the crashes observed on Mandarte may be explained by an inbreeding-stress threshold interaction, the apparent regularity of the Mandarte crashes, which occur at intervals of 9 and 10 years, cannot. With immigration rates of one per year, inbreeding levels increase rapidly after a population crash, so that within only one to four years, the population is again susceptible to a crash if a high stress winter occurs (e.g. Figure 20). As confirmed by the fact that none of the correlograms showed significant autocorrelation, true cyclicity was not explained by an inbreeding-stress interaction. 44 4.1.6 Heterogeneity in Survival Rates May Reduce Extinction When inbreeding depression occurred via a stress threshold with no background inbreeding depression, extinction rates declined from a maximum of 27% at 39%> inbreeding depression, to a low of around 8% at 80%> inbreeding depression (Figure 17c, III). A similar pattern occurred even with background inbreeding depression (Figure 17c, IV and V). The decline in extinction rate at high levels of inbreeding depression appears to be due to increased heterogeneity in survival rates at high inbreeding depression. The survival rates of the outbred individuals were high enough that they carried the population through high stress winters, preventing extinctions. Several authors have noted the stabilizing influence that individual variation can have in models. Kendall and Fox (2002) noted the stabilizing effect of variation among individuals in population viability models. Lomnicki's (1988) theory of unequal resource distribution was used to demonstrate increased stability in model populations with individual variation over those without (van Noordwijk 1994). Connor and White (1999) have also investigated the effects of individual heterogeneity in population viability. High survival rates in some individuals may help to prevent population extinction in models, but the application of this principle to inbreeding depression is uncertain. If levels of inbreeding in survival rates as high as those modeled here occur in the wild, it may be because outbred individuals do not have exceptionally high rates of survival, but rather that the survival rates of inbred individuals are exceptionally low. 4.1.7 Required Immigration Rate Increased With Inbreeding Depression With no inbreeding-stress interaction, and with a linear inbreeding-stress interaction, the number of immigrants required to reduce the extinction rate to 5% increased linearly with increased inbreeding depression (Figure 22). Under a linear interaction, immigration rates needed to be about 25% higher than with no inbreeding-stress interaction. When the inbreeding depression occurred as the result of an inbreeding-stress threshold, much higher rates of immigration were required to prevent extinction. At 50% inbreeding depression, the level of immigration required to reduce the extinction rate to 5% was 0.38, 0.50 and 1.25 under 'no interaction', 'linear interaction' and 'threshold interaction' respectively. This is equivalent to about two immigrants in five years if there is no inbreeding-stress interaction, one immigrant in two years if there is a linear interaction, and five immigrants in four years if there is a threshold interaction. 45 4.2 Conditionally Deleterious Alleles and the Potential for Purging Highly deleterious alleles are much more likely to be purged from a population's gene pool than mildly deleterious alleles. It seems unlikely that any allele deleterious enough to cause severe crashes in a population trajectory could persist for long. If highly deleterious alleles are purged rapidly from the population gene pool by natural selection, then frequent population crashes due to inbreeding-stress interactions may appear unlikely. However, environmental stochasticity may be regarded as the sum of many separate stresses, each of which affects different loci. If alleles are conditionally deleterious, affecting survival rates only rarely, they may not be quickly purged from the gene pool. Bijlsma et al. (1999) suggested that purging of deleterious recessive alleles may be specific to the environment in which the purging occurred, because additional load will become expressed under changing environmental conditions. Kristensen et al. (2003) found no inbreeding-stress interaction when Drosophila were tested in the environments in which they were reared, whereas there was a tendency toward an interaction when flies were exposed to a novel environment. Effects of inbreeding appear highly variable depending on the stress, the history of exposure of the population to that stress (i.e. the opportunity for purging of deleterious alleles) and the fluctuating genetic structure of the population due to drift, immigration, mutation and selection (Biljsma et al. 1999). Flux in the genetic constitution of a population may even mean that the same population will react differently to the same stress at a different time. Given that immigrants routinely reintroduce alleles to subpopulations (Keller et al. 2001), and that different stresses may act upon different loci, reducing purging of deleterious alleles, it seems plausible that the potential for inbreeding-stress thresholds could remain in a population over time, contributing to large declines in population size only when the portion of inbred individuals in the population is high. 4.3 Implications of Model Structure to the Generality of the Results Caution should be exercised in drawing generalizations from modeling explorations. In order to ensure that the effects of genetic stochasticity were not obscured, this model included low environmental stochasticity (±12.5%) on survival rates, 0% on reproduction rates and no environmental catastrophes. Using low environmental stochasticity almost certainly leads to underestimation of extinction rates. Higher environmental stochasticity would likely act additively with the moderate increase in C V N under a linear inbreeding-stress interaction to 46 increase extinction rate under a strong linear inbreeding-stress interaction. Higher C V N , or variability in population size, in itself can indicate increased vulnerability to extinction (Meffe et al. 1994, p. 223). Higher environmental stochasticity could also further increase extinction rates under a threshold interaction. Model results demonstrated that constant immigration reduced extinction rates below those observed under a Poisson distribution, suggesting that further increased variability in the immigration rate could also increase extinction rate. Other factors that would likely have changed model outcomes