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Mathematical models of life cycle evolution Scott, Michael Francis 2016

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Mathematical Models of Life CycleEvolutionbyMichael Francis ScottB.A. Biological Sciences (honours), University of Oxford, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Botany)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2016© Michael Francis Scott 2016AbstractIn this thesis, I investigate several aspects of life cycle evolution using math-ematical models.We expect natural selection to favour organisms that reproduce as oftenand as quickly as possible. However, many species delay development unlessparticular environments or rare disturbance events occur. I use models to askwhen delayed development (e.g., seed dormancy) in long-lived species can befavoured by selection. I find that long-lived plants experience ‘immaturityrisk’: the risk of death due to a population-scale disturbance, such as a fire,before reproducing. This risk can be sufficient to favour germination in thedisturbance years only. I show how demographic models can be constructedin order to estimate the contribution of this mechanism (and two othermechanisms) to the evolution of dormancy in a particular environment.All sexually reproducing eukaryotes alternate between haploid anddiploid phases. However, selection may not occur in both phases to thesame extent. I use models to investigate the evolution of the time spent inhaploid versus diploid phases. The presence of a homologous gene copy indiploids has important population genetic effects because it can mask theother gene copy from selection. A key innovation of my investigation is toallow haploids and homozygous diploids to have different fitnesses (intrin-sic fitness differences). This reveals a novel hypothesis for the evolution ofhaploid-diploid strategies (where selection occurs in both phases), where thegenetic effects of ploidy are balanced against intrinsic fitness differences.Many sex chromosome systems are characterized by a lack of recombi-nation between sex chromosome types. The predominant explanation forthis phenomenon involves differences in selection between diploid sexes. Idevelop a model for the evolution of recombination between the sex chro-mosomes in which there is a period of selection among haploid genotypesin pollen or sperm. I find that a period of haploid selection can also drivethe evolution of suppressed recombination between sex chromosomes, whichshould become enriched for genes selected in the haploid phase. This modelpredicts that the tempo of sex chromosome evolution can depend on thedegree of competition among haploids.iiPrefaceChapter 2 of this thesis has been published. Chapter 3 has been submittedand Chapter 4 is in preparation for publication. The contributions of thecandidate are as follows:Scott, M.F. and Otto, S.P. (2014) Why wait? Three mechanisms selectingfor environment-dependent developmental delays. Journal of EvolutionaryBiology 27:2219-2232.• The candidate designed and analyzed the models and wrote themanuscript. S.P. Otto guided and assisted with model analysis andedited the manuscript.Scott, M.F. and Rescan, M. (2016) Evolution of haplont, diplont orhaploid-diploid life cycles when haploid and diploid fitnesses are not equal.submitted• The candidate designed and analyzed the majority of models and wrotethe manuscript. M. Rescan designed and performed the explicit multi-locus simulations, wrote this portion of the manuscript and edited themanuscript. S.P. Otto supervised the project, edited the manuscriptand provided helpful input. D. Roze supervised the contributions ofM. Rescan and edited the manuscript.Scott, M.F. and Otto, S.P. (2016) The role of pollen and sperm competitionin sex chromosome evolution in preparation• The candidate designed and analyzed the models and wrote themanuscript. S.P. Otto assisted with model analysis and edited themanuscript.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Mathematical Modelling of Life Cycle Evolution . . . . . . 11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Utility of Mathematical Models in Evolutionary Theory 11.3 Evolutionary Invasion Analysis . . . . . . . . . . . . . . . . . 31.4 Life Cycle Variation . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Two Example Life Cycles . . . . . . . . . . . . . . . . 41.4.2 Aspects of Life Cycle Evolution Investigated . . . . . 62 Why Wait? Developmental Delays . . . . . . . . . . . . . . . 92.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Model Background . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Model and Results . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Mechanism 1: Low Seedling Survival in Some Envi-ronments . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Mechanism 2: Trade-Offs . . . . . . . . . . . . . . . . 192.4.3 Mechanism 3: Effects of Synchronization With Dis-turbances . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27ivTable of Contents3 Haploid-Diploid Life Cycle Evolution . . . . . . . . . . . . . 323.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Analytical Model . . . . . . . . . . . . . . . . . . . . 353.3.2 Multilocus Simulations . . . . . . . . . . . . . . . . . 383.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.1 Deleterious Mutations . . . . . . . . . . . . . . . . . . 393.4.2 Beneficial Mutations . . . . . . . . . . . . . . . . . . 453.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 The Role of Pollen and Sperm Competition in Sex Chromo-some Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Model Background . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1 Developmental Delays . . . . . . . . . . . . . . . . . . . . . . 685.2 Haploid-Diploid Life Cycles . . . . . . . . . . . . . . . . . . . 705.3 Sex Chromosome Evolution . . . . . . . . . . . . . . . . . . . 715.4 Extraordinary Sex Ratios: Revisited . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75AppendicesA Evolution of Developmental Delays Analysis . . . . . . . . 94A.1 Differences in Seedling Survival . . . . . . . . . . . . . . . . . 94A.2 Trade-Offs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.3 Approximating the Cycle Matrix . . . . . . . . . . . . . . . . 103B Further Analysis Of Haploid-Diploid Life Cycle Evolution 110C Evolution of Recombination Rate on Sex Chromosomes . 116C.1 Recursion Equations . . . . . . . . . . . . . . . . . . . . . . . 116C.2 Invasion of Recombination Modifiers . . . . . . . . . . . . . . 119vTable of ContentsC.3 Invasion of Modifiers that Increase Recombination from anInitially Low Level . . . . . . . . . . . . . . . . . . . . . . . . 121viList of Tables2.1 Default parameters used in numerical simulations . . . . . . . 253.1 Fitnesses of different genotypes. . . . . . . . . . . . . . . . . . 373.2 Life cycle assumptions used in various modifier models. . . . 37B.1 Fitnesses in discrete and continuous selection models . . . . . 112B.2 Fitnesses of mutated types when mutations occur at meiosis . 112C.1 Fitness of different genotypes. . . . . . . . . . . . . . . . . . . 116C.2 Marginal fitnesses of YA and Xa haplotypes . . . . . . . . . 117viiList of Figures2.1 Life-history parameters for which conditional germinationstrategies have a higher long-term growth rate. . . . . . . . . 182.2 Trade-offs between germination rates (g1 and g2) can favourintermediate strategies or specialization . . . . . . . . . . . . 212.3 The parameters for which conditional germination is expectedto evolve for various different disturbance cycle lengths . . . . 242.4 The germination rate in non-fire years that yields the highestlong-term growth rate . . . . . . . . . . . . . . . . . . . . . . 262.5 The effect of variability in disturbance-return interval on theevolution of conditional germination. . . . . . . . . . . . . . . 273.1 Models of haploid-diploid life cycles . . . . . . . . . . . . . . 363.2 Expected life cycle evolution where the strength of selectionagainst deleterious mutations and effective dominance is varied 423.3 Parameter space for which (a) deleterious mutations and(b) beneficial mutations favour haplont, diplont and haploid-diploid life cycles . . . . . . . . . . . . . . . . . . . . . . . . . 444.1 The zygotic sex ratio is biased by linkage between an XYsex-determining region (SDR) and a locus that experienceshaploid selection during pollen/sperm competition (A). . . . 574.2 A modifier that reduces the recombination rate between theA locus and the SDR can spread to fixation despite causingsex ratios to become biased. . . . . . . . . . . . . . . . . . . . 62A.1 Discrepancy between germination rate evolution for approxi-mated and non-approximated disturbance cycles. . . . . . . . 108A.2 The parameters for which conditional germination is expectedto evolve for various different disturbance cycle lengths: non-approximated . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.1 Expected life cycle evolution with selfing . . . . . . . . . . . . 115viiiList of FiguresC.1 Selection can favour increased recombination between the sex-determining region (SDR) and a selected locus that is closelylinked to the SDR . . . . . . . . . . . . . . . . . . . . . . . . 127ixAcknowledgementsThis thesis could not have been completed without the support of a fantasticgroup of collaborators, colleagues, and friends. First and foremost, I wouldlike to thank Sally Otto for providing me with outstanding academic train-ing, support, encouragement, and inspiration. I relocated over 7,000km towork with Sally and have never questioned this decision; she has exceededall expectations for scientific acumen and mentorship and is also a greatperson to spend time with, thank you.I have also benefited greatly from the consistent support and advicesupplied by my co-supervisor, Jeannette Whitton, and committee member,Quentin Cronk. Chapter 3 was written in collaboration with Marie Rescan,who is supervised by Denis Roze. Marie and Denis have been great people towork with on both scientific and personal levels. I am very grateful for theirhelp and expertise. I would also like to thank Simone Immler, with whomI have collaborated on a project not included in this thesis. Aside frombeing a great collaborator, Simone’s recent work and ideas are inspirationalin informing my most recent projects; I am happy to have been able to workwith her.Being a member of the Otto lab has been a particularly constructive andfun experience thanks to the other students and post-docs I have overlappedwith. For being fantastic and helpful people, thanks to Kazuhiro Bessho, FloDe´barre, Rich Fitzjohn, Aleeza Gerstein, Liz Kleynhans, Leithen M’gonigle,Karen Magnesson-Ford, Ailene McPherson, Jasmine Ono, Matt Osmond,Kate Ostevik, Matt Pennell, Carl Rothfels, Linnea Sandell, and NathanielSharp.I would also like to thank the outstanding community in the UBC bio-diversity building and Botany department. Several faculty and staff havehelped my education in various direct teaching and committee roles, includ-ing Michael Doebeli, Loren Rieseberg, Thor Veen, Jennifer Williams, andMike Whitlock. In addition, a long list of students and postdocs have hugelyenhanced my work life both socially and professionally, a big thank you toeveryone. Here, I will only single out those with whom I have shared anoffice (or been adjacent to for five years): Kim Gilbert, Carla Crossman,xAcknowledgementsAlana Schick, Philippe Fernandez Fournier, and Evan Hersh.Finally, I would like to thank all the friends that have made my yearsin Vancouver so amazing; it would be impossible to complete this thesiswithout your support and my life would also have been so much poorer.Because I can’t list everyone who should be thanked here, I will just thankmy roommates Sam Johns and Kyle Glenn for being great friends. Lastly,thank you to Kaeli Johnson, for being the best thing about the last fivefantastic years.xiFor my family: Mum and Dad,Jim, Sarah, and HarryxiiChapter 1Mathematical Modelling ofLife Cycle Evolution1.1 SummaryA central aim of evolutionary biology is to understand how and why thediversity of organisms that we observe came to be. A prominent aspectof biodiversity is that organisms exhibit a large number of adaptationsto different environments and interactions; remarkably, organisms alsodisplay a large amount of structural variation in their life cycles. Thatis, there is significant variation among species in the number and na-ture of life cycle stages between birth and reproduction, in the modeof reproduction, in the number of copies of genetic material, and inthe way genetic material is inherited between generations. Evolutionarytheory should give us insight into how this life cycle variation arose and why.In this Chapter, I first briefly discuss the role of mathematical modelsin investigating evolutionary problems and describe the logic of the tech-niques and methodology that I will use. Then, I outline the features of lifecycle evolution that are investigated in this thesis, using the example of twoorganisms that demonstrate variation in these life cycle aspects.1.2 The Utility of Mathematical Models inEvolutionary TheoryMost theories in evolutionary biology are given in verbal form. The mostfamous is the theory of evolution by natural selection itself, which Darwinexpressed as follows:Owing to this struggle for life, any variation, however slight andfrom whatever cause proceeding, if it be in any degree profitableto an individual of any species, in its infinitely complex relations11.2. The Utility of Mathematical Models in Evolutionary Theoryto other organic beings and to external nature, will tend to thepreservation of that individual, and will generally be inherited byits offspring. The offspring, also, will thus have a better chanceof surviving, for, of the many individuals of any species whichare periodically born, but a small number can survive. I havecalled this principle, by which each slight variation, if useful, ispreserved, by the term of Natural Selection.—Charles Darwin, On The Origin of Species Chapter IIIThis verbal theory constitutes a model about how the world works, con-sisting of some form of verbal ‘if. . . then. . . ’ statement through which wedescribe the logical consequences of some initial conditions. The same prin-ciple can be applied to simpler theories, such as ‘if you heat water to 100°C,then it will boil’. In this view, our representation of the world, or schema,consists of a series of theories and so theories are extremely common.It is often useful to make our theories formal using mathematical models.This approach has a long and successful tradition in evolutionary biology.Mathematical models played a key role in the modern synthesis, when theideas of natural selection and Mendelian genetics were reconciled (amongother advances, Mayr and Provine 1998). For example, in the first of a seriesof papers, Haldane (1924) used formal mathematical models to describe thechange in frequency of a trait under selection when traits are controlled by asingle Mendelian locus. Indeed, Haldane’s “Mathematical theory of naturaland artificial selection” forms the basis of the models of selection that I usein Chapters 3 and 4.Theories provide the framework into which observations can be placed.Empirical observation is ultimately the only way to determine whether aprocess occurs in nature. The role of mathematical models is often to for-malize the logic of our theories, providing an argument of the form ‘if A,then B’ or ‘A would promote B’ (Sober 2011). Some authors have arguedthat models themselves are often constructed and analyzed as a logical testof an idea, analogous to an experimental test (Caswell 1988, Servedio et al.2014). The questions addressed by these models may take the form ‘Am Icorrect in thinking that A would promote B due to an interaction with Cthat causes . . . ?’ (Kokko 2007, Chapter 1).Models can reveal features that might otherwise not be evident. Forexample, Haldane (1964) argues that his attempts to model natural selec-tion led him to the concept of mutation load and a method for estimatingthe mutation rate in humans. Anecdotally, I believe the role of mathemat-ical modelling in generating previously unexpected and unknown results is21.3. Evolutionary Invasion Analysisunder-appreciated. This is probably because models are usually presentedin a way that makes the conclusions most logical; this presentation is gener-ally decided upon after the results have been obtained. In Chapter 5, I willhighlight results from this thesis that were unexpected at the outset.The eventual success of a model is typically assessed by its usefulness.However, it can be difficult to evaluate the usefulness of a model directly,particularly when a model aims to advance our understanding of some pro-cess in a heuristic way. In many cases, models can appear to be caricaturesof reality, and yet still be extremely useful for advancing our understanding.This idea is well explicated in this satirical analogy by Lewis Carroll:“What a useful thing a pocket-map is!” I remarked.“That’s another thing we’ve learned from your Nation,” saidMein Herr, “map-making. But we’ve carried it much furtherthan you. What do you consider the largest map that would bereally useful?”“About six inches to the mile.”“Only six inches!” exclaimed Mein Herr. “We very soon got tosix yards to the mile. Then we tried a hundred yards to themile. And then came the grandest idea of all! We actually madea map of the country, on the scale of a mile to the mile!”“Have you used it much?” I enquired.“It has never been spread out, yet,” said Mein Herr: “the farmersobjected: they said it would cover the whole country, and shutout the sunlight! So we now use the country itself, as its ownmap, and I assure you it does nearly as well.”–Lewis Carroll, Sylvie and Bruno Concluded, Chapter XIOf course, the vastly detailed map created by Mein Herr will provide littleinsight. However, our experience will probably tell us that abstracted mapscan help us to understand the structure of the world by including only thekey details. The key details that are included in a model (or a map) dependon its purpose (Levins 1966). Like maps, models can provide us with veryuseful representations of the world and reveal features that might otherwisenot be evident (Hillis 1993).1.3 Evolutionary Invasion AnalysisIn this thesis, assorted evolutionary problems are addressed primarily us-ing the same technique: evolutionary invasion analysis (described in Kokko31.4. Life Cycle Variation2007, Chapter 7, and Otto and Day 2007, Chapter 12). This general ap-proach has a long history in population genetics (Fisher 1928, Nei 1969), lifehistory theory (Cohen 1966, Metz et al. 1992), and the evolution of socialinteractions (Hamilton 1964). Invasion analyses are typically used to ad-dress long-term evolutionary questions in which we wish to consider the fateof large number of possible alleles, each corresponding to a different trait,and evaluate the expected direction of evolution.An evolutionary invasion analysis considers whether a population that isinitially fixed for a particular allele can be invaded by a mutant allele thatspecifies a different trait value. We then infer how the trait is expected toevolve by determining which alleles can invade which populations. Thus,evolutionary invasion analyses proceed by considering a large number ofpairwise interactions between ‘resident’ and ‘mutant’ types. The ‘resident’is the allele that is initially fixed in the population and ‘mutant’ is the allelewhose invasion into the resident population will be evaluated.It is generally assumed that new alleles arise rarely; this assumption cangreatly simplify analysis, allowing us to examine more complex phenomena.Because new mutants rarely occur, it is generally assumed that the residentpopulation first reaches some long-term dynamical state (e.g., an equilib-rium) without the presence of mutant alleles in the population. Invasionof a mutant allele is then evaluated in the context of this resident popula-tion. A mutant allele invades successfully if it increases in frequency froman initially low level.While there might seem to be a prohibitively large number of pairwise in-teractions between residents and mutants to consider, types of successful orunsuccessful mutants can often be categorized. In a simple example of cate-gorization, mutants might always be able to invade residents if they confer ahigher trait value. In other cases, mutants that increase the trait value mayonly be successful under certain conditions. Therefore, categorization candivide parameter space into regions under which one evolutionary outcomeor another is expected. Categorizations are often used to make predictionsabout what trait values we expect to evolve in species with particular at-tributes.1.4 Life Cycle Variation1.4.1 Two Example Life CyclesTo illustrate variation in life cycles, we can compare the life cycle stages ofan angiosperm, Silene latifolia, and a green alga, Ulva lactuca. While these41.4. Life Cycle Variationorganisms are simply examples, they demonstrate many of the key life cyclefeatures examined in this thesis.White campion (S. latifolia) can often be found along roadsides acrossEurope and North America, growing to approximately waist height andbearing white flowers. Diploid S. latifolia plants are either male or female;each mature individual bears flowers with only male or female sexual organs.Meiosis occurs within each flower type, which halves the number of genomiccopies and yields haploid microspores (in males) or megaspores (in females).In male flowers, microspores mature into pollen grains and are presented topollinators, who may transfer them to a female flower on a different individ-ual. Once found on the receptive stigma of a female flower, these haploidpollen grains germinate and begin to grow as pollen tubes through the styletowards the mature female megaspores (female gametophytes). Many pollentubes can grow at the same time, each competing to fertilize the egg cellsof female gametophytes. A fertilized egg cell (zygote) will thus inherit onenuclear genome from the father and one from the mother. If the success-ful pollen tube had an X chromosome, the zygote will eventually developinto a female adult (with one maternal and one paternal X chromosome),whereas males develop from egg cells fertilized by pollen tubes that con-tain Y-bearing nuclei (diploid males have a maternally inherited X and apaternally inherited Y chromosome).Diploid Ulva lactuca green algae grow in rock pools and shallow subtidalareas and predominantly consist of green sheet-like thalli. Their overallappearance gives the species its common name, Sea Lettuce. Some cellsin the leaf-like thalli become reproductive and undergo meiosis to producefour spores, each bearing half the number of genomic copies (haploid). Thesespores are motile and, if successful, will settle on a rock and begin to growinto another lettuce-sized adult, this time a haploid. Haploid and diploidadults are difficult to distinguish morphologically. Reproductive cells ofhaploids produce motile gametes (also haploid) via mitosis. To form a newdiploid zygote, these gametes must fuse with a gamete released by anotherindividual of opposite ‘mating type’, where mating types are determined bythe haploid genotype. After fusion, a zygote will also settle on a suitablesubstrate and grow into a diploid adult, completing the sexual life cycle(Raven et al. 2005).Even in these highly simplified descriptions, S. latifolia and U. lactucaexhibit qualitative differences in their life cycles. Firstly, when zygotes of S.latifolia are dispersed in seeds, a fraction of seeds delay germination (remaindormant) for a short period (Purrington and Schmitt 1995). However, in U.lactuca, growth and development of zygotes is not delayed by environmental51.4. Life Cycle Variationconditions or time (Hoek et al. 1995). Secondly, the haploid phase of U.lactuca is as large and independent as the diploid phase and presumablyexperiences similar selection pressures; whereas the haploid phase of S. lat-ifolia is physically small and grows primarily within diploid tissue. Finally,S. latifolia has separate sexes in the diploid phase (and sex is determinedby