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Five studies in life history evolution Blachford, Alistair M. 2011

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Five Studies in Life History Evolution by Alistair M. Blachford B.Sc. (Hon) Biology, Queen’s University M.Sc. Zoology, University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Zoology) The University Of British Columbia (Vancouver) April 2011 c© Alistair M. Blachford, 2011 Abstract Assortative mating by fitness has the potential population-level benefit of reducing migration load during times of environmental stasis, while allowing introgression of immigrant genetic variation in the event of environmental change. Assortative mating by fitness was examined with respect to within-population spread of a recombination modifier under selective sweep and mutation- selection balance scenarios. Only the latter scenario boosted modifier frequency, given a strength of assortative mating unlikely to be present in most species. In a second attempt to identify a new general advantage for sexual reproduction, the focus was on how inter-individual reproduction might reduce noise in inheritance and increase the power of selection. Individuals can experience good and bad “luck” at various stages of their life history, in any habitat, and it was found that combining gametes from two separate experiences of this ecological noise could indeed reduce noise in inheritance. The puzzle of small mammal population density cycles was approached from an evolution- ary, rather than a population regulation perspective. An appropriate pattern of reproductive effort would seem key to survival through repeated population crashes to low numbers. Small mammals reproduce below their apparent potential through the decline and into the low phase of a cycle, and determining whether this reproductive pattern is adaptive is an important question. A standard cycling analytical model, the Rosenzweig-MacArthur, was carefully examined for the basis of this life history work, and found wanting even after considering several modifications. So an individual-based simulation was done. For simplicity and generality a novel mechanism was used: the “cumulative recent activity” of a population predicts several mortality causes, and has the property of delayed density dependence required to drive cycles. If animals cue from this quantity, then some controversy-causing experimental results might be explained. Branching theory and the simulation model showed that reproductive slowdown evolves under high mortality rates and, given a premium on short term persistence such as might exist at low numbers or densities, at low mortality rates. This explains the reproductive pattern observed in cycling mammals. The known reproductive suppression by stress physiology now appears to be adaptive, rather than inadvertent. ii Preface Chapter 1 has been published as: Blachford, A. and A.F. Agrawal (2006) Assortative mating for fitness and the evolution of recombination. Evolution 60(7): 1337–1343 Co-authorship: I conceived of the idea and recruited Aneil to help. The analytical work is his. We co-wrote the paper. The simulation model developed during the research effort for Chapter 1 has been published in part by inclusion in an open source population genetics simulator called NEMO (Guillaume & Rougemont 2006). Co-authorship: None. The simulation code was by Alistair Blachford. Chapter 2 has been published as: Blachford, A. and M. Doebeli (2009) On Luck and Sex. Evolution 63(1): 40–47 Co-authorship: Doebeli contributed many suggestions and some guidance. Blachford did the rest. Chapter 5 will be submitted for publication as: Blachford, A. and M. Doebeli (2011) Reproductive slowdown in cyclic populations. Co-authorship: Doebeli contributed many suggestions and some guidance. Blachford did the rest. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Assortative mating for fitness and the evolution of recombination . . . . . . . . . . 8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Model and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 On luck and sex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A brief example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Finding an approach — preliminary studies . . . . . . . . . . . . . . . . . . . . . . 32 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 iv Dissection of the Rosenzweig-MacArthur model . . . . . . . . . . . . . . . . . . . . . . 33 Modification of the Rosenzweig-MacArthur model . . . . . . . . . . . . . . . . . . . . 37 Dispersal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 The end of intrinsic versus extrinsic? — within-cycle phenotypic plasticity might not be cued by the environmental factors for which it is adapted . . . . . . . . . . . 45 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Cumulative recent activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Potential advantages of self-cueing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Potential limits to self-cueing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 Reproductive slowdown in cyclic populations . . . . . . . . . . . . . . . . . . . . . . 51 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 The lingering slowdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Prey perception of death rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Appendices A The ‘Loci’ population genetics simulator . . . . . . . . . . . . . . . . . . . . . . . . 76 B Methods used in ‘On Luck and Sex’ . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 C A simple model of predation-induced breeding suppression . . . . . . . . . . . . . . 98 D Breeding suppression in a Lotka-Volterra system . . . . . . . . . . . . . . . . . . . 100 E Deriving the Rosenzweig-MacArthur from reaction kinetics . . . . . . . . . . . . . 107 v F Three foraging models in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 G Calculation of fitness metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 H Simulation of slowdown evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 vi List of Tables Table 2.1 Variance in number of copies of a focal allele transmitted into the new adult generation (σ2) given only simple density dependence. . . . . . . . . . . . . . . 26 Table 2.2 Variance in number of copies of a focal allele transmitted into the new adult generation (σ2) given simple density dependence, variance in fecundity, and noise in survival. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 vii List of Figures Figure 1.1 The parameter space favouring recombination with assortative mating. . . . . . 16 Figure 2.1 A generic life history, with the sources of noise in inheritance we consider . . . 21 Figure 2.2 The specific life history used, showing the numbers at each stage. N is popu- lation size, λ is mean fecundity, v is the probability of survival to germination. The fecundity, survival and density dependent processes all produce noise. . . . 24 Figure 2.3 Relative noise in inheritance for selfing versus outcrossing given (a) simple density dependence only, no variance in fecundity, (b) simple density depen- dence plus variance in fecundity, and (c) all three variance sources: simple density dependence, fecundity, and survival . . . . . . . . . . . . . . . . . . . 27 Figure 2.4 σ2 of Selfer relative to σ2 of Crosser across each of the parameters λ , N and v. All three sources of noise are included . . . . . . . . . . . . . . . . . . . . . 28 Figure 2.5 σ2 of Selfer relative to σ2 of CrosserALT across each of the parameters λ , N and v. All three sources of noise are included . . . . . . . . . . . . . . . . . . 29 Figure 3.1 Proportion of population dying of starvation by different times for different basal requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Figure 3.2 The accelerating cost of finding non-regenerating sessile food . . . . . . . . . 42 Figure 5.1 Best litter size according to several fitness metrics and the simulation, for three sets of reproduction costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Figure 5.2 The reproductive effort that yields higher probability of survival in the short term is not always the same as the one that is best for the long term. These conditions are those at the centre of Figure 5.1a. . . . . . . . . . . . . . . . . . 59 Figure 5.3 Panel a shows a saturating functional response of a predator (Type II). Panel b shows how per capita prey death rate increases with lower prey density despite decreased capture rate per predator. . . . . . . . . . . . . . . . . . . . . . . . 60 viii Acknowledgements I thank the Department of Zoology for their full support, and for allowing me to take partial leaves of absence from my post as manager the Zoology Computing Unit for the first few years. I thank Andy LeBlanc and Richard Sullivan for shielding me from computer questions while I hid in my office spending holidays in “PhD-mode” during the last few years. Many thanks to my research committee: Michael Doebeli, Sally Otto, Dolph Schluter and Tony Sinclair for their guidance and encouragement. Michael enabled me to pursue this PhD, and provided financial support. I have come to regard him as an honest, bright, and knowledgeable friend. Sally’s astute insights and rapid and thorough editing helped out at several points. Aneil Agrawal was both co-author and somewhat of a mentor on my first ever paper. Huge thanks to Charley Krebs and Don Reid for organizing me into some International Polar Year projects on Herschel Island in the Arctic Ocean. I got to see lemmings and tundra voles in the wild, and to experience how incredibly hard-won are the pieces of truth gleaned by fieldwork, the pieces that I tried to puzzle together in the last few chapters. Rob Ahrens, Carl Walters and John Fryxell all offered great discussion and encouragement about my lemming cycle work. I thank all of the Doebeli lab members for discussion and feedback over the years, especially Jabus Tyerman, Jeff Fletcher and Rik Blok. Jabus and I talked about science and the world for hours and hours. Others with whom I talked at great length were Sam Yeaman, Brad Davis, Fred Guillaume and Crispin Jordan. Aleeza Gerstein, Bill Harrower, and lot of other people were all a boost to this caper at various stages. Rich Fitzjohn deserves special mention for more than bright comments on some points I bounced off him. He also enabled me to beat an important deadline by helping me with LATEX formatting puzzles. My parents knew I could do this, and that still helps. Last and most, I’d like to thank my wife Leanna, for putting up without me and accepting that I’m a nerd who needed to do this, and my son Laurian, for continually reminding me how cool the world is. ix Introduction My interests lie at the interface of ecology and evolution, and for my thesis I undertook studies in two areas: • the maintenance of sex, a long-standing evolutionary question, and • the basis of lemming cycles, one of the oldest and most tantalyzing of ecological questions. Here I provide the context of the research described in the following chapters, the process by which I chose to study each topic. Chapter 1 I think of evolution as a process in which genetic systems learn about what has been sufficient for persistence so far. The essential fact of sex is that it involves the combination of information from two separate experiences of what works in the environment, so the question of maintenance of sex then becomes a question about how that fact can provide an advantage. The idea in my thesis proposal was that every kind of environmental change of which I could conceive was spatially correlated: new predators/parasites/competitors, climate change — in all cases the change moves in over space, so that change experienced at one locale almost always has been experienced at other locales already. This means that immigrant individuals or gametes (e.g. pollen) can contain information appropriate for dealing with impending, or recently arrived, change. A paper had already shown (Ladle et al. 1993) that, in the absence of spatial heterogeneity in local conditions, and given appropriate host and parasite dispersal rates, adapted asexual host variants could arrive fast enough to outcompete sexual hosts adapting via production of suitable recombination products (i.e. Red Queen dynamics). The idea is called “metapopulation storage” of asexual variants. Intuitively, adding dimensions to the adaptation requirement would make the existence of suitable metapopulation storage much less likely. Spatial heterogeneity in local conditions would seem a common way for additional dimensions to be in play. 1 If no part of a metapopulation contains asexuals adapted to both the conditions of a specific locale and a newly arrived change, then asexual organisms must generate the suitable new variation through mutation within their locally-adapted genomes — a slow process. Sexual organisms have the potential to splice in immigrant genes suited to the change, while keeping the alleles adapted to local conditions — a much faster process — and my hypothesised advantage for sex. The problem for sexual systems in this scenario is “migration load”, i.e. the loss of local adapta- tion courtesy of the same process of incorporation of immigrant genes that might speed adaptation to change. This dashed my hypothesis until I realized that, in its simplest form, migration load assumes that immigrants have the same probability as locals of joining the gene pool. And yet, it is difficult to think of examples of species with absolutely no pre-mating choice, since choice of mate is so important. So my key idea for the first paper was that mating choice by fitness, rather than by trait, would be the perfect adaptive valve enabling introgression of migrant genes when novel vari- ation was potentially useful while reducing migration load during intervals between environmental change. Here is the essence of the idea described with two loci, each with two alleles. Assume that a genotype has one locus (A) for adaptation to local conditions, and a second locus (B) for adaptation to the environmental change. An immigrant genome would probably not be adapted to the same local environment, but might already be adapted to the change. So local genomes start out Ab and immigrant genomes might be either ab or aB. If one assumes AB to have the highest fitness, ab to have the lowest, and the others (Ab, aB) to have fitnesses between those extremes, then under assortative mating by fitness Ab and aB individuals mate with each other more often than do AB and ab individuals. Under these conditions, recombination drives linkage disequilibrium positive (not to zero), the optimum genotype is produced, and variance in fitness is increased. I needed help in attacking this topic, and knew that Aneil Agrawal was interested in the evo- lution of recombination modifiers, so I approached him with the notion that this scenario might benefit a hitch-hiking modifier. He listened to my explanation of how recombination could drive linkage disequilibrium positive, questioned it, became interested, and asked if he could write the paper with me. I learned a lot from Aneil. He brought analytical skills and deep understanding of the topic, and shepherded me through the series of successes and failures of working out something new and crafting a paper. Ironically, assortative mating for fitness does not benefit the spread of a recombination modifier under the scenario which prompted the study in the first place, of an allele for adaptation to an environmental change sweeping through the local population. As part of the study of assortative mating for fitness, I learned how to program in Java and wrote an individual-based population genetics simulator. I validated the simulator (called “Loci”) 2 by perfectly replicating the results of Iles et al. (2003). Individual-based models of evolution are noisy, and Aneil managed to push the analytics far enough that for the paper they were a cleaner way to present the nuances of the evolution of recombination under assortative mating for fitness. But “Loci” is very fast and, because of its speed, the recombination code has been “published” in a way — it was incorporated into the open-source simulator NEMO (Guillaume & Rougemont 2006) which is available at http://nemo2.sourceforge.net/. The documentation for my 1900 lines of code, in a form that is standard for programs written in Java, is Appendix A. I will provide the code to anyone who asks for it. The random number generation is done carefully, with the simulation switching between Poisson, Normal and exact calculation of Binomial variates as needed. The key to fast generation of recombinant chromosomes was to exploit bit-masking to avoid brute-force copying of individual loci from one chromosome to another. This pattern of research effort repeated a couple more times in this thesis. After extensive sim- ulation, which provoked ideas and yielded some insights, analytical work is what yielded results appropriate for a paper. Chapter 2 Having considered the assortative mating, I then returned to the idea in my proposal in order to include the elements of spatially heterogeneous local conditions, and local dispersal. The study of recombination modifiers did not cover population-level advantages gained by sexuals adapting to environmental change more quickly than asexuals. I wondered how to set up realistic hetero- geneity in local conditions, and decided on a fractal landscape. I wrote code to generate random fractal landscapes of specified “roughness” which wrap both horizontally and vertically. This pat- tern of local conditions looks believably natural . . . but the choice of any fractal landscape would be disturbingly arbitrary. I continued building a simulation model, all the while worrying about how to organize my runs and my findings. How could I enumerate all of the patterns of environ- mental change? There is the pattern of lengths of intervals between episodes of change, patterns of gradualness and severity of change, patterns of spatial correlation (invasion front versus isolated infections), etc.. There were too many possibilities. The study would have been tedious to do, and boring to read. And with so many contingencies to specify, how likely was it that I would find a nice rule of thumb? I am interested in generating simplifying rules of thumb, not in generating more data. As an alternative to studying a whole catalogue of patterns of environmental change I then thought of studying NOISE. It seemed to me that sexual reproduction could also average out the vagaries of individual experience, and that this noise-dampening might reduce noise in inheri- 3 tance, and improve response to selection. This idea was very appealing to me because, (1) it is a widespread fact of life that individuals can have good or bad luck regardless of how well-adapted their genotype may be, and (2) it was a different angle from the common view of sex being mainly a matter of recombination. After I had done a lot of hacking away at this notion using rules about variance and computer simulations, making little progress, Sally Otto asked if I had tried using moment generating functions (MGFs). She warned me that, in her experience, MGFs often only allowed limited progress toward a solution. After making limited progress, and being stuck for months, I found in some actuarial literature a handy rule about combining MGFs which allowed me to get around the obstacle, and Chapter 2 was born. I had found a way in which indeed there was an advantage to a process of reproduction that combined information from two individuals. Chapter 3 I had been stunned to learn how low lemming population densities can go during their famous cycles. A researcher in this department (Deb Wilson, PhD in 1999), even with expert help and advice, was unable to catch a single lemming of her study species during her first field season. And she caught only one individual of another species. To me this implied a huge evolutionary challenge and the existence of strong selection pressure for life history characteristics that mini- mize the probability of (local) extinction. I became interested in lemmings for the evolutionary lessons they might reveal, rather than for the population regulation lessons that have been sought for over 80 years. Reproduction is an obvious anti-extinction measure, so it was also surprising to learn that lemmings, which can reproduce so quickly that the Inuit call them “fall from the sky”, actually curtail reproduction below their potential during the decline phase, even though they do not run out of food. My first guess was that higher rates of reproduction, requiring greater amounts of foraging, might be so risky in the face of high predation rates that they yield diminishing returns. I found a paper by Oksanen and Lundberg (1995) which explored that same idea. But their 8-parameter model assumes that animals survive or starve according to their choice of foraging time (a case of living systems refusing to engage in homeostasis). I think starving animals will risk predation death when starvation death is otherwise certain. So I built a simpler (4-parameter) model in which the driving life history choice is not foraging time. Rather, it is the characteristic of interest: reproductive effort. The model, derived in Appendix C, assumes that greater litters require greater exposure to predation and confirms that the litter size which leaves the most recruits decreases as predation risk increases, which happens when there are more predators and/or less food. But that model is not dynamic, and these are cycling mammals being considered. So I used the 4 simplest of cycling models, the Lotka-Volterra, to explore whether natural selection would favour a morph that switches its reproductive effort to a lower level during the decline phase. The lessons from that model (Appendix D) are: 1. if predation losses rise supralinearly with reproductive effort, then the “switcher” morph is favoured. This result can be made obvious: if an increase in birth rate increases death rate by even more, then (birthrate - deathrate) decreases and it is disadvantageous. 2. An easy way to get supralinear predation losses is via the predator aggregation response, i.e. not only does increasing birthrate entail increasing death by predation, but predators will preferentially move into areas of greater prey productivity. If switchers and non-switchers are in neighbouring patches, and the predators favour patches with higher prey “availability” (density times average foraging amount), then the predators reduce the non-switchers to lower densities, and the relative numbers of the switchers rise. Point (2) caught my attention as the basis for an interesting spatial evolutionary game among “patches” of strategists. The strategies are different reaction norms of reproductive effort, and the patches result from local dispersal from isolated founding groups after deep crashes, or from cooperative signalling among neighbours. By reducing reproduction, both food consumption and predator interest in the locale could be reduced, perhaps easing the depth of the crash experienced in the patch and improving the odds that individuals there would be founders for the next increase phase. The unusual dispersal characteristics of lemmings added to the allure of this spatial game. Lemmings disperse least at high densities (i.e. remain within their “lifeboat” patches) and they disperse most at low densities, to repopulate the region from the numbers they retained through the crash. I set out to examine the well-known Rosenzweig-MacArthur model as a candidate either for an- alytical extension to capture dispersal effects, or for driving within-patch dynamics of an individual- based spatial simulation. My critique and work to improve it for my needs constitute Chapter 3. This chapter, unlike the others, is not packaged into publishable paper form. It documents care- ful consideration and construction of a tool I intended to use in addressing my research questions about lemmings. Instead of rendering some of that work publishable, such as the stability im- plications of a more realistic starvation term, I preferred to move on having concluded that the Rosenzweig-MacArthur would not be a simplifying starting point for my research question. 5 Chapter 4 One way to produce a model that is simple to convey (i.e. convincing) is to extend an analytical standard, like the Rosenzweig-MacArthur. Another is to build a simulation from just a few sim- ple rules. A common approach for a spatial model is to use cellular automata. My simulation could not be as simple as those because I wanted to model the evolution of a reaction norm and each individual would have a different state, namely a different set of genetic information. So I chose instead to use a free, cross-platform package for individual-based simulation called NetLogo (Wilensky 1999), in which a small amount of code suffices to generate interesting simulations. The inspiration for chapters 4 and 5, plus ideas for future work, resulted from my many explorations with Netlogo. The simplest way to drive cycles in the simulations was with food dynamics. But it is widely noted that lemming, vole and hare cycles are probably not (just) food driven. So any insights from such a model might be less than convincing. Driving an agent-based simulation with predation, on the other hand, requires that the mobile predators be modelled also, and there are many ways to do that. Predators are not accepted as a universal cause of cycles either. I finally realized that what was needed as a cycle driver was nothing more than reasonable dynamics of mortality rates. Built this way a model could be agnostic about particular circumstances, and be more general. It occurred to me that “cumulative recent activity” (CRA) of a population was a good predic- tor of mortality from both lack of food and presence of predators, and that greater reproductive effort would always means greater activity, if only to gather the greater amount of resources to be turned into offspring. What began as a modelling convenience took on greater meaning when I re- alized that if animals were to monitor CRA to control their life history changes then experimental manipulations of food and/or predators would have a lot less power than expected. The lack of experimental power due to such indirect cueing could explain much of the controversy in the field of small mammal cycles. I have written up these ideas in a form that could be published as a note in an ecology journal. Chapter 5 Despite the simplification allowed by the idea of cumulative recent activity, the complexities of modelling territoriality and dispersal remained, as did the noise levels inherent in any individual- based model. With the notion of modelling a “mortality-rate world” in mind, at the suggestion of my supervisor (Michael Doebeli) I stepped back to establish the aspatial case. Since my interest in lemmings from the start has been their anti-extinction adaptations, Michael suggested I consider using the results of branching theory as analytical tools. Branching theory arose out of considering 6 rates of extinction of family names, and is appropriate for considering rates of extinction of another kind of information — that which encodes a particular reproductive strategy. That line of explo- ration led to Chapter 5. As with Chapters 1 and 2, I had spent enormous amounts of time with computer simulations, and then fallen back to analytical models to obtain a simpler and cleaner foundation for a paper. The branching theory was used to compute one of two fitness metrics used in the study and a Netlogo simulation, based on the simple analytical model and using “cumu- lative recent activity” to drive mortality rates, was used to confirm that a reaction norm showing reproductive slowdown could evolve. This chapter is written in the form of a publishable paper. 7 Chapter 1 Assortative mating for fitness and the evolution of recombination Summary To understand selection on recombination, we need to consider how linkage disequilibria develop and how recombination alters these disequilibria. Any factor, including non-random mating, that affects the development of disequilibria can potentially change selection on recombination. Assor- tative mating is known to affect linkage disequilibria but its effects on the evolution of recombina- tion have not been previously studied. Given that assortative mating for fitness can arise indirectly via a number of biologically realistic scenarios, it is plausible that weak assortative mating occurs across a diverse set of taxa. Using a modifier model, we examine how assortative mating for fit- ness affects the evolution of recombination under two evolutionary scenarios: selective sweeps and mutation-selection balance. We find there is no net effect of assortative mating during a selective sweep. In contrast, assortative mating could have a large effect on recombination when deleterious alleles are maintained at mutation-selection balance but only if assortative mating is sufficiently strong. Upon considering reasonable values for the number of loci affecting fitness components, the strength of selection, and the mutation rate, we conclude that the correlation in fitness between mates is unlikely to be sufficiently high for assortative mating to affect the evolution of recombi- nation in most species. 