are population size and reproduction rates. Small population size is related to higher extinction rates in stochastic population models (Pimm et al. 1988). A lower carrying capacity would therefore lead to higher extinction rate. Furthermore, fluctuations in population size caused by catastrophes may reduce population size to the point where an "extinction vortex" drives the population extinct through the interaction of environmental stochasticity, demographic stochasticity, inbreeding depression, reduced effectiveness of natural selection and mutation (Gilpin and Soule 1986; Tanaka 2000). It is also important to emphasize that inbreeding depression will contribute to deterministic extinction if it reduces survival and reproduction rates to the level that the population cannot replace itself (Mills and Smouse 1994; Westemeier et al. 1998). Higher rates of reproduction would probably have made the model less susceptible to the deleterious effects of inbreeding depression (Mills and Smouse, 1994). Due to density dependent survival rates, only a portion of the juveniles survive regardless of their average inbreeding level. An increased rate of reproduction at low density, combined with an unchanged carrying capacity could lead to a higher juvenile mortality rate without impacting the population trajectory. If inbred individuals were included in the portion of the juvenile cohort dying due to density dependent factors, then a high rate of inbreeding depression in juvenile survival'rate could be accommodated without impacting the population trajectory or leading to an increase in the average inbreeding coefficients of adults. 4.4 Implications for the Management of Small Populations According to Hedrick et al. (1996), it's important to determine under what conditions genetic concerns are likely to influence population persistence. The conclusions of my model suggest that threshold interactions between inbreeding and environmental stress could contribute to 47 unpredictable catastrophic declines and extinction in small populations. Inbreeding therefore may be increase vulnerability to extinction even if inbreeding depression is not evident in the population over the short term. When a high stress event occurs, such as a very cold winter or a disease, most of the inbred individuals in the population may die due to compromised immune response or metabolic function, causing a sudden crash in numbers. If immigration is sufficient and a portion of the population is outbred, such an event may only contribute to a moderate chance of extinction. However, if immigration rate has dropped, a higher portion of the population may be inbred leading to a more severe crash and a higher extinction likelihood. The results of this model should not be interpreted as suggesting that high immigration rates will solve the genetic problems of small populations. This model does not account for the potential negative effects of immigrants, such as introducing disease and loss of local adaptation or the outbreeding depression found by Marr et al. (2002). Mills and Allendorf (1996) suggested that one immigrant per generation may be regarded as a minimum, but immigrants should not exceed 10 per generation. Keller et al. (2001), estimated that the generation time in the Mandarte song sparrow population is about 2 years. Therefore, according to the one-migrant-per-generation rule, the minimum annual migration rate to the Mandarte population should average 0.5 immigrants per year. While this rate would allow only 5% of runs to go extinct for the linear inbreeding-stress interaction at 50% inbreeding depression, and 5% extinction for inbreeding depression up to 75% under no inbreeding-stress interaction, it would not be sufficient to prevent extinctions in the case of an inbreeding-stress threshold. For inbreeding-stress threshold interactions, immigration rates of around 1.25 (five immigrants in four years) are required to lower the extinction probability to 5% in 230 year. Therefore these model results suggests a rate of about 2 to 3 per immigrants generation to avoid the potential for crashes related to an inbreeding-stress threshold-producing synergism. 4.5 Future Work There are many ways to explore the robustness of the results from a model, like this one, with many variables. However, the steepness of the inbreeding-stress threshold necessary to destabilize the population is an obvious initial area of investigation. Preliminary analyses show that a threshold declines much less steep than those presented here (over 30%> of the stress axis 4 8 instead of over 4% of the stress axis, T$ = 50) have the potential to cause crashes averaging 75% with a mean interval of 11.5 years. To explore the effect of purging of deleterious alleles, mutation rate and strength of selection, it would be useful to model inbreeding depression and genetic load with a simulated genome, rather than the inbreeding coefficient. Each genome would have, perhaps, 100 loci, and each locus would have neutral and deleterious alleles with associated selection coefficients and rates of mutation. It would be interesting to compare the predictions of such a model with analytical predictions based on estimated effective population size to determine the shortfalls of each approach. One empirical investigation that might disentangle the effects of stress and inbreeding is to monitor the fat load of inbred and outbred birds several times over winter. Also, perhaps in late summer after breeding, one could hold birds for about 12 hours in a cold box to monitor rate of weight loss across inbreeding levels. These investigations might help to link metabolic efficiency with inbreeding level. 