the X and Y chromosomes), whereas U. lactuca does not.1.4.2 Aspects of Life Cycle Evolution InvestigatedLife cycles are highly evolutionarily significant; we expect most of the struc-tural differences between life cycles to be important for individual survivaland/or reproduction (Roff 1992, Stearns 1992). In addition, the variationin the way genetic material is exposed to natural selection (e.g., how manycopies are present) and inherited (e.g., the asymmetrical inheritance pat-tern of XY sex chromosomes through males and females) will affect the wayselection manifests changes in the hereditary material through time (Al-tenberg and Feldman 1987). Thus, it is perhaps surprising that organismsdisplay such diverse life cycles. The evolutionary forces affecting some of thestructural aspects of the life cycle are explored theoretically in this thesis. Ipresent investigations into three aspects of life cycle evolution: developmen-tal delays, selection in both ploidy phases, and sex chromosome evolution,which are all evident in the life cycles of Silene latifolia and Ulva lactucadescribed above.Developmental Delays: Typically, we expect natural selection to favourorganisms that reproduce as often and as quickly as possible (Rees 1996).However, many organisms delay development and subsequently reproductionfor long periods (Tuljapurkar and Wiener 2000); a classic example is seeddormancy, as displayed by S. latifolia. In Chapter 2, we develop mathemati-cal models that reveal three mechanisms via which developmental delays canbe selectively favoured. One key novelty is that, unlike most previous mod-els, we allow adults to be long-lived (e.g., a perennial plant rather than anannual). This yields the insight that dormancy can be favoured in order tominimize ‘immaturity risk’, that is, death in a large-scale environmental dis-turbance such as a fire before reproductive maturity is reached (mechanism3 in Chapter 2), something that is not possible in a model of a short-lived,annual plant.Haploid-Diploid Life Cycles: While sexual reproduction in eukaryotesnecessitates an alternation between haploid and diploid phases, it is not61.4. Life Cycle Variationnecessary for both haploid and diploid phases to experience selection to thesame extent. For example, while U. lactuca appears to experience selectionsimilarly in the haploid and diploid phases, the diploid phase of S. latifo-lia is physically much larger and very different from the haploid phase. Theploidy level (diploidy or haploidy) affects how alleles are exposed to selectionbecause the presence of an extra genomic copy can ‘mask’ the fitness effectsof an allele (Fisher 1930). Thus, masking can alter individual fitness directlyand also alter the response to selection, affecting the frequency of alleles infuture generations (Crow and Kimura 1965, Otto and Goldstein 1992). InChapter 3, we evaluate whether life cycles evolve to expose either the hap-loid or diploid phase to selection. A key innovation in our model is that wefully explore fitness differences between haploids and homozygous diploids(‘intrinsic fitness differences’). This reveals that the balance between in-trinsic fitness differences and masking effects can favour haploid-diploid lifecycles (growth and development in both phases).Sex Chromosome Evolution: Finally, we consider the asymmetrical in-heritance patterns of sex chromosomes, such as the X and Y chromosomes ofS. latifolia. The presence of the Y sex-determining region specifies malenessand so the Y is always found in males, whereas the X is sometimes present inmales and in females but more often in females. One consequence of this in-heritance pattern is that associations can build up between male-beneficialalleles and the Y and between female-beneficial alleles and the X (Fisher1931, Bull 1983, Rice 1987). Suppressed recombination between X and Ychromosomes is thought to evolve in order to strengthen these associations(Charlesworth and Charlesworth 1980, Lenormand 2003, Otto et al. 2011,Charlesworth 2015). In Chapter 4, we investigate the spread of large effectmodifiers of recombination (such as fusions or inversions) that link haploid-expressed genes with the sex-determining region. We find that a period ofhaploid selection (e.g., pollen or sperm competition) can drive the evolutionof suppressed recombination between sex chromosomes.The studies in this thesis use mathematical models to investigate severalcomponents of life cycle evolution. The larger theory of life cycle evolutionincludes various other aspects, including the evolution of iteroparity (Cole1954, Charnov and Schaffer 1973, Tuljapurkar and Wiener 2000), age atfirst reproduction (Stearns 1992, Roff 1992, Charlesworth 1994), senescence(Medawar 1952, Partridge and Barton 1993, Rose 1994), mating systems(Emlen and Oring 1977, Barrett and Eckert 2012), sexual systems (Barrett71.4. Life Cycle Variation2002, Otto 2009), the number of sexes (Hurst and Hamilton 1992, Togashiand Cox 2011), and dispersal (Hamilton and May 1977, McPeek and Holt1992, Doebeli and Ruxton 1997). The overall aim of examining these prob-lems is that, by combining the theory developed for the evolution of differentaspects, we can better understand how and why complex life cycles (likethose described above) evolved.8Chapter 2Why Wait? ThreeMechanisms Selecting forEnvironment-DependentDevelopmental Delays12.1 SummaryMany species delay development unless particular environments or rare dis-turbance events occur. How can such a strategy be favoured over con-tinued development? Typically, it is assumed that continued development(e.g., germination) is not advantageous in environments that have low ju-venile/seedling survival (mechanism 1), either due to abiotic or competitiveeffects. However, it has not previously been shown how low early survivalmust be in order to favour environment-specific developmental delays forlong-lived species. Using seed dormancy as an example of developmentaldelays, we identify a threshold level of seedling survival in ‘bad’ environ-ments below which selection can favour germination that is limited to ‘good’environments. This can be used to evaluate whether observed differences inseedling survival are sufficient to favour conditional germination. We alsopresent mathematical models that demonstrate two other, often overlooked,mechanisms that can favour conditional germination in the absence of differ-ences in seedling survival. Specifically, physiological trade-offs can make itdifficult to have germination rates that are equally high in all environments(mechanism 2). We show that such trade-offs can either favour conditionalgermination or intermediate (mixed) strategies, depending on the trade-offshape. Finally, germination in every year increases the likelihood that someindividuals are killed in population-scale disturbances before reproducing;it can thus be favourable to only germinate immediately after a disturbance1A version of this chapter has been published. Michael F Scott and Sarah P Otto(2014) Journal of Evolutionary Biology, 27: 2219-2232.92.2. Introduction(mechanism 3). We demonstrate how demographic data can be used to eval-uate these selection pressures. By presenting these three mechanisms andthe conditions that favour conditional germination in each case, we providethree hypotheses that can be tested as explanations for the evolution ofenvironment-dependent developmental delays.2.2 IntroductionOne might expect organisms to reproduce as early as possible, yet manyorganisms delay development such that their eventual reproduction is alsodelayed, a strategy that should typically lead to a slower growth rate (Rees1996). This is the classic evolutionary problem posed by developmentaldelays (Tuljapurkar and Wiener 2000), such as seed, spore, and cyst dor-mancy in plants, fungi and bacteria (Cohen 1967, Ellner 1985a, Rees 1996),non-seed (‘prolonged’ or ‘vegetative’) dormancy in plants (Roerdink 1988,Gremer et al. 2012), and diapause in insects, crustaceans, sponges and fish(Tuljapurkar and Istock 1993, Evans and Dennehy 2005, Venable 2007). Inthis paper we consider the evolution of strategies that delay development ina manner that depends on environmental state in a demographically struc-tured population. First, we briefly review previous studies that explore theevolution of developmental delays and then place our work in this context.Two classic studies of seed dormancy in annual plants are the influentialtheoretical papers by Cohen (1966; 1967). Cohen (1966) constrained ger-mination rate to be the same in all years (constant germination strategy)but allowed the seed yield produced per germinating seed to vary acrossyears. The optimal germination strategy was found to depend on the vari-ation in yield across years. If, in some years, yield is lower than survivalin the soil, partial seed dormancy can evolve. Cohen (1967) considered adifferent scenario, in which germination strategy can vary according to theenvironment at the time of germination (state-dependent germination strat-egy, sometimes called ‘predictive germination’, Venable and Lawlor 1980). Ifseeds are able to perfectly predict eventual yield based on the environmentthey experience, germination should occur in ‘good’ years and dormancyin ‘bad’ years. If the yield cannot be accurately predicted at the time ofgermination, then the optimal germination rate in a particular perceivedenvironment depends on the distribution of yields that might actually oc-cur; this set can include some ‘good’ and some ‘bad’ yields, in which caseintermediate germination rates can again evolve. See the Model Backgroundsection for some mathematical details of these models.102.2. IntroductionRelated studies have modelled the timing of diapause in insects andcrustaceans in which the diapausing fraction can vary over a year in responseto temperature and day length cues (Cohen 1970, Taylor 1980, Hairstonand Munns 1984, Taylor and Spalding 1989, Spencer and Colegrave 2001).This is equivalent to an extremely plastic germination strategy, and thesestudies similarly find that populations should switch from non-diapausingto diapausing when the reproductive yield from breaking diapause is lowerthan the survival of a diapausing individual. For example, Taylor (1980)found that diapause should begin when the time until catastrophe (frost) isless than the time required to reach maturation and produce one offspringof diapausing age. This result assumes that the date of the first frost ispredictable. In reality, catastrophes do not reliably occur on the same dateeach year. Consequently, there is variation in reproductive yield on eachday, which can favour a mixed diapause strategy (Cohen 1970, Hairston andMunns 1984, Taylor and Spalding 1989, Spencer and Colegrave 2001).The above models correspond to annual plant and diapausing insect life-cycles in which only individuals of a single age class persist between years.This allows the demographic dynamics to be described by a single equation:the number of seeds, diapausing eggs, lavae, pupae or adults that overwinter.However, developmental delays are also common in species with overlappinggenerations. For example, while not explicitly comparing germination ratesin annuals and perennials, Baskin and Baskin (2014) find that the percent-age of tree or shrub species with some form of seed dormancy is generallysimilar to the percentage of herbaceous species with dormancy (figures 12.3and 12.4) in a review of over 13,000 species. With overlapping generations,demographic modelling becomes more complex because survival and repro-duction of each age (or stage) class must be considered. Conceptually, a keydifference is that lifetime reproductive output must be calculated over sev-eral time steps and so may include several environments and the particularorder of those environments.Nevertheless, there have been some studies that have considered theevolution of developmental delays in age- or stage- structured populationsexperiencing temporally varying environments. These studies generally con-sider environmental variation that affects fertility (seed yield) only (but seeKoons et al. 2008, discussed below) and assume that strategies do not de-pend on the environment. Roerdink (1988; 1989) modelled the evolutionof delayed reproduction in a predominantly biennial species that dies afterreproducing. Similarly, Tuljapurkar (1990a) presented a model for the evolu-tion of delayed reproduction in semelparous organisms and organisms witha very short adult life-span. Additionally, Tuljapurkar and Istock (1993)112.2. Introductionconsidered the evolution of a short developmental delay, e.g., diapause ininsects that can delay maturation for one year only. These studies haveshown that delays can evolve in a demographically structured population tobuffer against environmental variability in fertility, as in the unstructuredmodel considered by Cohen (1966).Developmental delays spread the reproductive effort from a seed/juvenilecohort over time, providing an ‘escape in time’ from environmental variation(Venable and Lawlor 1980). Iteroparity also spreads reproductive effort overtime, buffering against environmental variation even in the absence of de-velopmental delays (Tuljapurkar and Istock 1993, Tuljapurkar and Wiener2000). Developmental delays can evolve in an iteroparous population, pro-viding both forms of buffering, but only if mean seed (juvenile) survival ishigher than mean adult survival and thus seeds (juveniles) are able to ‘spreadthe risk’ more than iteroparity alone (Koons et al. 2008). Tuljapurkar andWiener (2000) also explored the evolution of both iteroparity and develop-mental delays, assuming a linear trade-off between adult survival and yearlyreproductive effort. They tended to find either the evolution of iteroparityor developmental delays, but other trade-off functions might generate simul-taneous selection for a mixture of iteroparity and developmental delay (assuggested by Wilbur and Rudolf 2006).The above studies for demographically structured populations all assumea constant strategy in all years, as in Cohen (1966). Here, we model theevolution of a state-dependent strategy in a demographically structured pop-ulation, that is, germination rate can be different in different environments.The case where cues allow the strategy to depend on the time of year hasbeen considered in models for the timing of diapause (Taylor 1980, Hairstonand Munns 1984, Spencer and Colegrave 2001). However, particular envi-ronments can provide cues that allow germination rates to vary in a state-dependent (not time-dependent) manner. Examples of state-dependent de-velopmental delays include seed germination responses to light and rainfall(Pake and Venable 1996, Evans et al. 2007) or spore germination responsesto heatshock (Perkins and Turner 1988), amino acid concentrations or host-specific substances (Cohen 1967). In a particularly clear example, smoke ortemperature cues from fires cause increased germination rates or release ofseeds from fruiting structures (‘serotiny’) in many species (including manyperennials, Keeley 1995). Treatment with smoke is estimated to increasegermination rates in over 2,500 species (Bradshaw et al. 2011) and up to1,200 perennials exhibit serotiny (Lamont et al. 1991, Lamont and Enright2000).For simplicity, we will use botanical terms (seeds, germination, etc.), al-122.3. Model Backgroundthough the models themselves can apply to other developmental delays thatdepend on environmental state. As discussed above and elsewhere (Rees1996, Evans et al. 2007) the evolutionary problem posed by dormancy isthat delaying development eventually delays reproduction and so reproduc-tive opportunities seem to be passed up. In this context, the problem ofconditional germination strategies is not ‘Why germinate in environment1?’ but ’Why forgo germination in environment 2?’.In this work, we investigate this problem and present three mechanismsgenerating selection that favours organisms that pass up germination op-portunities: (1) Avoiding germination in ‘bad’ environments that have lowseedling survival. (2) Avoiding costly physiological trade-offs between thegermination rates in different environments (in addition to the fundamental‘trade-off’ that seeds that germinate are no longer available to germinatein the future). (3) Minimizing the risk of experiencing a severe disturbancebefore reproducing (note that this requires state-dependent germination andperenniality).This provides a framework for researchers wishing to investigate theevolution of environment-dependent developmental delays. We provide athreshold level of seedling survival in ‘bad’ environments below which con-ditional germination should evolve. Thus providing a quantitative meansto test whether the most commonly envisaged mechanism can explain theevolution of conditional germination in a particular organism. If not (or ifthere are also physiological trade-offs or large-scale disturbances), we pointout that the other, less commonly discussed, mechanisms should be consid-ered. With demographic data for a particular species in different environ-ments, one can investigate whether these selective mechanisms should act bymanipulating the relevant parameters separately as we do here. For exam-ple, setting seedling survival in all environments to be equivalent eliminatesmechanism 1 and reducing the number of years required to reach maturitycan eliminate mechanism 3. We discuss some specific empirical data forthese mechanisms in more detail in the discussion section.2.3 Model BackgroundTo connect our model with previous results, we first provide a brief overviewof some key mathematical results. In the model by Cohen (1966), the num-132.3. Model Backgroundber of seeds (S) at time t is given byS[t] = S[0](∏i((1− g)sS + gyi)pi)t, (2.1)where g is the germination rate (assumed constant), sS is the survival ofseeds in the soil and pi is the proportion of the t years that has environmenti in which the environment-specific seed yield is yi. Increasing germinationrate will increase (decrease) growth rate if the derivative of the parentheticalterm with respect to g is positive (negative), where the sign of this derivativedepends on∑ipi(yi−sS)(1−g)sS+gyi . Dormancy may evolve if some years yield fewerseeds than would survive in the soil (yi < sS). For example, a populationthat germinates 100% of its seeds would go extinct if ever an extremely‘bad’ year (no seed set) were encountered, favouring the evolution of seeddormancy.Where environments vary over space, however, lineages can escape ex-tinction by surviving in ‘good’ environments and recolonizing. This has beencalled ‘escape in space’ via dispersal in contrast to ‘escape in time’ via dor-mancy (Venable and Lawlor 1980). MacArthur (1972, p.165-168) introduceda model with many patches and global dispersal among them, finding thatthe optimal strategy is the one that has the highest growth rate averagedover all patches. In this model, a proportion of the population experienceseach environment in each year and soS[t] = S[0](∑ipi((1− g)sS + gyi))t, (2.2)where pi is the proportion of the population that experiences environmenti with yield yi. In this model, changes in germination rate affect growthrate according to∑i pi(yi − sS), which must be positive in a populationcapable of growth, therefore seed dormancy should not evolve. These twomodels, with variability entirely temporal or spatial are extreme cases andintermediate scenarios have been considered by others (Levin et al. 1984,Cohen and Levin 1987, Klinkhamer et al. 1987, Wiener and Tuljapurkar1994), who also find that ‘escape in space’ via dispersal lessens the need for‘escape in time’ via dormancy.Closer to the models we consider, Cohen (1967) includes environment-specific germination into equation (2.1):S[t] = S[0](∏i((1− gi)sS + giyj)pij)t, (2.3)142.3. Model Backgroundwhere gi is the germination rate in a particular seed environment and pij isthe proportion of years that seeds are in environment i and yield yj seeds ifgerminated. Selection on the germination rate in a particular environment(gi) then has the same sign as∑ijpij(yj−sS)(1−gi)sS+giyj . Evolution of germinationrate in each environment therefore evolves in a similar manner to the overallgermination rate in the Cohen (1966) model. However, each seed environ-ment can have a different optimum. In a special case (termed ‘completeinformation’), the yield is reliably given by the seed environment (i), suchthat pij and yj can be replaced by pi and yi. In this case, the pure strate-gies of complete germination (gi = 1) and complete dormancy (gi = 0) arefavoured in ‘good’ (yi > sS) and ‘bad’ (yi < sS) environments respectively.MacArthur (1972) did not include environment-specific germinationrates into his model with purely spatial environmental variation. However,one can modify equation (2.2) to allow germination rate to vary along withthe environment that affects seed yield, such that g becomes gi. This mod-ification may seem equivalent to the ‘complete information’ case in Cohen(1967), but it also applies with uncertain assessment of yield if yi is definedas the average yield from seeds across environments – correctly or incorrectly– assessed as being in state i. Although the yield in each patch is uncer-tain, this uncertainty can be averaged across the patches in each year togive a particular yield for each seed environment. This model also predictscomplete germination in ‘good’ (yi < sS) patches and dormancy in ‘bad’(yi < sS) patches.In this study, we consider perennial species and assume that a fixed pro-portion of the population experiences each environment in each time stepin sections 1 and 2 (mechanisms 1 and 2), as in the annual plant modelby MacArthur (1972, p165-168). We use the approach explained above toinclude environment-specific germination rates. In the final section, we in-clude temporal variation where the whole population experiences the sameenvironment in each time step, as in Cohen (1966; 1967). In order to dealwith temporal variation in a demographically structured population we firstconsider strictly periodic disturbances to obtain some approximate analyti-cal results and then use numerical simulations based on the demography ofBanksia hookeriana (following Enright et al. 1998) to investigate the evo-lution of environment-dependent developmental delays with non-periodicdisturbances. For this section we consider the ‘complete information’ casebecause we focus on the effects of disturbance risk rather than uncertainassessment of yield. That said, when disturbances are non-periodic, weincorporate uncertainty in the ordering of environments even though the152.4. Model and Resultsdemographic parameters in each environment are constant.2.4 Model and ResultsWe evaluate the evolution of environment-dependent germination (condi-tional germination) with a variety of stage-structured models. All analyseswere conducted using Mathematica (Wolfram Research Inc. 2010), a file forreplicating our analyses is available on request. We considered environmen-tal variation that can affect all life-history parameters. In our notation forenvironment i, the survival of adult plants is sAi, seed survival is sSi, ger-mination rate is gi, post-germination seedling survival is sY i and each adultproduces bi seeds in each time step. We allow both seeds (S) and adults (A)to survive between time steps.2.4.1 Mechanism 1: Low Seedling Survival in SomeEnvironmentsIt is commonly thought that conditional germination evolves to avoid ger-mination in environments with low seedling survival (e.g., Lamont et al.1991, Lamont and Enright 2000, Midgley 2000, Keeley et al. 2011). To testthis mechanism we first modelled a population in which a random propor-tion of the population (pi) experience each environment in each time step(n∑i=1pi = 1, where n is the total number of environments), with no tem-poral autocorrelations in patch type (either because migration is global orpatches change randomly at each time step). Initially, we examine a density-independent growth model, but we then show that similar conditions arisewith a density-dependent model. The change in seed and adult populationsizes from time step t to time t+ 1 are described by the following