8 Introduction Evolutionary biologists have struggled to understand the ubiquity of recombination (Bell 1982; Barton and Charlesworth 1998; Otto and Lenormand 2002). Although group-level advantages of recombination have been identified, a more important challenge is in understanding the conditions that favor the evolution of recombination within a population (Feldman et al. 1997). Modifier models are used to study the evolutionary pressures on a gene that alters some aspect of the genetic system such as the rate of recombination. A modifier gene that increases the rate of recombination will tend to become associated with the haplotypes that are generated by recombination. The short- and long-term success of these haplotypes determine the modifier’s evolutionary fate (Lenormand and Otto 2000). If recombinant haplotypes are on average more fit than non-recombinant haplo- types then the modifier will experience a short-term fitness advantage. If the variance in fitness of recombinant haplotypes is greater than non-recombinant haplotypes then the modifier will have a long-term fitness advantage by better responding to selection. The distribution of haplotypes generated by recombination depends on the patterns of linkage disequilibria and how recombination modifies these disequilibria. With random mating, recombi- nation acts to reduce the magnitude of disequilibrium. Because disequilibrium is built by selection, it is usually disadvantageous to reduce it, at least in the short term. The evolution of recombination in a large panmictic population requires that epistasis be negative, but weak (Barton 1995). Under this condition, recombination enjoys a long-term advantage that is not overwhelmed by a short- term disadvantage. However, empirical data suggest that epistatic interactions are not confined to being negative and weak (de Visser et al. 1997; Elena and Lenski 1997; Whitlock and Bourguet 2000). Moreover, variation in epistasis can cause selection against recombination even if epistasis is, on average, weakly negative (Otto and Feldman 1997). The sensitivity to epistasis arises because epistasis is the only force generating disequilibrium when populations are large and randomly mating, as assumed by many classic models (Feldman et al. 1980; Kondrashov 1984; Charlesworth 1990; Barton 1995). However, non-random mating can change the sign and magnitude of disequilibrium as well as the effect of recombination on the disequilibrium. Consequently, the conditions favoring recombination may be quite sensitive to assumptions of breeding ecology (Charlesworth et al. 1979; Lenormand and Otto 2000). For example, a recent analysis by Roze and Lenormand (2005) shows that very low rates of sporophytic selfing can greatly increase the parameter space favoring recombination. Here we focus on assortative mating for fitness, which may be common in many species be- cause it can arise via a number of different mechanisms. Assortative mating can occur through male-male competition if the best males monopolize the best females. For example, in the water 9 strider Gerris lateralis, the biggest and most successful males spend a disproportionate amount of time guarding the largest and most fecund females (Rowe and Arnqvist 1996). Assortative mat- ing can also occur through female choice if the best females exhibit the strongest preference for the best males. In sticklebacks, for example, females in good condition show a stronger prefer- ence for brighter males than do females in poor condition (Bakker et al. 1999). Recent theory suggests assortative mating is expected in any species with either direct male-male competition or female choice (Fawcett and Johnstone 2003). Further, assortative mating may exist in species that are thought to epitomize random mating. Broadcast spawners might ‘mate’ assortatively if the timing of gametic release is slightly condition-dependent (e.g. individuals in worse condition shed gametes slightly later than individuals in good condition). Though assortative mating may be widespread, it is likely to be weak in many species. Here we investigate whether assortative mating alters the conditions favoring recombination and how much assortative mating is needed to do so. We expect assortative mating to affect the evolution of recombination because of its potential to alter linkage disequilibrium. This is most easily illustrated by considering a simple haploid two-locus, two-allele example in which all four genotypes are at the same frequency so that there is initially no linkage disequilibrium. If there is positive assortative mating for fitness then sexual unions between the extreme types (AB and ab) will occur less often than between the intermediate types (Ab and aB). In these latter unions, Ab and aB haplotypes are converted via recombination into AB and ab haplotypes, generating positive linkage disequilibrium. The effects of assorta- tive mating on linkage disequilibrium has been known for some time (Fisher 1918; Moran and Smith 1966; Vetta 1975) but there has been no attempt to understand how assortative mating might thereby affect the evolution of recombination. Model and results To investigate the importance of assortative mating for fitness on the evolution of recombination, we consider a simple deterministic haploid model with two diallelic fitness loci, A and B. The model follows the general approach of other modifier models of recombination (e.g. Barton 1995; Lenormand and Otto 2000; Otto and Nuismer 2004). The fitnesses of each of the four haplotypes are given by wAB = 1, wAb =waB = 1−s, and wab = (1−s)2+ε , where s is the selection coefficient and ε is epistasis. Alleles represented by lower case letters are deleterious, i.e. s > 0. Recombina- tion is determined by the modifier locus M, and the three loci are in the order MAB. The M locus affects recombination rate but not fitness (i.e. wMAB = wmAB = wAB, wMAb = wmAb = wAb, etc.). With two alleles at each locus, there are eight haplotypes. The frequency of the ith haplotype is xi. 10 The frequency of the ith haplotype after selection is x′i = xiwi/w̄ where w̄ = ∑xiwi. Mating occurs after selection. Random mating occurs with probability 1− t. Individuals mate assortatively (i.e. with another individual of the same fitness) with probability t. Haplotypes car- rying Ab or aB are assumed to have equal fitness (wAb = waB) whereas haplotypes carrying AB or ab are assumed to be different from Ab as well as from each other. Under these assumptions, it is straightforward to calculate the frequency of different mating pairs as is illustrated here by a few examples. The frequency of matings between haplotypes i jk and xyz is given by Fi jk∗xyz: FMAB∗MAB = (1− t)x2MAB+ t ( x2MAB xMAB+xmAB ) , FMAB∗mAB = (1− t)(2xMABxmAB)+ t ( 2xMABxmAB xMAB+xmAB ) , FMAb∗MaB = (1− t)(2xMAbxmaB)+ t ( 2xMAbxmaB xMAb+xmAb+xMaB+xmaB ) , FMAB∗Mab = (1− t)(2xMABxMab). As expected, the sum of all the Fi jk∗xyz is one. In this model, parameter t determines the strength of assortative mating and is equal to the correlation in fitness between mates. When t > 0, matings between the extreme types, AB and ab, occur less often than expected by chance whereas matings between the intermediate types, Ab and aB, occur more often than expected by chance. Although simplistic, this model is analytically tractable and captures the critical elements of assortative mat- ing for fitness. Following the standard rules of genetics, the distribution of offspring haplotypes is calculated for each mating pair. When two haploids unite to reproduce, their effective recom- bination rate is the average recombination induced by each parent’s allele at the M locus. The recombination rate induced by the ancestral allele m is rMA in the M–A interval and rAB in the A–B interval. The alternative modifier allele M increases these recombination rates to rMA + γMA and rAB + γAB, respectively. The distribution of haplotypes for the following generation is calculated from the weighted sum of offspring frequency distributions from all mating pairs. To study the evolution of the modifier, we must calculate the change in the frequency of the M allele, ∆pM, across one complete generation. To do so, the initial haplotype frequencies are rede- fined in terms of allele frequencies and disequilibria. For example, xMAB = pM pA pB + pMCAB + pACMB + pBCMA +CMAB and xmAB = (1− pM)pA pB + (1− pM)CAB− pACMB− pBCMA−CMAB where Ci j is the two-way disequilibrium between the ith and jth loci and CMAB is the three-way disequilibrium (Barton 1995). Calculating haplotype frequencies after one round of selection and reproduction, the exact change in the modifier frequency, ∆pM, is found to be: ∆pM = (CMA+CMB)s+(CMAB−CMA pb−CMB pa)(s2+ ε) 1− (pa+ pb)s+(CAB+ pa pb)(s2+ ε) (1.1) As expected, this result shows that modifier evolves only through its associations with the selected loci because we have assumed the modifier has no direct effect on fitness (though it is 11 easy to incorporate such an effect). To proceed, we employ the quasi-linkage equilibrium (QLE; Kimura 1965; Nagylaki 1993) to approximate the values for the associations. The QLE invokes a separation of timescales in which the disequilibria reach their steady state faster than allele fre- quencies change. Consequently, application of the QLE assumes that selection is weak relative to the ancestral rate of recombination, though numerical simulations indicate that the analytical approximations are surprisingly robust to this assumption. We find expressions for the evolution of the disequilibria (i.e. ∆CAB, ∆CMA , ∆CMB, and ∆CMAB), set these to zero, and solve for the steady state values of the disequilibria. Taylor series are used to find approximate steady state values for each disequilibrium measure. Specifically, we assume s, t, γMA and γAB are O(ξ ) and ε is O(ξ 2) where ξ << 1. The steady state values are found to be: qleCAB =VAVB ( (rAB− (pA+ pB−1)(2− rAB)s)t rAB(pA pb+ pa pB) + 2VAVBt2 (pA pb+ pa pB)3 + (1− rAB)ε rAB ) +o(ξ 2) (1.2a) qleCMA = γABVAVBVM rMArABrMAB × [( 2(pA+ pB−1) pA pb+ pa pB st− ε ) (1− rMA)s + ( 2(VA−VB) (pA pb+ pa pB)2 st+ (pA− pB) pA pb+ pa pB ε )] rMAt+o(ξ 4) (1.2b) qleCMB = γABVAVBVM rMBrABrMAB × [( 2(pA+ pB−1) pA pb+ pa pB st− ε ) (1− rMB)s + ( 2(VB−VA) (pA pb+ pa pB)2 st+ (pB− pA) pA pb+ pa pB ε )] rMBt+o(ξ 4)] (1.2c) and qleCMAB = γABVAVBVM rMArABrMAB ( 2(pA+ pB−1) pA pb+ pa pB st− ε ) +o(ξ 3) (1.2d) In these equations VX = pX px is the variance at locus X where pX and px are the frequencies of alleles X and x at this locus. The additional recombination parameters are defined as rMAB = 1− (1− rMA)(1− rAB) and rMB = rMAB− rMArAB. 12 These steady state association values are then used in approximating ∆pM: ∆pM ≈ (qleCMA+qle CMB)s+qle CMAB(s2+ ε)+o(ξ 5) ≈ γAB VMVAVBrABrMABκ θ +o(ξ 5) (1.3) where κ = ( 2(pA+ pB−1) pA pb+ pa pB st− ε ) and θ = (s2+ ε)+ s2 ( 1 rMA + 1 rMB −2 ) . The sign of κ indicates whether the modifier is associated with extreme or intermediate hap- lotypes. The modifier develops associations with particular haplotypes when the disequilibrium prior to mating deviates from qleCAB. It can be shown that κ reflects how selection perturbs CAB away from its steady state value. When κ < 0, selection pushes the disequilibrium above its steady state. Recombination then draws the disequilibrium back towards its steady state by generating intermediate haplotypes from the relative excess of extreme haplotypes. Consequently, a modi- fier that increases recombination becomes positively associated with the intermediate haplotypes it produces and negatively associated with the extreme haplotypes it destroys. Conversely, if selec- tion pushes the disequilibrium below its steady state as it does when κ > 0, the modifier becomes positively associated with the extreme haplotypes. Note that κ tends to be positive when beneficial alleles are common (pA, pB >> 0) but negative when they are rare. The factor θ describes the selective consequences for the M allele of being positively associated with the extreme haplotypes. The first and second terms of θ represent short- and long-term effects, respectively (Barton 1995; Lenormand and Otto 2000). The long-term effect of being associated with the extreme haplotypes is positive because there is more variance in fitness associated with these haplotypes than the intermediate haplotypes. The important implication of being associated with the extreme haplotypes is that the modifier is associated with the most fit haplotype, AB, which will eventually dominate the population. The strength of the longer-term effect depends on rMA and rMB because these values determine how long a modifier allele remains linked to the beneficial alleles that constitute the most fit genotype. The short-term effect is positive if the average fitness of the extreme haplotypes is better than the average fitness of the intermediate haplotypes (i.e. wAB +wab > wAb +waB), which is true when s2 + ε > 0. Combining both short- and long-term effects, θ is positive provided ε is not too strongly negative. As shown in [3], the sign of selection on the modifier is determined by the product of κ and θ as the sign of these two terms indicate, respectively, the sign of the association between the modifier and the extreme haplotypes and whether it is selectively advantageous to be positively associated with these haplotypes. With assortative mating (t > 0), but not random mating, the sign of κ can 13 change as the frequency of the alleles at the selected loci change. Thus it is possible for the modifier to be selected in one direction when beneficial alleles are rare and in the opposite direction when beneficial alleles are common. To examine the net change in the frequency of the modifier we integrate [3] over the course of simultaneous selective sweeps. Following Barton (1995), we use the relationship pA/pa = pB/pb = esT which reflects the logistic growth of the relative frequency of the beneficial allele. T measures time since the midpoint of the sweep (pA = pB = 1/2). The net change in the modifier is net∆pM ≈ +∞∫ −∞ ∆pMdT = γAB VMθ rABrMAB +∞∫ −∞ esT (e2sT st− esT ε− st) (1+ esT )4 dT =−γAB VMεθ6rABrMABs (1.4) This value is positive when λ < ε < 0, where λ is the value of ε for which θ = 0, λ =−s2(1/rMA+ 1/rMB− 1). The same result is obtained under random mating (Barton 1995). Thus, we find that assortative mating has no net effect on the evolution of recombination over the course of a selective sweep despite the fact that it alters the modifier’s rate of evolution in each generation. As shown by [3], when λ < ε < 0, assortative mating will increase the modifier’s rate of spread when the beneficial alleles are rare but decrease the rate when the beneficial alleles are common. These effects cancel each other out such that spread of the modifier is independent of assortative mating. While selective sweeps happen periodically, deleterious mutations constantly occur in all genes in all populations. In the selective sweep model, beneficial alleles move from rare to common and the effects of assortative mating cancel out in the process. In the mutation-selection balance model, beneficial alleles are always common. To incorporate deleterious mutation into the model, an additional step is added into the life cycle: selection, mating (and recombination), followed by mutation. The A and B alleles each mutate to their deleterious alternative states (a and b) at rate µ , where µ is O(ξ 3). There are no back-mutations at these loci or any mutations at the M locus. To determine the mutation-selection equilibrium with respect to the selected loci, we find expressions for ∆pA, ∆pB, and ∆CAB, set these to zero and solve for pA, pB, and CAB. Taylor series are used to find approximate equilibrium values for each. One approach to calculating selection on the modifier at mutation-selection balance is by using the definition βM ≡∆pM/VM where ∆pM, as defined in equation [3], is evaluated with the mutation- selection equilibrium values for pA, pB, and CAB. Doing so, we find βM ≈ γABθ ψrABrMAB (1.5) 14 where ψ = tµ ( 1− 3µ s ) − εµ 2 s2 + t2µ ( s rAB + ε 2s − 1 2 ) (1.6) which can be approximated more simply as ψ ≈ µ(t− εµ/s2) when t << ε . This result indicates that when assortative mating is very weak, t << |ε|(µ/s2), we recover Barton’s (1995) condition for the evolution of recombination under random mating, i.e. epistasis must be both weak and negative: λ < ε < 0. However, if assortative mating is sufficiently strong, increased recombination is favored unless epistasis is both strong and negative. That is, when ψ > 0, increased recombination is favored when ε > λ . In Figure 1.1, we show the parameter space favoring recombination. Simulations using the exact recursions confirm that the important analytical results (e.g. Figure 1.1). Additional simulations confirm that the key results of this model also apply when there are more than two loci or when assortative mating is modelled as increasing the probability of mat- ings between individuals of similar, rather than identical, fitness (i.e. wAb = waB is not required). Moreover, both simulations and analytical results show that these results also apply to diploids. Discussion A gene that modifies recombination experiences indirect selection as a consequence of how recom- bination alters linkage disequilibrium between selected loci. If epistasis is the only force generating linkage disequilibrium, the conditions favoring recombination are quite restrictive (Feldman et al. 1980; Barton 1995). This is because recombination with random mating reduces the linkage dis- equilibrium towards zero, typically opposing the work of selection. However, factors other than epistasis can generate linkage disequilibrium, including drift and non-random mating, and so can alter the conditions favoring recombination (Lenormand and Otto 2000; Barton and Otto 2005; Martin et al. 2005; Roze and Lenormand 2005). Assortative mating affects linkage disequilibrium and we have examined how it changes the conditions favoring recombination under two evolutionary scenarios: selective sweeps and mutation- selection balance. In our selective sweep model, we found that assortative mating has no net effect on the evolution of recombination over the course of the selective sweep, despite that fact that it alters the rate of change in the modifier’s frequency each generation relative to the random mat- ing expectation. This is because assortative mating has opposing effects on the modifier when beneficial alleles are common versus when they are rare. At mutation-selection balance, beneficial alleles remain common so a net effect of assortative 15   10-1 10-2 10-3 10-4 10-5 10-6 10-7 0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 0 !rec "rec !rec !rec !rec !rec !rec !rec !rec "rec "rec "rec Strength of Assortative Mating, t Epistasis, ! s = 0.003, µ = 10-6 s = 0.003, µ = 10-5 s = 0.03, µ = 10-6 s = 0.03, µ = 10-5 Figure 1.1: Parameter space favouring recombination with assortative mating. The shaded region in each indicates the region of parameter space where increased recombination is favored as calculated using Equation 1.6. With random mating (t = 0), increased recombination is favored in the range λ < ε < 0. If t >> |ε|(µ/s2), increased recom- bination is favored in the range ε > λ . Numerical simulations using the exact recuri- sions were performed for t ∈ 0,10−8,10−7,10−6,10−5,10−4,10−3,10−2,10−1 using 50 evenly distributed points between ε =−εMAX and ε = εMAX . Points show the parameter values found to favor recombination in these simulations. In this model, the maximum amount of epistasis possible with a monotonic fitness function is εMAX = s(1− s). Pa- rameter values used: rMA = rMB = 0.1, γMA = γAB = 0.01,top row s = 0.003, bottom row s = 0.03, left column µ = 10−6, and right column µ = 10−5. Weak assortative mating can influence selection on recombination when selection is weak and mutation rates are low. 16 mating can be observed. If t >> |ε|(µ/s2), recombination is favored under a much greater range of epistasis values than expected under random mating. With random mating, epistasis must be both weak and negative (λ < ε < 0); with assortative mating recombination is not only favored under these conditions but also if there is no epistasis at all or if epistasis is positive, regardless of its strength, i.e. ε > λ . The striking change in the range of epistasis values favoring recombina- tion caused by sufficiently strong assortative mating (Figure 1.1) is best understood by considering what epistasis does under the different mating systems. With random mating, epistasis plays two important roles with respect to the evolution of recombination. First, the sign of epistasis de- termines the sign of disequilibrium between selected loci and in so doing indirectly determines whether a modifier becomes associated with the extreme (AB and ab) or intermediate (Ab and aB) haplotypes. Second, epistasis determines whether the average fitness of the extreme haplotypes is better or worse than the average fitness of the intermediate haplotypes. These dual roles of epis- tasis usually result in selection against the modifier; the modifier is typically associated with the haplotypes that cause a short-term disadvantage because, with random mating, recombination acts to reduce the disequilibrium favored by epistatic selection. When there is assortative mating, epis- tasis loses one of its two roles — it no longer determines whether the modifier becomes associated with the extreme or intermediate haplotypes. At mutation-selection balance, assortative mating causes the modifier to become associated with the extreme haplotypes, regardless of the sign of epistasis, provided t >> |ε|(µ/s2). This association is always advantageous to the modifier in the long-term and is also beneficial in the short-term provided epistasis is not too negative (ε >−s2). Under random mating, the modifier becomes associated with the intermediate types when epistasis is positive, an association that is disadvantageous in both the short- and long-term. If t is not large relative to |ε|(µ/s2), then the conditions favoring recombination collapse back to the random mating expectation. How large a value of t might we expect to find in natural populations? First, we must recognize that in our model we considered fitness effects of only two loci and t was the correlation between mates with respect to these two loci. Thus, t should be thought of as the correlation between mates per locus pair. In reality, assortative mating does not occur at the level of each locus pair but rather at the level of the phenotype, which is fitness or some correlate of fitness. This phenotype will be controlled by many loci at mutation-selection balance, not just two, and the correlation at the phenotypic level will be distributed amongst all the locus pairs affecting the phenotype. Wright (1921) found that if a polygenic trait was affected by n loci and the correlation in the trait value between mates was ρ , then the correlation between mates per locus pair would be proportional to ρ/n, provided that ρ << 1. Assuming that 10% to 100% of all loci contribute to the fitness components that form the basis for assortative mating, n should 17 be in the range 103 to 104. The correlation in fitness between mates is unlikely to be more than a few percent in most species, i.e. ρ is O(10−2). Thus, we speculate that t is O(10−6) or O(10−5). Taking s in the range 10−3 to 10−2, ε in the range O(s2) to O(s), and µ in the range 10−5 to 10−6 then εµ/s2 is O(10−6) to O(10−2). Thus, we conclude that while assortative mating may affect the evolution of recombination in some species, it is unlikely to be of widespread importance. In extrapolating the results of a two-locus model to the whole genome, we are ignoring higher- order disequilibria (i.e. 3-, 4-, . . ., and n-way associations among selected loci) that could affect the evolution of recombination. These higher-order disequilibria are typically considered to be negligible (Barton 1995), at least with random mating. To ensure that higher-order associations were not having an unexpected influence when assortative mating occurs we performed computer simulations involving 1600 loci. Results from these simulations confirm that assortative mating does not favor recombination when t >> |ε|(µ/s2). Though we conclude that assortative mating is unlikely to affect recombination in most species, it is not possible to reach this conclusion by simply comparing the strength of the two factors that generate linkage disequilibrium (t and ε). Other parameters (i.e. µ , s) also affect the result in a non- intuitive fashion. If mutation rates were very low, then even small amounts of assortative mating would be sufficient to drastically change selection on recombination (Figure 1.1). That is, one could say that the failure of assortative mating to affect the evolution of recombination is because mutation rates are too high rather than because the correlation in fitness between mates is too low. Such a perspective is a reminder that even minor deviations from the simplifying assumption of random mating can be important for the evolution of recombination. While we have focused on the gene-level advantages to recombination, other authors (Davis 1995; Rice 1998; Jaffe 2000) have used computer simulations to show that assortative mating pro- vides a group-level advantage to a sexually recombining population over an asexual population. Indeed, our analytical approximations confirm this result (not shown). However, group-level ad- vantages to recombination do not translate into gene-level advantages. For example, recombining populations can be much fitter than non-recombining populations at mutation-selection balance when there is strong negative epistasis (Kimura and Maruyama 1966; Kondrashov 1982) but a modifier for increased recombination would be negatively selected under these conditions (Feld- man et al. 1980; Barton 1995). Nonetheless, the previously identified group-level advantage of assortative mating may have important implications for the reduction of mutation load (Rice 1998). 