5 Conclus ion In this study I investigated the potential for inbreeding to increase variability in the population trajectory of a small insular population. Increased stochastic variability in a population trajectory increases extinction rate (Meffe et al. 1994, p. 223), especially in small populations, and is therefore of interest in conservation biology. Evidence for inbreeding-stress interactions is accumulating (Bijlsma et al. 1999; Dahlgaard and Hoffmann 2000; Keller et al. 2002). Several published models have explored the effect of inbreeding depression on population viability (Haig et al. 1993; Mills and Smouse 1994; Jager et al. 2001) and population dynamics (Tanaka 1997). Shaffer (1987) suggested that synergism between genetic, environmental and demographic stochasticity could magnify their effects on a population. However, to my knowledge, the effect of an inbreeding-stress interaction on population dynamics has not been explored through population models. I explored three main inbreeding-stress interaction scenarios: 'no interaction', 'linear interaction', and 'threshold interaction' in which survival rates of inbred individuals were 49 greatly reduced beyond a stress threshold. I found that under constant immigration, even strong inbreeding depression, up to 80%, did not increase population fluctuations if there was no inbreeding-stress interaction. A linear inbreeding-stress interaction only increased extinction risk marginally. However, even under moderate inbreeding depression of 25%, the threshold interaction produced large fluctuations in population size over time and greatly increase the extinction rate. When background inbreeding depression of 25% at low stress was added to the threshold interaction, the extinction rate was reduced because the inbreeding depression at low stress removed inbred individuals, making the population less susceptible to high stress events. However, background inbreeding depression did not prevent large crashes in the population trajectory. I also varied the immigration rate under the three inbreeding-stress interaction scenarios. I found that the level of immigration required to prevent extinctions increased with the strength of inbreeding depression. The inbreeding-stress interactions also increased the rate of immigration needed to prevent extinctions. At 50% inbreeding depression, under a threshold interaction between inbreeding and stress, an immigration rate of 1.25 (5 immigrants in 4 years) was required to reduce the extinction rate to 5% of model runs in 230 model years. Under a linear interaction, an immigration rate of 0.50 (1 immigrant in 2 years) was required, and under no interaction, an immigration rate of 0.38 (about 2 immigrants in 5 years) was required to reduce the extinction rate to 5%. These results suggest that inbreeding-stress interactions have the potential to increase stochastic variability in a population trajectory and that higher levels of immigration may be required to reduce the risk of extinction if inbreeding depression is high or inbreeding-stress interactions occur. 50 Summary I investigated the potential for inbreeding to increase variability in the population trajectory of a small insular population. Increased stochastic variability in population size increases extinction risk, especially in small populations, and is therefore of interest in conservation biology. Inbreeding depression may vary with severity of environmental stress. I explored three main inbreeding-stress interaction scenarios: 'no interaction', a 'linear interaction', and a 'threshold interaction' in which the survival rates of inbred individuals were greatly reduced above a stress threshold. Due to uncertainty in parentage, the severity of inbreeding depression in wild populations is rarely known. I therefore explored a wide range of inbreeding depression in annual survival rate from 0% to 80%. I also explored the influence of immigration rate on population variability and extinction rate. I investigated these scenarios using an individual-based population model that traced a pedigree and determined an inbreeding coefficient for each individual. The model was parameterized using data from an island population of Song Sparrows (Melospiza melodia), monitored closely since 1975, with a continuous social pedigree from 1981. The population has varied from 4 to 72 adult females (mean: 42) and immigration has averaged about one bird per year. Most mortality in this population (and all mortality in my model) occurred in winter and appeared to be influenced by external stresses such as periods of extreme cold. Given constant immigration, little increase in variability in the population trajectory occurred as inbreeding depression increased from 0% to 80% with 'no interaction' between stress and inbreeding depression, and only slightly more with a 'linear interaction'. However, imposing a 'threshold interaction' between stress and inbreeding increased extinction rate markedly and led to regular crashes averaging 83% ± 1 1 % (mean ± SD) of population size at a mean interval of 11 ± 5 years. The immigration rate required to reduce the extinction rate to less than 5% increased with inbreeding depression, and with the level of the inbreeding-stress interaction. With inbreeding depression of 50%, an immigration rate of 1.25, or 5 immigrants in 4 years, was required given the 'threshold interaction'. By comparison, immigration rates of only 0.50 given a 'linear interaction', and 0.38 given 'no interaction', were required to reduce extinction rates to 5%. Under the 'threshold interaction', the highest extinction rates occurred when there was no inbreeding depression at low stress and high inbreeding depression at high stress. Moderate inbreeding depression with low external stress resulted in the gradual removal of inbred individuals from the population, thus reducing extinction. These results suggest that inbreeding-stress interactions have the potential to increase stochastic variability in a population trajectory and that higher levels of immigration may be required to reduce the risk of extinction if inbreeding depression is high or if inbreeding-stress interactions occur in nature. 51 a) Scenario I, No Interaction, Si = So*exp(-Bf) 0.6 B ro or To > '£ co 0.4 0.2 0.0 Q . cu O CU 1.0 0.8 0.6 0.4 0.2 0.0 b) Scenario II, Linear Interaction, Si = So*exp(-Lsf) c) Scenario III, Threshold Interaction, No Background, Si = So*L F 0.6 -ro 0C 0.4 H £ co 0.2 0.0 cu 1.0 0.8 -\ _ i n & 2 o.6 ^ jf t: 0.4 S> 0.2 -\ d) Scenario IV, Threshold Interaction, Constant Background, Si = So*exp(-Bf )*LF 0.0 F 1.0 c o '« 0.8 c/> & 2 0.6 H Q II i t: 0.4 S 0.2 0.0 e) Scenario V, Threshold Interaction, Linear Interaction Background, Si = So*exp(-Lsf )*LF 0) ro UL co 1.0 -g '(/) </> 0.8 -O) ir> Q . c s i 0.6 -cu o D H D) c 0.4 -TJ cu Q CU 0.2 -C 0.0 -f=0.00 f= 0.03125 f= 0.0625 f= 0.125 f=0.25 f= 0.50 A r 0.4 0.6 Stress 0.0 0.2 0.4 0.6 Stress 0.8 1.0 Figure 1. Survival curves for Scenarios I to V showing the manner in which survival rates decline as stress increases for individuals with a range of inbreeding levels (left), and inbreeding depression for an individual with f = 0.25, with arrows indicating the change in the curve as inbreeding depression, averaged over all levels of stress, increases (right). (LF = logistic function) 52 females males I : i I ! I 1 1975 1980 1985 1990 1995 2000 Year Figure 2. Number of adult song sparrows on Mandarte Island at the spring census from 1975 to 2002. 