recursionequations written in matrix form:(S[t+ 1]A[t+ 1])= TA(S[t]A[t]), (2.4)whereTA =n∑i=1pisSi(1− gi)n∑i=1pibin∑i=1pisY igin∑i=1pisAi . (2.5)162.4. Model and ResultsWe used the leading eigenvalue (λ) of the transition matrix, TA, to approxi-mate the long-term growth rate of the population of seeds and adults. Thenwe examined whether mutants that alter the germination parameters (gi)have an increased or decreased long-term growth rate. A small change inthe germination rate in environment j, gj , will affect the long-term growthrate, λ, according to∂λ∂gj=pjsSjn∑i=1pisAi + pjsY jn∑i=1pibi − pjsSjλ2λ−n∑i=1pisAi −n∑i=1pisSi(1− gi). (2.6)Unlike an annual plant version of the same model, equation (2.6) has termsfrom all n environments. That is, optimal germination rate in environmentj depends on the quality of the other environments that adults might sub-sequently experience when demographic structure is included. If equation(2.6) is positive for some environments (j) and negative for others, then con-ditional germination is expected to evolve. From this point on we will focuson the case where environments can be classified into two groups. Two isthe minimum number of environments required for conditional germination,in which dormancy is favoured in one environment but not another.In this section we demonstrate that differences in seedling survival canfavour conditional germination. For this purpose we define a ‘good’ (i = 1)patch as one in which seedling survival is higher than in the ‘bad’ (i = 2)patches (sY 1 > sY 2). Assuming that the population is capable of growth(λ > 1), germination rates in the ‘good’ environment should always bemaximized (mutants with higher g1 values always have high higher long-termgrowth rates). By contrast, germination rates in the ‘bad’ environments (g2)should sometimes evolve to be as high as possible and sometimes as low aspossible, with the transition occurring when the following condition holds:sY 2(b2sY 2 + sA2 − sS2)g1sS1(sY 1 − sY 2) + sY 2(sY 2(b2 − b1) + sA2 − sA1 + sS1 − sS2) − p = 0,(2.7)where we have specified that the ‘good’ (i = 1) environment is experience byp proportion of the population and the ‘bad’ (i = 2) environment by (1−p).See appendix A.1 for more details of our analysis. An example of how thelong-term growth rate (λ) changes on either side of this point is shown infigure 2.1A.Figure 2.1B illustrates the region in which conditional germination is172.4. Model and Resultsexpected to evolve, with germination only occurring in ‘good’ patches. Theproportion of ‘good’ patches (p) must be high enough and seedling survivalin ‘bad’ patches (sY 2) must be sufficiently low. When seedling survivalin both environments is equivalent (dashed line in figure 2.1B), conditionalgermination should never evolve in populations capable of growth (λ > 1). Ithas previously been noted that conditional germination should evolve whenestablishment ability in ‘bad’ environments is negligible (sY 2 = 0, Lamontet al. 1991). Equation (2.7) echoes this result but also shows a more generalcase, in which we indicate exactly how low seedling survival in ‘bad’ patches(sY 2) must be.Seedling Survival in 'Bad' Environments, sY2Long−Term Growth Rate, λl0 0.2 0.4 0.6 0.8 1.011.21.41.61.82.0ASeedling Survival in 'Bad' Environments, sY2Proportion 'Good' Environments, pl0 0.2 0.4 0.6 0.8 sY1 1.000.20.40.60.81.0BFigure 2.1: Life-history parameters forwhich conditional germination strategieshave a higher long-term growth rate. A) Thelong-term growth rate (λ) for a plant witha conditional germination strategy (dottedline, g2 = 0.1) and a plant without one(black line, g2 = 1). Conditional germi-nation confers a higher growth rate whenseedling survival is below the transitionpoint specified by equation (2.7) (circle).The grey region shows where conditionalgermination is expected to evolve. B) Whenthe frequency of ‘good’ environments (p)is high enough and the seedling survivor-ship in bad years (sY 2) is low enough, con-ditional germination should evolve (grey).If seedlings never establish in ‘bad’ years(sY 2 = 0, see arrow), a plant always fallsin the region favouring conditional germi-nation (grey). The dashed line indicateswhere seedling survival is equal across ‘good’and ‘bad’ years (sY 1 = sY 2), in whichcase conditional germination never evolves(see appendix A.1). Note that, even whereselection would favour germination if onlythe ‘bad’ environment were experienced (seewhite region along x-axis, p = 0), condi-tional germination can evolve. The otherparameters used are g1 = 1, p = 0.2, sY 1 =0.9, sS1 = 0.8, sS2 = 0.8, b1 = b2 = 4,sA1 = 0.6, sA2 = 0.6.In appendix A.1 we also show that the region in which conditional ger-mination should evolve expands when the seed bank is more persistent (sSiis larger), the proportion of the population experiencing ‘good’ conditions182.4. Model and Results(p) is larger and when germination rate (g1) and seedling survival (sY 1) ingood patches is higher. In contrast, this region will contract when adultsurvival (sAi) is higher, number of seeds produced (bi) is higher, or whenseedling survival in ‘bad’ years (sY 2) is higher.2.4.2 Mechanism 2: Trade-OffsIn the above model and MacArthur (1972), germination rates evolve tobe either maximized or minimized. We next include physiological trade-offs, which can allow intermediate germination rates to evolve even withpurely spatial variation in environments. This is true for both annual andperennial plant models (see appendix A.2 for a version of the MacArthur1972 annual plant model with trade-offs, which produces very similar resultsto the perennial model presented here). Trade-offs could exist between anyof the demographic parameters, see the discussion section for some examples,but to demonstrate the qualitative effects of trade-offs on germination rate,we incorporated a direct trade-off between germination rates in differentenvironments (g1 and g2) using a generic function (g1[g2]). For two typesof patches and global migration, the transition matrix describing changes inseed and adult populations then becomes:TB =(psS1(1− g1[g2]) + (1− p)sS2(1− g2) pb1 + (1− p)b2psS1sY 1g1[g2] + (1− p)sS2sY 2g2 psA1 + (1− p)sA2).(2.8)Here, we are particularly interested in cases where conditional germinationis expected to evolve where it wouldn’t without the trade-off. Therefore,we start by presenting the special case of (2.8) where seedling survival isconstant (sY 1 = sY 2), which never yielded conditional germination strategiesin the previous section.Our approach (details in appendix A.2) was to identify evolutionarilystable strategies (ESS) for germination rates (g1 and g2) where no mutantwould have a higher growth rate, λ. For model (2.8) with sY 1 = sY 2, asingular point occurs when:sS2sS2 − sS1g′1[g2]− p = 0 (2.9)where g′1[g2] is the first derivative of the trade-off function. In some cases,traits that maximize germination rate in one environment could also increasegermination rates in other environments (e.g., Simons 2014). If germinationrates are positively coupled in this manner (g′1[g2] > 0), the singular point192.4. Model and Resultsin (2.9) cannot be satisfied and germination rates should evolve to be high.However, in figure 2.2B, we plot equation (2.9) for a negative trade-off (wherephysiological constraints make it difficult to have simultaneously high germi-nation rates in all environments, g′1[g2] < 0). We next determined whetherthis singular point is a maximum or a minimum growth rate in order toassess whether germination rates are expected to evolve towards this pointor away (whether it is an ESS or evolutionary repeller). We found thatsingular point (2.9) changes from an ESS to an evolutionary repeller whenthe shape of the trade-off function transitions between concave (g′′1 [g2] < 0)and convex (g′′1 [g2] > 0), see figure 2.2.When trade-offs are concave (solid line in figure 2.2), seeds are ableto germinate reasonably well in both environments, and the germinationstrategy is expected to reach an intermediate ESS germination rate in bothenvironments, where the two germination rates satisfy equation (2.9). Ob-serving intermediate germination rates could then suggest the presence of atrade-off (e.g., Tonnabel et al. 2012, discussed below) or temporal variation(see next section).With a convex (dashed line in figure 2.2) trade-off, plants are expectedto specialize on germination in either environment 1 or 2. Thus, conditionalgermination can evolve as a means to specialize and avoid a costly trade-off.The germination strategy predicted to evolve with a convex trade-off de-pends on seed survival rates (sSi), the proportion of patch types 1 versus 2and any initial specialization. Importantly, though, even if survival and fer-tility are equal in all environments, conditional germination can still evolve,simply because the traits that allow good germination in one environmentprevent it in the other. Empirically then, trade-offs are likely present incases where little difference in demographic parameters can be detected.We next combine the effects of a trade-off with differences in seedlingsurvival (mechanisms 1 and 2). As in the previous section we arbitrarily as-sume that environment 1 has superior seedling survival (sY 1 > sY 2). WhensY 2 6= sY 1 the simple solution (2.9) no longer applies. We obtained a morecomplicated expression for the singular point (ESS or repeller) and plottedan example in figure 2.2C. What is apparent is that, decreasing seedlingsurvival in ‘bad’ environments (sY 2) increases the region of parameter spaceover which germination rates in the ‘bad’ environments should evolve to below.The above models ignore competition and assess growth rates of differentlife-history strategies. We next incorporated density dependence into thismodel by including a competition function that limits population size. Forexample, competition-related mortality might affect seedling survival such202.4. Model and ResultsGermination Rate 2, g2Germination Rate 1, g10.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0AProportion Experiencing Environment 1, pGermination Rate 1, g10.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0BProportion Experiencing Environment 1, pGermination Rate 1, g10.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0CFigure 2.2: Trade-offs between germinationrates (g1 and g2) can favour intermediatestrategies or specialization. Panel A) showsa concave (solid, s = 4/3) and convex(dashed, s = 3/4) example for the trade-offfunction using g1[g2] = 1− (1− (1− g2) 1s )s.Panels B) and C) show the evolutionarilystable germination strategy (solid) or the re-pelling strategy (dashed), with arrows repre-senting the expected evolutionary trajectoryfor germination rate for a given frequency ofenvironments 1 (p) versus 2 (1−p). Panel B)assumes sY 2 = sY 1 (corresponding to thedashed line in figure 1B), in which case thecurves are given by equation (2.9). PanelC) shows an example where sY 2 6= sY 1(sY 1 = 0.9, sY 2 = 0.4). Other parametersin B) and C) are as in figure 1.212.4. Model and Resultsthat the lower left element in matrix TB is multiplied by the logistic density-dependent function (1 − A[t]K ) where K is the population carrying capacityof adults. More generally, we multiplied seedling survival by an arbitrarycompetition function (comp[A[t]]) to re-write the transition matrix as:TC =(psS1(1− g1[g2]) + (1− p)sS2(1− g2) pb1 + (1− p)b2(psS1sY 1g1[g2] + (1− p)sS2sY 2g2)comp[A[t]] psA1 + (1− p)sA2).(2.10)We then conducted an evolutionary invasion analysis, in which a residentpopulation was allowed to reach an equilibrium size (assuming this to bestable) and then the invasion ability of a mutant with a different germinationrate was evaluated, as measured by the leading eigenvalue of TC for a raremutant (details in appendix A.2). If germination rates affect the number ofseedlings but not the nature of competition (i.e., comp[A[t]] is not a functionof g1 or g2), the results remain the same as above (for mechanisms 1 and 2),but with birth rates now multiplied by comp[A[t]].2.4.3 Mechanism 3: Effects of Synchronization WithDisturbancesHere we focus on a particular type of temporal variation in environment,such as large-scale disturbances like fire, which affect adult survival andpotentially germination rates across the entire population at the same time.Synchronizing germination to occur immediately after a disturbance thenmaximizes the number of years as an adult before experiencing the nextdisturbance. By contrast, plants that germinate in non-disturbance years1) have fewer chances to produce seeds before experiencing a disturbanceand 2) are more likely to die in a disturbance before producing seeds at all.We show that these costs of poor synchronization can be strong enough tocause plants to forgo germination in years without disturbances, even in theabsence of differential seedling survival or trade-offs.In this section, the notation for environment 1 (i = 1) is used for yearswith population-scale disturbances and environment 2 (i = 2) specifies life-history parameters in non-disturbance years. We assume that the popula-tion census is such that germination rate in disturbance years (disturbance-induced germination rate, g1) is measured for the first germination eventafter the disturbance (so that it can be affected by disturbances). With fire,for example, fire years (i = 1) would be associated with low adult survival(sA1) but potentially high seedling survival (sY 1) because seeds emergingafter the fire experience a low competition and high nutrient environment.222.4. Model and ResultsThe transition matrices describing changes in seed and adult populationsizes in non-disturbance and disturbance years are as follows:T1 =(sS1(1− g1) b1sS1sY 1g1 sA1), (2.11a)T2 =(sS2(1− g2) b2sS2sY 2g2 sA2). (2.11b)Firstly, we consider a disturbance cycle, in which disturbances occur every τyears. That is, we include a number of non-disturbance years (τ−1) followedby a disturbance year. To describe population size changes over the entirecycle we apply the disturbance year transition matrix (T1) and then iteratethe transition matrix τ − 1 times for non-disturbance years (T2). Usingstandard rules of matrix algebra,T2τ−1 ·T1 = A ·Dτ−1 ·A−1 ·T1 (2.12)where A is a matrix in which the columns are the eigenvectors of T2 and Dis a matrix in which the diagonal elements are the eigenvalues of T2.The logic of our analysis is similar to above. We evaluate whethermodifying the germination parameters increases or decreases the long-termgrowth rate, λ, given by the leading eigenvalue of the entire cycle matrix(equation 2.12). We provide the details of our approach in appendix A.3.While (2.12) accurately describes changes in the long-term growth rateover the entire cycle, it is quite complex to analyse. We thus used an ap-proximation to simplify the analysis. Specifically, we assume that D canbe approximated by omitting the smaller eigenvalue. This approximation ismost accurate when the difference between eigenvalues is large and/or whenthe number of years between disturbances is large (over time, the effects ofthe larger eigenvalue dominate, e.g., Otto and Day 2007, box 9.1). Care musttherefore be taken in interpreting the results when the cycle length is short,which is also when we find that conditional germination strategies tend tobe favoured. Thus, this approximation only serves as a guide to conditionsthat favour conditional germination; the accuracy of the approximation isdiscussed in appendix A.3.To distinguish synchronization effects from those already explored, wefocus on the case where there are no direct trade-offs between germinationrates (g1 and g2) and where seedling survival rates in disturbance and non-disturbance environments are equal (sY 1 = sY 2). We found that mutants232.4. Model and Resultswith higher disturbance-induced germination rates (g1) are expected to havehigher long-term growth rates, given that the population is able to grow innormal years (as assumed throughout this section). We therefore assumedthat disturbance-induced germination rate is high (g1 = 1) when analyzingthe evolution of the germination rate in non-disturbance years, g2. For verylong disturbance cycles (high τ), higher germination rates in non-disturbanceyears should also give higher long-term growth rates. However, when thedisturbance cycle is short enough (less than the critical value τc, see equationA.46), conditional germination, g2 < 1, is favoured.We took the derivative of τc with respect to life-history parameters in thedisturbance year to see the effect that the parameters have on the lengthof the disturbance cycles over which conditional germination is expectedto evolve. We found that increasing seed bank persistence through distur-bances (sS1) and increasing disturbance-induced germination (g1) increasesthe parameter space over which selection favours conditional germinationstrategies. However, increasing seeds produced in the disturbance year (b1)and adult survival through disturbances (sA1) decreases the range of distur-bance intervals for which conditional germination should evolve.Our results indicate that, conditional germination (g2 < 1) should evolvewhen adults that germinate in non-disturbance years risk death in a distur-bance before producing a significant number of seeds. By contrast, condi-tional germination should not generally evolve when disturbances have littleeffect on adult survival (sA1 is high) and when adults are guaranteed toproduce a large number of seeds even if they mature for the first time in thedisturbance year (b1 is high), see figure 2.3.Adult survival through disturbances, sA1Seeds produced in disturbance years, b 1τ = 2τ = 3τ = 4τ = 50.01 0.1 1.00.010.11.02.0 Figure 2.3: The parameters for which condi-tional germination is expected to evolve forvarious different disturbance cycle lengths(τ) based on our approximation, assumingadults reach reproductive maturity immedi-ately (in the time step after germination).See figure A.2 for comparison with a non-approximated model. Increasingly dark greyareas indicate where the germination ratein non-disturbance years (g2) is expectedevolve to be below one (conditional germi-nation) for cycle lengths of 2, 3, 4 and 5(lighter regions overlap darker regions). Inthe white region, conditional germination isnot expected to evolve for any cycle length,τ . Other parameters are g1 = 1, sY = 0.6,sS1 = sS2 = 0.9, b2 = 2 and sA2 = 0.7.242.4. Model and ResultsFigure 2.3 suggests that conditional germination should only evolve forrelatively short disturbance cycles. However, in the above models, organ-isms become reproductively mature after one year and so the advantages ofsynchronization are necessarily weak. We expand on these analytical resultsusing some numerical simulations that include more complex demography.Table 2.1: Default parameters used in numerical simulationsParameter Symbol Default Valuefire-induced germination rate g1 1normal germination rate g2 0-1∗seed survival sS2 = sS1 0.94adult survival sA2 p[age]†adult survival (fire) sA1 0.005seedling survival sY 0.042seed production b2 = b1 m[age]‡age at first reproduction A2 5age at max reproduction A3 15max seed age V m 15max adult age A4 40∗ Varied between 0 and 1 in steps of 0.05, the value yielding the highestlong-term growth rate (λ, leading eigenvalue of the transition matrix) wasrecorded.† For 1 < age < 25, p[age] = (1/f [age])/(1/f [age − 1]) where f [age] =69.03 log10[age] + 23.60. For 25 ≤ age, p[age] = (1/f [age])/(1/f [age −1])(1− 0.01(age− 24)).‡ For age < A2, m[age] = 0. m[age] = 200 when A3 ≤ age. For A2 ≤age < A3, m[age] =200(age+1−A2)1+A3−A2 .We based our simulations on those of Enright et al. (1998), using param-eters that approximately correspond to the demography of Banksia hooke-riana, an Australian shrub in the Proeaceae that retains almost all seeds onthe plant until immediately after a fire. The parameters are given in Table2.1. The major technical difference between our simulations and those ofEnright et al. (1998) is that we assume seeds remain in the seed bank afterplant death, whereas seeds died with the parent (but not in fires) in theoriginal model. This change allowed us to simulate the entire populationby multiplying by the appropriate matrix in (2.11) rather than tracking in-dividuals. We also allow a small fraction of adults to escape disturbancesin microclimates (sA1 = 0.005), this prevents complete population extinc-tion if ever two disturbance events occur in a row. We made two importantbiological modifications to expand on our analytical results: 1) we varied252.4. Model and Resultsthe number of years before maturity is reached to show that conditionalgermination should only evolve when there is a significant risk of death be-fore producing seeds, 2) we explored non-periodic disturbances (fires in thismodel) to show that the ‘synchronization effect’ continues to favour condi-tional germination. In all our simulations, there is no difference in seedlingsurvival (sY ) between environments (mechanism 1 absent).For particular fixed disturbance (fire) cycle lengths, we varied the num-ber of years to first reproduction (A2), from 1 to 3 to 5 years and recordedthe optimal germination rate in normal years (the g2 that yielded the highestlong-term growth rate, λ). The results are plotted in figure 2.4, which showsthat the advantage of conditional germination is increased when the num-ber of years to reproductive maturity is increased. This demonstrates that‘synchronization advantages’ favour conditional germination in this model,which was not originally made explicit in Enright et al. (1998).Disturbance−Return Interval, τNon−Disturbance Germination Rate, g 200.20.40.60.81.05 10 15 20 25 30Figure 2.4: The germination rate innon-disturbance (fire) years (g2) thatyields the highest long-term growthrate in our numerical analysis of alife history akin to Banksia hookeri-ana (Enright et al. 1998) for differentdisturbance-return intervals. The solidline is for the default parameters withan age of reproductive maturity of 5years, whereas the dashed and dottedlines are where age at first reproduc-tion (A2) was reduced to 3 and 1 years,respectively. Notice that when adultsbecome reproductively mature immedi-ately environment-dependent germina-tion never evolves (dotted line).For variable disturbance cycles, we next drew integer disturbance inter-vals from a Weibull distribution, see figure 2.5A. We varied the regularityof disturbances by using a shape parameter (β) of 1, 2 or 4, which representincreasing regularity of disturbances, starting from the exponential distribu-tion (β = 1, constant disturbance risk, β = ∞ corresponds to the periodiccase considered above). In figure 2.5B we plotted the germination strategyin non-fire years that gave the highest growth rate (averaged across repli-cate 100 draws of 20 disturbances) for various mean disturbance intervals.Figure 2.5B shows that, even when disturbance intervals are highly variable(β = 1), conditional germination (low g2) can be advantageous. We alsonote from figure 2.5B that variability tends to favour mixed strategies, with262.5. Discussiong2 values between zero and one, representing bet hedging between the longand short intervals.Disturbance−Return Interval, τProbability00.020.040.060.080.10 5 10 15 20 25 30ADisturbance−Return Interval, τNon−Disturbance Germination Rate, g 200.20.40.60.81.05 10 15 20 25 30BFigure 2.5: The effect of variability indisturbance-return interval on the evolutionof conditional germination. A) The shapeparameter affects the Weibull distributionused for the fire-return interval (τ). Themean in each case is 15 years between distur-bances. The dotted line shows the probabil-ity of selecting disturbance-return intervalswhen the Weibull shape parameter (β) is 1and is equivalent to a exponential distribu-tion with expected value 15. The dashedand solid lines are for β = 2 and 4 respec-tively and represent increasing regularitydue to an increasing hazard with time sincethe last disturbance. B) The solid, dashedand dotted lines show the corresponding av-erage germination rate in non-disturbanceyears (g2) that yielded the highest long-termgrowth rate in our simulations.2.5 DiscussionIn this paper, we explored three mechanisms by which a developmental