18 Chapter 2 On luck and sex Summary Sex has many costs with respect to asexual reproduction, so its ubiquity is a puzzle. There has been a continuing effort to identify general circumstances where aspects of sex generate an evo- lutionary advantage over asexual reproduction. Here we focus on the generality that individuals can experience good and bad “luck” at various stages of their life history regardless of genotype, and on the inter-individual nature of sex. Sexual outcrossing combines genetic information from individuals with potentially different experiences, so it is conceivable that sex might reduce the contribution of individual luck to noise in inheritance. In a simple way, we derive expressions for noise in inheritance in terms of some sources of within-generation ecological noise. We demon- strate that inter-individual reproduction can indeed dampen the effects of ecological noise better than lone-individual modes, but there are conditions under which it does not. Empirical and the- oretical work on plants, modelled here, suggest noise dampening conditions. Ecological noise dampening operates alongside other features of sex such as recombination and segregation and, since noise in inheritance weakens the role of selection in genetic change, we speculate that noise dampening may offer a benefit to be deducted from the costs of sex. We also suggest that the amount of selfing relative to outcrossing observed in natural populations may be influenced by the amount of individual-level ecological noise in a given habitat. Introduction Sexual reproduction incurs risks and costs including the making and fusing of gametes, the pro- duction of males, difficulty of finding a mate, risk of infection, risks of breaking up good gene 19 combinations, and non-optimal consequences of sexual selection (Crow 1999). Although particu- lar features of sex such as having two parents or having differentiation among the sexes are clearly beneficial under specific ecological circumstances, it has proven difficult to identify advantages which hold over a wide enough range of circumstances to justify the ubiquity of sexual repro- duction. The quest for general explanations is a matter of identifying a widespread circumstance paired with means by which known aspects of sex can generate an advantage relative to asexual reproduction. The advantage is usually sought in terms of how sexual reproduction might facilitate the maintenance and/or construction of genotypes that are enough better-adapted to pay for the costs of sex. This enduring puzzle has even forced consideration of the possibility that mainte- nance of sex may be a consequence of different mechanisms under different circumstances, acting independently and/or in synergy (West et al. 1999). Nature’s variety can make definition of categories difficult. In his review Kondrashov (1993) chose amphimixis versus apomixis because they are clearer categories than sexual versus asexual reproduction. His choice reflects, and has reinforced, the tendency of evolutionary theorists to view sexual reproduction as reproduction-with-recombination. As Kondrashov pointed out, some amphimictic species have neither sexes of individual nor sexes of gamete. We note that in most species with sexes, the gametes being fused are from different individuals. And most of the costs of “sexual reproduction” listed above have to do with its inter-individual nature. Here we focus on “inter-individualness” rather than recombination, and examine the former for implications with respect to the evolution or maintenance of sexual reproduction. Since different individuals necessarily occupy different locations in space and time, they can have differing experiences of the same habitat simply due to small-scale spatial heterogeneity in landscape and events. For example, in acquiring resources or incurring injury, individuals of both better- and worse-adapted genotypes can have both good and bad “luck”. This individual- level ecological noise is certainly widespread, and by contributing to noise in the transmission of alleles from one generation to the next it weakens the role of selection in genetic change. We wondered whether combining genes from individuals with separate experiences of ecological noise can “average out” luck and thereby reduce noise in inheritance. To our knowledge the relative degree to which reproductive modes compound within-generation individual-level noise has not been addressed before. What is needed for the comparison is expressions of noise in inheritance in terms of parameters describing the effects of ecological noise on the individuals of a population, under different reproductive schemes. Other contributors to noise in inheritance include genetic drift and Hill-Robertson interference, both of which have been well studied. Genetic drift (Wright 1945; Kimura & Maruyama 1963) is 20 adults offspring individuals new adults 2 1 3 Figure 2.1: A generic life history. The sources of noise in inheritance we consider are: 1) simple density dependence, 2) individual success in gathering reproductive resources, 3) spatial noise in offspring survival. caused by the fact that, in a finite population, the new adult generation contains only a sample of all of the offspring producible by the previous adult generation. This sampling may over-represent inferior genotypes and under-represent superior genotypes, slowing selection or even causing mal- adaptive changes in gene frequencies. Selection acting in different directions at correlated loci is another source of noise in inheritance with respect to any of the loci (Hill & Robertson 1966; Felsenstein 1974). The reduction of this Hill-Robertson interference by recombination can facil- itate adaptation (Barton 1995; Keightley & Otto 2006). Segregation reduces similar interference among whole chromosomes (Kirkpatrick & Jenkins 1989). We expect that any dampening of eco- logical noise would also facilitate adaptation. To derive expressions for noise in inheritance in terms of individual luck, we modelled a popu- lation of annual hermaphroditic plants which has a constant number of adults (N). Since plants are sessile, genotype-independent individual-level noise is easily visualised as a consequence of spatial heterogeneity in landscape and events at the scale of individual plants. (But we note that motile or- ganisms also experience ecological noise.) We consider three different reproductive modes: inter- individual fertilization (Crosser), self-fertilization (Selfer, amphimixis that is not inter-individual), and apomixis (Cloner, in which seeds are made by mitosis). A generic life history is diagrammed in Figure 2.1. In classical studies of drift, the N individuals of the new adult generation are each made from a pair of gametes drawn from an infinite gamete pool. This accomplishes two things: construction of offspring genotypes, and population regulation at constant size. Here we separate these steps, and model an explicit, finite, offspring pool (Figure 2.1) produced from the limited reproductive resources of all individuals. We then choose N offspring at random from the pool to form the new adult generation, effectively creating a mortality rate proportional to the size of the offspring pool. We refer to this step as “simple density-dependence” (Figure 2.1, source 1). It creates the sampling noise typically associated with genetic drift. The results using this structure are the same as the classical results when the pool of offspring individuals is very (infinitely) large, and into this generic structure it is easy to place additional sources of 21 individual-level noise. Such noise can arise from many different processes operating at various life stages. We lump these into luck in fecundity of parents (Figure 2.1, source 2), and luck in survival of offspring (Figure 2.1, source 3). As examples, one plant might have its stem and/or leaves damaged by a cow, impairing the acquisition of reproductive resources, while another plant might be well-fertilised by the cow. And for no fault of genotype, plants might be located next to inhospitable habitat such as barren ground or a lake, or not, thus creating noise in the survival rate of individual plants’ dispersed seeds. (Similar assumptions could be made for many other organisms.) Luck in fecundity was assumed to result from luck in the acquisition of reproductive resources and was modelled as variance in the number of ova produced by an individual. Luck in the survival of progeny was modelled as a probability that all of a plant’s seeds survived to (or died before) the simple density dependence stage. The density dependence is then a subsetting of the surviving offspring to yield the constant adult population size. Plants are able to alter the portion of acquired resources they allocate to ova and pollen — when they have more resources they usually allocate relatively more to female function (Sarkissian et al. 2001). We characterised the Crosser strategy as having variable allocation such that pollen pro- duction is the same for all plants, and the amount of reproductive resources each plant manages to acquire is reflected only in the number of ova each produces. Later we consider a strategy in which ova and pollen are produced in a fixed ratio. Selfers do not need to produce as much pollen as out- crossers, and the Cloners would not need any pollen. So, compared with outcrossers, one would expect optimally adapted asexual strategies to allocate proportionally more reproductive resources to ova. In this paper we examine only one factor, inter-individualness, and do not simultaneously account for reallocation of resources among strategies. Another way to think of this work is that we are implicitly considering the problem of maintenance of sex and a possible benefit of sex compared to strategies which are less inter-individual. (Reversion to at least partial selfing is a fre- quent evolutionary occurrence in plants (Harder & Barrett 2006).) We discount the possibility that a single mutation could simultaneously produce both reversion to selfing (or cloning) and optimal resource reallocation, and assume that the sex-to-asex mutation comes first, and that evolution of reallocation of reproductive resources comes later. Thus the strategies would have initially identi- cal resource allocation schemes. For these reasons, we model luck in fecundity as variance in the number of ova produced by a plant, for all reproductive strategies. An allele frequency (p) drifts because of variance in the change in p from one generation to the next. But instead of comparing variance in gene frequency, it is customary to compare the “variance effective population size”, Ne, for which p(1− p)/(2Ne) equals the observed variance of the allele frequency p. There is an extensive population genetics literature on Ne (Wright 1938; Kimura & 22 Crow 1963; Crow & Kimura 1970; Ewens 1982; Crow & Denniston 1988). We note emphatically that in the Ne literature variance in “reproductive success”, “family size”, “offspring number” and “successful offspring”, all refer to the variance of the distribution of “new adults” sensu Figure 2.1. This is not the same as the within-generation “luck in fecundity” or “variance in number of ova produced” with which we deal. Variance in “family size”, or in number of gametes which end up in new adults, already incorporates the consequences of all ecological noise-generating processes. This lumped result is a basic element of many Ne derivations. For example, it is well known that Crow and Kimura (1970) compiled expressions for variance effective population size for different modes of reproduction (p. 362). But in their “retrospective approach of defining the effective number” (p. 353) they start from the distribution of gametes contributed to the new adult generation — which as our results will show might already differ in variance among reproductive modes for individuals which are otherwise identical and under similar ecological circumstances. More recent work on Ne (Caballero 1994; Whitlock & Barton 1997; Wang & Caballero, 1999; La- porte & Charlesworth 2002) continues to skip within-generation noise as it deals with complexities such as population subdivision, sex and age classes, class-specific rates, and autosomal versus sex chromosome inheritance. To address our question about dampening of ecological noise we need expressions containing parameters describing sources of within-generation individual-level noise. Methods The particular life history we modelled is shown in Figure 2.2, along with approximate numbers of individuals at each stage. The negative binomial distribution was used to describe ova production because, unlike the Poisson distribution, its variance is not constrained to be the same as its mean. The negative binomial has two parameters: mean λ , and a parameter ω which influences variance. When ω is very large, the negative binomial is identical to the Poisson and has a variance of λ . As ω decreases, the variance of the negative binomial increases, allowing representation of greater noise caused by individual luck in fecundity. A greater or lesser proportion of a plant’s seeds may survive only by accident of the plant’s location. An extreme form of such luck was modelled by introducing a parameter v describing the probability that all seeds survive from time of production until germination, after which the density dependence is considered to act. With probability (1−v) all seeds from a plant are lost before germination. We denote as σ2 the variance in the number of copies of a focal allele from a given adult that make it into the next generation. That is a direct measure of noise in inheritance. We used moment generating functions (MGFs) to derive expressions for σ2 in terms of the parameters of the fecun- dity distribution (λ and ω) and probability of survival (v) for each of the three breeding strategies: 23 adults (N) adults (N) seeds (~N!) germinating seeds (~N!v) fec un dit y density dep. survival Figure 2.2: The specific life history used, showing the numbers at each stage. N is population size, λ is mean fecundity, v is the probability of survival to germination. The fecundity, survival and density dependent processes all produce noise. Crosser, Selfer, and Cloner. MGFs are a way to uniquely specify a probability distribution. There are established rules for producing the MGF of some compound distributions by transforming and combining the MGFs of their component distributions. And it is straightforward to derive the vari- ance of a distribution from its MGF. The method is to start with the MGF describing the number of ova produced by an individual, and then move step by step through the life history outlined in Figure 2.2 successively building the MGFs to describe: the number of gametes of an individual that make it into the offspring pool (seeds), the number of focal alleles in the offspring pool, the number of focal alleles that are in offspring which survive to germination and, finally, the num- ber of focal alleles which survive the density dependent mortality to make it into the new adult generation. From this final MGF the variance σ2 was then computed. The derivations assumed global pollination and that N is not very small. The analytical work was done using Mathemat- ica (Wolfram Research Inc. 2001), and the results were checked with numerical simulations using R (R Development Core Team 2006). A brief example follows. A Mathematica notebook fully describing the methods for all results, and the R script, are in Appendix B. A brief example Here we find σ2 for a Selfer, with noise in fecundity only. We use four rules about Moment Generating Functions (MGF): 1. MGFX+Y (t) = MGFX(t) ·MGFY (t) for independent X and Y 2. MGFaX(t) = MGFX(at) 3. MGFX+b(t) = ebt ·MGFX(t) We use the following result from Bowers et al. 1986, Chapter 11. Consider the random sum S = X1 + . . .+XN . If the Xi are identically distributed random variables with MGFX(t), N is a random number described by MGFN(t), and N and all Xi are mutually independent, then the MGF of S is: 4. MGFS(t) = MGFN( logMGFX(t)) 24 The mean of a distribution is the first derivative of its MGF evaluated at t = 0. The variance of a distribution is the second derivative of its Central Moment Generating Function (CMGF) evaluated at t = 0. Given µ as the mean of the distribution, CMGFX(t) = MGFX−µ(t) = e−µ t ·MGFX(t) which is an implementation of rule 3. The MGFs for the negative binomial distribution and the Bernoulli distribution are: mg f NegBin = ( w w+λ−et λ )w mg f Ber = (1− p)+ pet Fecundity, the distribution of seeds a Selfer puts into the offspring pool, is described by a negative binomial distribution, so mg f O f f = mg f NegBin. Simple density dependence means only 1/λ seeds becomes a new adult, and the number of those survival trials is determined by mg f O f f , so we use rule 4 to describe the distribution of seeds that make it to adulthood. mg f DensDep == mg f Ber |p=1/λ mg f AdSeed = mg f O f f |t= log(mg f DensDep) A Selfer seed can have 0, 1, or 2 focal alleles. The number of gametes that make it to adulthood is twice mg f AdSeed, and each of those gametes has either the focal allele or its homologue. mg f AdGam = mg f AdSeed |t=2 t mg f AlleleDraw = mg f Ber |p=1/2 mg f AdFocal = mg f AdGam |t= log(mg f AlleleDraw) mg f AdFocal describes the distribution of focal alleles making it into the new adult generation. As a check, we make sure the mean is 1 (constant population size): d dt (mg f AdFocal) ∣∣ t=0 It is. We now compute the variance, which is the σ2 we want. This is the value in Table 2.2, without noise in progeny survival (i.e. v=1). σ2 = d 2 (dt)2 (e −t ·mg f AdFocal) ∣∣∣ t=0 The result is 3/2+1/w. For the derivation of all results, see Appendix B. Results Our results are in the form of expressions for σ2, and our figures compare the relative size of σ2 for one reproductive strategy versus another. As a conceptual baseline we first derived σ2 for no variance in fecundity (all individuals produce exactly λ offspring), and no individual variance in seed survival. Table 2.1 shows the results for simple density dependence only. The expressions reflect the reality that the gamete/offspring pool produced by a finite population is finite — there is a dependence on mean fecundity. Note that σ2 increases with mean fecundity because the greater 25 Table 2.1: Variance in number of copies of a focal allele transmitted into the new adult gen- eration (σ2) given only simple density dependence. Crosser: 1− 1λ + 34λ (1− 13N ) Selfer: 1− 1λ + 12 Cloner: 1− 1λ N is the census size of the population, λ is mean fecundity. There is no variance in fecundity. the size of the finite offspring pool, the deeper the sub-setting of that pool via density dependence to create the new adults. Cloner has the least noise in inheritance, which is approached by Crosser at large mean fecundity. Segregation contributes to the σ2 of both Crosser and Selfer, but for λ > 1 (integer values, since each parent produces exactly λ offspring in this case), Selfer has the higher σ2 due to the greater variance in number of copies of focal alleles in Selfer offspring (many offspring have two copies or none). Since we are modelling a population at constant size, the expected number of copies of a neutral focal allele in the new adult generation is 1. So variation of Poisson magnitude would mean a σ2 that is also 1. For λ > 2 the Selfer variation cannot be that low, even in this (unrealistic) case without noise in either fecundity or survival. Table 2.2: Variance in number of copies of a focal allele transmitted into the new adult gener- ation (σ2) given simple density dependence, variance in fecundity, and noise in survival. Crosser: 1v (1+ 1 ω )+ 1 2 −(14 − 34N )(1v −1) − 1v ( 34ω (1− 1N )+ 12(1− 1Nλ )) Selfer: 1v (1+ 1 ω )+ 1 2 Cloner: 1v (1+ 1 ω ) N is the census size of the population, λ and ω are the parameters of the individual fecundity dis- tribution, v is probability a plant’s seeds are not all lost. Table 2.2 lists the results when all three sources of noise (Figure 2.1) are incorporated. It should be noted that increased individual-level noise (smaller v or ω) always increases noise in inheritance. Also, the two uniparental means of reproduction differ only in a very simple way, due 26 5 10 15 20 1 2 3 ! 2 (S elf er ) :  ! 2 ( Cr os se r) a mean fecundity ( " ) 5 10 15 20 b  4 "  2 "  1 "  0 var (fecundity) 5 10 15 20 c Figure 2.3: Relative noise in inheritance for selfing versus outcrossing given (a) simple den- sity dependence only, no variance in fecundity, (b) simple density dependence plus variance in fecundity, and (c) all three variance sources: simple density dependence, fecundity, and survival (v = 0.5). to the effects of segregation in Selfer. The σ2 for Crosser can be lower than that of either other strategy. Lower noise in inheritance (smaller σ2) means a greater role of selection in changing the frequencies of alleles within a population, which is advantageous to the extent that natural selection builds a better match of genotype to the local environment. (Obviously σ2 and response to selection are population-level attributes.) Figure 2.3 illustrates that, as more sources of noise are taken into account, the σ2 of Selfer increases relative to that of Crosser. The dip in relative advantage for Crosser at low mean fecundity (2a) is eliminated or reversed when variance in fecundity is included (2b). The greater the variance in fecundity (smaller ω in Table 2.2) the smaller the relative size of σ2 for the sexual strategy, especially at low fecundities. In addition, greater noise in survival boosts that relative advantage of outcrossing across all mean fecundities (Figure 2.3c). Figure 2.4 shows analytical and simulation results across ranges of mean fecundity, population size, and probability of offspring survival. It can be noted that the advantage of Crosser relative to Selfer does not decline with increasing population size. The relative advantage climbs as it becomes more likely that all of a given plant’s offspring be lost. But for small values of v the Crossers run out of other-ova spatial refuges for their gametes, their variances become more similar to that of a Selfer, and the relative advantage decreases. The analytical results are not accurate for small numbers of surviving individuals, as expected from the assumptions made in the derivation of σ2. We note that the allocation scheme used in Crosser is commonly both empirically observed (Andrieu et al. 2007; Méndez & Traveset 2003; Sarkissian et al. 2001) and theoretically predicted 27 mean fecundity ( ! ) " 2 (S elf er ) :  " 2 ( Cr os se r) N = 100, v = 0.5 5 10 15 20 1 2 3  4 ! 2 !  ! var (fecundity) population size (N) ! = 10, v = 0.5 20 40 60 80 100 prob. survival (v) 0 0.2 0.4 0.6 0.8 1.0 N = 100, ! = 10 Figure 2.4: σ2 of Selfer relative to σ2 of Crosser across each of the parameters λ , N and v. All three sources of noise are included. The lines are analytical results for 1-, 2- and 4-times Poisson variance in fecundity. The points are from variances of 50,000 simulation runs. For clarity the simulation results for Poisson variation are omitted. for other reasons (Sato 2004; Zhang & Jiang 2002; Greeff et al. 2001; Brunet 1992). But as a contrast with the plastic Crosser strategy, we also examined an extreme in which the ratio of ova to pollen produced is fixed, termed this strategy CrosserALT , and found this expression for v = 1: σ2 o f CrosserALT = 1+ 1 ω + 1 2 ( 3 2λ )(1− 1 3N ) We were unable to derive an expression for CrosserALT which includes the probability of progeny survival, so we used numerical simulations. Figure 2.5 illustrates the performance of CrosserALT across mean fecundity, population size and offspring survival, relative to the Selfer strategy toward which a plant could revert. It is notable that CrosserALT , unlike Crosser, does not dampen noise in fecundity. This is because a CrosserALT plant which produces more than the aver- age number of ova, can also fertilise more than the average proportion of all ova in the population. At the offspring stage, there is always more variance for a CrosserALT than for a Selfer. The final σ2 is always lower for CrosserALT only because simple density dependence adds more variance in the Selfer strategy due to the greater variance in number of copies of the focal allele in the offspring, mentioned above. CrosserALT does, however, generate the same relative advantages as Crosser at lower probabilities of progeny survival. 28 mean fecundity ( ! ) " 2 (S elf er ) :  " 2 ( Cr os se r) N = 100, v = 0.5 5 10 15 20 1 2 3  4 !  2 ! var (fecundity) population size (N) ! = 10, v = 0.5 20 40 60 80 100 prob. survival (v) 0 0.2 0.4 0.6 0.8 1.0 N = 100, ! = 10 Figure 2.5: σ2 of Selfer relative to σ2 of CrosserALT across each of the parameters λ , N and v. All three sources of noise are included. The points are from variances of 50,000 simulation runs. Discussion For both inter-individual and lone-individual means of reproduction, we have built forward through a life cycle to produce expressions for noise in inheritance that include parameters describing two sources of individual-level ecological noise. Our results (Table 2.2, Figure 2.4) show that the inter- individual Crosser strategy can dampen the effects of noise in fecundity and in progeny survival relative to the lone-individual Selfer and Cloner strategies. This means that under similar eco- logical conditions, otherwise identical individuals can generate different distributions of recruits (“successful offspring”, “family size”) or of “successful gametes”, depending on mode of repro- duction. This is of importance to calculations of variance effective population size, which often use the variance of the latter distributions. The noise-dampening properties of the Crosser strategy can be understood in the following ways. First, under the reproductive resource allocation scheme assumed for Crosser only female gametes are influenced by noise in fecundity — each individual contributes equally to the pollen pool, so the only pollination noise is in the draw from that pool to fertilise each ovum produced in the population. In Selfer, noise in fecundity hits male and female gametes equally since they are matched one-to-one. Second, noise in progeny survival is relatively well-buffered by the sexual strategy because pollination is an additional means of dispersal for alleles, and dispersal is a well- known buffer against spatial heterogeneity. The results show that, without proper attention to within-generation sources of noise, analytical and simulation models of evolution may overestimate the effectiveness of selection, and may hide differences among reproductive strategies. Simulations which construct the new adult generation 29 by drawing gametes from parents, with probability of choosing a given parent being weighted by its relative fitness, produce Poisson-distributed successful gametes (unless the population is growing very quickly). This accounts for the effects of relative fitness and drift, but not of ecological noise. Our results show that, given ecological noise, variance in transmitted alleles cannot be as low as Poisson variance (Table 2.2). Only Crosser can approach the latter, when both fecundity and population size are very large. We have confined this paper to illustrating a difference between reproductive modes. Work remains in quantifying the relative importance of noise dampening, and to what extent it can influ- ence the maintenance or invasion of sex at the within-population level of individual selection. The relative merits found for sexual versus asexual reproduction depend on many factors, including population size, the product of population size and mutation rate (Nµ), the presence of epistasis, the existence of linkage disequilibrium, the distribution of sizes of selection coefficients, and the substitution model used (Kim & Orr 2005). But some qualitative statements follow. The differential amplification of individual luck has consequences for adaptation. Casually phrased, adaptation is a matter of 1) producing the right variants, and 2) getting them to the right frequency. Gillespie (Gillespie 1994) has termed these “origination” and “fixation” processes re- spectively. Noise in inheritance impacts both. Variance effective population size varies inversely with σ2, and in the case of Crosser, it is ex- actly Ne = N/σ2 (Gillespie 2004 p. 49). With regard to “fixation”, in results from existing theory which combines the effects of selection and drift, Ne and s (the selection coefficient) usually appear multiplied together. The generation of homozygosity by selfing can have the effect of increasing the size of s (Lynch et al. 1995; Glémin 2003), facilitating selection. On the other hand, a larger Ne allows selection a role in changing the frequencies of alleles of smaller effects, both deleterious and beneficial. Work on the distribution of sizes of mutational effects finds that mutations become exponentially more frequent at smaller sizes of s (Lynch et al. 1999; Orr 2006). So it seems rea- sonable that conditions allowing selection a greater role in changing the frequencies of mutations of small effect could have significant consequences with respect to fitness. Indeed, the results of a previous theoretical study (Peck et al. 1997) of hermaphroditic plants which incorporated muta- tions of various effect sizes suggested that “even traits that have small effect on Ne may have large effects on fitness.” And differential amplification of ecological noise into σ2 affects Ne. Concerning “origination”, note that selection is strongest on alleles at intermediate frequencies (near 0.5) and it is vanishingly weak on alleles at very low frequencies, such as just after they appear by mutation. Even highly beneficial alleles can be lost at this stage, simply through the noise of individual luck. If beneficial mutations are rare, then they may limit adaptation to a 30 changing environment. Any dampening of the effects of individual-level noise would reduce the chance of throwing away any new allele just by accident, and let selection “see” farther into low allele frequencies, thus allowing a greater number of beneficial mutations to be recruited into a population. If individual-level noise is ubiquitous, and empirically observed sexual strategies dampen the effects of such individual luck and facilitate adaptation, then this noise-dampening is a general ben- efit which should be deducted from the costs of sex. The extent of this cost deduction awaits further study. We note that such noise-dampening is merely a consequence of the inter-individualness of sex, and that the effects reported here happen in addition to other sexual processes, such as recom- bination and segregation. Within the realm of plant reproduction, current theory about sex allocation in hermaphroditic plants (Klinkhamer et al. 1997; Cadet et al. 2004) takes into account the importance of plant size, the size of budget available for reproduction, and the frequencies of other strategies in the locale. Our results show that allocation schemes (e.g. Crosser and CrosserALT ) can differ with respect to noise-dampening, which suggests that this property should be considered when evaluating alloca- tion strategies on an evolutionary time scale. In nature, individual plants may engage in a mix of both outcrossing and selfing strategies and, within a given species, local populations may differ in the levels of partial selfing that is observed. Explanations have been sought in terms of balances or tradeoffs between the advantages of selfing, such as reproductive assurance, and its disadvantages, such as inbreeding depression. The present work suggests that another disadvantage of uniparental reproduction is relative vulnerability to individual-level ecological noise, and predicts that in habitats where there is more such noise, selfing should be less predominant. 