53 ) S9|BLuaj jo jequinN o CD o o CM O CM —I— o —I— CM —I— o o o o o (sjo|d xoq) s}|npv jo juapyjaoo 6u|p88jqu| CO d CD d O —T~ CM O ~I— o d o o o CM (35 o a> ro CD > TD +-> i n oo o oo a> m in CM d co" 3 s ° CD TD | ro o ® O i= s | s CD | § 3 £ o -s S ufs o d Q. CD X 8 E JD O C CD r i is s CD s~ I ' .£ ro CD CD CD D> ro ro . JD ro ro •> a <- >- CD > > CD ro tn 3 g '> CD t_ Q. C £ ro "ro > 3 3 TD CO 3 TD CD TD C CD CD tr ro TD c ro c o <n ro E CD 3 CO CD E 3 CO CD 3 O) CD CD i _ D) TD CD Q. CD c CD O se CD O O CO C TD CD CD t_ JD C C o cn c 5 o JO 00 00 CO CD _C 73 CD •a 5 o TD 3 O Q. X o JD TD c ro c CD <A JD CD CD k_ CO tn TD i JD TD CD L _ JD C o in i _ CO CD >. ro CD o "co JZ tn ) ja ju jM sno|A9Jd u| a jey I B A J A J H S l inpv 54 0.0 J 1 1 1 1 —^I 0.0 0.2 0.4 0.6 0.8 1.0 Inbreeding Coefficient, f Figure 4. Hypothesized decline in performance with inbreeding under reinforcing epistasis or a genotype-environment interaction. Lines (a) to (f) have a constant load of 6 = 0.30, but the location and steepness of the threshold can be changed (g) by manipulating the coefficients, as in a Taylor series. 55 4. Reproduction Number of independent young produced per female is a function of maternal inbreeding coefficient, maternal age and population density 5. Calculate inbreeding 6. Determine Winter Survival Adult survival probability is a function of age, inbreeding coefficient, sex and winter severity (stress level) Juvenile survival probability is a function of population density, inbreeding coefficient, sex and winter severity (stress level) Figure 5. Model sequence showing the order of events for one annual cycle. 56 0.5 -, i i i n l l 1 -1 0 1 2 3 4 5 Number of New Female Immigrants Poisson distribution with mean 0.82 I I Female immigrants Figure 6. The observed distribution of female immigrants arriving annually on Mandarte Island from 1975 to 2002 compared with a Poisson distribution with mean 0.82. 57 100 80 A 0 20 40 60 80 100 Total Number of Males Figure 7. The proportion of males that acquire territories declines slightly at high density and is well described by a fitted Type II functional response curve. The total number of adult males and the number of males holding territories was determine during the spring census each year on Mandarte Island. 58 10 n Figure 8. Scatter plot of residual versus predicted values from a multiple linear regression using the variables maternal age, population density and maternal inbreeding coefficient to predict the number of independent young produced per female annually. Data were from Mandarte, 1981 to 2001. The regression equation is: Y = 4.63 + AE - 0.047*NF - 3.42*fmalemai where Y is the predicted number of independent young, AE is the age effect, NF is the number of adult females in the population, and /maternal is the maternal inbreeding coefficient. Maternal age was entered as a categorical variable. The age effects, AE, were 0.000 for an age one female, 0.898 for age two, 0.342 for age three, -0.141 for age four, -0.431 for age five, -0.960 for age six, and -0.570 for age seven. The dotted lines delineate boundaries for stratification of predicted values by rounding to the nearest whole number. 59 a) Reproduction based on maternal age, population density and maternal inbreeding coefficient CU ro E CD LL CD Q. .—. O) ro c rj XI o (/) > CU c + CD XJ TJ C OJ CU o dap redi c Q. o cu XI E 3 14 - , 12 H 10 H 8 H 4 4 2H T 1 1 r -1 2 3 4 Predicted-reproduction Category b) Reproduction based on maternal age and population density 14 - , •2 12 cu LL CU a. ^ o) ro >- 6 10 x > XJ c a> f 1 6H X T 1 1 1 1 2 3 4 Predicted-reproduction Category — B o x Plot indicating 10th, 25th, 50th, 75th, 90th percentiles and outliers ( • ) • Mean number of independent young produced on Mandarte for each prediction catagory Figure 9. Box plots showing the observed distribution of independent young produced on Mandarte Island from 1982 to 2001, corresponding to the predicted-reproduction category (predicted number of young rounded to the nearest integer) from linear regressions of maternal age, population density, maternal inbreeding coefficient (a) or maternal age and population density only 60 Females Males 0 4 1 Age Class Figure 10. Average survival rate for each age class of male and female song sparrows on Mandarte Island from 1975 to 2002. Juveniles (age 0) are not sexed. 6 1 1.0 I 1 1 1 1 1 1 1 I 0 10 20 30 40 50 60 70 80 Number of Females in Previous Spring Figure 11. Observed density dependence in juvenile survival rate (recruitment to the breeding population from the age of independence to age one) showing regression line. 62 a) 0% Inbreeding Depression 100 80 60 40 20 0 b) 25% Inbreeding Depression 100 80 60 40 20 0 c) 50% Inbreeding Depression f=0 .00 f= 0.03125 f= 0.0625 f= 0.125 A =0.25 1.0 0.8 0.6 £ 0.4 > 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 B ro 0.6 0£ > 0.4 E w 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 d) 75% Inbreeding Depression 100 80 -60 -40 20 0 e) 98% Inbreeding Depression 100 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 n> ro 0.6 K > 0.4 E CO 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 Year 230 0.0 0.2 0.4 0.6 0.8 1.0 Stress Figure 12. Scenario I, No Interaction. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Inbreeding depression is the average across all levels of stress experienced by an age-one f= 0.25 male. Right: Survival curves corresponding to each trajectory showing adjustment of the intercept at zero stress for an age-one male. (Legend above) 63 a) 0% Inbreeding Depression 100 80 -I 60 40 20 H 0 f= 0.00 f= 0.003125 f = 0.0625 f = 0.125 f = 0.25 f'= 0.50 b) 25% Inbreeding Depression 100 -i d) 75% Inbreeding Depression 100 e) 98% Inbreeding Depression 100 -I Year 1.0 0.8 cu « r. ro 0.6 CC ro 0.4 E U CO 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.0 0.2 0.4 0.6 0.8 1.0 230 0.0 0.2 0.4 0.6 0.8 1.0 Stress Figure 13. Scenario II, Linear Interaction. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Inbreeding depression is the average across all levels of stress experienced by an age-one f= 0.25 male. Right: Survival curves corresponding to each trajectory showing adjustment of the intercept at zero stress for an age-one male. (Legend above) 64 a) 0% Inbreeding Depression 100 b) 25% Inbreeding Depression 100 -| d) 75% Inbreeding Depression 100 Year 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 B ra 0.6 Of > 0.4 2 3 CO 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 B « r. ra 0.6 a: "ra 0.4 I ZJ CO 0.2 0.0 230 0.0 0.2 0.4 0.6 0.8 1.0 Stress Figure 14. Scenario III, Threshold Interaction. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Inbreeding depression is the average across all levels of stress experienced by an age-one f = 0.25 male. Right: Survival curves corresponding to each trajectory showing adjustment of the intercept at zero stress for an age-one male. (Legend above) 65 a) 0% Inbreeding Depression 100 80 -j 60 40 20 0 f=0.00 f= 0.03125 f= 0.0625 f= 0.125 f=0.25 f=0.50 (Graph not available for Scenario IV due to background inbreeding depression) 1.0 0.8 0.6 or ra 0.4 I CO 0.2 0.0 b) 25% Inbreeding Depression 100 n 80 60 40 -20 -0.0 0.2 0.4 0.6 0.8 1.0 0 1.0 0.8 n ^ ro 0.6 a ro 0.4 | w 0.2 0.0 c) 50% Inbreeding Depression 0.0 0.2 0.4 0.6 0.8 1.0 1.0 d) 75% Inbreeding Depression 100 80 60 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 I'-ll1 1.0 0.8 ro 0.6 ro 0.4 E CO 0.2 0.0 e) 98% Inbreeding Depression 100 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 „ ~ ro 0.6 Qi ro 0.4 £ CO 0.2 0.0 Year 230 0.0 0.2 0.4 0.6 0.8 1.0 Stress Figure 15. Scenario IV, Threshold Interaction with Constant Background. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Inbreeding depression is the average across all levels of stress experienced by an age-one f= 0.25 male. Right: Survival curves corresponding to each trajectory showing adjustment of the intercept at zero stress for an age-one male. (Legend above) 66 a) 0% Inbreeding Depression 100 80 60 40 20 0 f=0.00 f= 0.03125 f= 0.0625 f= 0.125 f=0.25 f=0.50 (Graph not available for Scenario V due to background inbreeding depression) 1.0 0.8 0.6 a: 75 0.4 1 0.2 0.0 b) 25% Inbreeding Depression 100 0.0 0.2 0.4 0.6 0.8 1.0 d) 75% Inbreeding Depression 100 e) 98% Inbreeding Depression 100 30 80 Year 130 180 230 0.0 0.2 0.4 0.6 Stress Figure 16. Scenario V, Threshold Interaction with Linear Interaction Background. Left: Population trajectories with (dotted) mean line under a range of inbreeding depression severity. Inbreeding depression is the average across all levels of stress experienced by an age-one f= 0.25 male. Right: Survival curves corresponding to each trajectory showing adjustment of the intercept at zero stress for an age-one male. (Legend above) 67 a) Mean Population S ize 0.00 0.25 0.50 0.75 Inbreeding Depression for f= 0.25 1.00 b) Coefficient of Variation 1.0 > o c o ra ra > o 0.8 - \ 0.6 H n 0.4 0.2 H 0.0 III / IV 0.00 0.25 0.50 0.75 1.00 Inbreeding Depression for f = 0.25 O o or o c) Extinction Rate 0.6 0.5 - \ 0.4 H 0.3 - \ 0.2 - \ 0.1 0.0 0.00 III / \ IV 0.25 0.50 0.75 Inbreeding Depression for f= 0.25 1.00 d) Crash Severity 1.0 ro 0.8 H 0.6 H 0.4 0.2 H 0.0 III / / / / / /7 v / / r v / / 7 0.00 0.25 0.50 0.75 Inbreeding Depression for f = 0.25 1.00 e) >-Q. Period Length 16 14 12 H 10 8 6 H 4 2 0 0.00 0.25 0.50 0.75 Inbreeding Depression for f = 0.25 1.00 Scenar io I, No Interaction Scenario II, Linear Interaction Scenario III, Threshold Interaction Scenario IV, Threshold Interaction, Constant Background Scenar io V , Threshold Interaction, Linear Interaction Background Figure 17. Comparison of population mean (a), coefficient of variation (b), proportion of runs extinct (c), mean crash severity (d), and mean period length (e) for Scenario 's I to V . The curves were smoothed using a running average with a window 6% of the x-axis in width. 68 a) Scenario I, No Interaction 120 -i 100 H 80 H S 60 40 -[ 20 H 0.00 0.25 0.50 0.75 Inbreeding Depression for f = 0.25 1.00 b) Scenario II, Linear Interaction 120 - i 100 co 3 Q . O 0-0.00 0.25 0.50 0.75 Inbreeding Depression for f= 0.25 1.00 c) Scenario III, Threshold Interaction 120 0.00 0.25 0.50 0.75 Inbreeding Depression for f= 0.25 1.00 d) Scenario IV, Threshold Interaction, Constant Background 120 N CO c o ra u Q . O 0. 100 -80 -60 -40 20 0 y 0.00 0.25 0.50 0.75 Inbreeding Depression for f= 0.25 1.00 e) Scenario V, Threshold Interaction, Linear Interaction Background 120 -i 100 Q. O Q. 80 H 60 40 H 20 0.00 0.25 0.50 0.75 Inbreeding Depression for f = 0.25 1.00 Mean Minimum 25th Percentile Median 75th Percentile Maximum Figure 18. Profiles of population size and variability over inbreeding depression of 0% to 98% for and individual with / = 0.25 showing mean, median, minimum (excluding zero), maximum and quartiles. Each point is the average of 100 runs at each inbreeding depression increment. For example, the maximum is the observed maximum population size in each run, averaged over 100 runs. 69 a) Decrease from fall to spring CD O O T> CD CD 0.06 0.05 -\ 0.00 0.25 0.50 0.75 1.00 Inbreeding Depression for f = 0.25 0.00 0.25 0.50 0.75 1.00 Inbreeding Depression for f= 0.25 d) Run maximum in spring 0.00 0.25 0.50 0.75 1.00 Inbreeding Depression for f = 0.25 *= CD O o CD 0.00 0.25 0.50 0.75 Inbreeding Depression for f = 0.25 1.00 e) Variance in average inbreeding coefficient between years in spring 0.0012 0.0010 0.0008 CD O | 0.0006 -CO > 0.0004 -0.0002 0.0000 0.00 0.25 0.50 0.75 Inbreeding Depression for f= 0.25 1.00 Scenario I, No Interaction Scenario II, Linear Interaction Scenario III, Threshold Interaction Scenario IV, Threshold Interaction, Constant Background Scenario V, Threshold Interaction, Linear Interaction Background Figure 19. Comparison of average levels and variability of inbreeding, f, in the population between the five survival scenarios I to V. Average inbreeding level in spring does not include new immigrants or young, but average inbreeding level in fall does include immigrants and young. Each point on the curves is the average of 100 model runs. The curves were smoothed using a running average with a window 6% of the x-axis in width. 70 .) s 9 | B L U 8 j jo jaqiunN. o 00 o CD o o CN o CM i n CN ^ d • FT-] •HI 3 =H SH H K Z I H — I — E E H H-HlsOEH • H O T fa 41 d CO d CM d —I— o d (s}0|d xoq) juapyjeoQ 6u|paajqu| oo d CD d d - r -CM d o d o m CD co CD >-75 o o co o m i n - Si? o .2 o C CD CO c CD o i t CD CO o a> 0 j= g> 0 1 a CD CD C L £ s c ° ? o d Q_ -X ° o X I o CD CD CO — 0) CD CD 8 | | "S ^ CD 0 3 CD 10 t C « - CD • - O * -0 i_ 0 CD 0 CD CD JO CD 0 t CD > Z> > CD C CD o> fi c c 73 0 0 5 fi X I o LO CO g > 0 s CL CZ CD CD a> CD c o o CO 1_ 0 JZ 33 o JZ CO 0 ZS 73 CD CD CD O CO 0 "~ CD = E g fi ' i _ -»—• CD 3 != T3 8 « co o 2 fj *- E 3= 0 o o CD .2 — CD CO CD o cz 73 0 0 l _ X I cz Z> Q . o Q . o 5 3 C L "3 o 0 O ZS C CD C o c o zi X I CO 73 0 C L O i_ 73 CO 0 > 0 CD CZ c o .2 CM (0 O 0 CO fU . - O X ! 0 CO CD 0 LL 73 ) J9JUJAA snoiAajj ejey leAjAjns 8|ewaj 71 a) All Stress Levels, Model Years 30 to 229 3.5 3.0 2.5 2.0 1.5 | 1.0 o CU 0 0.5 0.0 C O o 1 1 1 1 1— 0.0 0.2 0.4 0.6 0.8 1.0 Overwinter Stress b) Low Stress (< 0.54), Inbreeding Coefficient 3.5 - i % o co 0 £ o cu (3 3.0 2.5 2.0 -1.5 -1.0 -0.5 0.0 y = -5.6743x+ 1.9083 r* = 0.0542 • • • • *•» • . • 0.00 0.04 0.08 0.12 Average inbreeding coefficient in fall c) Low Stress (< 0.54), Population Density 3.5 - i 3.0 H 2.5 2.0 1 1 . 