de-lay (e.g., seed dormancy) can be favoured in certain environments but notin others. This work builds upon the model of annual plants developedby Cohen (1967) but allows for demographic structure. While Cohen pre-dicted that optimal germination strategies would match the yields from anyone environment, demographic structure complicates the picture becauseyield must be calculated over multiple time steps and hence over multiple272.5. Discussionenvironments. We identified three mechanisms by which a developmentaldelay triggered by the state of the environment (conditional germination)can evolve.Mechanism 1: If seedling survival is sufficiently low in ‘bad’ environ-ments, it is optimal to limit germination to ‘good’ patches. In desert plants,seedling survival is much higher in years with high rainfall, and germinationrates are correspondingly higher when early season rainfall is high (Evanset al. 2007). Similarly, ‘classical disturbances’ (White and Pickett 1985),such as fires, create discrete patches in which resources are higher due to de-creased biological use, an ash-bed effect (Serrasolses and Vallejo 1999, Pausaset al. 2003) and/or increased decomposition. For non-annuals, only the ex-treme case in which seedling survival is impossible in ‘bad’ environments hasbeen formally considered (in the context of post-fire germination responses,Lamont et al. 1991). Empirically, the establishment ability of seeds germi-nating in post-fire environments is not always elevated, and establishment inother years is often not negligible (e.g., O’Dowd and Gill 1984, Cowling andLamont 1987, Brewer 1999, Quintana-Ascencio and Menges 2000, Liu et al.2005). As Bond and Wilgen (1996, p142) point out, it was not previouslyobvious whether reported differences in seedling survival are large enoughto select against germination in ‘bad’ years.We used a simple model lacking trade-offs and temporal variation to finda threshold level of seedling survival in ‘bad’ patches below which conditionalgermination is expected to be advantageous. The conditions for conditionalgermination to evolve via this mechanism are broader when ‘good’ envi-ronments are common, seed survival is high, adult survival is low and seedproduction is low. These results can thus guide empirical work to determinewhether demographic parameters would or would not favour conditionalgermination in a particular species.In sections 1 and 2, we used the simplifying assumption that a randomproportion of the (many) patches experience each environment in each timestep, with no reference to the previous environments experienced. Thus,after germination and seedling survival occurs in a particular environment,there is no link between the environment experienced at the time of germi-nation and the subsequent environments experienced by adults. We predictthat, in a spatially explicit model where the environment experienced acrossthe life span depends on the environment at the time of germination, lowadult survival and fecundity (not just low seedling survival) in ‘bad’ en-vironments could also favour conditional germination, assuming seeds canexperience different environments by delaying germination.We also incorporated intraspecific competition affecting seedling survival282.5. Discussionand found that our results were quantitatively altered but qualitatively un-affected. Similarly, the density independent annual plant model by Cohen(1966) was extended to include density dependence by Bulmer (1984), Ell-ner (1985a) and Ellner (1985b). In these annual plant models with temporalenvironmental variation, density dependence can exacerbate the effect of en-vironmental variation on germination fraction (or create temporal variationvia deterministic dynamics, Ellner 1987). In addition, we note that annualplant models show that spatial structure can introduce sibling competition,which can reduce the optimal germination fraction (Ellner 1987). Gremerand Venable (2014) find that annual plant models with density dependenceincluded predict germination fractions more accurately than density inde-pendent models. We caution that our model of competition was highlyidealized in order to make analytical headway. While density-dependentcompetition was experienced equally everywhere in our model, competitionshould be lessened in patches that have recently experienced low adult sur-vival. A more appropriate but complex model would be spatially explicitwith differences in seedling survival affected by competitive interactions onlywithin the same patch.Mechanism 2: Trade-offs can make it difficult to germinate equally wellin all environments, making conditional germination more likely to evolve.We considered a direct physical or developmental trade-off between germi-nation rates, such that a plant would have to decrease germination ratein environment 1 to increase the germination rate in environment 2. Thistrade-off is over-and-above the fact that seeds that germinate in one envi-ronment are unavailable to germinate in the future, which can also be seenas a form of trade-off that underlies all models of delayed development.Trade-offs are likely to arise whenever the features that protect seedsfrom the environment also alter their ability to germinate. For example,thickened seed coats or retention in cones may prevent germination in mostenvironments but allow seeds to survive fires and thus allow increased germi-nation in a post-fire environment. Indeed, many species with temperature-induced germination produce a mixture of seeds that are specialized foreither post-fire or for inter-fire germination (Keeley 1995). This suggeststhat individual seeds cannot do both well, which will generate a trade-offif the total number of seeds is limited. Previously, Tonnabel et al. (2012)considered a trade-off between seed production and maintenance (b and sShere). They assume seedling survival in ‘bad’ environments is negligible sothat selection should maximize germination in ‘good’ (post-fire) environ-ments only, which occurs at an intermediate (mixed) strategy with theirtrade-off.292.5. DiscussionTo demonstrate the effects of trade-offs on conditional germination weconsidered a direct trade-off between germination rates, which has not beenexplored before. We show that, with convex trade-off shapes (dashed linein figure 2.2), specialized germination strategies are favoured, even for pa-rameters that did not favour conditional germination in the model withouttrade-offs. By contrast, concave trade-offs (solid line in figure 2.2) can favoura mixed strategy with some germination in both environments, which max-imizes reproductive opportunities across all patches. Thus, intermediategermination rates can be favoured because of trade-offs, in addition to bethedging caused by temporal environmental variation (see next section).Mechanism 3: Limiting germination to disturbance events minimizes therisk of experiencing another disturbance before reproducing. The timing ofinsect diapause is thought to depend on the risk of seasonal disturbances(e.g., frost or drought) occurring before reproductive maturation is reached(Cohen 1970, Taylor 1980, Hairston and Munns 1984, Taylor and Spalding1989, Bradford and Roff 1993, Spencer and Colegrave 2001). We exploredsimilar risks in a model where germination strategy depends on environmen-tal state rather than time. We showed that conditional germination is morelikely to evolve when plants are prone to population-scale disturbances, pro-moting life-histories that are more synchronized with these disturbances. Asin our first model, conditional germination is more likely to evolve by thismechanism when seeds survive disturbances well but adults do not.Our analytical results indicate that conditional germination should onlyevolve if severe (detrimental to adult survival) disturbances can occur be-fore a significant number of offspring are produced (figure 2.3). In par-ticular, organisms that take multiple years to reach reproductive maturityshould have an increased risk of dying during disturbances before reproduc-ing. A previous model with pre-reproductive age classes by Enright et al.(1998) suggested that conditional germination strategies have higher long-term growth rates even without differences in seedling survival but the mech-anism favouring conditional germination was not discussed or made explicit.We produced a model based on that of Enright et al. (1998) but reduced thenumber of years until reproductive maturity to show that this eliminates thebenefits of conditional germination (figure 2.4). To our knowledge, avoidingdeath before reaching reproductive maturity has not previously been theo-retically investigated as an important driver for the evolution of conditionalgermination strategies, most likely because it requires a relatively complexdemographic model with environment-dependent germination.Interestingly, a synchronization advantage continues to favour condi-tional germination strategies even when the period between disturbances302.5. Discussionis variable. In this case, incomplete rather than complete disturbance-dependent germination strategies often have the highest long-term growthrate because they bet hedge (Philippi and Seger 1989) between experiencinglong and short intervals. This is an example of ‘germ banking’, as definedby Evans and Dennehy (2005), where unpredictable environmental varia-tion favours an intermediate strategy. Figure 2.5 shows that, even whenthe disturbance probability is exactly the same in each year (β = 1, ex-ponentially distributed disturbance intervals) and there is no difference inseedling survival, conditional germination is expected to evolve when plantstake multiple years to reach reproductive maturity. Demonstrating this caseexplicitly is significant because many types of disturbance are likely to benon-periodic (β = 1). For fires, a Weibull shape parameter of around 2 (seefigure 2.5A) has been estimated in some ecosystems (Polakow and Dunne1999, Moritz et al. 2008). Fire hazard is thought to increase with years sincea fire due to vegetation build up (Baeza et al. 2002, De Lu´ıs et al. 2004),causing a negative autocorrelation in fire intervals (Dodson et al. 2005) andmaking fires more uniformly spread over time (as in our periodic model). Onthe other hand, a positive temporal autocorrelation between disturbances(clumping, e.g., due to climate phenomena) would reduce the efficacy ofthe synchronization mechanism because disturbance risk is increased in theyears following a disturbance.In this paper, we determine the conditions under which these threemechanisms allow the evolution of environment-dependent germination. Wefirst explored the most commonly envisioned mechanism (mechanism 1,low seedling survival in ‘bad’ environments) and then show that trade-offsand synchronization effects (mechanisms 2 and 3) can favour environment-dependent germination even when there is no difference in seedling survival.These models provide a framework for exploring which mechanisms mightbe responsible for conditional germination in empirical systems. For exam-ple, we have shown that the fact that it takes several years for the Aus-tralian shrub Banksia hookeriana to mature greatly facilitates the evolutionof environment-dependent germination in this system (figure 2.4). Thus,by obtaining the required demographic parameters and using the models todetermine what conditions favour conditional germination, future empiricalwork promises to inform us why some species wait for particular environ-ments to continue development.31Chapter 3Evolution of Haplont,Diplont or Haploid-DiploidLife Cycles When Haploidand Diploid Fitnesses AreNot Equal3.1 SummaryMany organisms spend a significant portion of their life cycle as haploids andas diploids (a haploid-diploid life cycle). However, the evolutionary processesthat could maintain this sort of life cycle are unclear. Most previous modelsof ploidy evolution have assumed that the fitness effects of new mutationsare equal in haploids and homozygous diploids, however, this equivalencyis not supported by empirical data. With different mutational effects, theoverall (intrinsic) fitness of a haploid would not be equal to that of a diploidafter a series of substitution events. Intrinsic fitness differences betweenhaploids and diploids can also arise directly, e.g., because diploids tendto have larger cell sizes than haploids. Here, we include intrinsic fitnessdifferences into genetic models for the evolution of time spent in the haploidversus diploid phases, in which ploidy affects whether new mutations aremasked. Life cycle evolution can be predominantly determined by intrinsicfitness differences between phases, masking effects, or a combination of both.We find parameter ranges where these two selective forces act and show thatthe balance between them can favour convergence on a haploid-diploid lifecycle, which is not observed in the absence of intrinsic fitness differences.Specifically, haploid-diploid life cycles can evolve when diploids have higherintrinsic fitness but the net effect of new mutations favours haploidy.323.2. Introduction3.2 IntroductionSexual reproduction in eukaryotes requires an alternation of haploid anddiploid phases in the life cycle. Across taxa, there is a great deal of varia-tion in the amount of growth (and time spent) in each of the haploid anddiploid phases (see Valero et al. 1992, Klinger 1993, Richerd et al. 1993, Bell1994; 1997, Mable and Otto 1998, Coelho et al. 2007). Some organisms, in-cluding almost all animals, are diplontic (somatic development occurs onlyin the diploid phase) and others, including dictyostelid slime moulds, andsome green algae (e.g., Chara), are haplontic (somatic development occursonly in the haploid phase). However, a large and phylogenetically diversegroup of eukaryotes, including most land plants, basidiomycete fungi, mostbrown algae, red algae and some green algae, undergo some mitotic growthin both the haploid and diploid phases, which is referred to as a haploid-diploid life cycle here (sometimes called diplohaplontic or haplodiplontic) toavoid confusion with arrhenotoky (‘haplodiploid’ sex determination). Whileseveral theoretical studies have explored the conditions that should favourexpansion of the haploid or diploid phases, there are still relatively fewstudies that show how a haploid-diploid life cycle could be maintained byselection.A prominent theory for the evolution of either haplont or diplont lifecycles involves the direct consequences of ploidy level on the expression ofdeleterious mutations. The fitness effects of a deleterious mutation can bepartially hidden by the homologous gene copy in diploids, which is favourableif a heterozygote has a higher fitness than the average fitness of the two com-ponent haploids. Thus modifier models, in which the extent of haploid anddiploid phases is determined by a second locus, have found that diplonty isfavoured when deleterious mutations are partially recessive and haplonty isfavoured when deleterious mutations are partially dominant (Perrot et al.1991, Otto and Goldstein 1992, Jenkins and Kirkpatrick 1994; 1995). Asa consequence of mutations being partially concealed, an expanded diploidphase allows mutations to reach a higher frequency and thus increases mu-tation load (Crow and Kimura 1965, Kondrashov and Crow 1991). Modi-fiers that expand the diploid phase therefore become associated with lowerquality genetic backgrounds. These associations are broken apart by recom-bination and so diplonty is favoured over a wider parameter range whenrecombination rates are higher (Otto and Goldstein 1992).The evolution of life cycles in sexual organisms appears to be similarlyinfluenced by beneficial mutations. Using a numerical simulation approach,Otto (1994) and Orr and Otto (1994) show that diplonty is favoured during333.2. Introductionsweeps of beneficial mutations that are partially dominant. Increasing thelength of the diploid phase of the life cycle increases the amount of selec-tion experienced by heterozygotes and, with partial dominance, heterozy-gotes have higher fitness than the average fitness of the two componenthaploids. Conversely, haplonty is favoured when beneficial mutations arepartially recessive. Again, lower recombination rates between the life cyclemodifier and beneficial mutations broaden the parameter range over whichhaplonty is favoured because of associations between the modifiers expand-ing the haploid phase and higher quality genetic backgrounds that evolvewhen beneficial mutations are not masked.These models typically assume that the overall fitness of haploids ordiploids is the same. However, even with identical genomes, haploid anddiploid cells typically differ in size and often in shape (e.g., Mable 2001),and growth and survival often differs between haploid and diploid phases.The phase with higher fitness and the magnitude of fitness differences varieswidely and is heavily dependent on environmental context (Mable and Otto1998, Thornber 2006). In Saccharomyces yeast, differences between haploidand diploid growth rates measured by Zo¨rgo¨ et al. (2013) range from beingnegligible to substantial (one phase can have growth rates up to 1.75 timeshigher) in different environments. Similar differences in growth rate andsurvival are observed between haploid and diploid phases of the red algaeGracilaria verrucosa and Chondracanthus squarrulosus in some laboratoryconditions (Destombe et al. 1993, Pacheco-Ru´ız et al. 2011). In addition, thefitness effect of new mutations may be unequal when present in haploids or inhomozygous diploids, as reported by Gerstein (2012) and Zo¨rgo¨ et al. (2013).Therefore, following a series of substitution events, the overall (intrinsic)fitness of a haploid and a diploid should not be equal, as explored here.The models discussed above assume that selection is independent of thedensities of haploid and diploid individuals. These models also predict thateither haplonty or diplonty evolves but not biphasic, haploid-diploid lifecycles. Hughes and Otto (1999) and Rescan et al. (2016) consider density-dependent selection in which haploids and diploids occupy different ecolog-ical niches and show that haploid-diploid life cycles can evolve in order toexploit both the haploid and diploid ecological niches. In this study, we com-plement these studies by considering only density independent selection inorder to focus on intrinsic fitness differences between haploids and diploids.The effect of intrinsic fitness differences on the evolution of the life cyclemay seem obvious - selection should favour expansion of whichever phase(haploid or diploid) has higher fitness, as found by Jenkins and Kirkpatrick(1994; 1995). However, Jenkins and Kirkpatrick (1994; 1995) only consid-343.3. Modelered the case where the differences in intrinsic fitness is either much largeror much smaller than the genome-wide deleterious mutation rate. Here, weconsider the case where the two forces are of similar strength and quantifythe parameters (e.g., mutation rate) for which this is true. In addition, weconsider the effect of beneficial mutations on life cycle evolution when thereare intrinsic fitness differences between haploids and diploids. We show thathaploid-diploid life cycle can evolve even in the absence of density depen-dent selection due to a balance between intrinsic fitness differences betweenphases and the genetic effects of masking/revealing mutations. We alsoconsider branching conditions and find that, in haploid-diploid populations,sexually interbreeding mixtures of haploid and diploid specialists can befavoured (see also Rescan et al. 2016).3.3 ModelWe consider life cycle evolution using a modifier model in which the propor-tion of time spent in the haploid and diploid phases depends on the genotypeat a modifier locus. Selection on the modifier results from viability selectionon a set of L other loci. We first present a two-locus model, in which there isone viability locus and one modifier locus. We then extrapolate our resultsto the evolution of a modifier locus linked to many loci under selection;selection on a modifier caused by many loci is well approximated by thesum of the selective effect of each pairwise interaction considered separately(e.g., Jenkins and Kirkpatrick 1995, Otto and Bourguet 1999, Hough et al.2013), assuming that the viability loci are loosely linked, autosomal andnonepistatic and the modifier has a small effect. We then test this approachby comparing our results to an explicit multi-locus simulation. Finally, weshow that beneficial mutations can generate selection on the life cycle similarto that caused by deleterious mutations.3.3.1 Analytical ModelIn the modifier model presented here (figure 3.1b), zygotes are formed duringsynchronous random mating. The diploid genotype (ij) at the modifierlocus (MM , Mm, or mm) determines the timing of meiosis and hence theproportion of time each individual spends as a diploid (1 − tij) and as ahaploid (tij). Here, Sh and Sd represent selection acting across the genomedue to intrinsic fitness differences between haploids and diploids. As ourinitial focus will be on the selection experienced at each of L selected loci,we also define σh = Sh/L and σd = Sd/L as the intrinsic fitnesses per353.3. ModelSyngamy Selection1-dijdijMeiosis &RecombinationMeiosis &Recombination1n2nSyngamyMeiosis &Recombination1-tijtij2n1n(a) (b)Figure 3.1: Model (a) discrete selection and (b) continuous selection haploid-diploid life cycles.Single lines represent haploid phases and doubled lines indicate diploid phases. In (a), modifiedfrom Perrot et al. (1991) and Otto and Goldstein (1992), zygotes with the modifier genotype ijundergo selection as diploids with probability dij or undergo meiosis and recombination beforeexperiencing selection as haploids with probability (1−dij). In (b), after Jenkins and Kirkpatrick(1994; 1995) and Otto (1994), all zygotes with genotype ij experience viability selection as adiploid for a proportion (1 − tij) of their life cycle before undergoing meiosis and recombinationand then experiencing viability selection as a haploid for the remainder of the life cycle.viability locus. When σh > σd, haploids have higher fitness than diploidsand the fitness of diploids is higher when σd > σh. At each viability locus, weconsider a wild type and mutant allele (alleles A and a). The mutant allele ateach viability locus, a, can have a different effect on fitness when present ina haploid (sh) or in a homozygous diploid (sd). The fitness of heterozygousdiploids depends on the dominance of these mutations, given by h. Whenconsidering deleterious mutations, sh and sd are both negative, and whenconsidering beneficial mutations, sh and sd are both positive. The fitnessesof the various genotypes are given in table C.1. Recombination between themodifier and viability locus (at rate r) and mutation (from A to a, at rate µper viability locus) occur at meiosis followed by haploid selection and thengamete production. The frequencies of genotypes MA, Ma, mA and maare censused in the gametes (given by x1, x2, x3 and x4 respectively).Previous models have made various different life cycle assumptions, sum-marized in table 3.2. In ’discrete selection’ models, selection occurs once pergeneration and modifiers affect whether selection occurs during the haploidor diploid phase, figure 3.1a. On the other hand, ‘continuous selection’363.3. ModelTable 3.1: Fitnesses of different genotypes.Genotype FitnessA wA(tij) = exp[tijσh]a wa(tij) = exp[tij(σh + sh)]AA wAA(tij) = exp[(1− tij)(σd)]Aa wAa(tij) = exp[(1− tij)(σd + hsd)]aa waa(tij) = exp[(1− tij)(σd + sd)]models assume that selection occurs continuously throughout the life cycle,figure 3.1b. In addition, some models have assumed that mutations occurupon gamete production, and others assume that mutations occur at meio-sis. Thus, there are four possible life cycles, recursion equations for thesedifferent life cycles are provided in the appendix B. Generally, our results areunaffected by using these alternative models, these analyses can be found inthe supplementary Mathematica file (Wolfram Research Inc. 2010). How-ever, there are two cases in which life cycle assumptions qualitatively impactresults.Table 3.2: Life cycle assumptions used in various modifier models.Mutations at Mutations atGamete Production MeiosisDiscrete Selection(Figure 3.1a)Perrot et al. (1991)Otto and Goldstein (1992)Otto and Marks (1996)Rescan et al. (2016)Hall (2000)Continuous Selection(Figure 3.1b)Otto (1994)aOrr and Otto (1994)Otto (1994)aJenkins andKirkpatrick (1994; 1995)a Otto (1994) allows mutations to occur at both gamete production and meiosis.Firstly, Hall (2000) showed that ‘polymorphic’ haploid-diploid life cyclescan evolve if mutations occur at meiosis and selection is discrete. This lifecycle allows diploids to escape selection on new mutations for one generation,generating an advantage to diploids, which allows convergence to occur whendeleterious mutations favour haploids. As shown below, meiotic mutationdoes not favour haploid-diploid life cycles in the continuous selection model373.3. Model(figure 3.1b) because new mutations experience selection the instant theyappear in diploids.Secondly, alternative mating schemes have previously only been consid-ered by Otto and Marks (1996), who assume discrete selection and muta-tions at gamete production (and no differences in intrinsic fitness betweenhaploids and diploids). They found that haploidy is favoured over a largerparameter range when selfing, asexual reproduction or assortative matingis common. In the appendix, we include selfing into all four life cycle mod-els and show that this conclusion only applies when the fitness of haploidsand homozygous diploids are assumed to be equal (e.g., no intrinsic fitnessdifferences), otherwise additional effects of selfing are observed because self-ing generates homozygotes. Furthermore, even when there are no intrinsicfitness differences, we show that selfing can increase or decrease the param-eter range in which haploids are favoured when mutations occur at gameteproduction. This effect is presumably due to benefits that selfed diploidscan accrue following a period of haploid selection on new mutations andillustrates again that the impact of increased selfing on these models is notequivalent to reduced recombination.3.3.2 Multilocus SimulationsWe used individual-based simulations (C++ program available in the DryadDigital Repository) to test predictions from our analytical model when dele-terious mutations segregate at L loci. Each individual carries either one ortwo copies of a chromosome (depending on its ploidy level) represented by amodifier locus (located at the midpoint of the chromosome) and a sequenceof L bits (0 or 1) corresponding to the different loci.Mutations occur at a rate U per generation: the number of new muta-tions per chromosome is sampled from a Poisson distribution with parameterU and distributed randomly across the genome; alleles at mutant loci areswitched from 0 to 1 or from 1 to 0. Mutation and back mutation thusoccur at the same rate, but back mutations should generally have negligibleeffects under the parameter values that we use, as deleterious alleles remainat low frequencies. We assume that all deleterious alleles have the sameeffects on fitness (sd, sh, and h are constant) and that these effects multiplyacross loci: the fitness of a haploid carrying n deleterious alleles is given bywh = exp[Sh + shn], while the fitness of a diploid carrying nhe deleteriousalleles in the heterozygous state, and nho in the homozygous state is givenby wd = exp[Sd + nhehsd + nhosd].At the start of each generation, all N individuals are diploid. To produce383.4. Resultsthe 2N gametes that will form the diploids of the next generation, a diploidindividual is sampled randomly among all diploids of the previous genera-tion, and undergoes meiosis to produce a haploid; the number of cross-oversis sampled from a Poisson distribution with parameter R, while the posi-tion of each cross-over is sampled from a uniform distribution. If a randomnumber sampled from a uniform distribution between 0 and 1 is lower thanwd1−twht (where wd and wh are the fitnesses of the diploid parent and hap-loid offspring), divided by its maximal possible value, then the haploid isretained; otherwise another diploid parent is sampled, until the condition isfulfilled.At the beginning of the simulation, the modifier locus is fixed for anallele coding for an initial length of the haploid phase tinit (all simulationswere performed for tinit values of 0.1, 0.5 and 0.9) and all selected loci arefixed for allele 0. Then, deleterious mutations are introduced at rate U perchromosome (the length of the haploid phase being still fixed to tinit) untilthe population reaches mutation-selection equilibrium (after generally 2,000generations). After that, mutations at the modifier locus are introduced at arate mM per generation. When a mutation occurs, the length of the haploidphase coded by the mutant allele is sampled from a uniform distributionbetween told − 0.1 and told + 0.1, where told is the value of the parent allele;if the new value is negative or higher than 1, it is set to 0 or 1, respectively.We assume additivity among modifier alleles such that a zygote with allelest1 and t2 will have a haploid phase of length t = (t1 + t2)/2. Simulationsinitially lasted 100,000 generations, which was sufficient in most cases forthe average rate of diploidy to reach steady state, t¯. We categorized the lifecycle that evolved at the end of the simulation as haplont (t¯ > 0.95, whitecircles in figures 2 and 3b), diplont (t¯ < 0.05, black circles), or haploid-diploid (0.05 < t¯ < 0.95, green circles). In some cases, there was a repellingstate such that the population evolved to haplonty or diplonty dependingon tinit (red circles).3.4 Results3.4.1 Deleterious MutationsWe first find the frequency of deleterious mutations at mutation-selectionbalance (qˆa) when the modifier locus is fixed for a particular resident allele(MM fixed, so that the length of the haploid phase is tMM ). Assuming thatthe per locus mutation rate (µ) is small, terms of the order of the square of393.4. Resultsthe per locus mutation rate can be ignored, yieldingqˆa =µ exp[tMMsh]1− exp[tMMsh + (1− tMM )hsd] , (3.1)assuming there is some haploid or diploidy heterozygous expression so thedenominator isn’t near zero. When deleterious mutations are partiallymasked by the homologous gene copy in diploids (hsd/sh < 1), the fre-quency of deleterious mutations (qˆa) is higher when the diploid phase islonger (lower tMM ).Life cycle evolution is considered by introducing an allele (m) at themodifier locus that controls the timing of meiosis and evaluating whetherits frequency increases when rare. Mutants are able to invade when theleading eigenvalue of the system described by equations B.1c and B.1d, λl, isgreater than one. Jenkins and Kirkpatrick (1994) derive a version of λl whensd = sh, however, they only discuss per locus intrinsic fitness differences thatare of a much greater magnitude than the mutation load (|σd − σh|  µ).To investigate the interaction between these selective forces we first presentan approximation of λl in which the per locus fitness difference betweenhaploids and diploids (|σd − σh|) is of similar magnitude to the per locusmutation rate, O(2), the selective disadvantage of mutants (sd and sh) is ofa larger order of magnitude, O(), and linkage is loose (r of O(1)) yieldingλl ≈ 1 + (tMm − tMM )(σh − σd + 2(−sh)qˆa(hsdsh− 12))+O(3). (3.2)Because mutation rates are small, deleterious mutations are found at lowfrequencies, therefore life cycle evolution depends only on the fitness of het-erozygous mutants and not homozygous mutants (i.e., sd is always foundwith the dominance coefficient, h). Consequently, life cycle evolution de-pends only on the ‘effective dominance’, he = hsd/sh, rather than dominanceper se.Life cycle modifiers affect the amount of selection heterozygous zygoteswill subsequently experience as heterozygous diploids versus as the compo-nent haploid genotypes. Heterozygous diploids have higher fitness than theaverage of the two component haploids when deleterious mutations are effec-tively partially recessive (0 < hsd/sh < 1/2), favouring diploidy. Conversely,effectively partially dominant deleterious alleles (hsd/sh > 1/2) favour hap-loidy. The strength of this selection on the life cycle (caused by maskingalleles) depends on the equilibrium frequency of deleterious alleles, which isgreater when the diploid phase is longer (assuming 0 < hsd/sh < 1).403.4. ResultsUsing this approximation, haploid-diploid life cycles are evolutionarilysingular strategies when σh − σd = 2(sh)qˆa(he − 1/2). Without intrinsicfitness differences, there is no intermediate value of tMM that solves thiscondition, hence either haplont or diplont life cycles are favoured. Thus,whereas Hall (2000) shows that biphasic haploid-diploid life cycles can evolveif selection occurs once per generation (figure 3.1a) and mutations occur atmeiosis (as considered here), haploid-diploid life cycles in the continuousselection model (figure 3.1b) do not evolve in the absence of intrinsic fitnessdifferences.When diploids have higher intrinsic fitness (σd > σh), there are inter-mediate (biphasic haploid-diploid) singular strategies in the region wheredeleterious alleles favour haploidy. In this case, the strength of selection infavour of haploidy is strong when the diploid phase is longer (because dele-terious mutations reach higher frequencies) and can outweigh the intrinsicfitness differences. When the diploid phase is short, intrinsic fitness differ-ences dominate, favouring a longer diploid phase. This combination ensuresthat evolution converges towards a haploid-diploid life cycle (figure 3.2a).When haploids have higher intrinsic fitness (σh > σd), either haplonty ordiplonty is always favoured. Even if an intermediate singular strategies existsbecause deleterious alleles favour diploidy, this is a repelling point, such thateither haplonty or diplonty evolves. For these parameters, selection in favourof diplonty is stronger when the diploid phase is longer, favouring even longerdiploid phases (because the benefits of masking deleterious mutations isgreater). Conversely, intrinsic fitness differences dominate when the diploidphase is short, favouring longer haploid phases. Thus haplonty and diplontycan both be stable strategies (figure 3.2c).After convergence on a haploid-diploid strategy, we can then ask whetherthis singular strategy is evolutionarily stable. Using the same weak selectionapproximations as above, evolutionary stability is given by:δ2λlδtMm2∣∣∣tMm=t∗=2(−sh)(σd − σh)(hsd/sh − 1)(1− r)wa[t∗]wAa[t∗]wA[t∗]wAA[t∗]− (1− r)wa[t∗]wAa[t∗] , (3.3)where t∗ indicates the singular strategy for t, the length of the haploidphase. When convergence is stable (requiring that σd > σh and hsd/sh < 1,see below), the singular strategy is evolutionarily unstable (3.3 is posi-tive). Thus we expect weak disruptive selection after this singular pointis reached. Indeed, our multilocus simulations sometimes displayed branch-ing after 100,000 generations, such that there was a proportion t∗ of haploid413.4. Results(a)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.20.40.60.81.01. 0.75 0.5 0.25|sh|hsd/s hr (b)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.20.40.60.81.01. 0.75 0.5 0.25|sh|hsd/s hr(c)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.20.40.60.81.01. 0.75 0.5 0.25|sh|hsd/s hr (d)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.20.40.60.81.01. 0.75 0.5 0.25|sh|hsd/s hrFigure 3.2: Parameter space where haplont, diplont and haploid-diploid life cycles are favouredwhere the strength of selection against deleterious mutations (|sh|) and effective dominance hsd/shis varied. The top axis gives r, the relative growth rate of mutant haploids when deleteriousmutations have a fitness effect of sh. Background colors: prediction from the two-locus stabilityanalysis extrapolated to multiple loci. Circles: multilocus simulation results starting from threedifferent initial haploidy rates (tinit = 0.01, 0.5, or 0.99), with population size 20,000. White:evolution toward haplonty. Green: convergence stable haploid-diploid life cycles. Red: eitherhaplonty or diplonty is favoured, with a repelling state in between. Black and gray: evolutiontoward diplonty. (a) and (b): diploids have higher intrinsic fitness (Sh = 0, Sd = 0.025) (c) and(d): haploids have higher intrinsic fitness (Sh = 0.025, Sd = 0). Map length: R = 100 ((a) and(c)) and R = 0.35 ((b) and (d)). The dashed lines show where haplonty (above dashed lines)and diplonty (below dashed lines) are favoured when there is no difference in intrinsic fitness(Sh = Sd = 0). In (b) and (d), there is a repelling point between the dashed lines. Mutantschange the life cycle by a small amount (|tMm − tMM | = 0.001) and the genome-wide haploidmutation rate, U = 0.1.423.4. Resultsalleles (t1 = 1), and a proportion (1 − t∗) of diploid alleles (t2 = 0). In-creasing the number of generations always lead to branching when it wasnot observed by this time.The weak selection approximation above assumes that the recombinationrate is large relative to selection. Without intrinsic fitness differences, Ottoand Goldstein (1992) showed that haploidy is favoured over a larger rangeof parameter spaces when recombination rates are low because associationsbetween haploid-promoting modifiers and the high fitness, purged geneticbackgrounds they create are retained for longer. To consider tighter linkageand/or stronger selection we can use the more accurate expression of λlλl = exp[(tMm − tMM )(σh − σd)](1 +µK1K2K3), (3.4)whereK1 = 1− (1− r) exp[−(tMm − tMM )hsd]− r exp[(tMm − tMM )(sh − hsd)]+ (1− 2r){exp[(1− tMm − (tMm − tMM ))hsd + tMmsh]− exp[(1− tMm)hsd + tMmsh]}K2 = 1− exp[−(1− tMM )hsd − tMMsh]K3 = 1− (1− r) exp[(1− tMm)hsd + tMmsh],in which the per locus mutation rate (µ) is assumed to be small, so thatterms on the order of the square of the mutation rate can be ignored.Equation (3.4) shows that singular strategies can exist without intrinsicfitness differences when recombination rates are low, r < 1/2, see figures3.2b and 3.2d). As above, these singular strategies are always repellingpoints when σd = σh such that differences in intrinsic fitness are requiredfor haploid-diploid life cycles to evolve. Convergence upon a haploid-diploidlife cycle still requires that diploids have higher intrinsic fitness (σd > σh).However, as selection becomes less weak relative to recombination rates(such that the approximation in 3.2 is not appropriate), haploid-diploid lifecycles can evolve when hsd/sh < 1/2, see figure 3.2b. In addition, conver-gence stability requires hsd/sh < 1, such that the frequency of deleteriousmutations (qˆa) increases with the length of the diploid phase, see figure 3.3a.We next extend our two-locus result to consider deleterious mutationsacross L viability loci by assuming that these loci are loosely linked, auto-somal and nonepistatic. With these assumptions (e.g., Jenkins and Kirk-patrick 1995, Otto and Bourguet 1999, Hough et al. 2013, Rescan et al.433.4. Results(a)-0.05 0.00 0.05 0.10Sd-Sh0.20.40.60.81.01.2h sd/shHbL-3 -2 -1 1 2 3gHSd-ShL0.250.50.751.hFigure 3.3: Parameter space for which (a) deleterious mutations and (b) beneficial mutationsfavour haplont, diplont and haploid-diploid life cycles as a function of the difference in intrinsicfitness between haploids and diploids (Sd − Sh). (a) Shows the effective dominance of deleteriousmutations (hsd/sh) against intrinsic fitness differences (Sd − Sh), parameters and symbols as infigures 3.2a and 3.2c with |sh| = 0.4. (b) Regions in which particular life cycles are favoured in thepresence of beneficial mutations, evaluated using equation 3.11. g is the number of generationsbetween fixation events. Population size, N , is 20000.2016), invasion of a modifier of weak effect is given byλnet = 1 +L∑l=1(λl − 1). (3.5)In figures 3.2 and 3.3a we plot where this approximation predicts haplont,diplont or haploid-diploid life cycles to evolve for comparison to the explicitmulti-locus simulation (described above).Above, as in previous work, we consider the average dominance andselection coefficients (h, sd and sh). We can approximate the effect of smallamounts of variation (and covariation) among loci in these coefficients byperforming a Taylor expansion, as described in Lynch and Walsh (1998),Appendix 1. Because we have assumed that deleterious mutations are rare,sd is always found with h and we consider variation in sh and the compoundparameter hsd. Assuming that deviations between coefficients and theirmean value are of order  and that selection is weak (as assumed in equation443.4. Results3.2), yieldsλnet ≈1 + (tMm − tMM )(σh − σd + 2(−sh)Lqˆa(hsdsh− 12)+(1 + tMM )Lqˆa(−sh)µ2((1− tMM )(hsdshCov(hsd, sh)−Var(hsd))+ tMM(hsdshVar(sh)− Cov(hsd, sh))))+O(3)(3.6)Based on this analysis, variation in sh generally makes haplonty more stableto invasion (reduces λnet for tMM = 1, tMm < 1). Similarly, variation in hsdmakes diplonty more stable to invasion (where tMM = 0, tMm > 0). Positivecovariation between hsd and sh has the opposite effect. Yeast deletion dataindicate that the heterozygous effects of deleterious mutations may be muchless variable than their homozygous effects, due to a negative correlationbetween h and s (Phadnis 2005, Agrawal and Whitlock 2011, Manna et al.2011). Even if sd and sh are on average the same, it may thus be that thevariance of hsd is much lower than the variance of sh.3.4.2 Beneficial MutationsWhereas deleterious alleles are maintained at mutation-selection balance,beneficial mutations sweep to fixation. The time taken for a sweep to occurdepends on the length of the diploid phase; selective sweeps take longer inpredominantly diploid populations. During a selective sweep, heterozygotesare present in the population. Life cycle modifiers can affect whether het-erozygous zygotes subsequently experience selection as heterozygous diploidsor as haploids. Thus, the strength of selection exerted by beneficial muta-tions on modifiers depends on the time taken for fixation to occur, whichdepends on the life cycle of the current population. Therefore, as with dele-terious alleles, the direction of selection exerted by beneficial mutations de-pends on dominance. Here we evaluate how these genetic considerations areexpected to influence life cycle evolution and include differences in intrinsicfitness between haploids and diploids.We obtain analytical results using a quasi-linkage equilibrium (QLE)approximation, in which selection is assumed to be weak relative to re-combination so that linkage disequilibrium (D = x1x4 − x2x3) equilibratesquickly relative to the rate of change of allele frequencies (pA = x1 +x3 and453.4. ResultspM = x1 + x2). Assuming weak selection, O(), and low mutation rates,O(2), the leading order term for the quasi-equilibrium value of linkage dis-equilibrium (DˆQ) is given byDˆQ ≈ δt shrpM (1−pM )pA(1−pA)(1− pAhsdsh− (1− pA)(1− h)sdsh)+O(2),(3.7)where δt = (pM (tMm − tMM ) + (1 − pM )(tmm − tMm)) is the effect of themodifier on the length of the haploid phase (δt is positive if m increases thehaploid phase with tmm > tMm > tMM and negative if tmm < tMm < tMM ).Linkage disequilibrium is a measure of associations between alleles atdifferent loci. When D > 0, alleles A and M are more often found together,as are alleles a and m. When sh = sd and 0 < h < 1, as assumed inOtto (1994) and Orr and Otto (1994), equation (3.7) shows that m allelesthat increase the length of the haploid phase (δt > 0) are associated withthe beneficial mutation, a (DˆQ > 0). These associations are broken apartby recombination so associations are stronger (|DˆQ| larger) when the re-combination rate is low. Therefore lower recombination rates should favourhaplonty, as found numerically by Otto (1994) and Orr and Otto (1994).The change in the frequency of the modifier allele, m (∆qm) can then beexpressed as a function of linkage disequilibrium (DˆQ) and allele frequencies,pA and pM . Assuming that selection is weak and mutation rates are low,the leading order term of ∆qm is given by∆qm ≈ δtpM (1−pM )(σh − σd + sh(1− pA)(1− 2pAhsdsh− (1− pA)sdsh))+O(2).(3.8)Unlike deleterious mutations, beneficial mutations reach high frequencies inthe population, so the dynamics of the modifier depend on the fitness of bothheterozygous and homozygous mutants. Equation (3.8) shows that, whenfixed (pA = 0), a beneficial mutation with a different effect size in haploidsand diploids (sd 6= sh) affects life cycle evolution in a similar manner tointrinsic fitness differences (σd and σh). However, there is also transientselection on the life cycle that occurs during the fixation of a beneficialmutation. We isolate the transient selection on the life cycle from the effecton intrinsic fitnesses by considering the case where sd = sh = s so that∆qm ≈ δtpM (1− pM )(σh − σd + 2pA(1− pA)(1/2− h)s) +O(2). (3.9)463.4. ResultsEquation (3.9) demonstrates that, in the absence of intrinsic fitness differ-ences (σd = σh), haplonty is favoured during sweeps of partially recessive(h < 1/2) beneficial mutations and diplonty is favoured during sweeps ofpartially dominant (h > 1/2) beneficial mutations (as found numerically byOrr and Otto 1994).Whether life cycle evolution is dominated by differences in intrinsic fit-ness or transient selection generated by beneficial mutations depends on therate at which beneficial mutations occur and how long they segregate in thepopulation. The fixation time of beneficial mutations is different for differ-ent life cycles (longer when diploid phases are longer). We assume that themutant life cycle allele is rare or similar enough to that of the resident thatthe time taken to fix a beneficial mutation depends on the life cycle of theresident and then measure the transient selection on the modifier over theentire time course of the sweep using∫pM (1− pM )2pA(1− pA)pA(1/2− h)s dt. (3.10)This integral can then be evaluated assuming that a beneficial mutation willinitially be found at frequency 1/N , where N is the population size.Assuming that the rate of adaptation is limited by the rate of environ-mental change so that a beneficial mutation fixes every g generations andconsidering selection on the life cycle from all L loci, the average invasionfitness of a rare life cycle modifier per generation is∆q¯m ≈δtpM (1− pM )((Sh − Sd)− 1gln[1N+(N − 1)(h(1− tMM ) + tMM )N(1− h(1− tMM ))]/(1− tMM )),(3.11)where the last term accounts for the fact that the beneficial mutations occuronly once every g generations.As with deleterious mutations, there can be haploid-diploid life cycles(0 < tMM < 1) that are evolutionarily singular strategies. Assuming thatthe population size is large, mutants that increase the length of the haploidphase (δt > 0) can only invade a resident population that has a short haploidphase (tMM = 0) if beneficial mutations are partially recessive (0 < h <1/2). Similarly, mutants that decrease the length of the haploid phase (δt <0) can only invade a resident population that has a long haploid phase(tMM ≈ 1) if beneficial mutations are partially recessive (0 < h < 1/2).473.5. DiscussionTherefore, a haploid-diploid life cycle can only be convergence stable when0 < h < 1/2 (green in figure 3.3b). Figure 3.3b also shows the region inwhich both haplonty and diplonty cannot be invaded by small life cyclemodifiers, in which case the singular strategy represents a repelling point(red).When the rate of adaptation is not limited by the rate of environmentalchange, but by the rate of fixation of beneficial mutations, the time betweenfixation events depends on the occurrence of beneficial mutations (1/g) andtheir fixation probability (Pfix), which is given by 2s(tMM + (1 − tMM )h).Fixation probability decreases when the diploid phase is longer because ben-eficial mutations are partially hidden by the extra chromosomal copy indiploids. Under mutation-limited adaptation g can be replaced in equation(3.11) by g/Pfix. In this case, haploid-diploid life cycles are never main-tained by selection. Thus, beneficial mutations can only favour haploid-diploid life cycles if the rate of adaptation is not mutation-limited.3.5 DiscussionEmpirical evidence suggests that the fitness effects of new mutations arenot generally the same in haploids and diploids (Gerstein 