31 Chapter 3 Finding an approach — preliminary studies Introduction This chapter presents several theoretical explorations of the hypothesis that cycling small mammals are phenotypically plastic, and are evolved to alter their behaviour, notably their reproductive rate, at different stages of the cycle. Two of my early studies, one a non-dynamic one, and the other based on the neutral cycles of a Lotka-Volterra system, are presented in Appendices C and D. These both supported the notion that reproductive slowdown can be adaptive, but they were both too simplistic to be convincing. They assumed that all mortality was by predation, and thereby lacked generality since cycles sometimes happen without significant predation. Nor did they include considerations of space, the importance of which is hinted by the unusual “pre-saturation dispersal” that is common to cycling lemmings, voles and snowshoe hares. Most convincing would be a simulation model in which one could actually watch a pattern of reproductive slowdown evolve. It is not the only way to verify a computer simulation, but a powerful way is to build the simulation so that it can be collapsed to cases which correspond with known analytical results or properties. For robustness and wide applicability I wanted to model with the fewest universal ingredients, just: lemmings, their food, and space. This required a model for local dynamics of only lemmings and food, which could be extended to incorporate dispersal and space via computer simulation. The Rosenzweig-MacArthur (R-M) pair of differential equations has often been used to describe cycling population dynamics, so I selected it as a candidate analytical system. I then carefully considered the biology that it either assumes or implies, since I would have to build that biology into the individuals simulated in my computer model, the individuals that will potentially alter their phenotype during a population cycle and disperse in space. Even if it turned out that I had to make alterations to the R-M model, it might be easier to make an argument that extends an 32 accepted system, rather than to invent a different “wheel”. Although all models contain simplifi- cations, indeed their purpose is to simplify, I found that the R-M implies some biology which is quite unrealistic, especially for populations with high-amplitude cycles. To incorporate the realism of a reaction norm of reproductive effort would have been inconsistent with the lack of reaction norm in things like searching intensity, which should vary considerably between high and low food abundance. As described below, I began patching this model for my purposes, but the modifica- tions would have been extensive enough that I would have lost any advantage of building from a widely-known standard. Dissection of the Rosenzweig-MacArthur model The standard Rosenzweig-Macarthur model is: dN dt = r N ( 1− N K ) − ( aN 1+ahN ) P dP dt = ( aN 1+ahN ) bP−d P This system of two first-order ordinary differential equations describes the rate of change of the densities of resource (N) and consumer (P), which vary continuously as real numbers. Since pro- ducer and consumer populations are composed of individuals, the densities in the model are really average densities, unless the populations are assumed to be very large. All sub-processes are as- sumed to be continuous so, for example, the events of reproduction and predation are modelled in the R-M as processes which, on average, are gradually happening all the time. Feedbacks are instantaneous, so a change in density immediately changes terms on the right-hand side. The growth (positive) term in the producer equation of the R-M model provides for logistic growth of the producer in the absence of the consumer. This pattern of growth is a robust result of opposing effects of density. If greater density speeds the increase of density at one rate, and speeds the decrease of density at a second, different rate, the net result is logistic growth. The biological validity of this growth term in a model of voles and food is contested by Turchin and Batzli (2001). They argue that a logistic term is only appropriate if the growing ability of the plant can be significantly reduced by grazing and that, while this might indeed be the case for some lemming food (e.g. mosses), it is not the case for the grasses which voles eat. They write that in the opinion of most botanists, grass regrowth is better modelled as “linear-initial”, with new shoots springing readily from energy stored underground. But Turchin and Batzli’s argument does not hold up if voles always forage the way I saw them do. 33 Thanks to Charley Krebs and Don Reid, I was able to observe lemmings and voles in the real world — on Herschel Island in the Arctic Ocean, as part of an International Polar Year project. Tundra voles always act as though they are potential prey. When feeding, they dart out from a place of cover (e.g. underneath a piece of driftwood), nip off a piece of grass, and run back to cover. There they eat the grass shoot from base to tip. Then they dart out to nip off another shoot, return, and repeat the procedure. Given this modus operandi it would seem that the utility of the grass to the voles varies with shoot length. If a field of mowed grass is regrowing linear- initially, there is at first only a tiny piece of green shoot on each grass blade. For a vole foraging as observed, this means a great many trips from cover with very little payoff for each trip. As the bite sizes increase in size, the grass becomes more efficiently exploitable by the voles. So, even if the grass grows linear-initially, I think it is reasonable that the net utility to the voles accelerates for a while. In other words, the regrowth of food value from the voles’ point of view is logistic, even if the regrowth of above-ground biomass is linear-initial. If the R-M had some means of deducting foraging costs, a food growth term could remain linear-initial and the cost term could change with food abundance. Since it does not contain a cost-of-foraging term, it seems a reasonable simplification that the growth term represent the growth of net utility to the animals rather than just the amount of food that is above ground. So I would argue that a logistic term is more appropriate even for grass-eating voles. The next two terms of the Rosenzweig-MacArthur model are built from the consumer “func- tional response”, which is the per capita rate of consumption of food as a function of food density. In the R-M the functional response is the Holling Type II, a function which accounts for the fact that food must be handled, so consequently the maximum consumption rate occurs when all for- aging time is spent handling (none searching). The parameters of the Type II as used here are a, the search rate in area per time, and h, the handling time per food item found. While searching, food is found at a rate of aN (area per time × food items per area), so the average time to find a food item is 1/(aN). Let foraging time be defined as the sum of two components, searching time and handling time. The Type II functional response is the mean proportion of foraging time spent searching, 1/(aN) 1/(aN)+h = 1 1+ahN multiplied by the mean rate at which food is discovered while searching (aN). The way the R-M is written, it is assumed that the consumers spend all of their time foraging (all of each time step dt). No time is spent for reproducing or self-maintenance, e.g. for feeding young or resting. So what we have is a population dynamics model which, although containing the detail of handling time for each food item, does not account for the fact that animals do more than just forage. This 34 means inaccuracy unless non-foraging time is trivial in comparison with total handling time. (In “foraging arena” theory, the total time spent foraging, i.e. for which the Type II pertains, is varied as a consequence of predation risk faced while foraging.) In the Type II functional response, the search rate a and handling time h are constant parameters which do not vary with food density. The constant handling time means that consumers never high- grade their food during times of plenty, nor spend more time extracting all benefit possible from food during times of scarcity. The constant search rate implies two things. First, consumers are never satiated and their rate of intake is limited only by handling time. Second, as food density falls, so does the per capita consumption rate of the consumers — the consumers never alter either parameter in order to maintain a target net intake rate. In other words, homeostasis, the hallmark of living systems, is completely absent from this model which is capable of representing cycling dynamics and therefore widely differing conditions. A lack of homeostasis could be argued as realistic in cases where animals have no target intake rate, and simply maximise intake rate (constant maximal search rate) and produce another offspring just as soon as they have accumulated sufficient resources. But most animals produce offspring in reproductive bouts (e.g. in litters). Reproductive bouts are typically longer than foraging bouts, and the R-M bases changes in food density on the shorter foraging bout dynamics. This means that food availability can change during a reproductive bout. Having chosen a litter size, an animal cannot increase it during a reproductive bout just because more food is available — the amount of food an animal can use does not increase just because food density increases. And yet that, and the converse, is what is modelled here. The implication to note is that search rate will not always be constant and can be expected to be lower than maximum if the maximum possible food intake rate is not required. In general, I think it is more common that animals make life history choices which determine an intake rate goal, and then homeostatically seek to maintain that. There is another problem with search rate. The Type II functional response is a mean field expression which assumes a “well-stirred” mix of predators and prey. This requires that preda- tors and/or prey are constantly globally-dispersing. Note that in Holling’s (1959) original “disk” experiments, the predator was a human hand that could search any part of the table surface with equal (no) cost. In the case of small mammals eating plants, neither predator nor prey are globally dispersing, and areas local to where a predator is feeding become depleted of plants. To remain realistic under those conditions, the Type II prey density would need to be the density of the areas likely to be explored by the animal, including previously exploited areas if the animal is inclined to revisit some areas. This requires knowledge of an animal’s foraging patterns. Alternatively, if previously visited areas are assumed to be depleted, the constant search rate used should be NEW- 35 areas-searched-per-time. This again requires knowledge of an animal’s foraging patterns. So as soon as space is considered, issues of scale arise, and use of the Type II is assuming some very specific things about the foraging patterns of real herbivores in a spatial context. Now to consider the next term in the model. In the growth term of the consumer, the harvested food is converted into consumer density at a fixed discount rate (b). This means that the net benefit to consumer density of a given amount of harvested (captured) food is always the same. The consumer that eats an apple every minute gets the same net benefit from each apple as it would get from just one apple that required a whole day of searching to find. (Remember that the model assumes that all time is spent either searching or handling.) What is implied is that searching is free of cost in both energy consumption and mortality risk, so that there is never anything to deduct from the food value of a prey item — there is only a fixed conversion (digestion) efficiency. Finally let’s consider the loss term in the consumer equation, −d P. This is often described as the rate at which consumer density decreases when there is no food. But this is incomplete. It is actually the rate at which consumer density drops at all times, regardless of how much food there is. So in the R-M system, which describes the dynamics of food and consumer, there is no death by starvation! The only deaths are occurring at a constant rate due to some other process. I don’t mind so much that there is some unspecified mortality because, after all, the resources which allow logistic growth of the food are not specified either. But to make one trophic connection, from harvesting food to increasing consumer biomass, and not make the other trophic connection — lack of food leading to starvation death of consumers — seems inconsistent. In the R-M, when food density is zero, the number of consumers just fades away in an exponential decay. This means that the individuals in the tail of the distribution somehow survive a long while with no food, implying significant variation in the quality of individuals. One way to make the death term look reasonable is to suppose that the densities of the producers and consumers are biomass per area, rather than individuals per area. Then the loss term of the consumer can be interpreted as a constant loss rate of biomass (e.g. to metabolism), and the gain term is the intake rate of biomass. But this perspective implies that the Type II functional response is the capture rate per gram of consumer. Individual-based simulation would then be in units of mass, and the noise resulting from such stochastic simulations would be difficult to map to a real- world analog. Dispersal over space would be in terms of grams, rather than individuals. The biomass perspective that could make the death term seem reasonable would make other parts of the model look odd. In summary, the Rosenzweig-MacArthur describes partially coupled rates of change of mean densities (individuals per area) of resource and consumer. The consumers spend all of their time 36 foraging and exhibit no homeostasis with respect to food intake rate. They are never satiated — maximum consumption is limited only by handling time. The consumers search without cost in either energy or mortality risk, and the system is assumed to be aspatial (well-stirred). Consumers never starve, they die at a constant rate regardless of food abundance, due to some unspecified process(es). Modification of the Rosenzweig-MacArthur model Rather than immediately discarding it, I first tried to fix the greatest shortcomings of the Rosenzweig- MacArthur and adapt it for individual-based simulation. There is a literature on means to exactly and efficiently stochastically simulate chemical re- actions which, at a mass action scale, are modelled as systems of ordinary differential equations. These simulate chemical reactions molecule by molecule (analogous to animal by animal), so that at the lowest scale, reactions are step-wise rather than continuous. Rik Blok explained this methodology to me. He uses it as a systematic way to create differential equation models with sound mechanistic underpinnings. Note that birth, death and predation are all step-wise processes at the lowest level. Also, it is well known that the Type II functional response is homologous to the Michaelis-Menten expression for reaction kinetics in which two reactants (e.g. predator and prey) combine to form an intermediate complex (prey being handled) which breaks down after a certain (handling) time. I created a system of chemical reactions which, at a mass-action scale, result in exactly the R-M system. This gave me a structure with which I could experiment to find ways of correcting some of the shortcoming of the R-M which I noted above. In chemical reactions, all molecules of a given kind are identical and stateless. Representation of different states must be done by using different kinds of molecules, each kind having different properties. For example, a predator which could be in either a searching or a handling state must be modelled as two different entities, searchers and handlers, with a searcher “reacting” with a prey molecule to become a handler, and the handler “decaying” after a certain time to become a searcher again. Here is the system of reactions from which the R-M results when searchers (S) and handlers (H) are grouped together as predators (P), and prey items are the reactants N: 37 N→ 2N at rate α } together result in logistic growth of N 2N→ N at rate β S+N→ H at rate a H→ S at rate (1−b)/h } equivalent to: H→ (1+b)S at rate 1/h H→ 2S at rate b/h S→ /0 at rate d H→ /0 at rate d This gives the standard R-M system (see Appendix E). Note that in the above predators always die at the same rate regardless of what they are doing (S or H). My first tweak was intended to correct the most glaring problem with R-M — the lack of any death by starvation. If we assume that predators do not starve while they are eating, only while they are searching, then we can modify the above by imposing a mortality only on searchers (by removing the last reaction from the system). Then the consumer equation looks like this: dP dt = ( aN 1+ahN ) bP− ( 1 1+ahN ) d P = ( abN−d 1+ahN ) P The parameter d is now a “death rate while searching”. Consumers die at rate d P when food density is zero, just as in the original R-M, but at higher food densities the “death” rate is less. Although the food dependence of the death rate appears more like the result of a starvation process, this is only a phenomenological patch — there is no mechanistic rationale for a starvation rate that varies directly with the proportion of foraging time spent searching. The term may better suit other losses while searching such as accidental death (e.g. falling off cliffs) and predation. Note again that in the low-level “reactions”, all searchers have exactly the same state, so that the decay (mortality) reaction means that, at any time, a searcher has probability d of being about to starve. It does not matter how many captures that predator has recently made because there is no memory of state (satiated / hungry) for any predator. This probability is constant, so it does not matter how much food is around, the proportion of searchers about to starve is always the same. Instead of a constant death rate for all predators regardless of food density, we have a constant death rate for all predators while they are searching, regardless of food density. Not much better. I did the stability analysis, and incorporation of such a cost makes the system more likely to be stable. Why not carry the implications of Type II foraging through the whole model to accomplish both 38 the biomass conversion into the next trophic level, and to govern consumer death by starvation? If the consumers live by the Type II, they should reproduce and die by the Type II. A more mechanistic treatment of starvation would be to consider the proportion of individuals which fails to intake food at a high enough rate to keep ahead of self-maintenance costs. For this we need to know the distribution of foraging success for individuals foraging in a Type II manner. Such individuals are finding food items at a Poisson rate of aN and then handling each for time h. I simulated this using R (fresp2.R listed in Appendix F) and deduced the distribution of number of food items found and handled within a fixed time period T . The mean and variance are: mean = aN 1+ahN T variance = aN (1+ahN)3 T The mean is famous, but not so the variance. Note that the variance decreases rapidly with increas- ing food density or, conversely, will only be significant at low food densities. It is at these low food densities that we would want to know the proportion of the distribution that ends up below the minimum intake requirements of individual consumers. If we knew the relationship between the size of this tail and food density, then we could make a more sensible starvation mortality term. But this gets very messy very fast. Captures are discrete events, so the intake rate before a success is zero. Clearly an organism has a certain buffer against starvation to carry it through these gaps between meals. How big a buffer? Over what time period do we assess the rate of capture? Buffers get used up, so right away we are into questions of individual state. Even if all individuals have exactly the same maximum buffer size, since the duration of gaps between food captures varies continuously, we have a continuous range in states of individuals (assuming depletion of the buffer during searching is continuous). One way to side-step buffer size is to just play with rate. If over time the intake rate is less than the ongoing costs, then the thing is bound to die. So how does that look? So if the time period of assessment (T) is doubled, then so should be the deducted costs. How does this look for different T? I looked at this with starvecurve.R (listed in Appendix F), which gives the proportion of individuals whose capture rate is less than their minimum self-maintenance needs m∗T . As one would expect, the curve is sigmoidal. Larger T makes the live-or-die transition sharper, because over a longer run the good and bad luck in searching averages out and there is relatively less variance among individuals in the number of captures they make over a longer period. There is no simple algebraic expression for the cumulative normal distribution, so I used a sim- ple sigmoidal expression comprising the variance and mean of the Type II as found via fresp2.R. A sigmoidal death term that fits fairly well can be made with just one more parameter than stan- 39 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  10  m= 0.15 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  50  m= 0.15 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  100  m= 0.15 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  1000  m= 0.15 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  10  m= 0.3 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  50  m= 0.3 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  100  m= 0.3 a=0.02 h=1 N pr op or tio n sta rv ing 0 5 10 15 20 25 30 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 T=  1000  m= 0.3 a=0.02 h=1 N pr op or tio n sta rv ing Figure 3.1: The proportion of the population that is starving versus food density (N) at dif- ferent times (T , rows) for different rates of required food intake (m, columns). Black curve: simulation results; red curve: analytical approximation. The black vertical line marks x, the food density at which half of the population starves. 40 dard R-M. The parameter d is replaced by m, the minimum intake rate to avoid starvation and T , the time period over which the evaluation is made. The latter can probably be thought of as being related to the time the organism’s anti-starvation buffer can last. This approach seems reasonable if starvation happens on a time interval shorter than appreciable change in food density. Let hal f be the food density at which the mean of the Type II exactly equals the minimum intake require- ments of a consumer. At that food density half of the population is catching enough food to avoid starvation. Let varhalf be the variance of the Type II, evaluated at food density half , and for the time period T (which is in units of handling time). In the plots of Figure 3.1, the black line is the actual proportion of the population in the starving tail, and the red line is the expression: 1 1+ varhalf (N−half )/2 which would appear to be a decent starvation term to use. For comparison, the blue line is propor- tional to the cost-while-searching loss term calculated earlier. The sigmoidal shape of the mortality rate has interesting implications for stability and cyclic dynamics from this system. Perhaps an ap- proach to further analysis would be to use a sigmoidal death term of the form − ( 1 1+ tN−x ) P as I did in starvecurve.R, and see happens as t goes from a value of 1 to values larger. In that transition the death rate changes from a constant 0.5 to a sigmoidal curve ranging between 1 and 0 and going through 0.5 at N = x. The reproduction term should be similar in construction to the starvation term, except that it would involve the other tail of the foraging-success distribution. Only those animals that intake enough to cover both self-maintenance plus the reproduction costs can reproduce. (This symmetry might allow simplification by combining the repro and starvation terms.) The reproduction term should also account for foraging costs. Earlier I patched the R-M to include a cost-while-searching. But we need a cost-of -searching, especially if we are considering varying searching rate during a cycle. Which we are, because that is how to represent differing reproductive rate strategies in the R-M. So let’s look at cost-of -searching in a spatial context. In the well-mixed case modelled by the Type II, food is encountered at a constant Poisson rate of aN (and that is how I simulated it in fresp2.R). But without global mixing, the encounter rate with uneaten food is affected by the proportion of time spent re-searching areas in which food has already been eaten. I thought I 41 Figure 3.2: Two views of how the cost of finding food rises if the search is a random walk on a two-dimensional plane, and the food is sessile and not regenerating. The top panel shows the number of search steps required versus area searched. The bottom panel shows the number of steps since encountering a new area, versus area already visited. 42 would have a look at what this proportion might be, and how it might change with decreasing food density, i.e. as food gets eaten. A random walk is one possible search pattern, and can be taken as a null hypothesis about how lemmings forage. If one considers the cost of searching to be the cost of locomotion, i.e. the cost of the number of steps taken, then a linear cost function assumes that to search twice the amount of new area per time takes exactly twice the number of steps. This kind of cost function is only possible under a perfect “mark-and-cover” (or “visit-and-remember”) type pattern in which no area is ever accidentally revisited, or some other scheme by which the proportion of revisits remains constant as the number of visited sites increases. Unfortunately the literature is pretty thin on rate of cover by a random walk. Almost all work is on total cover time, i.e. amount of time or number of steps before all areas are visited. One exception (Barnes and Feige 1996) addresses rate of cover, but only gets an upper bound on it. My simulations (walker.R listed in Appendix F) show linear at start, then increasingly non-linear as many places are visited (see Figure 3.2). That is for a random walk. My intuition is that, on one hand, animals are smarter than a random walk but, on the other hand, revisits are more probable if the search path is constrained to return to the starting point (e.g. a burrow or place of cover), or constrained by area (e.g. by a territory). So I don’t know what exponent to use, but the lower bound is 1 and it is probably not at the lower bound. As cost-of-searching becomes increasingly non-linear, it would make sense for an animal to move locations to forage in an area with a lower density of visited areas. If territoriality or other barriers to dispersal prevent this, then the animal is faced with rapidly increasing costs per food item found. It would be interesting to know the implications for (cycling) population dynamics of this non-linear rise in costs with decreasing food density, but I did not tackle that question. I also note that this explicit consideration of foraging pattern is a general way of arriving at distributions determining what portion of a population starves or gathers enough to reproduce. Earlier the explicit foraging pattern was whatever magic generated a Poisson encounter rate, and the resulting distribution was that of a Type II. Now the assumption is random walk, and info about the distribution is in Figure 3.2. Combining these with the distribution of states of animals, i.e. the extent of their anti-starvation buffers, would be satisfyingly mechanistic. I’m not sure how to approach the math, and this is not really what I set out to do. Dispersal Dispersal plays a key role in my hypothesis about what lemmings are up to. The term −d P might be a net emmigration. But lemmings exhibit an unusual “pre-saturation” dispersal in which 43 dispersal rates are higher at low densities than at high densities. I have done this, adding such a term to the Rosenzweig-Macarthur, starting from the elemental reactions. The lemmings were of two categories high- and low-dispersers, and the more they encountered each other the more likely they were to become low-dispersers. I did a stability analysis of the resulting system and found that pre-saturation dispersal makes it less likely to be stable. Lessons The Rosenzweig-Macarthur is just too far off the mark to be a reference system for local dynamics for my lemming simulations. It is awkward that I have to map the strategies I wish to explore — fixed versus plastic reproductive rate — onto search rate in the R-M. I know of no literature reporting observed changes in search rate during a lemming cycle, so I would have to argue to biol- ogists that, you see, the search rate is just a mathematical proxy for the reproductive rate strategies I would like to examine. Search rate governs the rate of extraction of resources, which automati- cally turn into the density gain in consumers. Then I have the situation where I am supposing that the animals are plastic in that they can reduce their search rate according to a cue during a cycle but, amazingly, they are not plastic in that they increase their search rate for food to counteract decreasing food density and avoid starvation. Then there are the implications of the other simplifi- cations that underlie the R-M, explained earlier. I could see how to profitably use the R-M model analytical system, and moved on to other approaches to studying the life history changes that occur in cycling mammals. 44 Chapter 4 The end of intrinsic versus extrinsic? — within-cycle phenotypic plasticity might not be cued by the environmental factors for which it is adapted Introduction In many regions lemmings, voles and snowshoe hares undergo regular high-amplitude cycles in population density on a period spanning many animal lifetimes. The quality of the animals alive at any time changes throughout a cycle (Krebs & Myers 1974, Erlinge et al. 2000, Gilg 2002, Schaffer & Tamarin 1973, Mihok & Boonstra 1992, Carey & Keith 1979, Norrdahl & Korpimäki 2002). Despite over 80 years of extensive research, there is still no general consensus about what determines the various features of these small mammal cycles, and indeed this area of research is infamous for level of controversy (Hudson & Bjørnstad 2003). Considering the relative simplicity (few components) of the high latitude ecosystems in which these phenomena occur, an ecologist might view this enduring puzzle as both important and somewhat exasperating. In a review Batzli (1992) listed 23 hypotheses for microtine (lemming and vole) cycles, and a few have been put forward since. They span virtually all parameters of an animal’s life, including quantity and quality of predators, food and parasites, spatial heterogeneity with respect to the above, climate, within-cycle genetic feedback, stress and maternal effects. In broad terms the hypotheses can be divided into two groups: intrinsic causes arising from factors within a population (i.e. the animals do it to themselves), and extrinsic causes of the cycle (i.e. the animals get it done 45 to them). Particular researchers have tended to favour the relative importance of either intrinsic or extrinsic factors. Adherents of each perspective have good reasons not to be shaken from their bias. The in- trinsic view can point to experiments in which cycles occurred regardless of food and/or predator manipulation (food: Klemola et al. 2000, Ostfeld et al. 1993; predation: Graham & Lambin 2002, Korpimäki et al. 1994, Norrdahl & Korpimäki 1995, Marcström et al. 1988), as well as obvious changes in the characteristics of the animals throughout a cycle and the degree to which animals are known to influence each other. Levels of stress hormones in the animals have been observed to vary with cycle phase (Boonstra et al. 1998a, Sheriff et al. 2009), and stress is known to im- pair reproduction and survival (Christian & Davis 1964, Wingfield & Sapolsky 2003) and to have consequences across generations (Boonstra & Hochachka 1997, Meaney 2001). Intrinsic stressors include territoriality and violent interactions (e.g. infanticide: Caley & Boutin 1985), and interfer- ence with the dispersal of conspecifics (Hestbeck 1982). These facts clearly demand examination of the role of intrinsic factors, and consideration that predators, for example, might simply be riding on an intrinsically-caused phenomenon. On the other hand, there is also evidence for the importance of extrinsic factors in driving the cycles, both from some field experiments and from mathematical study of observed population density patterns of microtines and their predators (Turchin et al. 2000). Investigators leaning to- ward this view can point out that the changes in the animals throughout a cycle can be interpreted as responses to extrinsic factors, and that snowshoe hares, unlike microtines, are not territorial so that any explanation based on that within-population interaction cannot be general across small mammals. The extrinsic camp have built analytical models (Hanski & Korpimäki 1995, Turchin & Batzli 2001, Gilg et al. 2003), comprised of food, lemmings, specialist predators and generalist predators which mimic the cyclic pattern of vole densities. Among serious investigators a high level of controversy is less likely a case of stubbornness and more likely due to variety in sign and significance of experimental results. What might cause such variety in results? Here I consider the notion that experimenters might usually be manipulating factors of only indirect, or partial, relevance. I hypothesize that, although the animals must be adapted for the extrinsic factors that make up their environment, the plasticity in their life history characteristics need not be cued from those extrinsic factors — the animals could be cueing off themselves. I argue that such a mechanism is a sufficient and general means for effecting life history changes in a few-component ecosystem. Such indirect cueing would blur the distinction between intrinsic and extrinsic causes as separate mechanisms behind small mammal population cycles, and would decrease the experimental power of manipulating extrinsic factors. 