0 -o cu O 0.5 -0.0 -ro ? o y = -0.0194x + 2.3346 r2 = 0.7021 20 40 60 Number of adult females —r~ 80 d) cu High Stress (> 0.54), Inbreeding Coefficient 3.5 | y = -12.195x+ 1.3414 r2 = 0.2545 o C O o o cu 0 0.00 0.04 0.08 0.12 Average inbreeding coefficient in fall e) High Stress (> 0.54), Population Density 3.5 - , y = -0.0142x+ 1.2863 ^ = 0.4316 20 40 60 80 Number of adult females 100 Figure 21. Scatter plots showing the relation of the geometric population growth rate ( X*) to overwinter stress (a), and to the inbreeding level of the population and population density at low stress (b and c) and high stress (d and e). This example shows model output for a single run under Scenario III, the threshold interaction with no background inbreeding depression, at 50% inbreeding depression for f= 0.25. The dotted lines delineate the division between population growth (X > 1) and population decline {X < 1), as well as "low stress" (< 0.54) and "high stress" (> 0.54). *The geometric growth rate (X) is the number of females in year t+1 divided by the number of females in year t. 72 a) 0% Inbreeding Depression for f = 0.25 1.0 ti 0.8 H b) 25% Inbreeding Depression for f= 0.25 LU a 0.6 H or o o '•c 0.4 0.2 0.0 0.0 0.5 1.0 1.5 Immigration Rate c) 50% Inbreeding Depression for f = 0.25 1.0 1.0 1.5 Immigration Rate 2.0 0.5 1.0 1.5 Immigration Rate d) 75% Inbreeding Depression for f = 0.25 1.0 0.0 0.5 1.0 1.5 Immigration Rate 2.0 e) 98% Inbreeding Depression for f= 0.25 0.0 0.5 1.0 1.5 Immigration Rate Scenario I, No Interaction Scenario II, Linear Interaction • Scenario III, Threshold Interaction Scenario IV, Threshold Interaction, Constant Background Scenario V, Threshold Interaction, Linear Interaction Background Figure 22. Extinction rate decreases as immigration rate increases from an average of zero to three immigrants per year for a range of levels of inbreeding depression. Each point is the proportion of 100 runs that went extinct before model year 230. 73 1.0 - , 0.2 H o.o -\ 1 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Immigration Rate — Scenario I, No Interaction Scenario II, Linear Interaction — Scenario III, Threshold Interaction - Scenario IV, Threshold Interaction, Constant Background Scenario V, Threshold Interaction, Linear Interaction Background Figure 23. Decline in crash severity as immigration rate (mean number of immigrants per year) increases from zero to three for inbreeding depression of 50% for f= 0.25. Crash severity was defined as the percent change in population size ((maximum - minimum) / maximum) for a large scale population fluctuation. At very low immigration most runs go extinct early, so mean crash severity could not be calculated. 74 Literature Cited Arcese P. 1987. 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Example of a Threshold Interaction: Low Inbreeding Depression at Low Stress, but High Inbreeding Depression Beyond a Stress Threshold A recent study on heat shock proteins provides a possible mechanism for a threshold relationship between fitness and inbreeding. Kristensen et al. (2002) found that inbred Drosophila larvae produced much higher levels of a heat shock protein, Hsp70, relative to outbred larvae under low heat stress. At higher temperatures, levels of Hsp70 became comparable in inbred and outbred larvae, and at very high temperatures, production of Hsp70 by inbred larvae fell below outbred levels. Heat shock proteins (Hsps) are understood to help to refold and dispose of malformed proteins resulting from heat denaturation. Kristensen et al. (2002) suggests that Hsps may also help to refold proteins malformed by deleterious point mutations. Inbred individuals would be more likely to be homozygous for such mutations. The relationship between Hsp levels and fitness is complex because Hsps increase tolerance of severe heat shocks, but even low levels of Hsps can also reduce reproductive performance and growth rate. However, these observations indicated that inbred individuals may be able to compensate for compromised metabolic function by producing buffering agents at low stress levels. This could allow inbreeding depression to go undetected until a high stress situation is encountered. Then a threshold and decline to very low fitness may occur as inbred individuals are unable to produce sufficient Hsps to maintain functioning metabolic proteins, and homeostasis breaks down (Kristensen et al. 2002). 80 Appendix 3. Parameterization The objective for these analyses was to update the parameters by incorporating the most recently collected data, including 2001. The aim was not to redo previous statistical tests and analyses but merely to acquire the mean values necessary to represent the population. All statistical analyses were performed in SAS, version 8.0, (SAS Institute, 1999). A. Immigration From 1975 to 2002 there were 31 immigrants, an average of about one immigrant per year. Of these, 23 were female (74%) and 8 were male. Of the 8 male immigrants, only 4 produced offspring, and only one produced offspring in his first summer on Mandarte, whereas 21 of the 23 female immigrants produced independent young in their first summer on Mandarte. We therefore focus primarily on female immigration. Over the 28 years from 1975 to 2002, female immigration was distributed as follows: in 12 observation years zero immigrants arrived, in ten years one immigrants arrived, in five years two immigrants arrived, and one year three immigrants arrived. This amounts to an immigration rate averaging 0.82 per year (23 immigrants / 28 study years = 0.82 immigrants per year) or about 4 female immigrants every 5 years. The frequency distribution of the number of female immigrants arriving in each year was well approximated by a Poisson distribution with mean 0.82 (see Figure 6). B. Territoriality I used Mandarte data from 1975 to 2001, omitting 1980 when no spring census was conducted to determine the number of territory holders for the breeding season. A linear regression with the intercept set to the origin showed that 82.6% of males had territories (R-squared = 0.9471). However, a Type II response curve (Begon et al. 1990, p. 313) provided an improved fit with more evenly distributed residuals than a linear relationship (Figure 7). C. Reproduction (Production of Independent Young) Because the pedigree is unbroken from 1981, I used data from 1982 to 2001 to parameterize reproduction. I ran a multiple linear regression to predict the number of young produced per 81 female based on the density of adult