2012, Zo¨rgo¨ et al.2013). We show that, when the average fitness effect of new deleterious mu-tations is unequal in haploids and diploids, whether deleterious mutationsfavour haploidy or diploidy depends on their effective dominance (hsd/sh).Most mutation accumulation studies in Saccharomyces yeast estimate eitherthe average heterozygous (hsd) or haploid (sh) effect of mutations on fitness(Wloch et al. 2001, Zeyl and DeVisser 2001, Joseph 2004, Hall et al. 2008),from which effective dominance could be estimated. However, because theexpectation of a ratio is not generally equal to the ratio of expectations,estimates of effective dominance would be more accurate if calculated fromthe same strains. In such a study, Korona (1999) took relevant haploid anddiploid fitness measures but did not estimate effective dominance. In addi-tion, Szafraniec et al. (2003) found deleterious mutations affected haploidfitness more strongly than diploid fitness but they caution that the hap-loid spores were required to germinate, which may have biased their fitnessmeasurements in favour of diploids. Thus, further empirical estimates ofthe effective dominance of deleterious mutations would better inform ourunderstanding of how life cycles are impacted by deleterious mutations.Haploid and diploid phases can also differ in their intrinsic fitnesses(Thornber 2006, Zo¨rgo¨ et al. 2013). While life cycle evolution depends on483.5. Discussionthe effective dominance when there are no intrinsic fitness differences, largedifferences in intrinsic fitnesses favour expansion of the phase with higherfitness (Jenkins and Kirkpatrick 1994). In this study, we show how life cy-cles are expected to evolve when life cycles experience selection due to bothdominance and intrinsic fitness differences. To leading order, these selectiveforces both contribute when intrinsic fitness differences are similar in mag-nitude to the haploid genome-wide mutation rate. For example, figure 3.3Ashows how life cycles are expected to evolve when the deleterious mutationrate per haploid genome (U) is 0.1, approximately equal to estimates ofthe deleterious mutation rate in Amsinckia and Arabidopsis plants (Schoen2005, Halligan and Keightley 2009). Figure 3.3A suggests that these forcesare of similar strength when the intrinsic fitness difference between haploidsand diploids (Sd − Sh) is between 2% and 5%. Estimates of the deleteriousmutation rate per haploid genome vary across studies and organisms (Halli-gan and Keightley 2009). For deleterious mutation rates that are a factor flarger, the scale of the x-axis on this figure can be multiplied by f to deter-mine when selection on the life cycle due to deleterious mutations should beapproximately the same strength as selection due to differences in intrinsicfitness. We note that mutation rate estimates in yeast and Chlamydomonas(Morgan et al. 2014) are lower but are typically calculated per mitotic cell di-vision. However, the relevant mutation rate for models of life cycle evolutionis per sexual cycle (i.e., per meiosis), which has been estimated to involvetens of thousands of mitotic generations in natural S. cerevisiae populations(Magwene et al. 2011).In laboratory environments, substantial differences in fitness betweenhaploid and diploids phases of Saccharomyces yeast and algae have beenobserved in some environments (Mable and Otto 1998, Destombe et al. 1993,Pacheco-Ru´ız et al. 2011, Zo¨rgo¨ et al. 2013). However, measuring the fitnessof yeast in natural environments is challenging. Some demographic studiesof natural red algae populations of Mazzaella flaccida and Chondrus crispushave shown that diploids have moderately increased survivorship relative tohaploids (Sd − Sh ≈ 0.1, Bhattacharya 1985, Thornber and Gaines 2004).Other studies have found no difference in survivorship, perhaps becausethere is limited power to detect smaller differences in mortality in the field(e.g., Engel et al. 2001, Thornber and Gaines 2004). Overall, estimates of themagnitude of intrinsic fitness differences are still uncertain, partly becauseexisting algal studies do not compare survivorship of isogenic haploids anddiploids, which would be required to remove the effect of masked mutationsin heterozygotes.For haploid-diploid life cycles to evolve by selection, individuals with493.5. Discussionlonger diploid phases must be favoured in predominantly haploid popula-tions and individuals with longer haploid phases must be favoured in pre-dominantly diploid populations. Previous models predicting the evolutionof biphasic haploid-diploid life cycles have posited indirect benefits fromdecreasing senescence by reducing phase-specific generation time (Jenkins1993), reducing the frequency of sexual reproduction (Richerd et al. 1993),or exploiting more ecological niches (Bell 1997, Hughes and Otto 1999, Res-can et al. 2016). However, haploid-diploid life cycles are not a unique way ofaccessing these benefits. For example, diplont or haplont species can reducegeneration times or the frequency of sexual reproduction without evolvinghaploid-diploid life cycles. Similarly, differentiated life cycle stages (Steen-strup alternations), phenotypic plasticity or genetic polymorphism can allowdiplontic or haplontic species to exploit multiple ecological niches withouttying growth form to the sexual cycle. Here, we use a population geneticmodel to show that haploid-diploid life cycles can evolve as a direct conse-quence of ploidy if the intrinsic fitness of haploids and diploids is not equal.In species where intrinsic fitness differences and genome-wide mutationrates are of a similar magnitude to one another, haploid-diploid life cyclescan only evolve according to the model presented here if diploids have higherintrinsic fitness than haploids and deleterious/beneficial mutations favourhaploidy. In this case, the frequency of deleterious mutations (or time takenfor beneficial mutations to fix), and thus the strength selection in favour ofhaploidy, is largest in predominantly diploid populations and weakest in pre-dominantly haploid populations generating the type of frequency-dependentselection needed for haploid-diploid life cycles to evolve. In theory, a diploidintrinsic fitness advantage may be particularly likely due to several previ-ously proposed hypotheses. Firstly, Orr (1995) showed that diplonty canprotect organisms from partially recessive somatic mutations (e.g., mask-ing potentially cancerous mutations that arise during development). Al-though Orr (1995) did not explicitly explore whether haploid-diploid lifecycles could evolve, considering somatic mutations that are partially re-cessive in his model generates a diploid advantage of the type consideredhere (see Mathematica file). Secondly, Haig and Wilczek (2006) proposedthat, when diploid growth is partly provisioned by the female haploid (e.g.,if diploids grow on haploids), paternally expressed genes will favour greaterfemale allocation to his diploid offspring, improving the fitness of that phase.Given that deleterious mutations are typically partially recessive (Sim-mons and Crow 1977, Agrawal and Whitlock 2011, Manna et al. 2011), theregion in which a haploid-diploid life cycle evolves is unlikely to be commonlyencountered, except in two circumstances. First, if mutations are more dele-503.5. Discussionterious in homozygous diploids than in haploids (sd > sh), haploid-diploidlife cycles can be favoured when deleterious mutations are partially recessive(figure 3.2a). Second, when recombination rates are low, the region in whichhaploid-diploid life cycles are favoured moves into the zone where deleteriousmutations are partially recessive (figure 3.2b).A previous investigation by Otto and Marks (1996) found that haploidywas also favoured by recessive deleterious mutations when selfing, asexual re-production or assortative mating is common (similar to low recombination).These results were interpreted via the fact that these mating schemes partlycause the effective recombination rate to be reduced, e.g., recombinationhas no impact in a selfed, homozygous individual. However, this analysisassumed that homozygotes and haploids have equal fitness, thus increasedhomozygosity had no direct impact on fitness. Here, we show that, whenhaploids and diploids have unequal fitness and/or when new mutations oc-cur during the life cycle (e.g., at meiosis), the net effect of selfing can favourhaploidy or diploidy (Appendix B). We also note that the frequency of dele-terious mutations, and thus their relative impact on life cycle evolution, isalso decreased with increased selfing because they are exposed to selectionin the homozygous state (Appendix B). Thus, if the fitness of haploids andhomozygous diploids differs, we caution against generally predicting thathaplont and haploid-diploid life cycles should be more common in specieswhere selfing, asexual reproduction and assortative mating are frequent. Forexample, this may explain why a survey by Mable and Otto (1998) foundno correlation between haploidy and the estimated degree of sexuality inprotists or green algae.When the balance between intrinsic fitness differences and the effectof mutations favours convergence on haploid-diploid strategies, disruptiveselection then arises such that polymorphisms can evolve with alternativealleles coding for longer haploid and longer diploid phases (i.e., a polymor-phic strategy of specialists). In our simulations, a single modifier locus isable to confer fully haplont or diplont life cycles, polymorphism at this locustherefore means that these specialists life cycles can be relatively common(along with the life cycle of the heterozygote at the modifier locus). If ge-netic control of the life-cycle instead involves many modifier loci, each ofwhich was limited to a having a small effect on the length of the haploidphase, a higher proportion of intermediate phenotypes would be observedin a population experiencing disruptive selection due to mating and recom-bination. This is especially true when modifier loci are loosely linked be-cause associations between alleles at different loci (linkage disequilibria) aresmall when recombination is large relative to selection (e.g, Otto and Day513.5. Discussion2007, equation 9.45). Disruptive selection was also observed in a density-dependent model where haploids and diploids occupy different niches withor without deleterious mutations (Rescan et al. 2016). Temporal variabilityof niche sizes can, however, stabilize obligatory alternation between phases(Rescan et al. 2016). Thus, for haploid-diploid life cycles to be favoured overa polymorphic population of specialist haploids and diploids appears to re-quire constraints on the genetic architecture underlying life cycle variationor external variability.It is intuitively and empirically reasonable that haploids and diploidsshould both differ in intrinsic fitness and in the extent to which new muta-tions are masked/revealed to selection. Here, we find the conditions underwhich these selective forces are approximately balanced and show that thissuggests a new hypothesis for the evolution of haploid-diploid life cycles.A significant strength of this hypothesis is that haploid-diploid life cyclesevolve in species undergoing an alternation of haploids and diploid phaseswithout positing any extrinsic benefits.52Chapter 4The Role of Pollen andSperm Competition in SexChromosome Evolution4.1 SummaryTo date, research on the evolution of sex chromosomes has focused on sexu-ally antagonistic selection, which has been shown to be a potent driver of thestrata and reduced recombination that characterize many sex chromosomesIn this study, we expand our view of the forces driving sex chromosomeevolution by considering also selection among haploids, which is likely tooccur predominantly among male gametes in angiosperms and animals, i.e.,during pollen or sperm competition. We find that suppressed recombina-tion is favoured on the sex chromosomes, even without selective differencesbetween male and female diploids. Reduced recombination is favoured be-cause it creates a stronger association between haploid beneficial alleles andthe male determining region (Y or Z), which experiences haploid selectionmost often. Similarly, reduced recombination creates linkage between alle-les selected against in the haploid stage and the female determining region(X or W). In XY systems, these associations also result in biased sex ra-tios at birth. Overall, we predict that whether and how fast recombinationsuppression evolves on the sex chromosomes can depend on the degree ofhaploid competition, not just on selective differences between the diploidsexes. Based on our models, sex chromosomes should become enriched forgenes that experience haploid selection, as is expected for genes that ex-perience sexually antagonistic selection. Thus, we generate a number ofpromising predictions that can be evaluated in emerging sex chromosomesystems.534.2. Introduction4.2 IntroductionIn organisms with diploid genetic sex determination, recombination is typ-ically suppressed between the X and Y chromosomes or Z and W chromo-somes. Suppressed recombination appears to begin near the sex-determiningregion (SDR) and then expand to include larger segments of each sex chro-mosome (Bergero et al. 2007, Nam and Ellegren 2008, Lemaitre et al. 2009,Wang et al. 2012, Charlesworth 2013). In the absence of recombination, thesex-limited chromosome (Y or W) accumulates deleterious mutations (in-cluding gene losses) within the non-recombining region and ‘genetic degen-eration’ occurs (Rice 1996, Charlesworth and Charlesworth 2000, Bachtrog2006, Marais et al. 2008). Thus, the selective forces driving reduced recom-bination on sex chromosomes are fundamental to our understanding of sexchromosome evolution.Typically, selective differences between males and females have beenevoked to explain the suppression of recombination around established sex-determining regions (Fisher 1931, Bull 1983, Rice 1987). Charlesworth andCharlesworth (1980) showed that loci where males and females differ in equi-librium allele frequency due to selection (for example, sexually antagonisticselection) should evolve complete linkage with the sex-determining locus viatranslocations or fusions. More recently, Lenormand (2003) demonstratedthat sex differences in allele frequencies at equilibrium are not required inorder to favour reduced recombination with the sex-determining region. Infact, recombination suppression can evolve around the sex-determining re-gion even if selection favours the same allele in both sexes as long as thatallele is favoured more strongly in one sex than the other. In essence,these studies have demonstrated that suppressors of recombination can befavoured because they strengthen the association between the sex in whichan allele is favoured and the chromosome that is present in that sex more of-ten, e.g., between male beneficial alleles and the Y or Z and between femalebeneficial alleles and the X or W (Otto et al. 2011).While differences in selection between the diploid sexes has attracted themost theoretical and empirical attention, the haploid gametes/gametophytesproduced by males and females also experience different selective environ-ments from one another, with particularly intense competition typically oc-curring among pollen and sperm (Mulcahy et al. 1996, Bernasconi 2004,Joseph and Kirkpatrick 2004). To the extent that pollen and sperm successreflects differences in their haploid genotypes, selection among these ga-metes/gametophytes is qualitatively distinct from selection among diploidmales. That is, diploids cannot be assigned fitness values that also account544.2. Introductionfor the fitness of their haploid gametes (Immler et al. 2012). In plants, selec-tion among haploid male gametophytes is thought to be pervasive (Skogsmyrand Lankinen 2002, Moore and Pannell 2011, Marshall and Evans 2016); inArabidopsis, 60-70% of all genes are expressed during the haploid phase(Borg et al. 2009), and pollen expressed genes exhibit stronger signatures ofpurifying selection and positive selection (Arunkumar et al. 2013, Gossmannet al. 2014). For agricultural breeding, pollen has been exposed to a varietyof selection pressures in vivo and in vitro, including temperature (Hedhlyet al. 2004, Clarke et al. 2004), herbicides (Frascaroli and Songstad 2001),metals (Searcy and Mulcahy 1985), water stress (Ravikumar et al. 2003),and pathogens (Ravikumar et al. 2012), resulting in an increased frequencyof resistant genotypes among the diploid sporophytic offspring. In animals,expression during the haploid sperm stage is traditionally thought to be sup-pressed (Hecht 1998), although recent evidence suggests that the extent andselective importance of postmeiotic gene expression may be underestimated(Zheng et al. 2001, Joseph and Kirkpatrick 2004, Vibranovski et al. 2010,Immler et al. 2014).In this study, we include selection during the haploid phase in mod-els for the evolution of recombination around the sex-determining region(XY or ZW). Specifically, we include a period of selection among the ga-metes/gametophytes produced by one sex, e.g., competition among pollenor sperm but not among ovules or eggs. Thus, we investigate whethersex differences in the selective environment experienced by haploid ga-metes/gametophytes can drive the evolution of suppressed recombinationon sex chromosomes, as with sex differences in diploid selection. One com-plication is that haploid selection can cause zygotic sex ratios to becomebiased. For example, a high fitness allele that becomes more associatedwith the Y than the X will cause Y-bearing pollen/sperm to outcompeteX-bearing pollen/sperm. Thus, increased fertilization success of Y-bearingpollen/sperm will lead to an excess of male zygotes, even if X-bearing and Y-bearing pollen/sperm are produced in equal proportions by males. Tighterlinkage with the sex determining region allows greater differences in fitnessbetween X- and Y-bearing pollen/sperm to evolve and thus greater sex ratiobiases, figure 4.1. Some models that include haploid selection have foundthat mothers experience selection to equalize zygotic sex ratios (Hough et al.2013, Otto et al. 2015). Here, we find that a period of selection among hap-loid pollen/sperm favours suppressed recombination on the sex chromosomesdespite causing biased zygotic sex ratios.554.3. Model Background4.3 Model BackgroundRecombination evolution on sex chromosomes is usually modelled by con-sidering a locus under selection, the sex-determining region, and anotherlocus that modifies the recombination rate between them, where modifiersinclude inversions, fusions, hotspot changes, and changes to genes involvedin double strand breaks and recombination repair. Thus, a general modelincludes three loci and the recombination rates between them, which is typi-cally too complex to interpret without further simplifying assumptions (Ottoand Day 2007). Lenormand (2003) assumed that the recombination ratesbetween these loci are large relative to selection, such that the linkage dis-equilibrium between loci equilibrates on a faster timescale than changes inallele frequencies (a ‘quasi-linkage equilibrium’ approximation). This analy-sis is most appropriate for selected loci that are far from the sex-determiningregion on sex chromosomes and when modifiers of recombination are weakand loosely linked (e.g., autosomal modifiers of recombination machinery).Secondly, Charlesworth and Charlesworth (1980) assumed that the selectedlocus is initially autosomal and then considered fusions with (or translo-cations to) the sex-determining region, where their analysis assumed theserearrangements became closely linked to the selected locus. Their modelalso corresponds to modifications on sex chromosomes (e.g., inversions) thatchange the recombination rate with the sex-determining region from a veryhigh to a very low level. Finally, Otto (2014) considered modifiers of re-combination between the sex-determining region and selected loci when thelinkage between them is initially very tight.Here, we study recombination evolution in a manner akin toCharlesworth and Charlesworth (1980) and Otto (2014) except that we in-clude a period of selection among haploid male gametes/gametophytes. Themodel of Lenormand (2003) is very general and allows a period of haploidselection (assuming weak linkage); he recognizes but does not discuss thepotential of such sex-specific haploid selection to favour suppressed recom-bination on sex chromosomes. Here, our goal is to complete the set of recom-bination evolution analyses that include a period of haploid pollen/spermcompetition and explicitly describe why loci that experience haploid selec-tion can drive the evolution of reduced recombination near sex-determiningregions. Models where haploid selected loci and the sex-determining regioncan become tightly linked are particularly significant because strong associ-ations between haploid selected alleles and the sex-determining region (thatcan build up when linkage is tight) will cause zygotic sex ratios to becomebiased, figure 4.1.564.4. Model0.001 0.01 0.05 0.50.4250.450.4750.50.525Recombination rate, rZygoticSexRatio,FemalesHMales+FemalesLFigure 4.1: Here, we assume that the population is fixed for a particular modifier of recombinationsuch that all individuals have the same recombination rate, r. We then allow the A locus to reachan equilibrium frequency and calculate the zygotic sex ratio. Alleles with high fitness duringpollen/sperm competition typically become associated with the Y, causing sex ratios to becomemale-biased (solid and dashed lines). However, female biased sex ratios can arise if the haploid-beneficial allele is also strongly female-beneficial, causing it to become associated with the X(dotted line). The parameters used in this plot are: solid line (wmij = wfij = wij , waa = 1,wAa = 0.97, wAA = 0.91, wa = 0.9, wA = 1) dashed line (wmij = wfij = wij , waa = 1,wAa = 1.12, wAA = 1.24, wa = 0.8, wA = 1), dotted line (wmaa = 1, wmAa = 0.94, wmAA = 0.8,wfaa = 1, wfAa = 1.14, wfAA = 1.2 , wa = 0.9, wA = 1).4.4 ModelWe consider a modifier model in which the recombination rate betweena locus under selection (selected locus, A, with alleles A and a) andthe sex-determining region (SDR) depends on the genotype at the mod-ifier locus (M, with alleles M and m). In our model, male haploidgametes/gametophytes experience selection according to their genotypeat the A locus (see table C.1) before random mating with female ga-metes/gametophytes. The resulting zygotes develop as males or females de-pending on their genotype at the sex-determining region. Diploid genetic sexdetermination systems are either male heterogametic (females XX and malesXY) or female heterogametic (females ZW and males ZZ). There are there-fore two asymmetries in the model, the sex in which haploid selection occursand the sex that is heterogametic. For simplicity, we primarily describe XYsex determination with male gametophytic selection (pollen/sperm compe-574.4. Modeltition), although we also present results for ZW sex determination and malegametophytic selection.After a period of selection among diploid males and females (ta-ble C.1), meiosis with recombination occurs to produce haploid ga-mete/gametophytes. Because females are homozygous at the SDR (withXY sex determination), the only recombination event of consequence in fe-males is between the A and M locus, which occurs at rate Rf . In males,recombination similarly occurs between the selected locus A and the mod-ifier locus M at rate Rm. Recombination can also occur between the SDRand the A locus in males, this recombination rate is controlled by the modi-fier locus and is given by rij , where ij is the genotype at the M locus (MM ,Mm, or mm), allowing this recombination rate to evolve. Double recombi-nation events in males occur at rate χij , such that any ordering of the locior type of modifier (genic, inversion, fusion) can be modelled with appropri-ate choices of χij , rij , and Rm. We track the frequencies of MA, Ma, mAand ma genotypes among female eggs/ovules, male X-bearing sperm/pollen,and male Y-bearing sperm/pollen separately to allow sex-specific allele fre-quencies