46 Cumulative recent activity In the next chapter it is shown that mortality rate (the probability of dying per time) should be a key determinant of reproductive effort during a cycle. This implies that well-adapted plasticity should involve a means to perceive death rate. Obviously the rate itself cannot be “seen” and the animals must either construct an estimate (from direct witnessing of mortality events combined with an estimate of population numbers) or monitor a correlate of death rate. One correlate that can be used is the recent history of the prey population itself. To justify this idea I assert two ecological rules: 1. All animals of a population within a locale have a similar individual death rate. 2. To the extent that a population is author of its own circumstances, death rate for the individ- uals in a locale rises with the “cumulative recent activity” of the local population. The first rule is justified by the fact that ecological mortality processes have spatial extent. Consider starvation, predation, and disease. If food in an area is depleted, then all of the local individuals are more likely to starve (although territoriality can modify this to some extent). If predators are in the area, predation risk exists for all of the prey individuals living there. Local dispersal of pathogens also implies that all of the individuals in an area face similar risks of exposure to diseases. The second rule results from the facts that organisms have effects on their environment that do not instantly “heal”, and that individuals of a given kind share food preferences, predators and diseases. “Cumulative recent activity” is a metric of activity history. The components of “activity” are the number of individuals and the mean activity level per individual. Since food is required to drive animal activity, perhaps food consumption is a good measure of “activity”. Some examples of the lag inherent in the words “cumulative recent” follow. Since consumed food takes a while to regenerate, food-just-consumed is not a good indicator of present food levels. It is the history of food consumption that determines current food levels. Food previously consumed and not yet regenerated must be taken into account. (The lower the productivity of the environment, the longer the time window.) Predators take time to realize that their prey have become more available in the (focal) locale. After their aggregation response the predators’ numerical response then further increases predation mortality in the area. When prey become less available (fewer and/or less active) it then takes a while for predators to decide to leave this formerly-good locale, or to starve. Integration of information over a period of time is a well-known feature of the stress physiology of animals, and and could provide a means of constructing a state corresponding to “cumulative recent” given inputs (stressors) corresponding to “activity”. Such inputs would need to contain information about both population density and the typical level of activity of each animal. Perhaps 47 olfactory inputs could provide both, but in microtines territorial interactions are another possibility for gaining information about density. A state of high level of stress is known to reduce repro- duction (Wingfield & Sapolsky 2003) which, as the next chapter shows is an adaptive life history response during the parts of a cycle in which animals have been shown (Sheriff et al. 2009) to be most stressed. Given the insight that reproductive slowdown itself is adaptive, rather than an inadvertent side effect of measures to increase short term survival, stress physiology seems to be a perfect candidate for controlling phenotypic plasticity via cumulative recent activity. Potential advantages of self-cueing An advantage of self-cueing is that it can “predict” mortality from several different sources at once. If predation, starvation and disease all tend to increase with cumulative recent activity, then animals do not need separate perceptual apparatus for detecting each of those differing potential sources of mortality. Recent activity of others need not be the only input to an animal’s phenotype-determining physiology. If, additionally, animals could detect the assessments that had been made by other individuals (i.e. their stress level) then an information sharing system is produced. Since different individuals experience different trajectories through the environment, obtaining information about the state of conspecifics both dampens spatial noise and increases the area of spatial perception. If stress physiology is intimately bound to reproductive physiology, then the perceived level of stress in another individual is an honest signal — it is not possible for an animal to “cheat” by signalling high stress to cause others to slow their reproduction while it does not slow its own reproduction. In an evolutionary game the strategy which is best often depends upon what com- petitors are doing, and this information would be available via monitoring conspecifics. Honest signals also provide a potential basis for cooperation among animals in a region. Self-cueing permits strategies that are not possible by only monitoring extrinsic factors. For example, it is conceivable that a local area with the same extrinsic cues is best exploited differently during the increase phase than during the decline and low phases. During the pre-saturation dis- persal of the increase phase it may be advantageous to quickly deplete an area’s resources in order to generate more dispersers to invade the ample remaining “green pastures”. During the decline and at the start of the low phase of a cycle, under identical local conditions, but with few remaining green pastures, it may be a better strategy to conserve local resources and reduce local activity in order to boost the odds of survival through the population low. Such a strategy is only conceivable given a proxy for landscape-scale senses, and given a basis for cooperation among genotypes in the locale, both of which can be provided by self-cueing. 48 Potential limits to self-cueing The effectiveness of cueing from cumulative recent activity depends entirely on the predictability of responses by mortality agents. It is possible that in more complex ecosystems a given population is less of a determiner of its own circumstances, or at least that there is more noise in the response of mortality processes. For example, where there are alternate prey and shared predators, cumulative recent activity of the focal population itself is not as effective a predictor of predator buildup, because predator numbers can be influenced by other prey populations each of which may be differentially affected by environmental conditions. Likewise, food could be impacted by other consumers. Implications Some form of delayed density dependence is required for the generation of population density cycles. Delayed density dependence requires that there be memory in the system, somewhere. Since “current activity” has density as a component, the quantity “cumulative recent activity” is delayed density dependent because it is a memory of previous “current activities” discounted by an environmental regeneration rate. Cumulative recent activity can be used to generate density cycles by means of death rates that are agnostic about the source of mortality (as is done in the next chapter). This agnosticism about source of mortality is not just a means to side-step debate about causes, or to make a more phenomenological model that can be generalized across, for example, both microtines and showshoe hares. In some ways it may be more accurate because it can be tuned to cover the multiple mortality sources that are always present in any system. A cycle model explicitly and mechanistically including only predation includes many approximations, including the lack of importance of all other mortality sources. The existence of an indirect means of predicting the environment might explain some of the variation in experimental findings. For example, although there are many studies which indicate the importance of predators in microtine cycles, especially in Fennoscandia, there are also ex- amples of cycles occurring in the apparent absence of significant predation mortality (Graham & Lambin 2002). One explanation of predator-free cycles, or at least of predator-free reproductive slowdown, is that the animals’ anti-predation phenotypic plasticity is simply not cued by the preda- tors themselves. If the cue were the animals’ own cumulative recent activity, then the life-history changes would happen in the animals even if the predators were temporarily (e.g. experimentally) missing. The idea that cumulative recent activity might be a sufficient cue for cyclic small mammals 49 is premised on the notion that they are to a large extent the author of their own circumstances. The latter in effect proposes that intrinsic versus extrinsic is a chicken-and-egg situation rather than a means of identifying different classes of population-regulating mechanisms. For example, even when deaths are primarily by predation one could say that it is the prey, whose life history strategy creates circumstances suitable for the buildup of predators, who do it to themselves. And self-cueing of phenotypic plasticity appropriate for the environment further blurs intrinsic/extrinsic distinction. Ergon et al. (2001) found that voles reciprocally swapped between areas with asynchronous cycles — so that all voles in each area were then immigrants — adopted the life-history traits similar to the voles that had been at the destination area. This experiment demonstrated both the existence of phenotypic plasticity and the importance of cues in the environment. They considered this as confirmation of the importance of extrinsic causes without commenting on the nature of the cues, which quite possibly could have been signals left by the former inhabitants of the environment (an intrinsic cause). Leading representatives of each of the intrinsic and extrinsic views (Krebs 1996, Stenseth 1996, respectively) have stated that probably both intrinsic and extrinsic factors are important, but what is not clear is how they interact. I suggest that they interact via a stress physiology (Christian & Davis 1964, Boonstra et al. 1998a, Wingfield & Sapolsky 2003, Charbonnel et al. 2009, Sherrif et al. 2009) that is adapted to both: (1) boost short term survival of individuals, and (2) construct a record of cumulative recent activity that functions to cue life history characteristics appropriate for increasing the probability of leaving descendants (next chapter). 50 Chapter 5 Reproductive slowdown in cyclic populations Summary A common feature of mammalian population density cycles is a reproductive slowdown during the high-mortality decline phase (Erlinge et al. 2000, Krebs & Myers 1974, Cary & Keith 1979, Stefan & Krebs 2001) which lingers well into the low density phase (Boonstra et al. 1998b). Ecologists have struggled to explain this pattern (cf. Norrdahl & Korpimäki 2000, p.150; Sheriff et al. 2009, p.1255). We show that the decline-phase slowdown is consistent with classical life history theory (Fisher 1930, Charnov & Schaffer 1973, Charlesworth & León 1976). On the other hand, field ecologists have uncovered a reaction norm of a basic life history trait that operates on a trans- lifetime timescale and has not been explicitly addressed by theory. Using a model premised on reproduction entailing elevated risk, we explain the reproductive slowdown pattern by showing that it is always adaptive at high mortality rates, and adaptive under some circumstances at low mortality rates. We further describe how the stress physiology that has been studied for its impact on reproduction (Boonstra et al. 1998a, Wingfield & Sapolsky 2003, Sherrif et al. 2009) might be a proximate mechanism adapted for effecting reproductive effort appropriate for the environment- caused mortality rate. Introduction Lemmings do not commit mass suicide by jumping off cliffs. But aspects of their life history still puzzle ecologists despite more than 80 years of extensive research (Elton 1924, Batzli 1992, Batzli 51 1996, Hudson & Bjørnstad 2003). In many regions lemmings undergo regular high-amplitude cycles in population density. These cycles last much longer than the expected lifespan of the individual animals. Along with the density cycle is a regular pattern of change in the characteristics of the animals living during each stage (Krebs & Myers 1974, Erlinge et al. 2000, Gilg 2002). Cycles of other small mammals, such as snowshoe hares and voles, share the same properties (Schaffer & Tamarin 1973, Mihok & Boonstra 1992, Carey & Keith 1979, Norrdahl & Korpimäki 2002). Here we consider only the within-cycle change of one characteristic: reproductive output. During the increase phase of a cycle (rising population density) there are multiple bouts of reproduction each year, large litter sizes and rapid sexual maturity. High densities can appear so suddenly that the idea of lemmings falling from the sky is common to arctic legends and early natural histories. But from the peak and into the low phase these same populations breed less often, have smaller litters, and mature more slowly (Schaffer & Tamarin 1973, Erlinge et al. 2000, O’Donoghue & Krebs 1992, Stefan & Krebs 2001) — typically without food limitation (Krebs et al. 1986, Klemola et al. 2000, Ostfeld 1994). We refer to this suite of changes as a “reproductive slowdown”.It is very important to note that the period of time during a cycle in which reproductive rate is restrained is longer than the lifespan of individuals (Boonstra 1994). In other words, this is not a case of individuals delaying reproduction until a more opportune time (Kokko & Ranta 1996, Fuelling & Halle 2004). Nor is it a case of waiting, in a declining population, to have offspring later when they will comprise a larger fraction of the population. Rather, it is a case of multiple cohorts of individuals producing fewer (or far fewer) offspring than they could, under conditions where individuals are unlikely to survive long. It is not purely a case of breeding suppression (Ylönen 1994, Hik 1995) to avoid predation, as is evidenced by the different qualities of animals from the increase and decrease phases of a cycle being retained when they are taken into the lab (Mihok & Boonstra 1992, Sinclair et al. 2003). The hypothesis (Chitty 1967, Krebs 1978) that animals from different phases have different genotypes is considered disproven (Chitty 1996), so the differing phenotypes must be the result of plasticity cued or caused by changing aspects of the environment during a cycle. A life history theory retrospective summarizes: “If adult mortality rates increase, the optimal age at maturity decreases” (Stearns 2000, p.480) — and yet delayed maturity is one aspect of the reproductive slowdown during the high mortality rates of the decline phase. Literature on mammal cycles has presumed that “If the future breeding possibilities are low relative to current breeding options, the animal should invest in current reproduction despite a high predation risk, whereas, in an opposite situation, the animal should invest in survival (and future reproduction) at the cost of current reproduction.” (Norrdahl & Korpimaki 2000, p.150). This belief underlies such puzzles 52 as “. . . the question remains as to why snowshoe hares would decrease their reproductive output when their chance of survival also decreases” (Sheriff et al. 2009, p.1255). But the life history rules just quoted are not generally true. The pattern of adult mortality alone is insufficient to determine reproductive effort — what matters is the ratio of adult survival probability to juvenile survival probability (Charnov & Schaffer 1973). It is also crucial whether the population is increasing or decreasing (Fisher 1930, Charlesworth & León 1976). The decline phase of mammal cycles violates the conditions under which the above life history rules hold. We apply life history theory to explicitly address the adaptedness of a trans-lifetime pattern of reproductive slowdown during small mammal cycles. Life history theory has mainly concerned itself with patterns of changing costs and mortality across stages within the lifetime of an organism, seeking optimal equilibria or evolutionarily stable strategies. But in the cycles addressed here, the patterns of mortality span multiple lifetimes of individuals — and during some of those lifetimes mortality rates are so high that no strategy can avoid population decline. We work from a simple model in which reproduction entails elevating individual death rate above the basic death rate, that of a non-reproducing adult. We use two different fitness metrics, and calculate the reproductive effort that maximizes each metric in the short term at a given basic death rate 0→ 1. Since death rate is not constant within a cycle, animals’ life histories should be adapted to temporal trajectories of basic death rate. To address this aspect, we construct a simple individual-based simulation which internally generates cycles in death rate, and in which mutation and selection result in the reaction norm of reproductive effort as a function of death rate. We then discuss how, in the light of our findings, stress physiology (Boonstra et al. 1998a, Wingfield & Sapolsky 2003, Sherrif et al. 2009) seems well-adapted for effecting appropriate level of reproductive effort. Methods The process of reproduction entails two costs in terms of increased mortality rates. One is the ele- vation of the mortality rates of parents and juveniles above the mortality rate of a non-reproducing adult. For example, pregnant or lactating females require more food, which can mean more expo- sure to predation. Juveniles have elevated food requirements because they are still growing, are less experienced with the landscape, and thus are also more vulnerable to predation. The second cost is the effect of reproductive activity on the environment, which can raise the minimal mortality rate proper. An example is the attraction of a greater number of predators to a locale in which the mean prey activity level is higher. Consideration of these two costs forms the basis of our explanations. We use the following model of a generalized mammal. Let a reproductive period consist of 53 two time steps. During the first step the parent is pregnant and then nursing, and its death rate is d (1+ck) where d is the death rate for a non-reproducing adult, k is the litter size, and c is the cost- per-offspring during this period. Larger litters mean higher risk to the parent. If the parent does not survive this period, then all offspring die also. During the second step the juvenile offspring (and, for simplicity, the recovering parent) suffer a higher mortality rate of d (1+ j). There is no difference in the quality of juveniles from litters of differing size. We consider that d, c and j are constants dictated by the environment, whereas a reproductive strategy, defined by k, can vary. The probability of survival to weaning is (1− d(1+ ck)) at which point (1+ k) individuals start the second time step, which they survive with probability (1−d(1+ j)). So the mean number of recruits from a bout of reproduction is (1−d(1+ck)) (1+k) (1−d(1+ j)). This is one fitness metric — the mean rate of population increase — that k could potentially evolve to maximize. In the decline phase, populations can drop to levels below detection (Krebs & Myers 1974, Steen & Haydon 2000), so local extinction seems a possibility. One might therefore expect per- sisting genotypes to have life history features that minimize the probability of extinction during the crash and low phases of a population density cycle. This is a motivates a second fitness metric — the probability of survival of an individual’s lineage. Reproductive effort might evolve to maximize this probability, ensuring that the genes encoding the strategy best avoid extinction. We explain this metric more fully below. In an environment where every individual has an independent probability d of dying during any time step, a genotype represented by a single individual has probability (1− d) of surviving a single time step. A genotype represented by two individuals survives a time step unless both individuals die, so the resulting survival probability is (1−d ∗d). In the absence of reproduction, the more individuals that carry a genotype, the higher the probability that the genotype survives to any future time. But this does not mean that the production of more individuals always favours survival of the genotype, because of reproduction having costs in terms of elevated death rates. The simplest model of reproduction with mortality costs would mirror the above example and be a parent producing a single offspring with a potential result of two individuals. Using our model assume that c = 0 so that both parent and offspring suffer a higher death rate d(1+ j) for a single time step. The probability that both parent and offspring die is d(1+ j)∗d(1+ j). This is greater than the probability that a non-reproducer dies during that time step for d > 1/(1+ j)2. Unless the cost of reproduction decreases to zero as d approaches 1, a biologically unrealistic scenario, then there is a range of high d < 1 at which reproduction decreases probability of survival of the genotype. The method of comparing strategies just used is only strictly valid at the end of one time step. 54 Even this simplest of possible reproduction strategies actually produces a distribution of possible outcomes at that time, with probabilities of leaving 0, 1, or 2 individuals. Although the proportion of time that 0 individuals are left by the reproducer (both die) may be greater than the proportion of time that 0 are left by the non-reproducer (it dies), sometimes there are 2 survivors, which can then both reproduce. What we need to know for times beyond the end of a single reproductive period is whether reproduction by any survivors can eventually slow the overall loss rate to below that of the non-reproducer. And depending on parameters, it can. What we have is a branching process, for which there are methods for calculating probabilities of extinction and probabilities of survival of lineages to given future times. The probability of survival to a future time is the probability that the number of individuals in the lineage at that time is not zero. For this we need a description of the distribution of possi- ble numbers of descendants at that time, and a probability generating function (PGF) is a useful description. From the PGF of the distribution describing the number of recruits, f (z), one can obtain the probability of having a certain number of “tips” in the descendants tree resulting from a single branching. In particular, f (0) is the probability of having zero tips. Each one of the tips then branches and, assuming each does so independently and according to the same recruit distri- bution, the PGF for the number of descendants after a second round of reproduction and survival is g(z) = f ( f (z)), and the probability of extinction by that time is g(0). We constructed a PGF ac- cording to the above model for a generalized mammal. The resulting general PGF to be evaluated for probability of extinction is a function of d, c, j plus the dummy variable z. In only certain cases can the composite PGF f ( f ( f (. . .))) be collapsed into a closed form. We could not collapse this case, so we used f (0) for one bout, f ( f (z)) evaluated at z=0 for two bouts, etc. In cyclic populations d is not constant for long, so since this metric assumes constant d we calculate only near-term results. We used Mathematica to calculate the outcomes of all pos- sible parameter settings over many time steps, and then chose the strategy that yielded the highest probability of survival to each time, under each environment (d,c, j). Litter size was assumed to be physiologically constrained to 10 or less, c was assumed less than or equal to j (since a par- ent pregnant with one offspring is likely to be less handicapped than is a juvenile), and juveniles were assumed to be never more than 60% worse off than non-reproductive adults ( j ≤ 0.6). A non-reproducer (litter size choice of zero) survives a reproductive bout (two time steps) with a probability of (1−d)2. See Appendix G for the Mathematica notebook used. Next we used an individual-based simulation to explore the evolution of a reproductive strat- egy where d changes over time. The animals (genotypes) were free to evolve a norm of reaction of reproductive effort versus current mortality rate (d). The cycles in mortality rate were driven 55 using the “cumulative recent activity” (CRA) of the population, a concept fully described in the previous chapter. CRA is a predictor of many source(s) of mortality. In our simulation greater reproductive effort, and greater numbers of animals, generate greater “recent activity”, and the environment “heals” from the animals’ activity at a constant rate. CRA provided a simple way to create the delayed density-dependent mortality required to produce population cycles — mortality rate at a given density during the decrease phase was higher than at the same density during the increase phase. Moreover, mortality rates continued to cycle from low to very high even as repro- ductive output evolved, which allowed selection to continue to act on the entire reaction norm of effort versus death rate. If high death rate caused population numbers to drop below a threshold (50), surviving individuals were cloned fairly to raise numbers to the threshold. If the population went extinct, the last threshold-number of genotypes to be alive were resurrected to continue the evolutionary process. These low-number measures can be viewed as metapopulation rescue by immigrants from adjacent areas in which conditions are similar. The animals had a “chromosome” of 5 genes for determining the litter size to use within each of 5 equal subranges of mortality rate ( [0,0.2), [0.2,0.4), etc. ). Litter size was initialized at 1 across all death rates, and the genes were subject to both mutation and free recombination. Mutation was bidirectional in units of 1 offspring, and in each run mutation rate was set at 0.01 per gene for the first 5000 generations, and then switched to 0.001 for 15,000 more generations. The program is written in Netlogo (Wilensky 1999), a free, cross-platform package for agent-based simulation, and it is provided along with further details in Appendix H. Results In the three panels of Figure 5.1 we show the litter size that maximizes the fitness metrics for each of three sets of costs of reproduction (c=0.02 j=0.12, c=0.05 j=0.30 and c=0.1 j=0.6). These combinations were chosen arbitrarily such that a parent pregnant with six offspring had the same survival handicap as a juvenile. Other combinations of cost parameters investigated produce the same qualitative patterns of both metrics — even minimal and equal costs of c=0.01 and j=0.01. Mean population growth rate, the first fitness metric, is given by the red line. Without the imposed physiological maximum litter size (10) it would rise quickly to infinity as death rate decreases. The probability of persistence of an individual’s lineage, the second fitness metric, has the property that it changes with the time horizon at which it is evaluated (black lines). A strategy that is more risky over the short term may have a better longer term payoff. For this reason we plot this metric after 1, 2 and 10 bouts of reproduction. At high death rates, both fitness metrics indicate that the optimal litter size should decline 56 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 a c= .02   j= .12 litt er  si ze 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 b c= .05   j= .30 litt er  si ze 1 bout 2 bouts 10 bouts for survival after for growth rate simulation 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 c c= .10   j= .60 death rate litt er  si ze Figure 5.1: Best litter size according to several fitness metrics and the simulation, for three sets of reproduction costs c and j. Population decline is unavoidable in the shaded regions. 