females, the female's inbreeding coefficient, / , and the female's age as a categorical variable: Y =4.63 + AE - 0.047*NF - 3.42*fmaiemU where Y is the predicted number of independent young, AF is the age effect, NF is the number of adult females in the population, and / m a t e r n a l is the maternal inbreeding coefficient. The age effects, AF, were 0.000 for an age one female, 0.898 for age two, 0.342 for age three, -0.141 for age four, -0.431 for age five, -0.960 for age six, and -0.570 for age seven. Inbreeding depression averaged 31.0% over all ages in the Mandarte data, but the 95% confidence intervals stretched from zero to one. To remove the effect on inbreeding depression in reproduction, I also ran a multiple linear regression based on maternal age and population density only: Y = 4.43 + AE - 0.046*NF The age effects, AE, were 0.000 for an age one female, 0.899 for age two, 0.325 for age three, -0.142 for age four, -0.416 for age five, -1.015 for age six, and -0.501 for age seven. These analyses were not corrected for heteroskedasticity because they were not being used in statistical tests. They also did not account for repeated measures on each female, but this was deemed unnecessary for the purpose of parameterizing the model. D. Inbreeding Coefficient, f This recursive approach to pedigree calculation (Mrode 1996) is somewhat simplified by calculating one generation at a time (See Appendix 4, ModelCode.doc, Sub ComputeFcoef). All individuals alive in the pedigree are coded from 1 to n, with older individuals listed first. A square, (0 to n by 0 to n) matrix listing the individuals along row 0 and column 0 provides a framework for describing the coancestry or coefficient of kinship of each bird with every other bird. The coefficient of kinship is the same as the inbreeding coefficient of the offspring produced by mating the two individuals (Falconer 1989). Because the matrix is symmetrical, only the upper portion above the diagonal and the diagonal itself have to be calculated. The lower portion, below the diagonal is filled in as the upper portion is calculated. 82 The off-diagonal element of the matrix, ay, equals the numerator of the coefficient of kinship between animals i and j. The diagonal element is: an, = 1 +fi where f, is the inbreeding coefficient of individual i. If neither parent is known, the inbreeding coefficient,/, is assumed to be zero: al3 = 0 a,i = 1 If both parents, the sire, s, and dam, d, of animal i are known and i is younger than j, then: a,j = Q.5(ajs + ajd) The diagonal element is: a n= 1 + Q.5(asd) The inbreeding coefficients are then taken from the diagonal of the matrix by subtracting one. E. Survival Rates Factors known to influence annual survival rates include exogenous (externally generated) effects such as winter severity, and endogenous (internally generated) effects such as individual age, sex and inbreeding level. The effects of age, sex and inbreeding were analyzed separately and incorporated additively in the model. Average survival probability for each age-sex class were calculated using the Mandarte data set from year 1975 to 2001 (Table 5). The 95% confidence intervals were calculated for a binomial distribution (Figure 10). Table 5. Average rate of survival for each age and sex class from 1975 to 2002. Total Bird Standard Lower Upper Sex Age Years Survival Rate Error 95% Cl 95% Cl Juvenile 0 2343 0.395 0.010 0.375 0.415 1 416 0.563 0.024 0.515 0.610 Female 2 230 0.639 0.032 0.577 0.701 3 145 0.572 0.041 0.492 0.653 4+ 131 0.397 0.043 0.313 0.481 1 545 0.615 0.021 0.574 0.656 Male 2 330 0.658 0.026 0.606 0.709 3 213 0.638 0.033 0.574 0.703 4+ 243 0.519 0.032 0.456 0.581 83 Adult survival rates are not density dependent, but juvenile survival probability declines linearly with population density, represented by the number of females at the spring census (Figure 11). Linear regression of recruitment rate versus female density showed that 39.8% of the variation in juvenile survival rate could be explained by population density. I recalculated inbreeding depression for adults and juveniles using a discrete time proportional hazards model (PROC GENMOD, link = cloglog) as per Keller (1998). Survival rate calculations involving inbreeding levels used data from 1981 to 2001 using birds with four known grandparents or those whose parent was an immigrant. Confidence intervals overlapped for adults and juveniles, so they were assumed to have the same degree of inbreeding depression in annual survival rate for the purpose of the model. None of the rates of inbreeding depression were significantly different from 25%. Appendix 4. Compact Disk SurvivalCurves.xls Logistic Threshold Taylor Series Threshold Alternate Threshold (Eqn. 15) Model Code, doc Code for Form 1 Code for Module 1 Model Program in Visual Basic 6.0 Appendix 5. Alternative Survival Curve An alternate method of incorporating a threshold in the survival curves, which is perhaps more in keeping with Crow and Kimura's (1970) explanation of the influence of reinforcing epistasis on inbreeding depression, is as follows: Si = So.exp(~(Bf + Lsf + f(vS)n + s(2vj)m) (18 ) where n and m control the slope of the threshold and are set to about 18, perhaps approximating 18 interacting loci. The parameter v controls the location of the threshold and is varied between 0 and 10 to increase inbreeding depression due to a threshold interaction (see 84 SurvivalCurves.xls, Alternate Threshold). When v=0, there is no truncation. Parameters B, f, s and L function as in Equation 12. The logistic formulation was used for in this thesis because it offered easy control of curve shape. It also allowed for greater spread of in inflection points for a range of inbreeding depression which may be more representative of the Taylor series approach (see Background, Equation 5). Appendix 6. Model Notes Model Note 1 The fact that female survival rates were adjusted, but male rates were adjusted only by the same increment, causes male survival rates to increase very slightly (for example, from 0.60 to 0.62) as inbreeding depression increases. This causes a slight excess of males at high inbreeding levels, but does not affect model performance because model mechanisms are based on female numbers, not male numbers. The savings in computer run time was worth this slight approximation. An increase in number of males could decrease the population average inbreeding coefficient slightly, but this effect would be difficult to perceive and would not affect major model results. 