and disequilibria. The recursion equations describing the change ingenotype frequencies after a single generation of this life cycle are providedin the Appendix C.In our first analysis, we assume that selection is weak relative to theinitial recombination rate (rMM ), such that allele frequency differences be-tween males and females are small. We then evaluate the spread of modifiersof recombination (m) that cause recombination rates to become very small(assuming rMm, χMm, and Rm are all small). These modifiers could betranslocations or fusions from autosomes to sex chromosomes or, if the se-lected locus (A) begins on the sex chromosome, inversions or expansions ofthe non-recombining region. We assume that chromosomes are still able todisjoin regularly from their homologs during meiosis.In our second analysis, following Otto (2014), we assume that the Alocus begins at equilibrium and in tight linkage with the SDR (rMM andχ are on the order of a small term, ). We then consider whether anymodifiers can invade that increase this recombination rate slightly (wherethe change in recombination rate, rMm − rMM , is on the order of ). Therecombination rate between the modifier locus and these sex chromosomeloci (Rf and Rm) is not constrained. This analysis focuses on the finalstages of sex chromosome evolution, asking when complete recombination isfavoured or not.584.5. Results4.5 ResultsConsidering a population originally fixed for the M allele at the modifierlocus, the frequency of the A allele among X-bearing eggs/ovules, X-bearingsperm/pollen, and Y-bearing sperm/pollen is given by pXf , pXm, and pY mrespectively. The spread of rare mutants that change the recombination ratecan be evaluated using the leading eigenvalue, λ, of the system described byequations (A1c), (A1d), (A2c), (A2d), (A3c), and (A3d).Complete suppressors of recombination (rMm = 0) that are closely linkedto the A locus (Rf = Rm = χ = 0) experience the strongest selective force.These modifiers can bring either the A or the a allele into tight linkage witheither the X or Y chromosome. Thus, the invasion of these mutants can beevaluated separately and is given by λij , where ij is the haplotype at thenewly linked SDR and A loci.The spread of modifiers that create tight linkage between the Y and Aallele is given byλY A = w¯mYA/w¯m (4.1)where w¯mYA is the marginal fitness of YA haplotypes and w¯m is the meanfitness of males, see table C.2. Such modifiers will spread if λY A > 1, whichis true when w¯mYA > w¯m.Invasion of modifiers that create a strong linkage between the X and aallele is determined by the largest solution to the characteristic polynomial(C.4). For such modifiers, the leading eigenvalue λXa is greater than one ifw¯mat,fXa /w¯f + (w¯mat,mXa /w¯m)(w¯pat,fXa /w¯f ) > 2 (4.2)where w¯f is the mean fitness of females and w¯i,jXa indicates the marginalfitness of Xa haplotypes when inherited from the mother (i = mat) or father(i = pat) and found in offspring of sex j. This condition demonstrates thatthe newly formed sex chromosome is able to invade if its marginal fitness ishigher than average (once appropriately weighted across carriers of maternaland paternal copies).Here, we consider the case where the A locus is initially at an intermedi-ate frequency maintained by selection. Polymorphisms can be maintained bya combination of sexually antagonistic selection, ploidally antagonistic selec-tion, and/or overdominance (Immler et al. 2012). We write λY A in terms ofthe difference in fitness between haploid genotypes (δH = wA−wa) and thedifference in equilibrium allele frequency between Y-bearing pollen/spermand ovules/eggs (δ = pˆY m − pˆXf ) where the caret indicates an equilibrium594.5. Resultsfrequency. We can then write equation (4.1), for the invasion of modifiersthat bring the A allele into tight linkage with the Y chromosome, asλY A = 1 +rMMwmAapˆY mw¯m(δ + VmδH/w¯H) (4.3)where Vm = pˆY m(1 − pˆY m) is the variance among Y-bearing pollen/spermand w¯H = (pˆY mwA + (1 − pˆY m)wa) is the mean fitness of haploid malegametes/gametophytes. If there is no selection among haploid genotypes(wA = wa), equation (4.3) is equivalent to equation (A3) in Charlesworthand Charlesworth (1980), in which case these tightly linked YA haplotypesinvade if the A allele is more common in males than females (pˆm− pˆXf > 0),as expected if the A allele is beneficial in males with sexually antagonisticselection. We also find an additional term, demonstrating that tight linkge isalso favoured when the A allele is beneficial during haploid selection (wA >wa), even in the absence of frequency differences between males and females(pˆY m = pˆXf ), i.e., even when there is no difference in selection betweendiploid males and females.Here, in order to solve (C.4) for λXa, we will assume that linkage isinitially loose between the SDR and A locus (rMM = 1/2), such that segre-gation in males is random and pˆXm = pˆY m = pˆm. In Appendix C we presentequivalent results for cases where we do not assume that recombination isinitially free (rMM < 1/2). We will further assume that selection is weak,such that the difference in frequency between A alleles in males and females(δ = pˆm − pˆXf ) and the difference in fitness between haploid genotypes(δH = wA − wa) are small (δ and δH of order 2). Ignoring terms of order3 and higherλXa = 1 +13wmAa2(1− pˆm)w¯m (δ + VmδH) . (4.4)Thus, the same conditions that favour linkage between the Y and the Aallele, favour linkage between the X and the a allele. Specifically, when thea allele is more common in females (δ > 0, e.g., a is a female beneficial allele)and when the A allele is favoured during haploid competition (δH > 0). Inthe special case where there is no difference in selection between male andfemale diploids (wmij = wfij = wij), we can find an exact expression for λXAby solving for pˆm and pˆXf without assuming that selection is weak, whichconfirms the expectation from (4.4) that linkage between the X chromosomeand alleles deleterious in haploid pollen/sperm is favoured by selection, seeAppendix C.604.5. ResultsIt may not be intuitively obvious why an association with the allele thatis less fit during haploid selection should be favoured. This result comesfrom the fact that the a allele is initially maintained at an equilibrium fre-quency by selection. At equilibrium, selection against a in haploid malegametes/gametophytes must be balanced by selection in favour of A in fe-male and/or male diploids. The X chromosome is found in males less oftenthan an autosomal or loosely linked locus and therefore experiences haploidselection less frequently. Thus linkage between the a locus and the X isfavoured because it allows the a allele to experience haploid selection lessoften. Similarly, equation (4.3) indicates that linkage between the Y, whichexperiences haploid selection most often, and a haploid beneficial allele isfavoured.As with previous analyses (Charlesworth and Charlesworth 1980,Charlesworth and Wall 1999, Lenormand 2003), we find that the strength ofselection in favour of recombination modifiers is strongest on Y chromosomesbecause these are always found in only one sex whereas the X will some-times be found in males and sometimes in females. In particular, (4.3) and(4.4) differ by a factor of 1/3 once we account for the difference between theprobability of linkage arising with the A allele, pm, or the a allele, (1− pm).However, mutations causing linkage with the Y (e.g., fusions) should alsoarise at a lower rate because there are three times as many X chromosomesas Y chromosomes in the population, such that the overall establishmentrate of recombination modifiers is the same on the X and Y (Pennell et al.2015).The tight linkage case considered above is the best case scenario for gen-erating selection in favour of recombination suppressors. For a few parame-ters, Charlesworth and Charlesworth (1980) find numerically that recombi-nation suppressors spread, but at lower rates, if Rm and Rf are larger. Here,we find analytical results by assuming the recombination rates between theA locus, the M locus, and the SDR are small (χ, Rm and Rf or order 3).Neglecting terms of order 4 and higher, the growth rate of such mutants(λi˜j) is.λY˜ A ≈λY A − (1− pˆm)wmAaRmw¯m− χMm (4.5a)λX˜a ≈ λXa − pˆm3(2wfAaRfw¯f+wmAaRmw¯m)− χMm3(4.5b)614.5. Results0 1000 2000 3000 4000 5000 60000.0.51.GenerationFrequencyofYhaplotype,Y im0 1000 2000 3000 4000 5000 60000.0.51.GenerationFrequencyofXhaplotype,X imFigure 4.2: Here, we iterate the recursion equations C.1, C.2, C.3 to track the change of genotypefrequencies among X-bearing female haploids (Xfi ), X-bearing male haploids (Xmi ), and Y-bearingmale haploids (Ymi ), respectively. Across this plot, X-bearing haploids in males and females havevery similar haplotype frequencies so we plot Xmi only. We assume that the population initiallyhas loose linkage between the A locus and the SDR (rMM = 0.5, where M is initially fixed) andallow allele frequencies to reach a polymorphic equilibrium. We then introduce a modifier allelem that reduces the recombination rate between A locus and the SDR (rMm = rmm = 0.01);in generation 0, m is at frequency 0.01 and in linkage equilibrium with M . We assume that theM locus lies between the A locus and the SDR such that χij = (rij − Rm)/(1 − 2Rm), whereRm = Rf = 0.005. Fitness parameters are as in the solid line in figure 4.1. That is, there areno differences in selection between diploid sexes and selection is ploidally antagonistic with Afavoured by haploid selection, thus pˆXf = pˆXm = pˆYm initially, see Appendix C Lines show thefrequencies of the A allele (dashed), the a allele (dotted) the recombination suppression mutant,m (solid). Due to continuing recombination between the A locus, M locus, and the SDR, aparticular haplotype does not fix on the Y chromosome, as is the case when rij = 0 (see AppendixC). However, after recombination has evolved to a lower level, the haploid beneficial allele (A,dashed lines) becomes more common on the Y and less common on the X.where λij corresponds to the tight linkage results (4.3) and (4.4). Theadditional terms in (4.5) illustrate that the spread of linked haplotypesis slowed when the alternative A allele recombines onto the modifier andSDR background (recombination rate Rm or Rf ) or when the modifier re-combines onto the opposite sex chromosome (which occurs at rate χMm inmales). In figure 4.2, we track the spread of a recombination modifier whereRm, Rf , χ, rMm 6= 0, such that both M alleles and both A alleles can re-combine onto both sex chromosomes. As predicted from equation (4.5), arecombination suppressor increases in frequency and the X and Y chromo-somes become associated with the a and A alleles, respectively.We derive equivalent results for ZW sex chromosome systems (wheremales are ZZ and females are ZW) with a period of haploid selection amongmale gametes/gametophytes. We again consider invasion of a modifier thatcreates tight linkage between the A locus and the M locus (rMm, χ, Rm andRf of order 3) in a population in which linkage is initially loose between624.5. Resultsthe SDR and A locus (rMM = 1/2). Here, we present λWa and λZA underthe same assumptions as (4.5), where the difference in frequency of theA allele between males and females and the difference in fitness betweenhaploid genotypes are small (δ = pˆZm − pˆWf where δ and δH are of order2), yieldingλW˜a ≈1 + rMMwfAa(1− pˆf )w¯f (δ + VfδH)−pˆfwfAaRfw¯f− χMm (4.6a)λZ˜A ≈1 + 13wfAa2pˆf w¯f(δ + VfδH)− (1− pˆf )3(2wmAaRmw¯m+wfAaRfw¯f)− χMm3(4.6b)where we discard terms of O(4). λZA and λWa show that, when the A alleleis more common in males (δ > 0), linkage between the male Z chromosomeand the A allele and linkage between the female specific W chromosome andthe a allele are both favoured. In addition, linkage is favoured between theZ and the allele favoured during haploid selection (A if δH > 0) and betweenthe female specific W chromosome and the allele with low haploid fitness (aif δH > 0). Thus, recombination suppression allows an association betweenthe chromosome that is present in males most often (Z) and alleles favouredduring pollen/sperm competition.Finally, we evaluate the evolution of recombination during the finalstages of sex chromosome evolution by considering the evolution of smallamounts of recombination around the sex-determining region (SDR). Con-sidering diploid selection alone, Otto (2014) demonstrated that a smallamount of recombination around the SDR can be maintained by selection.This result is counterintuitive because, as discussed above, linkage allowsassociations to build up between the sex-determining region and selectedloci. Because these associations arise due to selection, they are generallyfavourable and one would expect that breaking them apart by recombina-tion would be detrimental. However, particular forms of selection, combinedwith the asymmetrical inheritance patterns of sex chromosomes can favourloosely linked modifiers that increase recombination around the SDR.With tight linkage between the SDR and a selected locus (A), the Ychromosome always becomes fixed for one allele or the other, see Appendix634.5. ResultsC. Without loss of generality, we will assume that selection on the Y favoursthe A allele, which becomes fixed on the Y. X chromosomes will thereforebe paired with a YA haplotype in diploid males; this alters selection ex-perienced by X chromosomes found in diploid males versus those found indiploid females. For example, the A locus will never be homozygous forthe a allele in males but could be in females. When there is a polymor-phism maintained (such that recombination can have an effect), the X caneither be fixed for the a allele or be polymorphic. In either case, the ef-fect of increasing the recombination rate with the sex-determining region isto produce more Ya and XA haplotypes among pollen/sperm. Ya haplo-types always have low fitness given that the Y was originally fixed for theA allele. However, the XA haplotypes produced by recombination can befavoured because they are found in male gametes/gametophytes. X-bearingmale gametes/gametophytes first experience pollen/sperm competition andthen produce females in the next generation. Thus, the XA haplotypes pro-duced by recombination do not experience the same selective environmentas average X chromosomes. Certain selection regimes favour XA haplotypesin pollen/sperm (even if the X is fixed for the a allele), which can favourmodifiers that increase recombination around the sex-determining region.With diploid selection only, increased recombination around the SDR canonly evolve if selection in females favours the A allele (which is fixed on theY) because XA pollen/sperm produced by recombination will next be foundin a female (Otto 2014). For this to occur, selection in males must be over-dominant (a necessary but not sufficient condition). With overdominancein males, the a allele has the highest fitness on the X chromosome in malesbecause it is always paired with an A allele on the Y. Thus, the a allele canbe maintained (or even fixed) on X chromosomes and yet the A allele can befavoured during selection in females. However, with pollen/sperm competi-tion, it is possible for a small increases in recombination to be favoured un-der a wider variety of selective regimes in diploids, including overdominance,underdominance, sexually antagonistic selection and ploidally antagonisticselection. In Appendix C, we show that the evolution of increased recombi-nation requires either that the a allele is favoured by selection on the X inmales (wmAa > wmAA or that the A allele is favoured during selection amonghaploid male gametes/gametophytes (wA > wa). If the A allele is selectedagainst on male X chromosomes, it is possible for it to be favoured on femaleX chromosomes and yet still maintain the a allele. In addition, XA haplo-types produced by recombination will be found in pollen/sperm and thusexperience haploid selection before becoming diploid females. Therefore, ifhaploid selection favours the A allele, XA haplotypes in pollen/sperm can644.6. Discussionhave high fitness.Given that XA pollen/sperm have an advantage through haploid com-petition and/or high fitness in female diploids, the fitness advantage ofXA pollen/sperm must outweigh by the cost of producing low-fitness Yapollen/sperm. Thus, increased recombination around the SDR only evolvesin particular regions of parameter space (figure C.1). In addition, the evo-lution of increased recombination requires that the modifier is sufficientlyloosely linked to the SDR (Rf and Rm are sufficiently large), e.g., autosomalmodifiers. This allows the modifier to gain a short term advantage beforerecombining onto a different background. If Rf and Rm are small, the modi-fier remains linked to the haplotypes it creates (XA and Ya), such modifiersnever spread because increased recombination breaks down the associationbetween alleles that are favoured on average in a sex and that sex.4.6 DiscussionEven in predominantly diploid organisms such as animals and angiosperms,there is considerable potential for selection among haploid male gametes(sperm/pollen). Here, we demonstrate that linkage between the diploidsex-determining region (XY or ZW) and a locus that experiences haploidselection is typically favoured by selection. Thus, along with selective dif-ferences between diploid sexes, selection among haploids could be a potentdriver of recombination suppression on sex chromosomes.In ZW sex determination systems, the sex ratio among diploids is unaf-fected by selection among male haploids. However, in XY sex determinationsystems, the number of males and females in each generation depends onthe frequency of X and Y gametes after haploid selection. Despite this, wefind that selection on recombination modifiers is not primarily driven bybalancing the sex ratio of diploids. In fact, the evolution of recombinationsuppression should lead to Y-bearing gametes/gametophytes that have highfitness during haploid selection. Thus, we predict that sex ratios at birthcan become male biased in the early stages of sex chromosome evolution.Biased flowering sex ratios, especially male-biased sex ratios, are com-mon among dioecious plants (Field et al. 2013). However, in Rumex, moreintense pollen competition appears to result in more female biased sex-ratiosamong the progeny (Conn and Blum 1981, Stehlik and Barrett 2006, Fieldet al. 2012). This phenomenon may reflect the accumulation of deleteriousmutations on the Y-chromosome following recombination suppression (Lloyd1974, Stehlik and Barrett 2005), as suggested by the prevalence of female654.6. Discussionsex ratio bias in species with heteromorphic rather than homomorphic sexchromosomes (Field et al. 2013). Thus, the net effect of experimentallymanipulating the intensity of haploid selection may depend on the stage ofsex chromosome degeneration, as well as the alleles associated with the Y.The increasing availability of sex-linked markers should allow sexes to beidentified before reproductive maturity in plants, thus allowing changes inthe sex ratio to be directly evaluated across haploid and diploid phases inspecies with differing degrees of Y chromosome degeneration.The emergence of both haploid expression profiles (Joseph and Kirk-patrick 2004, Borg et al. 2009) and a larger number of sex chromosome sys-tems (Ming et al. 2011, Charlesworth 2013; 2015, Bachtrog et al. 2014, Vicosoand Bachtrog 2015) provides an excellent opportunity to evaluate whethersex chromosomes are enriched for genes selected during the haploid phase,as predicted by our models. If possible, a stronger signal of association withsex-determining regions should occur among loci explicitly shown to exhibitvariation in haploid competitive ability (Travers and Mazer 2001) or lociwhere mutants affect fitness in both haploid and diploid phases (Murallaet al. 2011). Finally, we predict that the strength of haploid competitionpartly determines whether and how fast recombination suppression evolves.Evaluating a related hypothesis, Lenormand and Dutheil (2005) correlateheterochiasmy (differences in autosomal recombination between sexes) withthe degree of sex specific haploid selection, using outcrossing rate as a proxyfor male haploid selection. We would predict a similar pattern for recom-bination suppression around sex-determining regions. Estimates of pollenlimitation could also be used as proxy for the intensity of haploid competi-tion (Vamosi et al. 2006, Friedman and Barrett 2009).As in a previous analysis by Otto (2014), we find that a small amountof recombination can be selectively maintained around the sex-determiningregion. Otto (2014) considered only diploid selection and found that over-dominance in males was required for recombination to be selectively main-tained. Here, we include a period of selection among haploids and findthat increased recombination can be favoured with various forms of selec-tion among diploids, including sexually antagonistic selection and ploidallyantagonistic selection (figure C.1), as long as the allele fixed on the Y isfavoured in haploids and/or females. However, increased recombination isnever favoured when modifiers of recombination act locally, such that theyare also closely linked to the sex-determining region. Therefore, while thesedynamics may influence the maintenance of small amounts of recombinationaround sex-determining regions when polymorphisms with the right form ofselection arise (e.g., within the coloured regions in figure C.1), suppressed664.6. Discussionrecombination will be favoured in most circumstances.Meiotic drive is not exactly equivalent to a period of haploid selection.In particular, meiotic drive can only occur in heterozygotes and often in-volves an interaction with a separate susceptible/resistant locus (Bull 1983).However, meiotic drive is also usually sex specific, either acting during sper-matogenesis in males or polar body formation in females. In this respect, weexpect loci experiencing meiotic drive to behave similarly to those experienc-ing haploid selection. In particular, we predict that selection should favourlinkage between alleles favoured by drive and the sex chromosome for the sexin which drive occurs (e.g., with the Y or Z when drive occurs during sper-matogenesis). Despite theoretical interest in related topics (Feldman andOtto 1989, Haig 2010, Brandvain and Coop 2012, Patten 2014, Rydzewskiet al. 2016), such as the evolution of recombination between an X chromo-some that experiences drive and another selected locus (Feldman and Otto1989, Rydzewski et al. 2016), this process has yet to be explicitly modelledand is worthy of future exploration.Overall, as well as providing several predictions, our model offers a newperspective on drivers of sex chromosome evolution. Traditionally, sex differ-ences in selection are thought to provide the raw material driving recombi-nation suppression on sex chromosomes. However, even where diploid sexesexhibit very few morphological or ecological differences, the selective envi-ronment of their haploid gametes may be very divergent. We have shownthat this condition - differences in fitness among pollen or sperm - shouldalso favour suppressed recombination. Consequently, our view of sex chro-mosome evolution is expanded to incorporate the degree of sex specific se-lection in haploids along with that in diploids.67Chapter 5ConclusionsIn this Chapter, I briefly review some of the results presented in this thesisand highlight instances where the results obtained were not apparent atthe outset. Finally, following Chapter 4, I further discuss the relationshipbetween haploid selection (e.g., pollen/sperm competition) and sex ratios.5.1 Developmental DelaysA traditional explanation for the evolution of developmental delays is thatsome environments are unsuitable for growth and thus continuing develop-ment ‘does not pay’ (Cohen 1967, Levins 1968, Schaal and Leverich 1981).We model the evolution of seed dormancy (a developmental delay) in Chap-ter 2, in which we consider non-annual plants. Perhaps unsurprisingly, theoptimal germination rate for non-annuals depends on the environments thatmight subsequently be experienced, not just the environment in