57 with increasing death rates. This is consistent with classical results showing that reproductive effort should decline in shrinking populations, and as the ratio of juvenile survival to adult survival decreases (Fisher 1930, Charlesworth & León 1976, Charnov & Schaffer 1973, Kokko & Ruxton 2000). In the shaded region of the plots there exists no litter size that can yield an expected number of recruits (including the parent) greater than one, i.e. it is the region of unavoidable population extinction for constant mortality. In our model the juvenile : adult survivorship ratio is (1−d(1+ j))/(1−d), which decreases rapidly at high d, and decreases at lower values of d for higher j (compare panel c to a). Thus ecologists should not be puzzled about reproductive restraint under conditions in which adult survival is very poor. If the process of reproduction itself entails elevating individual mortality rates, as in our model, then such a reproductive slowdown is a way of decreasing both the rate of population decline and the probability of extinction of an individual’s lineage. However, the two fitness metrics indicate very different patterns in litter size at low death rates. Lengthening the time horizon at which the probability of persistence is assessed reaches an asymptotic pattern quickly — there is little difference between the best strategy as measured at 2 versus at 10 reproductive bouts. But the asymptotic pattern does not converge to the same litter sizes that would result in maximum growth rate of the population. Instead, there is a decline in best litter size according to the second metric with decreasing mortality rates, which is not in agreement with classical life history theory. (This part of the pattern is not a matter of allocation of more of an individual’s resources to reproduction as mortality rates increase, since we model no budget constraint.) The intuitive explanation for the results using probability of persistence is that at low mortality rates, since survival is quite certain already, engaging in ambitious reproductive effort incurs needless risk. Interestingly, smaller litter size can improve the probability of persistence over the short term but decrease it over the long term (Figure 5.2). This pattern holds for any starting time and holds regardless of initial population numbers. As time horizon increases, at low death rates the differ- ences among values of metric two for different litter sizes becomes very slight. In other words, although the best litter size is as plotted, the amount by which it is best wanes as the time horizon lengthens. Thus we would expect stronger selection for the smaller litter sizes prescribed by the second metric when there is a premium on short-term survival. This can happen when numbers are so low that persistence is more important than rate of growth in numbers. Both fitness metrics describe an optimal litter size at a certain death rate given that the death rate remains constant into the future. This is clearly not the case during a small mammal population density cycle. Some assurance is needed that the reproductive slowdown picture does not change 58 1 2 3 4 5 6 0.20 0.25 0.30 0.35 0.40 0.45 # of reproductive bouts p ro b a b ili ty  o f s u rv iv a l for differing litter sizes  for d=0.5  c=0.02  j=0.12 litter size 5 litter size 10 Figure 5.2: The reproductive effort that yields higher probability of survival in the short term is not always the same as the one that is best for the long term. These conditions are those at the centre of Figure 5.1a. in qualitative ways when plotted against death rate given the trajectory that death rate will have into the future (spanning several individual lifetimes). This trajectory is difficult to predict since it depends not only on the (evolving) pattern of reproductive effort of the focal animals but also on any co-evolving patterns in the mortality processes. Our evolutionary simulation is very simple and is just one example of a model with internally generated cycles in death rate. The results of the simulation runs are plotted in Figure 5.1 as points, with blue lines between those to aid in visualizing the pattern. The standard error bars of these means of 100 runs are plotted also, but they do not extend beyond the edges of most of the plotted points. At high death rates the simulation results all show a reproductive slowdown with increasing death rate, in agreement with both fitness metrics. In all cases the slowdown is greater than that predicted by metric one, and by metric two calculated at a time horizon of 10 bouts. Death rates remained in this region for only 1 or 2 bouts. In none of the sets of runs does the litter size evolve to the physiological maximum and, in the first two sets of runs, reproductive effort evolves to a maximum at intermediate death rates. In total they support the relevance of fitness metric two, the probability of persistence of an individual’s lineage. 59 {r -> .25, K -> 20, a -> 0.5, h -> .5, b -> 0.3, d -> 0.39} prey density prey density capture rate per predator prey death rate per predator prey predator prey death rate a b Figure 5.3: Panel a shows a saturating functional response of a predator (Type II). Panel b shows how per capita prey death rate increases with lower prey density despite de- creased capture rate per predator. Discussion Using a simple and general model we have shown that, according to two fitness metrics, reproduc- tive effort should be suppressed at high per capi a death rates. We have also shown that a pattern of reproductive slowdown evolves in a simulation that internally generates cycles in death rate. The reaction norm that results from the simulation confirms the adaptedness of reproductive restraint at high death rates, and hints at the importance of maximizing probability of persistence at low death rates. The lingering slowdown The suite of changes in life history characteristics that we refer to as “reproductive slowdown” extends from the peak of a cycle, through the decline, and well into the low density phase. Given the high mortality rates during the decline phase (measured at 58% over a reproductive bout for snowshoe hares, Sheriff et al. 2009, Krebs et al. 2001), the decline phase slowdown seems suffi- ciently explained. But the cause of the lingering of slowdown into the low phase of the cycle is not as obvious. We note that “low phase” refers to low density of the focal population, rather than low numbers of predators or low mortality rate. If mortality rate remains high into the low phase then we would have a ready explanation. To consider death rate at low densities we will first assume that mortality is mainly caused by predation. This is accepted in the case of snowshoe hares but remains controversial with voles and lemmings. Assuming specialist predators — or generalist predators with little in the way of 60 alternate prey — a saturating (Type II) functional response of the form aN/(1+ ahN) (Figure 5.3a) is a reasonable description of the capture rate per predator at a given local prey density N (a is the area searched per time, and h is the time to handle each captured prey). Each capture is a prey death out of a population of N prey, so dividing the capture rate by N gives the per capita prey death rate per predator. Per predator, or per given number of predators, the prey death rate increases quickly (hyperbolically) as prey density decreases (Figure 5.3b). This happens because at lower prey densities, despite the drop in capture rate per predator, the prey harvest is divided among fewer prey. So high mortality can be caused by either high predator densities or low prey densities. Unless predator numbers drop just as quickly as prey numbers do, and without a time lag, predation mortality of prey tends to rise as prey densities drop — and can remain high into the low phase even as predators become rare. Following our results, a continuing reproductive slowdown during this part of the low phase is adaptive. If, instead of predation, we consider starvation to be the main mortality process, a population might crash due to over-harvesting of food on a spatial scale exceeding that of the dispersal capa- bilities of the animals. If so, then even once at low densities the remaining animals will still face the problem of foraging at low food density until the food regenerates. In a low-productivity environ- ment the regeneration could take a while. Note that starvation and predation processes would work synergistically to maintain high death rates into the low phase, conditions under which slowdown should be maintained as well. But there is evidence that reproductive slowdown continues even after mortality rates ease (Boonstra et al. 1998b). This is explainable using the work we have presented if selection acts to increase the probability of persistence (second fitness metric) at this stage in a cycle. A premium on short term survival under realistic conditions can produce this kind of selection. In our simple model there is no space — it is a well-mixed, or mean-field model — and even then two sets of simulation runs clearly show a pulldown of the selected reproductive effort at low death rate (Figure 5.1a,b). Our explanation for this is that lineage persistence is important in the shorter term at the low numbers to which populations crash. Probability of persistence at low local density (rather than low numbers) could exert strong selection pressure if there is any kind of spatial priority effect. Spatial modelling is beyond the scope of this paper. However, we note that a simple priority effect results if the animals are basically mining low productivity food during the increase phase: subsequent immigration by dispersers generated from riskier larger litters elsewhere may not sufficiently compensate for the meanwhile productivity of an earlier colonizer that took fewer risks in order to ensure establishment. Lemmings, voles and snowshoe hares all exhibit greater dispersal at low than at high densities, which is uncommon among animals in general, and is 61 suggestive of the importance of quickly finding open space. Full consideration of space may reveal other reasons for selection to put a premium on short-term survival at low densities, which would explain the virtue of reproductive restraint even under low death rates later in the low phase. To summarize, lingering reproductive slowdown after predators drop in number is explainable by: (1) the lingering of high per capita prey mortality at low prey numbers and, after mortality rate drops, (2) the existence of selection to minimize the probability of (local) extinction. Prey perception of death rate If death rate is a key determinant of appropriate reproductive effort, then well-adapted plasticity requires a means to perceive the current death rate or a proxy. Predators are not evenly spread over the landscape. They occur as a scattered integer number of individuals, and prey can perceive only a limited area of a landscape. Just because a single predator happened into its perceptual range, a prey individual might falsely conclude that the world is full of predators and that mortality rate is high. This is a maladaptive conclusion if it leads to reproductive slowdown under inappropriate conditions. But if the individual were to regularly perceive predators, then it could more safely conclude that predation risk is high. Any input other than a witnessing of multiple predation events is a measurement of a correlate of predation death rate. Whether other animals regard the environment as very dangerous is also relevant information. Apart from the direct or indirect sensing of the presence of mortality agents (predators, lack of food, diseased individuals) there is the possibility of predicting the existence of higher mortality rates by monitoring the “cumulative recent activity” of one’s own population, as was described in the previous chapter. Such prediction requires indicators of both the local density and the mean level of activity of neighbours. The above means of either sensing or predicting per capita death rate all rely on integration of information over a period of time. Since this capability is a feature of the stress physiology of animals, and a high level of stress is known to reduce reproduction (Wingfield & Sapolsky 2003) we suggest that stress physiology has a central role in the matching of reproductive effort to environmental conditions. It is possible that the stress system take any of the following as inputs (stressors): sightings of predators or predation events, stress levels in other animals, territorial interactions or other indicators of density, and activity level of conspecifics. Stress physiology is a leading proximate explanation of the entire reproductive slowdown — from the peak, through the decline, and into the low phase — but its impairment of reproduc- tion has thus far been viewed as a deleterious and inadvertent effect of maintaining physiological homeostasis, boosting anti-predator vigilance, and pre-programming offspring for better survival 62 (Boonstra et al. 1998a, Wingfield & Sapolsky 2003, Charbonnel et al. 2009, Sherrif et al. 2009). Given our results showing the adaptedness of “impaired reproduction” during the parts of a cycle in which animals are stressed, we suggest that stress physiology is a remarkable adaptation for increasing the probability of leaving descendants by both: (1) boosting short term survival of indi- viduals, and (2) determining the appropriate pattern of reproductive effort versus per capita death rate. Summary During small mammal population density cycles, which have periods of many animal lifetimes, there is a suite of regular changes in the life history characteristics of the animals alive during each phase of a cycle. During the decline and into the low density phase there are changes that restrain the high reproductive potential of these animals. Using a model in which parents and juveniles suffer mortality rates elevated above that of a non-reproducing adult, we show that reproductive slowdown is adaptive under mortality rates so high that population decline is unavoidable anyway. This fact was not known in the ecological literature. We calculate “probability of persistence of an individual’s lineage” according to our model, and note that this fitness metric predicts reproductive slowdown at low death rates if there is a premium on short term persistence, such as might exist at low numbers or at low local densities given a spatial priority effect. To our knowledge this is a finding not present in the life history literature. Our simple aspatial individual-based simulation shows the evolution of a reproductive slow- down at high death rates, and also lends some support to the “persistence” fitness metric. We discuss means by which animals might sense or predict death rate in their environment in order to effect appropriate life history changes, and point to stress physiology as being a good candidate mechanism. 63 Conclusions This thesis comprises a wide range of studies in life history evolution. I first consider how selection might act to enhance recombination, a sub-process of sexual reproduction, if organisms were to choose mates according to their fitness rather than according to particular traits. I then address whether the mere inter-individualness of sex is advantageous, and ecology enters the picture via the fact that different individuals have different experiences of the same habitat. At the most inclusive level, I consider how much effort should even be put into reproduction in the context of one of the oldest puzzles in ecology, that of small mammal cycles. Here I briefly review the separate findings. Recombination appears to have population-level advantages with respect to asexual reproduc- tion, for example by generating genetic change quickly (by shuffling alleles rather than creating novelty by mutation) or by increasing the contrast between better and worse genotypes to facilitate the action of selection. But population-level advantages can exist even when selection is acting to decrease the rate of recombination within a population. Modifier models offer a rigorous ap- proach to examining when recombination is favoured within a population, and in Chapter 1 that method showed the favourable conditions to be: rather strong assortative mating while at mutation selection balance. The attractiveness of assortative mating as a possible recombination-favouring process was its potential ubiquity. But the strength of assortment required is probably not found in most species. Increasing the contrast in fitness among individuals is one way that sex can facilitate the ac- tion of selection in increasing the level of adaptedness of a population. Another way is by de- creasing the noise in inheritance relative to the signal of selection. Chapter 2 considered whether inter-individual production of offspring can dampen the genetic effects of “luck” (individual-level, within-generation, ecological noise) better than lone-individual modes of reproduction — and found that it can. The dampening of noise in fecundity does not happen under all conditions, but the conditions under which it does happen are realistic ones for the model system (hermaphroditic plants). Dampening of noise in survival always happens in this system, because inter-individual 64 reproduction (via pollen) amounts to a form of dispersal. This dampening of the noise generated from individual luck is a new insight. It reveals a benefit which can offset some of the cost of sex, and predicts that in natural situations with greater amounts of ecological noise there should be a relatively greater amount of outcrossing. Chapter 3 then discusses several steps in the quest for a general, tractable and convincing way to address the pattern of reproductive slowdown that is observed in small mammal cycles. Two models of predation-induced breeding suppression were constructed, both assuming that the greater foraging time required for greater reproductive effort means greater mortality from preda- tion. The first (Appendix C) describes conditions favouring reproductive suppression in a simple, non-dynamic model and the second (Appendix D) found conditions under which a reproductive slowdown is advantageous in a Lotka-Volterra system. Each offered some insights, but neither is general or very convincing because the role of predation is controversial in lemming and vole cycles and the neutral cycles of the Lotka-Volterra systems are not realistic. Because it has stable limit cycles, the Rosenzweig-MacArthur (R-M) system was then closely examined as a candidate system for the local dynamics of a spatial individual-based model in which pattern of reproductive effort could be allowed to evolve. It was thought that this could be quite convincing if the animals were programmed to act in ways that biologists would agree is reasonable. But several assumptions of the R-M are not reasonable for a system in which food is cycling between abundant and scarce. In particular, it assumes lack of plasticity in search rate across food abundance, i.e. consumers do not search harder to avoid starvation. It also assumes that, even after searching a long time for an item of scarce food, that the consumer gains exactly the same net benefit by consuming it. A few modifications were developed, but the R-M was discarded as an appropriate foundation for the modelling goals here. If reproductive slowdown and population density decline still happen in an experiment in which predation mortality is manipulated to be insignificant, then it is not necessarily correct to conclude that predation is not a driver of the cycles . . . it could be that predation is simply not a proximate cue. Chapter 4 argues that the “cumulative recent activity” (CRA) of a population is a predictor of mortality rates from many causes, and would therefore be useful as a general proximate cue for life history changes. Proximate cueing from CRA would reduce the experimental power of manip- ulations of extrinsic factors such as predation and food, and could explain some of the controversy about causes of small mammal cycles. Stress physiology is a good candidate for a system that can accumulate information over time to assess mortality risk, including CRA. Since cumulative recent activity incorporates delayed density dependence it can be used as the basis of models of population density cycles that are not specific about mortality causes. 65 The trans-lifetime reproductive slowdown that is a feature of small mammal population den- sity cycles extends from the peak phase, through the decline and into the low phase. In Chapter 5 a model premised on reproduction entailing elevated risk was used to show that slowdown is al- ways adaptive when mortality rates are so high that population decline is inevitable. This explains the reproductive slowdown during the decline phase, which had been a puzzle. Since cycling small mammal populations can crash to very low densities, the “probability of persistence of an individual’s lineage” was calculated using the model plus some methods from branching theory. This fitness metric predicts reproductive slowdown at low death rates if there is a premium on short term persistence, such as might exist at low numbers or at low local densities given a spatial priority effect. This is a finding not present in the life history literature. The existence of such conditions could be the adaptive basis of the lingering of reproductive slowdown long into the low phase. A simple, aspatial individual-based model allowed simulated animals to evolve a reaction norm of reproductive effort versus per capita mortality rate. Starting from a strategy of identical low effort for all values of mortality rate, the simulated animals evolve a reaction norm similar to that observed during small mammal population cycles. 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(1938) Size of population and breeding structure in relation to evolution. Science 87:430–431 Wright, S. (1945) The differential equation of the distribution of gene frequencies. Proc. Natl. Acad. Sci. USA 31:382–389 74 Ylönen, H. (1994) Vole cycles and antipredatory behaviour. Trends Ecol. Evol. 9: 426–430 Zhang D.Y., & X. H. Jiang (2002) Size-dependent resource allocation and sex allocation in herba- ceous perennial plants. J. Evol. Biol. 15:74–83. 75 Appendix A The ‘Loci’ population genetics simulator The following pages describe my simulation code in a standard JavaDocs format, so that a pro- grammer can see how it is put together. It runs quickly even for large population sizes, and takes particular care in how random number generation is done. The code itself is available upon request. 76  Package  Class Tree Deprecated Index Help  PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    All Classes SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD Class Loci java.lang.Object   Loci public class Loci extends java.lang.Object The source code for these objects is available upon request from Alistair Blachford ( alistair@zoology.ubc.ca ). Loci reads the command-line run parameters, creates an instance of Deme and acts as metronome for the run. Usage: java -jar Loci.jar maxGen k nAttribs delMutRate benMutRate minRecRate modRecRate selRate epistasis assort initq maxGen maximum number of generations to run k the population size to use nAttribs the number of loci on a chromosome. Limitations on number are determined by class Bion delMutRate per locus forward (deleterious) mutation rate benMutRate per locus backward (beneficial) mutation rate minRecRate the minimum recombination rate between loci. Presence of mutation modifier alleles will elevate this. modRecRate the amount by which recombination rate is increased in an individual homozygous for the modifier selRate selection coefficient. Exact meaning depends on fitness function being used epistasis carefully peer at fitness function to see what this is. It warps the effects of increasing number of deleterious alleles. assort assortativeness of mating by fitness (not by genotype) initq the initial frequency of deleterious alleles across all fitness loci. Used only to create all individuals at time 0. Output: Except for usage errors, all output is from class Deme. Sample Run: java -jar Loci.jar 5 10000 64 .001 0 .01 0 .01 0 0 .5 1 -0.00188 -0.00753 0.50275 0.53361 0.01567 0.49972 1226 0 12695 84 0.52683 0 2 0.00032 0.00128 0.50030 0.53492 -0.00827 0.49780 1290 0 12544 83 0.50059 1 3 -0.00219 -0.00876 0.50090 0.53583 -0.00322 0.49645 1213 0 12581 84 0.49866 0 4 -0.00066 -0.00263 0.50610 0.53752 -0.00799 0.49393 1257 0 12667 81 0.49858 0 5 -0.00395 -0.01582 0.50800 0.53887 0.00304 0.49193 1276 0 12561 81 0.51757 1 Output fields are described in class Deme. Constructor Summary Loci()   Method Summary static void main(java.lang.String[] args)   Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait  Constructor Detail Loci public Loci() Method Detail main public static void main(java.lang.String[] args)  Package  Class Tree Deprecated Index Help  PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    All Classes SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD 77  Package  Class Tree Deprecated Index Help  PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    All Classes SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD Class Deme java.lang.Object   Deme public class Deme extends java.lang.Object A place with potentially interbreeding, constant population. Currently there is no dispersal, and fitness() is determined entirely by number of deleterious alleles, i.e. there is no requirement to match any particular local environmental conditions. Usage: created and called from class Loci Output: generation LD correlAB avgRec avgFit corr qav totmut totben totrec maxDel myLD numSelf generation completed bouts of viability selection, (assortative)mating, recombination, forward and backward mutation LD linkage disequilibrium between first and second fitness loci correlAB correlation between alleles at first and second fitness loci avgRec the average frequency of the recombination modifier avgFit the average fitness of all k individuals in the local population corr the realized correlation in fitness between all mums and dads qav the average frequency of deleterious alleles across all fitness loci totmut total number of forward mutation events this generation totben total number of* backward mutation events this generation totrec total number of recombination events this generation maxDel the maximum number of deleterious alleles found in any individual this generation. Of interest with respect to epistasis in the fitness function. myLD of all of the left/right chromosome pairings that matter with respect to recombination between the first and second fitness loci, this is the proportion that is repulsion pairings (Ab/aB). This is calculated from just the parents, and makes better sense to me than an LD at the haploid/chromosomal level. numSelf individuals are not allowed to mate with themselves. This is the number of selfings that were avoided. Sample Run: The command-line arguments are described in class Loci. java -jar Loci.jar 5 10000 64 .001 0 .01 0 .01 0 0 .5 1 -0.00188 -0.00753 0.50275 0.53361 0.01567 0.49972 1226 0 12695 84 0.52683 0 2 0.00032 0.00128 0.50030 0.53492 -0.00827 0.49780 1290 0 12544 83 0.50059 1 3 -0.00219 -0.00876 0.50090 0.53583 -0.00322 0.49645 1213 0 12581 84 0.49866 0 4 -0.00066 -0.00263 0.50610 0.53752 -0.00799 0.49393 1257 0 12667 81 0.49858 0 5 -0.00395 -0.01582 0.50800 0.53887 0.00304 0.49193 1276 0 12561 81 0.51757 1 Field Summary  double assort           the correlation in fitness between parents private  double avgFit  private  double avgRec   double benMutRate           per locus back mutation rate  Bion bion  (package private)  double correlAB  private  int coupling   double[][] cumDist  private  double[] cumFit  private  Bion dad  private  Bion[] dads  78  double delMutRate           per locus mutation rate  int demeGen  private  int drop   double epistasis   double initq           initial average frequency of deleterious alleles across fitness loci  boolean[] isHit  private  double[] juvCumFit  private  Bion[] juveniles           pre-reproductives private  double juvSumFit  private  int k           the constant population size private  Bion[] locals           les habitants private  double localSumFit  private  int maxDel   double minRecRate           per meiosis probability of recombination between any two adjacent loci, without modifiers  double modRecRate           the increment in rec rate for homozygote for modifier private  Bion mum  private  Bion[] mums   int nAttribs           the number of loci in play, including recombination modifier  double[] np           pre-computed binomial means private  int numDel  private  int numJuveniles  private  int numLocals  private  int numRec  private  int numSelf  (package private)  double pA  (package private)  double pAB  (package private)  double pB   double[] prob   MersenneTwisterFast rand  private  double recRate  private  int repulsion   double selRate           selection coefficient, exact meaning depends on function used in fitness()  double[] sqrnpq           pre-computed binomial std dev (package private)  int tab  (package private)  int taB  (package private)  int tAb  (package private)  int tAB  (package private)  int totben  static int[] totm  (package private)  int totmut  static int[] totr  79 (package private)  int totrec   Constructor Summary Deme(MersenneTwisterFast rand, int k, int nAttribs, double delMutRate, double benMutRate, double minRecRate, double modRecRate, double selRate, double epistasis, double assort, double initq)   Method Summary  void fitness(Bion bion)           Whatever fitness function is desired is plugged in here.  void flip()           juveniles become potentially-reproducing adults private  void heapsort(Bion[] ra, int n)           Heapsort kludged from Numerical Recipes in C. Used to sort mums and dads by fitness to generate assortativeness.  