85 Model Note 2 I used an estimated average coefficient of variation, CVe , to minimize variability in the estimate due to small mean population sizes. Although the estimated average coefficient of variation is not exactly the same as the average coefficient of variation , CV, the values are within 1% of each other. Therefore the CVe is simply referred to as the mean coefficient of variation in population size, C V N , in the thesis text. In order to calculate the CVe for a set of 100 model runs with the same parameter settings, I calculated the mean population size, /u, in terms of the number of adult females, over the 200 year analysis window, and the variance in population size, cr , over the same interval. Then I averaged the means and the variances for the 100 runs to get the mean, JJ, and mean variance, cr2 . The CVe for that parameter combination was then calculated as: CVe = V?//7 (19) The average coefficient of variation should really be calculated as follows: CV = cTTJi ( 20 ) However, small values of ju in a few runs could skew the CV upward. To avoid skewing CV upwards, an alternative would have been to calculate the average CV as: CVa=a/ju (21) This approach would have avoided potential problems due to small ju without the following potential problem: In using Ver2" rather than <T, I inflated estimate of average CV slightly because: E(x2) > (E(x))2 (Recall that Var(x) = £( x2) - E( x ) 2 > 0 ) JE(x2) > E(x). Fortunately, even for the most variable population trajectories (e.g. Scenario III with ID = 50%) the difference between CV and CVe is less than 1%. This is because the variance in means between model runs with the same parameter combination was low, as was the covariance between the means and standard deviations of the runs (Lynch and Walsh 1998, p. 818-21). 86 Appendix 7. Glossary Catastrophe A sudden collapse in the population size caused by extreme environmental events such as flood, drought, fire or epidemic. Catastrophes may be considered the extreme end of the environmental stochasticity distribution (Shaffer 1987; Lande 2002). Coefficient of Variation in Population Size, CV^ The coefficient of variation in population size over time is a measure of the variability in the population trajectory. Details on the calculation of C V N are in Appendix 6, Model Note 2. Deterministic variability Variability which affects the mean population size, or the expected value of the population. Demographic stochasticity Stochasticity caused by random variation in individual fitness that is independent among individuals. These fluctuations are inversely proportional to population size (Shaffer 1987; Lande 2002). Demographic stochasticity refers to fluctuations in a small population due to the random fortunes of each of its members in reproduction and survival (Caughley 1994). In large populations, dynamics are driven by the law of averages, but in small populations, demographic stochasticity becomes important. Demographic stochasticity is due to the discrete nature of populations (May 1973 p. 9, 32-33). Environmental stochasticity Stochasticity caused by changes in physical or biological factors that affect the fitness of all individuals in a population in a similar fashion. The magnitude of these fluctuations may be unrelated to population size (May 1973p. 34-35; Shaffer 1987; Lande 2002). Environmental stochasticity generally dominates demographic stochasticity in populations of more than 100 animals (Goodman 1987). 87 Exogenous Caused by factors (such as food or a traumatic factor) or an agent (such as a disease-producing organism) from outside the organism or system (Begon et al. 1990). Endogenous Caused by factors inside the organism or system (Begon et al. 1990). Genetic stochasticity Stochasticity caused by variation in individual or population fitness due to changes in population gene frequency via random genetic drift. Genetic drift occurs in all finite populations and is exacerbated by small population size, high variance in reproductive success among individuals and inbreeding (Shaffer 1987; Lande 2002). Inbreeding depression Reduced fitness of offspring of related parents relative to outbred offspring from unrelated parents. Inbreeding depression, S = (Outbred Fitness -Inbred Fitness)/Outbred Fitness. Inbreeding depression varies from 0 to 1 and is usually referred to as a percent. Because the level of inbred fitness varies with the degree of inbreeding, a given 5 must be qualified by stating the associated inbreeding coefficient,/(Keller and Waller 2002). Inbreeding depression is a function of inbreeding load, B, and inbreeding coefficient, f (see also Eqn. 3, p. 8): S=l-e~Bf (19) Jensen's inequality For any concave function, F, of a random variable, x: E(x) < F(E(x)) where E(x) is the expected, or average, value of x and F(E(x)) is the value of the function at the expected value of x (Karlin and Taylor 1975). Picture F(x) as a dome-shaped curve such that the maximum value of F(x) occurs at x = a. The average value, E(x), of two points, x = a + 1 and x = a -1, will be x = a. However, the average value of the function, F(E(x)), at x = a +1 and x = a -1 will be less than F(a). To find the average value of the function, one must sum and average the value of the function at each point: E(F(x)) = (F(a+l) + F(a-l))/2 88 Population trajectory The number of adult females over time. Stochastic variability Variability which affects the distribution of future population values, and the population variance, but does not affect the mean population size. Stochasticity is variability in a complex system due to numerous interacting factors which is generally not practically predictable (Shaffer 1987). Stress threshold The stress level above which survival rates decline rapidly for an individual with a given inbreeding coefficient. A p p e n d i x 8. S t a t i s t i c a l S i g n i f i c a n c e o f C o r r e l o g r a m R e s u l t s The correlograms regressed a 200-year section of the trajectory from the analysis window against the same trajectory, offset by the specified number of year. For a single test to be significant, the absolute value of V must exceed 0.138. (Zar 1999, p. 381, Appl 10, 198 degrees of freedom). This was a t-tailed test to determine whether the correlation coefficient was different from zero, with a Type I error level of 0.05. The assumption of bivariate normality was not met because the distribution of population sizes was skewed at many values of inbreeding depression. However, the test statistic was not very close to the significance level of r > 0.138. Therefore I am fairly confident that the lack of normality did not alter the outcome of the test. 89 

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