void init()           Generate pre-computed values, and populate deme with Bions.  void nextGeneration()           does viability selection, mating, and stats printout private  Bion pickParent()           The fitter are more likely to be chosen as parents.  int searchCDF(double[] cumArray, int topIndex, double target)           Binary search to find Cumulative Density Function array element >= target used for picking parents according to fitness, and for exact evaluation of the Binomial distribution  Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait  Field Detail nAttribs public final int nAttribs the number of loci in play, including recombination modifier k private final int k the constant population size selRate public final double selRate selection coefficient, exact meaning depends on function used in fitness() epistasis public final double epistasis assort public final double assort the correlation in fitness between parents initq public final double initq initial average frequency of deleterious alleles across fitness loci delMutRate public final double delMutRate per locus mutation rate benMutRate public final double benMutRate per locus back mutation rate minRecRate public final double minRecRate per meiosis probability of recombination between any two adjacent loci, without modifiers 80 modRecRate public final double modRecRate the increment in rec rate for homozygote for modifier sqrnpq public final double[] sqrnpq pre-computed binomial std dev np public final double[] np pre-computed binomial means cumDist public final double[][] cumDist prob public final double[] prob demeGen public int demeGen numDel private int numDel maxDel private int maxDel bion public Bion bion numLocals private int numLocals numJuveniles private int numJuveniles numRec private int numRec locals private Bion[] locals les habitants juveniles private Bion[] juveniles pre-reproductives mums private Bion[] mums dads private Bion[] dads mum private Bion mum dad private Bion dad 81 drop private int drop cumFit private double[] cumFit juvCumFit private double[] juvCumFit localSumFit private double localSumFit juvSumFit private double juvSumFit avgFit private double avgFit avgRec private double avgRec rand public MersenneTwisterFast rand totr public static int[] totr totm public static int[] totm tAB int tAB tAb int tAb taB int taB tab int tab totmut int totmut totben int totben totrec int totrec correlAB double correlAB pAB double pAB pA double pA 82 pB double pB recRate private double recRate coupling private int coupling repulsion private int repulsion numSelf private int numSelf isHit public boolean[] isHit Constructor Detail Deme public Deme(MersenneTwisterFast rand,             int k,             int nAttribs,             double delMutRate,             double benMutRate,             double minRecRate,             double modRecRate,             double selRate,             double epistasis,             double assort,             double initq) Method Detail init public void init() Generate pre-computed values, and populate deme with Bions. fitness public void fitness(Bion bion) Whatever fitness function is desired is plugged in here. searchCDF public int searchCDF(double[] cumArray,                      int topIndex,                      double target) Binary search to find Cumulative Density Function array element >= target used for picking parents according to fitness, and for exact evaluation of the Binomial distribution pickParent private Bion pickParent() The fitter are more likely to be chosen as parents. make array of cumulative fitnesses of all local bions choose uniform random number, scale to total, and then binary search for the individual whose fitness puts cumulative fitness over the scaled random number. heapsort private void heapsort(Bion[] ra,                       int n) Heapsort kludged from Numerical Recipes in C. Used to sort mums and dads by fitness to generate assortativeness. nextGeneration public void nextGeneration() does viability selection, mating, and stats printout flip public void flip() 83 juveniles become potentially-reproducing adults  Package  Class Tree Deprecated Index Help  PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    All Classes SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD  Package  Class Tree Deprecated Index Help  PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    All Classes SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD Class Bion java.lang.Object   Bion public class Bion extends java.lang.Object The individual reactive biological thingy. Diploid hermaphrodite version, with one recombination modifier locus at the end of the chromosome. I have coded 4 different versions of Bion with different speed and flexibility tradeoffs: BionBits Each chromosome is a single long int which means that the total number of loci is 64, including the locus of the recombination modifier. This is the fastest implementation, partly because bit-masking is used in generating recombinant chromosomes. Restricted to two alleles per locus. BionArray Each chromosome is an array of integers. This is the least refined of the versions. I gave up on it early when I saw how slow it was going to be, and how much memory it gobbles up. I tried it because of the potential flexibility: any number of loci, and any number of alleles per locus. BionList Each chromosome is a linked list of nodes, one node for each "different" allele. The idea here was to cut down on the amount of storage needed by only keeping track of the nodes with information. In the two-allele case I am dealing with, near mutation-selection balance, I keep track of only the deleterious alleles, since they are relatively rare. For example, at a mut-sel balance of 0.1, chromosomes are only a tenth of the size they would be if they had to store all loci. Although storage requirements were indeed dimminished, the processes of recombination and mutation are more complex. That complexity plus the overhead of the linked lists both cost in speed. BionPlus In a final rewrite, I coded each chromosome as an array of long integers. Although all alleles are stored all the time, each locus is just a bit, rather than one whole node on a linked list. Bit masking can be used to speed recombination (in 64-locus chunks), and there is no built-in limit to the number of loci that can be represented. This runs at more than twice the speed of BionList, and requires much less memory. This is the best written code, since I put more time into it once I saw its speed potential. This is the version documented here. Bions are instantiated by Deme, and Loci is the class that runs the whole simulation. I use MersenneTwisterFast. Field Summary private  double benMutRate 84  private  int chunks   int del   int delleft  private  double delMutRate   int delright  private  Deme deme   double fit  private  boolean[] isHit   int lab  private  long[] left  private  double minRecRate  private  double modRecRate  private  int nAttribs   int rab  private  MersenneTwisterFast rand   int rec  private  int recloc  private  long recMask  private  long[] right   int tab   int taB   int tAb   int tAB   Constructor Summary Bion(Deme deme)   Method Summary private  long[] getGamete(Bion bion)           Do recombination, and then mutation to generate a gamete.  int getNumEvents(int n, double p)           Choose the appropriate means, pre-computed or not, of generating the number of random events and then return the appropriate value.  void getRec()           Calculates how many recombination modifiers are present.  void init()           Build initial left and right chromosomes and set info fields.  Bion mateWith(Bion other)           grab a gamete from each parent, and set info fields private  void mutateBen(long[] chromo, double mutRate)           uses the global (all-loci) benMutRate private  void mutateDel(long[] chromo, double mutRate)           uses the global (all-loci) delMutRate  int poisson(double lambda)           found Poisson recipe somewhere: [Calculate exponential.] Set p = exp(-mu), n = 0, q = 1. private  void sumAB()           Calculate bits of information having to do with allele correlation and match- up. private  void sumDel()           Calculate number of deleterious alleles on each chromosome, and the total.  java.lang.String toString()   Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait  85 Field Detail nAttribs private final int nAttribs chunks private final int chunks left private long[] left right private long[] right recloc private final int recloc recMask private final long recMask delMutRate private final double delMutRate benMutRate private final double benMutRate minRecRate private final double minRecRate modRecRate private final double modRecRate fit public double fit del public int del rec public int rec delleft public int delleft delright public int delright tAB public int tAB tAb public int tAb taB public int taB tab public int tab lab 86 public int lab rab public int rab deme private Deme deme rand private final MersenneTwisterFast rand isHit private boolean[] isHit Constructor Detail Bion public Bion(Deme deme) Method Detail init public void init() Build initial left and right chromosomes and set info fields. All subsequent chromosomes originate from parents. getNumEvents public int getNumEvents(int n,                         double p) Choose the appropriate means, pre-computed or not, of generating the number of random events and then return the appropriate value. For trials n >= 50 Poisson or Normal approximations to the Binomial are used, depending on p, otherwise exact Bionomial distribution is used. poisson public int poisson(double lambda) found Poisson recipe somewhere: 1. [Calculate exponential.] Set p = exp(-mu), n = 0, q = 1. 2. [Get uniform variable.] Generate a random variable u, uniformly distributed between 0 and 1. 3. [Multiply.] Set q = q*u. 4. [Test against the exponential.] If q is greater than or equal to p, set N = N + 1 and return to step 2. getGamete private long[] getGamete(Bion bion) Do recombination, and then mutation to generate a gamete. mutateDel private void mutateDel(long[] chromo,                        double mutRate) uses the global (all-loci) delMutRate mutateBen private void mutateBen(long[] chromo,                        double mutRate) uses the global (all-loci) benMutRate mateWith public Bion mateWith(Bion other) grab a gamete from each parent, and set info fields toString public java.lang.String toString() sumDel private void sumDel() 87 Calculate number of deleterious alleles on each chromosome, and the total. This is the information used to calculate fitness. sumAB private void sumAB() Calculate bits of information having to do with allele correlation and match-up. getRec public void getRec() Calculates how many recombination modifiers are present. used for warping the basal minRecRate  Package  Class Tree Deprecated Index Help  PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    All Classes SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD 88 Appendix B Methods used in ‘On Luck and Sex’ The following pages detail the methods used in Chapter 2. They are: • a Mathematica notebook describing the work with moment generating functions • the R code used to compare with the analytical results 89 ü moment generating functions We use four rules about Moment Generating Functions (MGFs):     1. MGFX + Y HtL = MGFX HtL * MGFY HtL    2. MGFa X HtL = MGFX Ha tL    3. MGFX + b HtL = ‰b t * MGFX HtL And we use the  result from BOWERS et al. 1986, Chapter 11 : For  the  random  sum  S = X1 + ... + XN  if  the  Xi are  identically  distributed  random  variables  with MGFX HtL, N is a random number with MGFN HtL,  and N and all Xi are mutually independent, then the MGF of S is:     4. MGFS HtL = MGFN HlogMGFX HtLL The mean of a distribution is the first derivative of its MGF evaluated at t=0.  The variance of a distribu- tion is the second derivative of its Central Moment Generating Function (CMGF) evaluated at t=0.  Given m as the mean of the distribution,     CMGFX HtL = MGFX-m HtL = „-m t * MGFX HtL which is an implementation of the third rule (subtracting the mean m). The MGF of the negative binomial distribution, parameterized for l and w, and of the Bernoulli distribution: mgfNegBin := J wÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ w + l - ‰t l Nw mgfBer := H1 - pL + p ‰t A Bernoulli trial (coin flip) as to whether an allele is the focal one or its homologue is described by: mgfAlleleDraw := mgfBer ê. p Ø 1 ê 2 ü gametes of Selfer, Crosser and CrosserALT Here we want to look at the variance in number of gametes put into offspring, for the  strategies Selfer, Crosser and CrosserALT. ü Selfer mgfSelfGam := mgfNegBin ê. t Ø 2 * t FullSimplify@D@mgfSelfGam, tD ê. t Ø 0D 2 l varSelfGam = FullSimplify@D@‰-2 l t * mgfSelfGam, 8t, 2<D ê. t Ø 0D 4 l Hw + lL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅw lucksex.nb 1 Printed by Mathematica for Students 90 ü Crosser With Crossers we assume that reproductive resource allocation is such that each plant produces the same amount of  pollen,  and  differential  resource  stores affect  only the  number  of ova each plant  produces. Thus, each of the N individuals has chance p=1/N of fertilizing each of the approximately N*l ova. Total  number  of gametes  going into  offspring  is the number  of pollens  fertilizing  other  ova (mgfFert- Other), plus (#ova + #pollens accidentally selfing own ova), described by mgfOwn. mgfOtherOva describes the number of trials for non-self  pollination, and that number is (N-1) draws from the negative binomial fecundity distribution. mgfOtherOva := mgfNegBinHN-1L mgfFertOther := mgfOtherOva ê. t Ø Log@mgfBerD ê. p Ø 1ê N mgfOvaPolDraw describes a draw, each draw counting one ovum plus the outcome of a Bernoulli  trial of p=1/N, the probability of this plant fertilizing this ovum. The plant's #ova determines the number of such draws. mgfOvaPolDraw := ‰t * mgfBer ê. p Ø 1 êN mgfOwn := mgfNegBin ê. t Ø Log@mgfOvaPolDrawD mgfCrossGam := mgfFertOther* mgfOwn FullSimplify@D@mgfCrossGam, tD ê. t Ø 0D 2 l varCrossGam = FullSimplify@D@E-2 l t * mgfCrossGam, 8t, 2<D ê. t Ø 0D l H2 H1 + NL w + H3 + NL lL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅN w lucksex.nb 2 Printed by Mathematica for Students 91 ü CrosserALT For the CrosserALT's we assume that pollen production is proportional to ova production and, in particular, that for each ovum produced, the plant makes K pollens.  So the number of pollens produced by an individual is  K*#ova.   Number of  pollens  produced  population-wide  is K*sumOva,  so p=(K*#ova)/K*sumOva)  is the probability that a given ovum in the population is fertilized by pollen from the focal individual.  The distribu- tion of successful pollens from an individual is Binomial(p=#ova/sumOva,  n=sumOva), allowing random self- ing, and the total number of gametes from an individual is [ #ova + Binomial(p=#ova/sumOva,  n=sumOva) ] where both "#ova" are the same draw from the fecundity distribution. To build the MGF we must have just one instance of the fecundity draw, #ova. So we will approximate the successful pollen distribution with Binomial(p=1/N, n=N*#ova).  In this case the probability stays fixed and the number of trials varies with #ova, so we have a random sum of (N*#ova) Bernoulli trials of p=1/N.  Equivalently, we have a random sum of #ova elements, where each element is a sum of N Bernoulli trials. For each ovum we get the ovum plus N shots at fertilization. MGF for   1 + (N Bernoulli trials) : mgfOvaPol := ‰t * mgfBerN ê. p Ø 1ÅÅÅÅ N mgfCrAltGam := mgfNegBin ê. t Ø Log@mgfOvaPolD FullSimplify@D@mgfCrAltGam, tD ê. 8t Ø 0<D 2 l varCrAltGam = FullSimplify@D@E-2 l t * mgfCrAltGam, 8t, 2<D ê. 8t Ø 0<D l J5 - 1ÅÅÅÅN + 4 lÅÅÅÅÅÅÅÅw N After some rearrangement, varSelfGam = 4 l H 1 + lÅÅÅÅÅw L varCrossGam = 4 l H 1 + lÅÅÅÅÅw L + l I1 - 1ÅÅÅÅÅN M varCrAltGam = 4 l H 1 + lÅÅÅÅÅw L - l I1 - 1ÅÅÅÅÅÅN M I2 + 3 lÅÅÅÅÅÅÅÅw M ü focal alleles in offspring We have MGFs for number of gametes put into the offspring pool, but want to know the distributions of focal alleles there. lucksex.nb 3 Printed by Mathematica for Students 92 mgfSelfFocal := mgfSelfGam ê. t Ø Log@mgfAlleleDrawD D@mgfSelfFocal, tD ê. t Ø 0 l varSelfFocal = FullSimplify@D@E-l t * mgfSelfFocal, 8t, 2<D ê. t Ø 0D 3 l ÅÅÅÅÅÅÅÅ2 + l2 ÅÅÅÅÅÅw mgfCrossFocal := mgfCrossGam ê. t Ø Log@mgfAlleleDrawD FullSimplify@D@mgfCrossFocal, tD ê. t Ø 0D l varCrossFocal = FullSimplify@D@E-l t * mgfCrossFocal, 8t, 2<D ê. t Ø 0D l HH2 + 4 NL w + H3 + NL lL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ4 N w mgfCrAltFocal := mgfCrAltGam ê. t Ø Log@mgfAlleleDrawD D@mgfCrAltFocal, tD ê. t Ø 0 l varCrAltFocal = FullSimplify@D@E-l t * mgfCrAltFocal, 8t, 2<D ê. t Ø 0D 1 ÅÅÅÅ4 l J7 - 1ÅÅÅÅN + 4 lÅÅÅÅÅÅÅÅw N ü focal alleles in new adults ü Selfer Seeds survive to adulthood with probability 1/l under simple density dependence.  In a Selfer a seed can have 0, 1, or 2 focal alleles.  It's easier to get the seeds to adulthood, and then figure out the number of focal alleles after that. mgfDDep := mgfBer ê. 8p Ø 1 ê l< mgfSelfAd := mgfNegBin ê. t Ø Log@mgfDDepD mgfSelfAdGam := mgfSelfAd ê. t Ø 2 * t mgfSelfAdFocal := mgfSelfAdGam ê. t Ø Log@mgfAlleleDrawD FullSimplify@D@mgfSelfAdFocal, tD ê. t Ø 0D 1 varSelfAdFocal = FullSimplify@D@E-t * mgfSelfAdFocal, 8t, 2<D ê. t Ø 0D 3 ÅÅÅÅ2 + 1 ÅÅÅÅw lucksex.nb 4 Printed by Mathematica for Students 93 ü Crosser mgfCrossAdFocal := mgfCrossFocal ê. t Ø Log@mgfDDepD FullSimplify@D@mgfCrossAdFocal, tD ê. t Ø 0D 1 varCrossAdFocal = FullSimplify@D@E-t * mgfCrossAdFocal, 8t, 2<D ê. t Ø 0D 2 w + H3 + N + 4 N wL l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ4 N w l ü CrosserALT The next line neglects random selfing in CrosserALT by assuming that no seed contains 2 focal alleles. mgfCrAltAdFocal := mgfCrAltFocal ê. t Ø Log@mgfDDepD FullSimplify@D@mgfCrAltAdFocal, tD ê. t Ø 0D 1 varCrAltAdFocal = FullSimplify@D@E-t * mgfCrAltAdFocal, 8t, 2<D ê. t Ø 0D 1 + 1ÅÅÅÅw + -1 + 3 N ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ4 N l After some rearrangement, varSelfAdFocal = 1 + 1 ÅÅÅÅÅ w + 1 ÅÅÅÅÅÅ 2 varCrossAdFocal = 1 + 1 ÅÅÅÅÅÅ w + 1 ÅÅÅÅÅÅ 2 - 3 ÅÅÅÅÅÅÅÅÅÅÅÅ 4 w  ikjj1 - 1ÅÅÅÅÅÅÅN y{zz - 1ÅÅÅÅÅÅ2  ikjj1 - 1ÅÅÅÅÅÅÅÅÅÅÅN l y{zz varCrAltAdFocal = 1 + 1 ÅÅÅÅÅÅÅ w + 1 ÅÅÅÅÅÅ 2 3 ÅÅÅÅÅÅÅÅÅÅ 2 l  ikjj1 - 1ÅÅÅÅÅÅÅÅÅÅÅÅÅ3 N y{zz ü Making Table 2: fecundity, progeny survival, density dependence ü Cloner and Selfer All or no seeds from a plant survive to germination. Specifically, proportion v of plants get a draw from the fecundity distribution (NegBin) to germination. Because of simple density dependence, seeds survive from germination to adulthood with probability 1/(lv). mgfCloneGerm := mgfBer ê. 8t Ø Log@mgfNegBinD, p Ø v< mgfDDepSurv := mgfBer ê. 9p Ø 1ÅÅÅÅÅÅÅÅ l v = mgfCloneAd := mgfCloneGerm ê. t Ø Log@mgfDDepSurvD FullSimplify@D@mgfCloneAd, tD ê. t Ø 0D 1 lucksex.nb 5 Printed by Mathematica for Students 94 varCloneAdSurv = FullSimplify@D@E-t * mgfCloneAd, 8t, 2<D ê. t Ø 0D 1 + w ÅÅÅÅÅÅÅÅÅÅÅv w mgfSelfAd := mgfCloneAd mgfSelfAdGam := mgfSelfAd ê. t Ø 2 * t mgfSelfAdSurv := mgfSelfAdGam ê. t Ø Log@mgfAlleleDrawD FullSimplify@D@mgfSelfAdSurv, tD ê. t Ø 0D 1 varSelfAdSurv = FullSimplify@D@E-t * mgfSelfAdSurv, 8t, 2<D ê. t Ø 0D 1 ÅÅÅÅ2 + 1 + 1ÅÅÅwÅÅÅÅÅÅÅÅÅÅÅÅÅv Limit@%, v Ø 1D 3 ÅÅÅÅ2 + 1 ÅÅÅÅw ü Crosser mgfOtherPlants := HmgfBer ê. p Ø vLHN-1L mgfOtherOvaSurv := mgfOtherPlants ê. t Ø Log@mgfNegBinD mgfFertOtherSurv := mgfOtherOvaSurv ê. t Ø Log@mgfBerD ê. p Ø 1 ê N mgfOvaPolDraw := ‰t * mgfBer ê. p Ø 1 êN mgfOwn := mgfNegBin ê. t Ø Log@mgfOvaPolDrawD mgfYesNo := mgfBer ê. p Ø v mgfOwnSurv := mgfYesNo ê. t Ø Log@mgfOwnD mgfCrossGamSurv := mgfOwnSurv * mgfFertOtherSurv mgfCrossFocalSurv := mgfCrossGamSurv ê. t Ø Log@mgfAlleleDrawD mgfCrossAdSurv := mgfCrossFocalSurv ê. t Ø Log@mgfDDepSurvD FullSimplify@mgfCrossAdSurvDJ1 + v J-1 + 2w I N v wÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 - ‰t + 2 N v w MwNN-1+Nikjj1 + v ikjj-1 + 4w J- N v2 w lÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + ‰2 t - 2 v H1 + N + 2 N v wL l + 2 ‰t H-1 + H1 + NL v lL Nwy{zzy{zz FullSimplify@D@mgfCrossAdSurv, tD ê. 8t Ø 0, p Ø 1 ê N, h Ø 1 ê2<D 1 varCrossAdSurv = FullSimplify@D@E-t * mgfCrossAdSurv, 8t, 2<D ê. 8t Ø 0, h Ø 1ê 2, p Ø 1 êN<D 2 w + H3 + N + H3 + N + 3 H-1 + NL vL wL l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ4 N v w l FullSimplify@varCrossAdSurv ê. v Ø 1D ã varCrossAdFocal True lucksex.nb 6 Printed by Mathematica for Students 95 After some rearrangement, varCloneAdSurv = 1 ÅÅÅÅÅ v  ikjj1 + 1ÅÅÅÅÅÅw y{zz varSelfAdSurv = 1 ÅÅÅÅÅ v  ikjj1 + 1ÅÅÅÅÅw y{zz + 1ÅÅÅÅÅÅ2 varCrossAdSurv = 1 ÅÅÅÅÅ v  ikjj1 + 1ÅÅÅÅÅw y{zz + 1ÅÅÅÅÅÅ2 - ikjj 1ÅÅÅÅÅÅ4 - 3ÅÅÅÅÅÅÅÅÅÅÅÅÅ4 N y{zz ikjj 1ÅÅÅÅÅv - 1y{zz - 1ÅÅÅÅÅv  ikjj 3ÅÅÅÅÅÅÅÅÅÅÅ4 w  ikjj1 - 1ÅÅÅÅÅÅÅN y{zz + 1ÅÅÅÅÅÅ2  ikjj1 - 1ÅÅÅÅÅÅÅÅÅÅÅÅN l y{zzy{zz lucksex.nb 7 Printed by Mathematica for Students 96 # R code used to simulate sigma^2 for Selfer, Crosser and CrosserALT # Resulting variance is p*q*2N*sigma^2. alleles<- 0:1   # the 2 alleles possible p<- .5          # initial frequency of the 1 allele s<- .5          # probability of a plant not losing all seeds N<- 100         # population size m<- 5           # mean individual fecundity w<- 5           # "omega" parameter of fecundity distn (neg binom) allstats<- NULL alldata <- NULL for (N in c(100,70,40,20,10)) for (m in c(3,5,10,15,20)) for (w in m/c(.01,1,3)) {       # to get 1x, 2x and 4x Poisson variance  all<- NULL  for (t in 1:50000) {   # replications   repeat {     chromo <- rbind( sample(alleles, N, replace=T, c(1-p, p)),         sample(alleles, N, replace=T, c(1-p, p)) )     startcnt<- sum(chromo)     if(startcnt==2*N*p) break   # only proceed if have the right start freq   }   fecundity<- rnbinom(N, w, mu=m) # each plant's draw from the fecundity distn   numsurv<- rbinom(1, N, s)       # number plants whose seeds are not all lost   if (numsurv>0) {     surv<- sample(1:N, numsurv)   # index of the above plants     sumova<- sum(fecundity)       # used for pollination prob weightings     ova<- NULL     for (i in surv) ova<- c(ova, sample(chromo[,i], fecundity[i], replace=T))     numova<- length(ova)  # number of surviving seeds to this point     #pollen<- sample(chromo, numova, replace=T)                   # Crosser     pollen<- sample(c(chromo[1,],chromo[2,]), numova,                   replace=T, prob= c(fecundity,fecundity)/sumova) # CrosserALT     newchromo<- rbind(ova, pollen)     if (numova>N) chromo<- newchromo[,sample(1:numova,N)]     else chromo<- newchromo     endcnt<- sum(chromo)     all<- rbind(all,c(N,m,w,startcnt, endcnt))   }   else {     all<- rbind(all,c(N,m,w,startcnt, 0))   }  }  allstats<- rbind(allstats, c(N,m,w,length(all[,2]), var(all[,5])))  alldata <- rbind(alldata,all) } 97 Appendix C A simple model of predation-induced breeding suppression Oksanen and Lundberg (1995) built a non-dynamic foraging time model to look at predation- induced reproductive suppression such as that described by Hik (1985). They compare the value of two strategies, reproducing and non-reproducing in generating representatives in the next breeding period. Their model incorporates differential: vulnerability to predation while foraging, mainte- nance requirements, and ability to convert foraging time into resources for maintenance. It is a complex model, and the paper contains several mistakes. The model contains the odd rationale that increasing time spent foraging decreases the chance of dying from starvation. Animals do not choose an amount of time to forage, and then die if that estimate proves insufficient. They will assume the risks of additional foraging if it means certain death by starvation if they do not. The complexity of the 8-parameter model by Oksanen and Lundberg (1995) prompted me to build the following, simpler analytical model to examine the possibility of situations in which maximal reproduction is maladaptive. Like their model, this one is not dynamic. It considers a single reproductive period, during which predation density and food availability are assumed not to change significantly. Let e−k f be the probability of being alive at the end of a reproductive period, where f is the foraging rate (area/time) and k is a factor which converts foraging rate into predation-mediated mortality rate. Reproducing is a better strategy than not-reproducing when e−k fr(1+ p)> e−k fn (C.1) where fr and fn are the reproductive and non-reproductive foraging rates, respectively, and p is 98 the number of progeny: the payoff for reproducing. Let fn = m/g where m is the maintenance requirement (resources/time) and g is the resource density (greenstuff, resources/area). Let fr = (m+ c p)/g where c is the per-progeny resource requirement per time incurred by a reproducer. Then we have e −k(m+c p) g (1+ p)> e −k m g e −k(m+c p) g (1+ p) e −k m g > 1 e −ck p g (1+ p)> 1 (C.2) The LHS is the product of a part that decreases with p and a part that increases with p. It is maximized with respect to p when the (derivative of loss part)÷(loss part) = -(derivative of gain part)÷(gain part), i.e. when −ck g =− 1 (1+ p) p = g− ck ck (C.3) So when g is less than ck there is no level of reproduction (positive p) that has an advantage over not reproducing. Both low resource density (g) and high predation mortality rate (k) can produce such a situation. A further requirement for favoured reproduction is that the maximal p of (C.3) make the LHS of (C.2) greater than one. Substitution yields (e ck g −1) ( g ck ) > 1 (C.4) In contrast with Oksanen and Lundberg (1995), these conditions for assessing the value of repro- duction are more simple (only three parameters), and the environmental conditions (g and k) are quite explicit. 99 Appendix D Breeding suppression in a Lotka-Volterra system 100 Non-dynamic models show that, under certain conditions, breeding suppression can lower rate of population decline — but the systems of interest to me are of small mammal density cycles, which require a dynamic model for full explanation. Ruxton and Lima (1997) use the neutral cycles of a Lotka-Volterra framework to examine breeding sup- pression. They use a single type of prey that passes from reproducing-vulnerable into non-reproducing-invulnerable (to predation) as predator densities rise. In effect, a portion of the overall population enters a refuge. This does indeed model breeding suppression, but at the population level. At the individual level it is unclear how this could work from an evolutionary perspective — the non-reproducers would be losing out to the reproducers. The latter could keep producing more of their genotypes, which then enter the refuge pool to be saved for later. I also explored what could be learned using the neutral cycles of a Lotka-Volterra system, and assuming that all mortality is from predation. The standard Lotka-Volterra predator-prey system is: ẋ = r x− a x y (D.1) ẏ = b a x y − d y where x and y are the densities of prey and predator, respectively, r is the intrinsic rate of increase of the prey, a is the rate at which captures per predator per time increase with prey density, b is the conversion efficiency of prey captures into predator density, and d is the rate at which predator density declines in the absence of prey. In equations (D.1) the linear functional response ax is a function of the absolute prey density x. Let’s assume that this response is more realistically a function of the perceived (by the predator) density of the prey, which would be the absolute density moderated by a factor having to do with activity level. Let this activity level be foraging rate, and assume that foraging rate varies directly with the intrinsic rate of increase, r. Then the perceived 101 prey density would be proportional to rx and the system is of the form ẋ = r x− a r x y (D.2) ẏ = b a r x y − d y I added a second prey type so that I could experiment with the coexistence of two different strategies with respect to r: ẇ = rw w − a rw w y ẋ = rx x− a rx x y (D.3) ẏ = b a rw y + b a rx x y − d y In a standard Lotka-Volterra system the morph with the smaller r declines relative to the other morph (Figure D.1). But if a greater r costs in greater vulnerability, as in equations (D.3), the situation can be different. Figure D.2 shows that the morph with the smaller r both increases more gradually and declines more gradually with respect to the other morph, but that both morphs persist. Parameter values in both figures are a=.1 (Fig. D.1), a ∗ rx=.1 (Fig. D.2), b=.15, d=.15, rw=.1, rx=.12. Prey w is the thick solid line, prey x is the thick dashed, and the predators are the thin solid line. Figure D.1 Figure D.2 50 100 150 200 250 5 10 15 20 25 50 100 150 200 250 5 10 15 20 25 Both morphs persist indefinitely, i.e. neither form takes over, when the functional response is exactly proportional to r. I experimented with the exponent e in the following equations as a way of modifying the degree to which r influences capture rate. 102 ẇ = rw w − arwew y ẋ = rx x− arxe x y (D.4) ẏ = b arwe w y + b arxe x y − d y I concluded that if the capture term is supralinear in r (e > 1), then the morph with the smaller r increases relative to the morph with the higher intrinsic rate of increase. If the capture term is sublinear in r then the morph with the smaller r is gradually excluded. Figure D.1 is the case where e=0, Figure D.2 is for e=1, and Figure D.3 is for e=1.2. Figure D.3 50 100 150 200 250 5 10 15 20 25 Now let’s assume that the two morphs occupy separate adjacent spatial patches on a scale small enough that predators can compare patches. Predators will target one area in preference to the other if perceived prey density seems to be higher there. This aggregation response will increase the impact of r in the capture term. (And over longer time, greater resource depletion in the spatial patch with higher r will also have an effect on foraging rate and capture.) Let’s use a patch effect in which, where the perceived density of prey is proportional to rw e w and rxe x, the patch preference is the relative perceived density, e.g. rwew/(rwew+ rx e x). Then the system becomes 103 ẇ = rw w − a rwe w y ∗ rwe w/(rwew + rxe x) ẋ = rx x− a rxe x y ∗ rxe x/(rwe w + rxe x) (D.5) ẏ = b arwe w y ∗ rwe w/(rwew + rxe x) + b arxe x y ∗ rxe x/(rwew + rxe x)− dy and the output is: Figure D.4 (e=1.3) Figure D.5 (e=0.6) 50 100 150 200 250 5 10 15 20 25 50 100 150 200 250 5 10 15 20 25 The patch (aggregation) effect is very effective at keeping both morphs in the game, re- gardless of the value of e. The patch effect also introduces the dampening effect of density dependence since it in effect creates a density target relative to the other patch. The “microtine syndrome” is one of initially explosive growth followed by a reproduc- tive slowdown later in the cycle. So what if morph w has a lower r only when its density is decreasing, rather than continually through the cycle as modelled above? I wrote a 4th order Runge-Kutta routine to use instead of Mathematica’s NDSolve to facilitate intro- ducing a switch in rw according to any arbitrary rule. Figure D.6 shows the result of using equations (D.3), which lack a patch effect, and having rw = rx = .12 whenever density w is increasing, rw = .10 whenever density w is decreasing, and e=1. 104 Figure D.6 50 100 150 200 250 5 10 15 20 25 30 The switching strategy causes rapid damping by decreasing the average rate of increase of the prey while the predators are increasing during the decline phase of the cycle. Why does switching work? The two prey equations can be written as ẇ = w (rw − arwe y) (D.6) ẋ = x (rx − arxe y) The expressions within parentheses are equations of straight lines with the predator density y as the independent variable and the respective r’s as the intercepts. The line with the greater r has both the greater intercept and the greater negative slope, so the lines always cross, regardless of the value of e, as shown in Figure D.7. Figure D.7 1 2 3 4 5 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 For any predator density (position along the x-axis) it is better to have the strategy with the upper of the two lines. Here this means the more positive growth rate when the population is increasing, and the less negative growth rate when the population is decreasing, which is exactly the switching strategy used in Figure D.6. When e 6= 1 the intersection point of 105 the two lines is not exactly on the x-axis, so the switching rule would have to be modified. But as long as predator densities fluctuate to either side of the intersection point, there exists a switching rule (e.g. according to a critical density of predators) which will work better than either constant reproductive rate. Figure D.8 (e=0.6) Figure D.9 (e=1.3) 1 2 3 4 5 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 1 2 3 4 5 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 Finally, let’s model the switching strategy and the patch effect together. When the switching strategy described above is used in combination with equations (D.5) the morph with the “microtine syndrome” (the switcher) effectively excludes the morph with the fixed maximum reproductive rate, even when the relationship between r and probability of capture is substantially sublinear. This is shown in Figures D.10 and D.11. Figure D.10 (e=1) Figure D.11 (e=0.4) 50 100 150 200 250 5 10 15 20 25 30 50 100 150 200 250 5 10 15 20 25 30 The purpose of this exploration with the Lotka-Volterra equations was to examine, in a dynamic system, the plausibility of an adaptive advantage associated with switching to a lower-than-maximal reproductive rate. I concluded from this preliminary work that such an advantage can exist, and that further exploration of this idea, this potential explanation for the “microtine syndrome”, was warranted. 106 Appendix E Deriving the Rosenzweig-MacArthur from reaction kinetics 107 **** Standard Rosenzweig-MacArthur **** Define N: food item S: searching predator H: handling predator P: all predators The Rosenzweig-MacArthur system is: dNdt = N Ha - N bL - P J a N1+a h N N dPdt = P b J a N1+a h N N - P d With molecules and animals, i.e. for an individual-based model, we are confined to integer  numbers.   So  handlers  have  to  take  one  of  two  pathways  to  reverting  to searchers, producing either 1 or 2 searchers, balanced so that (1+b) searchers are produced at the appropriate rate (1/h): 1)  N Ø 2N  @rate a 2)  N + N Ø N  @rate b  (together 1 and 2 give logistic growth) 3)  N + S Ø H  @ rate a (searchers finding food become handlers) 4)  H Ø S @ rate (1-b)/h 5)  H Ø 2S @ rate b/h 6)  P Ø .  @ rate d  (predator death)         1      2    3          4          5      6 dNdt = aN   -bN*N -aNS dSdt =            -aNS   ((1-b)/h)H    2(b/h)H dHdt =             aNS  -((1-b)/h)H    -(b/h)H dPdt=                            (b/h)H         -dP dNdt = a N - b N N - a N S; dSdt = -a N S + HH1 - bL ê hL H + 2 Hb ê hL H; dHdt = a N S - HH1 - bL ê hL H - Hb ê hL H ; dPdt = dSdt + dHdt - d P; FullSimplify@HH1 - bL ê hL H + 2 Hb ê hL HD FullSimplify@-HH1 - bL ê hL H - Hb ê hL HD FullSimplify@dSdt + dHdt - d PDH1 + bL H h - H h b H h - d P Assume handling is fast (big, so h small) and dHdtã0 H0 = FullSimplify@Solve@dHdt ã 0 ê. S Ø HP - HL, HDD::H Ø a h N P 1 + a h N >> Printed by Mathematica for Students 108 FullSimplify@dNdt ê. 8S Ø P - H< ê. H0D:N - a P 1 + a h N + a - N b > FullSimplify@dPdt ê. 8S Ø P - H< ê. H0D:-d P + a b N P 1 + a h N > **** A small modification **** Assume animals die of starvation while looking for food, not while handling/eating it. 1)  N Ø 2 N  @rate a 2)  N + N Ø N  @rate b  (together 1 and 2 give logistic growth) 3)  N + S Ø H   @ rate a (searchers encountering food become handlers) 4)  H Ø S @ rate (1-b)/h 5)  H Ø 2 S @ rate b/h 6)  S Ø .  @ rate d  (death while searching)         1    2    3         4          5     6 dNdt = aN -bN*N -aNS dSdt =            -aNS  ((1-b)/h)H  2(b/h)H -dS dHdt =             aNS -((1-b)/h)H  -(b/h)H dNdt = a N - b N N - a N S; dSdt = -a N S + HH1 - bL ê hL H + 2 Hb ê hL H - d S; dHdt = a N S - HH1 - bL ê hL H - Hb ê hL H; dPdt = dSdt + dHdt; Assume (1/h) is fast and dHdt ã 0 2   app3c.nb Printed by Mathematica for Students 109 H0 = FullSimplify@Solve@dHdt ã 0 ê. S Ø HP - HL, HDD; FullSimplify@dNdt ê. 8S Ø P - H< ê. H0D FullSimplify@dPdt ê. 8S Ø P - H< ê. H0D:N - a P 1 + a h N + a - N b > : H-d + a b NL P 1 + a h N > So dNdt = N Ha - N bL - P a N 1 + a h N dPdt = P b a N 1 + a h N - P d 1 + a h N Higher N now causes lower death rate This formulation makes clear sense when it is noted that 1 ê H1 + a h NL is the proportion of searchingand handling time that is spent searching while finding food at rate aN and starving at rate d Thandling = h; Tsearching = 1 ê Ha NL; FullSimplify@Tsearching ê HTsearching + ThandlingLD 1 1 + a h N app3c.nb   3 Printed by Mathematica for Students 110 Appendix F Three foraging models in R These models were used for the studies described in the third chapter: • fresp2.R simulates a Type II functional response in which a predator encounters food items at a Poisson rate of aN and then handles them for 1 unit of time. It allowed deduction of what is the variance of Type II foraging. • Given the variance in foraging success, plus some assumptions about mininal food intake rate required to avert starvation, one can simulate the proportion of the population that would starve at a given food density. This is what is simulated by starvecurve.R. • It seemed to me that the net gain from a food item would be less at low food density because of the added energy spent in finding it. If one assumes foraging on a plane by means of a random walk, it does, as can be seen with walker.R. The code listings follow. ################################################################# # Program: fresp2.R # This assumes non-depleting, randomly distributed prey. # Time is in units of handling time. When a predator encounters a prey # it then stops looking for 1 time unit. This program calculates the # number of prey consumed over a given total time T. By modelling the # Type II functional response like this one get the mean= aNT/(1+ahN) # and can deduce that the variance is aNT/(1+ahN)ˆ3 # Using variance one could decide what proportion of predators fail to # meet minimum needs, and should starve. (As discussed in Chapter 3.) 111 a<- 0.02 # area searched per time unit h<- 1 # handling time per prey caught N<- 10 # prey density in units of number per area tvec<- c(10,25,50,100,250,500,1000) # total time T numtrials<- 1000 obs<-NULL trials<-NULL for (i in 1:numtrials) { # number of replicates rounds<-rexp(3000,a*N) # compute search times rounds<- rounds + h cumrounds<- cumsum(rounds) # total time to end of each handling for (j in tvec) { # for each total time duration obs<-c(obs,sum(cumrounds<j)) # num rounds managed in that interval } trials<-rbind(trials,obs) # each row a replicate, each column a T obs<-NULL } print(apply(trials,2,mean)) print(apply(trials,2,var)) print(apply(trials,2,var)/apply(trials,2,mean)) par(mfcol=c(4,2)) for (k in 1:length(tvec)) { hist(trials[,k], breaks=15, main=paste("a=",a," N=",N," T=",tvec[k]," ",numtrials," trials")) } frame() ################################################################# # Program: starvecurve.R # This produces the first figure in Chapter 3. # 112 # work out proportion of population starving versus food density # proportion starving is proportion with Type II foraging input less than # self-maintenance requirements. # # Work it out for different evaluation periods T. I am expecting # sigmoid curve with 2 parameters: food density at which half die, and # the steepness of the change in starvation rate (which I think is # going to depend on T). T would have something to do with the buffer # each animal has, and it should be small compared with the speed of # change in food density. par(mfcol=c(4,2)) N<-seq(0,30,.1) for(m in c(.15,.3)){ for(T in c(10,50,100,1000)){ mn<- a*N*T/(1+a*h*N) va<- mn/(1+a*h*N)ˆ2 half<- m/(a*(1-h*m)) plot(N,pnorm(m*T,mn,sqrt(va)),ylim=c(0,1), main=paste("T= ",T," m=",m,"a=0.02 h=1"), ylab="proportion starving",type="l") # using the standard dev at half: lines(N,1/(1+sqrt(m*T/(1+a*h*half)ˆ2)ˆ((N-half)/2)),lty=2,col="red") lines(N,1/(1+a*h*N),lty=2,col="blue") abline(v=half) } } ################################################################# # Program: walker.R # This is useful for examining how much a herbivore needs to walk to # find new patches of food. Encounters with food are thereby NOT a # Poisson process as is assumed by the Type II functional response. # As food gets more rare, it becomes more energetically expensive to # get, and so less valuable per food item. The Rosenzweig-Macarthur # assumes that a captured food item always has the same value to the 113 # consumer, even as food becomes more rare, which is not the case here. N<-100 steps<-0 nodes<-0 m<-matrix(rep(1,N*N),ncol=N) x<-N/2 y<-N/2 coords<-c(x,y) walk<-c(0,0) xlast<-x ylast<-y par(pty="s") plot(1,1,xlim=c(1,N),ylim=c(1,N),type="n") flag<- T #should draw the line, not a wraparound for(i in 1:(10*N*N)) { if (runif(1)>.5) # move in x- or y-direction? {if (runif(1)>.5) x<-x+1 else x<-x-1} # right or left? else {if (runif(1)>.5) y<-y+1 else y<-y-1} if (x<1) {x<-N flag<- F} if (x>N) {x<-1 flag<- F} if (y<1) {y<-N flag<- F} if (y>N) {y<-1 flag<- F} steps<-steps+1 if ( m[x,y] > 0 ) { nodes<-nodes+1 m[x,y]<-0 } coords<-rbind(coords,c(x,y)) walk<-rbind(walk,c(steps,nodes)) if (flag) segments(xlast,ylast,x,y) 114 flag<- T xlast<-x ylast<-y } ans<-scan() plot(walk[,2],walk[,1],xlim=c(0,N*N), xlab="nodes visited",ylab="# of steps",pch=".") abline(0,1) par(mfrow=c(2,1),mar=c(5,4,1,1)+.1,pty="m") plot(walk[,2],walk[,1],xlab="nodes visited",ylab="# of steps",pch=".") lines(walk1[,2],walk1[,1]) lines(walk2[,2],walk2[,1]) lines(walk3[,2],walk3[,1]) yup<-walk[2:100001,2]>walk[1:100000,2] ww<-walk[yup,] len<-length(ww[,2]) plot(ww[2:len,2],ww[2:len,1]-ww[1:(len-1),1],pch=".", xlab="nodes visited",ylab="steps since last new node") yup<-walk1[2:100001,2]>walk1[1:100000,2] ww<-walk1[yup,] len<-length(ww[,2]) points(ww[2:len,2],ww[2:len,1]-ww[1:(len-1),1],pch=".") yup<-walk2[2:100001,2]>walk2[1:100000,2] ww<-walk2[yup,] len<-length(ww[,2]) points(ww[2:len,2],ww[2:len,1]-ww[1:(len-1),1],pch=".") yup<-walk3[2:100001,2]>walk3[1:100000,2] ww<-walk3[yup,] len<-length(ww[,2]) points(ww[2:len,2],ww[2:len,1]-ww[1:(len-1),1],pch=".") 115 Appendix G Calculation of fitness metrics The following Mathematica notebook describes how the fitness metrics were calculated for ‘Re- productive slowdown in cyclic populations’, including how the “probability of persistence” fitness metric was constructed. It is an application of branching theory, using probability generating func- tions (PGFs). 116 This notebook presents the code that was used to calculate both fitness metrics in Chapter 5. THE  MODEL:  Reproduction  starts  with  a  parent  being  pregnant  and  then  nursing. Together these last 1 time step, during which the parent suffers a higher mortality rate, d(1+k*c), where k is the litter size and c is the cost per offspring. That is a Bernoulli trial where the parent either survives or does not. Then there is a split- ting into (k+1) tips.  These become separate Bernoulli trials of survival for 1 time period at elevated death rate d(1+j), followed by an optional wait time of t time steps at mortality rate d. At that point we have the distribution of individuals we need, the reproduction distribution of the Galton-Watson process or, as I call it in Chapter 5, the recruit distribution. Wait time is a complication that was not needed to make any point in the manuscript.  It only came into play at a few of the very highest death rates. The  probability  generating  functions  for  constant  random  variable,  Bernoulli  and Poisson: Gcon := zconst Gber := H1 - pL + p z Gpoi := EHl Hz-1LL To recover the probability P(X=k) one takes the kth derivative divided by k! evalu- ated at dummy variable z=0.  So, just as a test, the probability of drawing 2 from a Poisson distribution with a l=2 isHD@Gpoi, 8z, 2<D ê 2!L ê. 8l Ø 2, z Ø 0< 2 ‰2 N@%D 0.270671 Clear@c, d, j, k, p, t, zD Gwean := Gber ê. p Ø 1 - d H1 + k cL Gsplit := Gcon ê. const Ø H1 + kL Grep := Gwean ê. z -> Gsplit Grepsurv := Grep ê. z Ø IGber ê. p Ø H1 - d H1 + jLL H1 - dLtM Grepsurv1 := Grepsurv Grepsurv2 := Grepsurv ê. z Ø Grepsurv Grepsurv3 := Grepsurv ê. z Ø Grepsurv2 Grepsurv4 := Grepsurv ê. z Ø Grepsurv3 Grepsurv5 := Grepsurv ê. z Ø Grepsurv4 Grepsurv6 := Grepsurv ê. z Ø Grepsurv5 Grepsurv7 := Grepsurv ê. z Ø Grepsurv6 Grepsurv8 := Grepsurv ê. z Ø Grepsurv7 Grepsurv9 := Grepsurv ê. z Ø Grepsurv8 Grepsurv10 := Grepsurv ê. z Ø Grepsurv9 Grepsurv11 := Grepsurv ê. z Ø Grepsurv10 Mean, variance and lower bound at large n: m := D@Grepsurv, zD ê. z Ø 1 v := D@Grepsurv, 8z, 2<D + D@Grepsurv, zD - HD@Grepsurv, zDL2 ê. z Ø 1 lb := H1 - mL mn+1 ë v Printed by Mathematica for Students 117 m ê. t Ø 0 mm := m ê. t Ø 0H1 - d H1 + jLL H1 + kL H1 - d H1 + c kLL The death rate d beyond which the mean number of recruits, including the parent, is less than 1 : Solve@1 == H1 - d H1 + jLL H1 + kL H1 - d H1 + c kLL ê. 8c Ø .02, j Ø .12, k Ø 10<, dD88d Ø 0.60132<, 8d Ø 1.12487<< Solve@1 == H1 - d H1 + jLL H1 + kL H1 - d H1 + c kLL ê. 8c Ø .05, j Ø .3, k Ø 10<, dD88d Ø 0.496026<, 8d Ø 0.939872<< 2   reproslonb.nb Printed by Mathematica for Students 118 Solve@1 == H1 - d H1 + jLL H1 + kL H1 - d H1 + c kLL ê. 8c Ø .1, j Ø .6, k Ø 10<, dD88d Ø 0.382735<, 8d Ø 0.742265<< Calculation of the first fitness metric, the mean population growth rate. The file- name ends with the value of c: str = OpenWrite@"growthrate02"D ktop = 10; c = .02; j = .12; t = 0; For@d = .01, d § .99, d += .01, kmax = 0; gmax = 0; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, growth = mm ê 1; If@growth > gmax, gmax = growth; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDD Close@strD str = OpenWrite@"growthrate05"D ktop = 10; c = .05; j = .3; t = 0; For@d = .01, d § .99, d += .01, kmax = 0; gmax = 0; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, growth = mm ê 1; If@growth > gmax, gmax = growth; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDD Close@strD str = OpenWrite@"growthrate1"D ktop = 10; c = .1; j = .6; t = 0; For@d = .01, d § .99, d += .01, kmax = 0; gmax = 0; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, growth = mm ê 1; If@growth > gmax, gmax = growth; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDD Close@strD We are using a deathrate of d(1+handicap), so there is a maximum d for which the deathrate is < 1 for any given handicap. Since the handicap increases with litter size k, we need to be sure that we are not artificially restricting strategies to low k at high d -- hence the (kmax+1) in the conditional below.  The filename is a concate- nation of "best", the value of c, "." and the number of reproductive bouts: reproslonb.nb   3 Printed by Mathematica for Students 119 str = OpenWrite@"best02.1"D ktop = 10; c = .02; j = .12; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H1L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv1 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD str = OpenWrite@"best02.2"D ktop = 10; c = .02; j = .12; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H2L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv2 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD str = OpenWrite@"best02.10"D ktop = 10; c = .02; j = .12; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H10L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv10 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD str = OpenWrite@"best05.1"D ktop = 10; c = .05; j = .3; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H1L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv1 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD 4   reproslonb.nb Printed by Mathematica for Students 120 str = OpenWrite@"best05.2"D ktop = 10; c = .05; j = .3; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H2L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv2 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD str = OpenWrite@"best05.10"D ktop = 10; c = .05; j = .3; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H10L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv10 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD str = OpenWrite@"best1.1"D ktop = 10; c = .1; j = .6; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H1L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv1 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD str = OpenWrite@"best1.2"D ktop = 10; c = .1; j = .6; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H2L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv2 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD reproslonb.nb   5 Printed by Mathematica for Students 121 str = OpenWrite@"best1.10"D ktop = 10; c = .1; j = .6; t = 0; ForAd = .01, d § .99, d += .01, kmax = 0; gmax = H1 - dL2 H10L; For@k = 1, k § ktop, k++, If@d < H1 ê Max@1 + c * k, 1 + jDL, given = H1 - Grepsurv10 ê. z Ø 0L; If@given > gmax, gmax = given; kmax = kDDD If@d < H1 ê Max@1 + c * Hkmax + 1L, 1 + jDL, WriteString@str, d, " ", kmax, " ", AccountingForm@gmaxD, "\n"DDE Close@strD 6   reproslonb.nb Printed by Mathematica for Students 122 Appendix H Simulation of slowdown evolution This is the individual-based model that was used to verify that a pattern of reproductive slowdown can evolve. Cycles of death rate are generated internally using the “cumulative recent activity” of the population, as described in Chapter 4. It is written in Netlogo, which is cross-platform and free from http://ccl.northwestern.edu/netlogo/. Most Netlogo models are spatial, but this model is not. Immediately following is an image of the interface for this model, which is run by simply setting the costs of reproduction, pressing Setup, and then pressing run. All of the program code is then listed. I would be glad to provide the model, reproslo4.nlogo, to anyone who asks for it. 123 124 ;; Model reproslo4.nlogo ;; Intended to correspond to the math.  Litter size choice only (no wait time). extensions   [array table] breed        [lemmings lemming] lemmings-own [state litter chromo mate] globals      [numTraits numPhases act activity maxActivity newActivity actArray actIdx   mutRateArray sdArray minArray maxArray idx   arb theirs theirsCnt wholist dict drateArray deadArray deadIdx numDead] to setup   clear-all   set maxActivity 3000   set numTraits 1   set numPhases 5   set mutRateArray array:from-list (list .01 0 0); trait-specific mutation rates   set sdArray  array:from-list [ 1   1 0.01]   set minArray array:from-list [ 0   0  0  ]   set maxArray array:from-list [10 100  .99]   set actArray array:from-list n-values 100 [0]   set drateArray array:from-list n-values numphases [0]   set actIdx 0   set deathRate 0.2   set geneHisto? false   no-display ; model has no space, so why slow it down ...    set dict table:make ; this table is just so can specify gene in natural way   table:put dict "L1" 0 set L1 1   table:put dict "L2" 1 set L2 1   table:put dict "L3" 2 set L3 1   table:put dict "L4" 3 set L4 1   table:put dict "L5" 4 set L5 1    ; the rest is to speed free recombination, and mutation      set arb 199  ; yeah I just picked an arbitrary prime number      set wholist []      foreach n-values arb [?] [        ifelse random 2 = 0          [ set wholist fput false wholist ]          [ set wholist fput true  wholist ]      ]      set theirs array:from-list wholist      set theirsCnt 0 ; index into the array    ask patches [set pcolor white]   setup-dead   setup-lemmings   draw-plots end ;[globals] deadArray deadIdx numDead to setup-dead 125   set numDead 50   ;create 2-D array of arrays (array of chromosomes) to store latest dead genotypes   set deadArray array:from-list n-values numDead [0]   set deadIdx 0   while [deadIdx < numDead] [     array:set deadArray deadIdx array:from-list n-values (numTraits * numPhases) [0]     set deadIdx deadIdx + 1   ]   set deadIdx 0 end to setup-lemmings   create-lemmings 600   [     set color black     set size 0.7     setxy random-xcor random-ycor     set chromo array:from-list (list L1 L2 L3 L4 L5)     set state "par"     set litter L1   ] end to go   if ticks = 5000 [array:set mutRateArray 0 0.001] ; reduce mutation to lessen noise   if ticks = 20000 [ stop ]   set newActivity 0    ask lemmings [      if state = "juv" [set act 1 + juvCost]      if state = "par" [set act 1 + litter * offCost]      ifelse (random-float 1) <= (deathRate * act)        [croak] ; improper (approximate) non-update of newactivity here        [set newActivity newActivity + act]      if state = "juv" [set state "par" set litter 0]      if litter > 0 [ ; was parent at start of time step and wants to reproduce        hatch litter [          set state "juv"          set litter 0          ;automatically inherits *pointer* to chromo.  Need separate copy.          set chromo array:from-list array:to-list chromo          set mate one-of other lemmings with [litter > 0]          if mate != nobody [crossWith [chromo] of mate]          mutate          ;setxy random-xcor random-ycor ;no space in this model          ]        set state "juv" ; parent enters recovery period        set litter 0       ]   ]   if not any? lemmings [rescue]   while [ count lemmings < (numDead / 2) ] [     ask lemmings [ 126       hatch 1 [         set state "par"         set chromo array:from-list array:to-list chromo         ;setxy random-xcor random-ycor       ]     ]   ]   if (count lemmings < numDead) [     ask n-of (numDead - count lemmings) lemmings [       hatch 1 [         set state "par"         set chromo array:from-list array:to-list chromo         ;setxy random-xcor random-ycor       ]     ]   ]   set activity (activity - 1000) ; environment heals (from perspective of lemmings)   if activity < 0 [set activity 0]   set activity activity + newactivity   set deathRate activity / maxActivity   set idx floor (deathRate * numPhases)   if idx >= numPhases [     ;show "activity > maxActivity"     set idx (numPhases - 1)   ]   array:set drateArray idx ((array:item drateArray idx) + 1)    ask lemmings with [state = "par"] [; assess life-history strategy given current deathRate      set litter array:item chromo (idx * numTraits)   ]    tick   draw-plots end to draw-plots   set-current-plot "population"   plotxy ticks count lemmings   if (ticks > 200) [ set-plot-x-range (ticks - 100) ticks ]    if ticks mod 10 = 0 [     if any? lemmings [       set L1 mean [array:item chromo  0] of lemmings       set L2 mean [array:item chromo  1] of lemmings       set L3 mean [array:item chromo  2] of lemmings       set L4 mean [array:item chromo  3] of lemmings       set L5 mean [array:item chromo  4] of lemmings     ]    set-current-plot "experience of deathRates"   clear-plot 127   let deathIdx 0   while [deathIdx < numPhases] [     plotxy (deathIdx / numPhases) array:item drateArray deathIdx     set deathIdx deathIdx + 1   ]   ]    if geneHisto?     [ set-current-plot "histogram of gene"       histogram [array:item chromo table:get dict gene] of lemmings     ] end to croak  ;; the way out for all lemmings   if count lemmings <= numDead     [ ;die leaving a ghost which can be resurrected     array:set deadArray deadIdx array:from-list array:to-list chromo     set deadIdx deadIdx + 1     if deadIdx >= numDead [set deadIdx 0]     ]   die end to rescue ;; metapopulation rescue from extinction using last living genotypes   create-lemmings numDead [     set chromo array:from-list array:to-list array:item deadArray deadIdx     set deadIdx deadIdx + 1     if deadIdx >= numDead [set deadIdx 0]     set color black     set size 0.7     ;setxy random-xcor random-ycor   ] end to crossWith [otherChromo] let recRate 0.5   let numLoci array:length chromo   let crossAfter []   ifelse recRate >= 0.5   [     foreach n-values (numLoci - 1) [?] [       if array:item theirs theirsCnt [set crossAfter fput ? crossAfter]       set theirsCnt theirsCnt + 1       if theirsCnt = arb [set theirsCnt 0]     ]     set crossAfter reverse crossAfter   ]   [   let unhit (numLoci - 1)              ; cannot do a crossover after the last locus   let crossEvents random-poisson unhit * recRate   ; should be random-binomial   if crossEvents > unhit [ set crossEvents unhit ]   let unhitArray array:from-list n-values unhit [?] ; unhit is the number, NOT the 128 right-most index   let hit 0   let tmp 0   while [crossEvents > 0] [     set hit random unhit      ; choose a random index of unhitArray     set crossAfter sentence crossAfter array:item unhitArray hit     set crossEvents crossEvents - 1     set unhit unhit - 1  ; OK, now it is the rightmost index     set tmp array:item unhitArray unhit      ; swap hit item with rightmost item not- yet-hit     array:set unhitArray unhit array:item unhitArray hit     array:set unhitArray hit tmp   ]   set crossAfter sort crossAfter ; now an ordered list of crossover points   ]   ;show crossAfter ;an ordered list of crossover points   let mine true   if random 2 = 0 [ set mine false ]   foreach n-values numLoci [?] [     if not mine [array:set chromo ? array:item otherChromo ?]     if not empty? crossAfter [       if ? = first crossAfter [ ; cross over to other chromosome         set mine (not mine)         set crossAfter but-first crossAfter       ]     ]   ] end to mutate   ; allows trait-specific mutation rates, assumes nothing before (left of) traits on chromosome foreach n-values numTraits [?] [   let mutEvents random-poisson (numPhases * array:item mutRateArray ?)   ; should be random-binomial   if mutEvents > 0 [ ; then there is something to do for this trait     let unhit numPhases                               ; the number of phase-specific genes coding for the trait     if mutEvents > unhit [ set mutEvents unhit ]      ; Poisson is unbounded on high side     let trait ?     let unhitArray array:from-list n-values unhit [? * numTraits + trait] ; unhit is the number, NOT the right-most index                    ; this is the array of the actual indices in chromo     let hit 0     let tmp 0     let hitGene 0     let sd 0     while [mutEvents > 0] [       set hit random unhit      ; choose a random index of unhitArray       set hitGene array:item unhitArray hit ; the location hit in the chromosome       set mutEvents mutEvents - 1 129       set unhit unhit - 1  ; OK, now it is the rightmost index       set tmp array:item unhitArray unhit      ; swap hit item with rightmost item not- yet-hit       array:set unhitArray unhit hitGene       array:set unhitArray hit tmp       ; do the mutation       set sd array:item sdArray trait        ; get rid of call to random normal       ifelse array:item theirs theirsCnt         [array:set chromo hitGene (array:item chromo hitGene + sd)]         [array:set chromo hitGene (array:item chromo hitGene - sd)]       array:set theirs theirsCnt (not array:item theirs theirsCnt)       set theirsCnt theirsCnt + 1       if theirsCnt = arb [set theirsCnt 0]        ;array:set chromo hitGene (array:item chromo hitGene + random-normal 0 sd)       ; must not be less than min       if array:item chromo hitGene < array:item minArray trait [         array:set chromo hitGene array:item minArray trait ]       ; must not be more than max       if array:item chromo hitGene > array:item maxArray trait [         array:set chromo hitGene array:item maxArray trait ]     ]   ] ] end 130


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