the yearthat germination occurs (equation 2.6). However, it might not be immedi-ately obvious that this allows dormancy to evolve in environments that arenot ‘bad’ per se.Firstly, we show that dormancy can evolve in environments whereseedling survival is low (mechanism 1), even if this environment is not in-trinsically ‘bad’. Consider the case where the population would grow if itexperienced a particular environment (environment 2) all of the time. Ifthe population also experiences another environment (environment 1) withhigher seedling survival, it can be favourable to evolve dormancy in envi-ronment 2 so that seeds can germinate in environment 1 in future years.Initially, I anticipated that dormancy would only evolve in environmentswith negative growth rates because we generally expect immediate devel-opment to maximize growth rate in all favourable environments (Bulmer1985, Philippi and Seger 1989, Rees 1996). However, this heuristic is pri-marily based on models of short-lived organisms; by explicitly modellingdemographically structured populations we were able to modify our intu-ition, finding that decreased seedling survival can be sufficient to favourdormancy. We also note that dormancy can evolve in favourable environ-685.1. Developmental Delaysments if there are physical or developmental trade-offs that make it difficultto have germination rates that are equally high in all environments.Secondly, when large disturbance events occur, we found that dormancycan evolve in apparently favourable environments in order to avoid ‘imma-turity risk’: the risk of dying in a disturbance before reaching reproductivematurity. We might expect this risk to be significant; previous models pre-dict that insects should enter diapause once there is a significant risk offailing to reproduce before a catastrophe occurs (e.g., a winter frost, Cohen1970, Taylor 1980, Hairston and Munns 1984, Taylor and Spalding 1989,Bradford and Roff 1993, Spencer and Colegrave 2001). On the other hand,before constructing these models, it was not clear that immaturity risk couldbe significant when the probability of a disturbance is the same in every year.In this case, it seems intuitively reasonable that all years have the same im-maturity risk for newly germinating seeds. However, even with a constantrisk of disturbance, strategies where germination occurs immediately afterthe previous disturbance will maximize the age at the time of the next dis-turbance, relative to strategies in which germination occurs indiscriminately.Overall, I think one of the key features of this study is that we demon-strate how the mechanisms we present can be isolated. For example, remov-ing differences in seedling survival eliminates mechanism 1 and reducing thenumber of years required to reach maturity can eliminate mechanism 3. Thisapproach can be applied to carefully collected demographic data, e.g., wherestimulated seeds or young seedlings are transplanted into different environ-ments to measure the survival rates if the germination rate was the same.Thus, the contribution of these mechanisms to the evolution of dormancyin a particular environment could be estimated. However, most currentlyavailable demographic data does not include transplants into environmentsin which germination does not occur (e.g., Enright et al. 1998, used as anexample in Chapter 2).Incorporating temporal variation in the environment into a demographi-cally structured population model is challenging and generally requires extraapproximations or assumptions (Tuljapurkar 1990b). For example, to ob-tain analytical results in Chapter 2, we assume that temporal variation inthe environment is cyclical. Similarly, approximations are often required toconsider how the evolution of life history traits (e.g., dormancy) are affectedby density dependence in non-demographically structured populations (e.g.,annual plants) if there is temporal environmental variation (Bulmer 1984,Ellner 1985a;b, ?). A challenge for future research will be to incorporatedensity dependence into a model with a temporally varying environmentand demographic structure. This problem is intuitively important, e.g., the695.2. Haploid-Diploid Life Cyclesovershadowing of younger plants by older plants is likely to be a key factor inthe evolution of dormancy but is not explicitly included in current models.5.2 Haploid-Diploid Life CyclesPloidy significantly affects the way in which alleles are exposed to selection.Intuition commonly suggests that diploidy is favoured in order to mask theeffect of deleterious alleles. For example, H.J. Muller is said to have an-nounced that he wasn’t concerned about the mutagenic effects of pepper bystating “that’s why we’re diploid” (Kirkpatrick 1994). On the other hand,because deleterious alleles are liable to be removed by selection in haploids,the frequency of deleterious mutations is typically lower in haploid popula-tions (Crow and Kimura 1965). Thus, it is not immediately clear how toweigh the immediate masking of deleterious mutations with the change inallele frequency in subsequent generations. Evolutionary invasion analyses(in the form of ‘modifier models’) have clarified this problem by specificallyevaluating the success of mutations that alter whether selection occurs pre-dominantly in the haploid or diploid phase (Perrot et al. 1991, Otto andGoldstein 1992, Jenkins 1993, Jenkins and Kirkpatrick 1995, Hall 2000).Unlike previous models, in Chapter 3, we allowed haploids and homozy-gous diploids to differ in intrinsic fitness and considered the interaction be-tween intrinsic fitnesses and the masking effects of ploidy. At the outset ofthis project, I considered the possibility that the balance between these twoforces could favour life cycles that have both haploid and diploid phases.As predicted, our results show that intermediate haploid-diploid life cyclescan evolve if diploids have higher intrinsic fitness and deleterious muta-tions favour haploidy. However, unexpectedly, we found that the reversesituation - where haploids have high intrinsic fitness and deleterious mu-tations favour diploidy - does not favour convergence upon haploid-diploidlife cycles. This is because the frequency of deleterious alleles is highest inpredominantly diploid populations. Therefore, selection due to deleteriousalleles is strongest when diploidy is common, which prevents convergenceupon a haploid-diploid strategy if deleterious alleles favour diploidy. Thus,an intrinsic diploid advantage is a strong condition for haploid-diploid lifecycles to have evolved via the mechanism we present. This condition can beexamined by measuring fitness components of isogenic haploids and diploidsin a natural environment.Taken together, these models have greatly clarified how haploidy anddiploidy influence life cycle evolution. The main contribution of Chapter705.3. Sex Chromosome Evolution3 is to remove the assumption that haploids and homozygous diploids areequivalent. From this, we found a novel hypothesis for the evolution ofhaploid-diploid life cycles.5.3 Sex Chromosome EvolutionOne feature that characterizes many sex chromosome systems is that, alongmost of the length, recombination with the opposite sex chromosome hasbeen lost (Skaletsky et al. 2003, Bergero et al. 2007, Nam and Ellegren 2008,Lemaitre et al. 2009, Wang et al. 2012, Wright et al. 2014). The primaryexplanation for this phenomenon involves sexually antagonistic selection be-tween diploid sexes (Fisher 1931, Bull 1983, Rice 1987, Charlesworth 2013;2015). For example, reduced recombination allows a a stronger associationbetween male beneficial alleles and the Y and between female beneficial al-leles and the X. However, the haploids produced by males and females alsoexperience very different selective environments; particularly intense selec-tion occurs among pollen and sperm (Mulcahy et al. 1996, Bernasconi 2004,Joseph and Kirkpatrick 2004). Lenormand (2003) indicates (among otherresults) that sex-specific haploid selection could favour weak recombinationsuppressors. However, this possibility is not commonly cited as an impor-tant driver of sex chromosome evolution. In Chapter 4, I wanted to examinewhether selection during the haploid phase could allow strong suppressors ofrecombination (e.g., inversions or fusions) to spread. An important aspectof our study is that, while we tend to think of physical size as a proxy forthe importance of haploid and diploid phases, we might want to alter ourperspective to think about how much selection occurs during the haploidstage.Before conducting this study, we expected sex ratio selection to be animportant driver of recombination evolution. The sex of offspring in an XYsex determination system is determined by the chromosome carried by thesuccessful pollen/sperm (after haploid selection). Thus, strong associationsbetween haploid beneficial alleles and the Y (which can build up when re-combination is strongly suppressed) will cause sex ratios to become malebiased. In general, there is strong selection selection in favour of balancingsex ratios (Fisher 1930, Hamilton 1967). Therefore, we expected that thebiasing of sex ratios might prevent recombination suppression. However, ourresults suggest that sex ratio selection has little impact on the evolution ofrecombination suppression. This can be understood via the fact that sex ra-tios are affected by pollen or sperm competition, which is a male-like period715.4. Extraordinary Sex Ratios: Revisitedin which selection tends to maximize siring success rather than balance sexratios (Otto et al. 2015).We found that selection among haploid genotypes in pollen or sperm candrive the evolution of suppressed recombination between sex chromosomes.Our result presents a number of promising avenues for empirical investiga-tion. In particular, we predict that sex chromosomes will become enrichedfor genes that experience haploid selection. This prediction can be examinedby finding the genomic locations of genes that potentially experience hap-loid selection (Joseph and Kirkpatrick 2004, Borg et al. 2009) and/or thoseshown to be essential in both phases (Muralla et al. 2011). More generally,we would expect the rate of sex chromosome evolution to reflect the degreeof pollen or sperm competition, for which we might be able to use a proxylike pollination syndrome or outcrossing rate (e.g., Lenormand and Dutheil2005).5.4 Extraordinary Sex Ratios: RevisitedIn Chapter 4, we show that sex chromosomes should evolve to become linkedto alleles that are selected in the haploid phase, resulting in biased sex ratios.Generally, any sex-linked gene that harbours genetic variation in haploidfitness should cause sex ratios to become biased. Sex ratio bias caused bypollen competition has previously been discussed in the context of Y-linkeddeleterious mutations, which are thought to build up after recombinationsuppression evolves (Lloyd 1974, Stehlik and Barrett 2005). Sex ratios canalso become biased due to meiotic drive; in a classic paper, Hamilton (1967)showed that X- or Y-linked alleles that experience meiotic drive will bias sexratios. He assumed that driving alleles are under directional selection andspread to fixation but such alleles can also be maintained at intermediatefrequencies by selection (Feldman and Otto 1989, Holman et al. 2015). Whensex ratios are biased, other loci are expected to evolve to restore equal sexratios. Indeed, alleles that negate the effect of sex-linked meiotic driversand restore equal sex ratios have been identified (Stalker 1961, Smith 1975).A similar process occurs with cytoplasmic male sterility alleles (that causebiased sex ratios) and nuclear ‘restorer’ genotypes (Frank 1989).When sex ratio bias occurs due to haploid selection, a natural class of sexratio ‘restorers’ exist because haploid selection often occurs in a context thatis determined by the diploid parents. For example, the intensity of pollencompetition can be manipulated by altering style length (Travers and Shea2001, Lankinen and Skogsmyr 2001, Ruane 2009), delaying stigma receptiv-725.4. Extraordinary Sex Ratios: Revisitedity (Galen et al. 1986, Lankinen and Madjidian 2011) and/or delaying pollentube growth in the pistil (Herrero 2003). Where the X and Y have fixedfitness differences, Hough et al. (2013) demonstrated that mothers shouldgenerally evolve to balance sex ratios by reducing the intensity of haploidcompetition. However, reducing competition among haploids also reducesthe potential for harmful deleterious mutations to be purged. When delete-rious mutations are included, the optimal intensity of haploid selection canreflect a balance between maximizing offspring fitness and equalizing sexratios.As part of a collaborative project (Otto et al. 2015), I considered theevolution of the haploid ‘selective arena’ in cases where the X chromosomeharbours a polymorphism that affects haploid fitness. Mothers again primar-ily evolve to restore equal sex ratios. However, modifying haploid selectionalso affects the X-linked genotypes that are inherited by offspring. Specif-ically, increasing the intensity of haploid selection increases the proportionof daughters (all progeny of X-bearing sperm/pollen are female) that inheritthe allele with high haploid fitness. If this allele has high fitness in daugh-ters, mothers can be selected to increase the intensity of haploid selection;otherwise, decreased selection among haploids is favoured. Thus, becausealtering haploid selection intensity affects the alleles that are inherited bydaughters, mothers can favour slightly biased sex ratios. In addition, I foundthat stronger sex ratio biases can be favoured by paternal manipulations ofthe haploid ‘selective arena’ because fathers are strongly selected to maxi-mize their own siring success (above selection to equalize the sex ratio).Several aspects of the relationship between haploid selection (e.g., pollenor sperm competition) and sex ratios remain to be explored. For example,new sex-determining systems (particularly transitions between male and fe-male heterogamety) can be favoured in order to restore equal sex ratios inpopulations that have a sex ratio bias (Bull 1983, Kozielska et al. 2010,U´beda et al. 2015). Based on the results of Chapter 4, we would expectthat sex ratio biases would occur via associations between sex-determiningloci and loci that experience haploid selection. However, these associationsshould also select against transitions between sex-determining systems, ashas been found with sexually antagonistic selection (van Doorn and Kirk-patrick 2007; 2010). It is not clear how the spread of new sex determinationsystems would be influenced by the combination of sex ratio biases andfavourable associations between haploid selected loci and sex-determiningregions. 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PLOS Genetics9:e1003388.93Appendix AEvolution of DevelopmentalDelays AnalysisA.1 Differences in Seedling SurvivalHere we provide an outline of our proofs; we also provide a Mathematica(Wolfram Research Inc. 2010) file, which can be used to re-derive our resultsand contains additional details.The transition matrix for the case of two environments can be writtenTD =(psS1(1− g1) + (1− p)sS2(1− g2) pb1 + (1− p)b2psS1sY 1g1 + (1− p)sS2sY 2g2 psA1 + (1− p)sA2), (A.1)which is a specific form of the more general matrix:TE =(a[g1, g2] bc[g1, g2] d). (A.2)Writing (A.1) in this more general form simplifies the presentation belowand allows more general insights to be obtained. The long-term growth rate(λ[g1, g2]) is given by the larger root of the characteristic polynomial for thismatrix,λ[g1, g2]2 − a[g1, g2]λ[g1, g2]− d λ[g1, g2] + d a[g1, g2]− b c[g1, g2] = 0. (A.3)Dynamics of Germination Rate in ‘bad’ Patches, g2The effect of a small mutation on the growth rate is obtained by differenti-ating this polynomial with respect to g1 or g2. For the g2 case, re-arranginggives∂λ[g1, g2]∂g2=λ[g1, g2]∂a[g1,g2]∂g2− d∂a[g1,g2]∂g2 + b∂c[g1,g2]∂g22λ[g1, g2]− d− a[g1, g2] . (A.4)94A.1. Differences in Seedling SurvivalWe can then make a simplification in cases where seed survival (a[g1, g2])and seed germination (c[g1, g2]) rates are linear functions of g2 and g1, asin equation (A.1), so that ∂a[g1,g2]∂g2 is proportional to∂c[g1,g2]∂g2. Specificallywe assume that ∂c[g1,g2]∂g2 = β∂a[g1,g2]∂g2, where β is the proportionality con-stant (β = −sY 2 in equation A.1). With this simplification, equation (A.4)becomes∂λ[g1, g2]∂g2=∂a[g1,g2]∂g2(λ[g1, g2]− d+ b β)2λ[g1, g2]− d− a[g1, g2] . (A.5)A potential ESS germination rate (g2) occurs if∂λ[g1,g2]∂g2= 0. This conditionrequires that either∂a[g1, g2]∂g2= 0 (A.6)orλ[g1, g2] = d− b β. (A.7)For the parameters in (A.1), ∂a[g1,g2]∂g2 = −(1−p)sS2, which is negative. Hencesolution (A.6) does not provide a relevant ESS. However, solution (A.7) doesand can be re-written in terms of the original parameters as presented inequation (2.7).Special Case sY 1 = sY 2Here we show that a conditional germination does not evolve when sY 1 = sY 2(dashed line in figure 1B). Assuming that sY 1 = sY 2 = sY and re-arrangingthe transition point in equation (2.7) givesb1 =(1− p)(sS2 − sA2 − b2sY ) + p(sS1 − sA1)psY. (A.8)Substituting this point into transition matrix TD in equation (A.1) givesthe population dynamics at this point. We then calculated the eigenvalues(λ1 and λ2) for this new transition matrix in order to assess whether thelong-term growth rate could be at or above replacement (λ ≥ 1). Theseeigenvalues are:λ1 = (1− p)sS2 + psS1 (A.9a)λ2 = (1− p)(sA2 − g2sS2) + p(sA1 − g1sS1), (A.9b)95A.1. Differences in Seedling Survivalboth of which must be less than 1. Note that the parameters in (A.9) arerates or proportions that must be between 0 and 1. Thus the long-termgrowth rate when this transition point occurs is below replacement rate(λ < 1) when sY 1 = sY 2.Furthermore, it can be shown that increasing the germination rate in‘bad’ patches (g2) has a positive effect on growth rate (∂λ[g1,g2]∂g2> 0) whensY 2 = sY 1, assuming that the population is able to grow (λ > 1). This proofuses equation (A.5), which can be written as:∂λ[g1, g2]∂g2= (λ[g1, g2]− d+ b β)∂a[g1,g2]∂g22λ[g1, g2]− d− a[g1, g2] . (A.10)The fraction in equation (A.10) is always positive, assuming that λ[g1, g2] >1. We also show that the remaining part (λ[g1, g2]−d+b β) is always positivewhen sY 1 = sY 2 and λ[g1, g2] > 1. The details of this proof can be found inthe supplementary Mathematica file. Hence, we show that, when seedlingsurvival in ‘bad’ environments (sY 2) is near its maximum value (assumingsY 1 ≥ sY 2), increasing germination rate in ‘bad’ environments (g2) shouldincrease the long-term growth rate. While the transition to evolution favour-ing conditional germination cannot occur within self-sustaining populations(λ > 1) with sY 1 = sY 2, the transition (7) can occur with lower survivalrates (e.g., as shown in figure 1B). In addition, we can show that there is,at most one positive transition point satisfying (7) as the juvenile survivalrate in normal years (sY 2) is varied and that increasing g2 will decrease thelong-term growth rate for values of sY 2 below this point (see supplementaryMathematica file).Dynamics of Germination Rate in ‘good’ Patches, g1The same analysis for mutations to germination rate in ‘good’ patches (g1)shows that g1 is always expected to increase when the population growthrate is at or above replacement (λ ≥ 1) and sY 1 ≥ sY 2. An invasion analysis,as conducted above, yields the solutionλ[g1, g2] = d− b γ, (A.11)which is analogous to equation (A.7), except that γ = −sY 1 (whereas β =−sY 2). We substituted the parameters from (A.1) back into this equation96A.1. Differences in Seedling Survivaland re-arranged to get:b1 =((1− p)(sS2sY 1 + g2sS2sY 2 − b2sY 12 − g2sS2sY 1 − sA2sY 1) + psS1sY 1 − psA1sY 1)psY 12(A.12)We then substituted this point into the transition matrix in equation (2.4)and obtained the following eigenvalues:λ1 = (1− p)(sS2(1− g2(1− sY 2sY 1)))+ psS1 (A.13a)λ2 = (1− p)(sA2 − g2sS2sY 2sY 1)+ p(sA1 − g1sS1), (A.13b)neither of which can be greater than or equal to 1 (assuming sY 1 ≥ sY 2).That is, there is never a transition point at which ∂λ[g1,g2]∂g1 = 0 assumingthat λ[g1, g2] > 1 and sY 1 ≥ sY 2. Therefore, unlike germination rate in‘bad’ patches (g2), which can undergo a transition for sufficiently small sY 2,there is no transition point for germination rate in ‘good’ patches, g1.We can also show that the sign of ∂λ[g1,g2]∂g1 is positive assuming thatsY 1 ≤ sY 2 and λ[g1, g2] > 1 (see supplementary Mathematica file). Hence,increasing the germination rate in ‘good’ patches, g1, is always expected toincrease the long-term growth rate given that the population is able to grow.Effect of other parameters on the size of the region in whichconditional germination evolvesWe re-write transition point (7) in terms of b1 (called b1crit for critical b1value) here:b1crit =psS1g1(sY 1 − sY 2) + sY 2((1− p)sS2 − (1− p)sA2 − (1− p)b2sY 2 + p(sS1 − sA1))psY 22.(A.14)As shown in the supplementary Mathematica file, conditional germination isfavoured for values of b1 below this point but not above. Thus any parameterthat decreases b1crit will decrease the parameter space over which conditionalgermination evolves. Taking the derivative of b1crit with respect to sA1, sA297A.1. Differences in Seedling Survivaland b2 gives:∂b1crit∂sA1= −(1sY 2), (A.15a)∂b1crit∂sA2= −((1− p)psY 2), (A.15b)∂b1crit∂b2= −((1− p)p), (A.15c)which are all negative, indicating that increasing adult survival or the num-ber of seeds produced in ‘bad’ patches will decrease bfcrit and thereforerestrict the conditions under which conditional germination evolves. In con-trast, the effect of changes in sS1, sS2, sY 1 and g1 on b1crit are given by:∂b1crit∂sS1=g1sY 1 + (1− g1)sY 2sY 22, (A.16a)∂b1crit∂sS2=(1− p)psY 2, (A.16b)∂b1crit∂sY 1=g1sS1sY 22, (A.16c)∂b1crit∂g1=sS1(sY 1 − sY 2)sY 22, (A.16d)which are all positive (assuming that sY 1 > sY 2). Hence, increasing seedsurvival, seedling survival in ‘good’ patches, or the germination rate in‘good’ patches all broaden the conditions under which conditional germi-nation evolves. Increasing p also generally increases b1crit but this proof alsorequires the assumption that the population can grow in ‘bad’ patches. Theeffect of a change in p on b1crit is∂b1crit∂p=sA2 + b2sY 2 − sS2p2sY 2. (A.17)For the population to be able to grow in the absence of ‘good’ patchesrequires that the leading eigenvalue with p = 0 be greater than one, whichin turn implies thatb2 >1− sA2 + (1− g2)sA2sS2 − (1− g2)sS2g2sS2sY 2. (A.18)98A.2. Trade-OffsRearranging (A.18) in the form of equation (A.17) gives:sA2 + b2sY 2 − sS2p2sY 2>(1− sS2)((1− sA2) + g2sS2)p2g2sS2sY 2. (A.19)The right hand side of equation (A.19) is always positive, thus ∂b1crit∂p mustalso be positive, indicating that increasing the proportion of ‘good’ patches,p, broadens the conditions under which conditional germination is expectedto evolve.A.2 Trade-Offsequation (2.8) is a form of the more general transition matrixTF =(a[g2] bc[g2] d). (A.20)The effect of a small mutation altering the germination rate on the long-termgrowth rate (dλ[g2]dg2 ) is now given bydλ[g2]dg2=λ[g2]da[g2]dg2+ bdc[g2]dg2 − dda[g2]dg22λ[g2]− d+ a[g2] . (A.21)In this case, da[g2]dg2 is not always proportional todc[g2]dg2. To clarify this, wewrite da[g2]dg2 anddc[g2]dg2in terms of their original parameters (from equation7):da[g2]dg2= −((1− p)sS2 + psS1g′1[g2]) (A.22a)dc[g2]dg2= (1− p)sS2sY 2 + p(sS1sY 1g′1[g2]). (A.22b)The simplification that dc[g2]dg2 is proportional toda[g2]dg2requires that seedlingsurvival rates are constant (sY 1 = sY 2). If we make this simplification thensolutions (A.6) and (A.7) again describe the potential ESS (where dλ[g2]dg2 =0). The proof used above to show that there is no relevant ESS solutionfrom equation (A.7) when sY 1 = sY 2 continues to apply with trade-offs.Equation (A.6) now does yield a potential ESS, which occurs at equation(2.9).99A.2. Trade-OffsEvolutionary StabilityTo determine whether the singular point (2.9) represents a maximum or aminimum growth rate we take the second derivative of the characteristicpolynomial (the roots of which yield the long-term growth rate) and evalu-ated it at the singular point:d2ψdg22∣∣∣∣dλ[g2]dg2=0= 0, (A.23)where ψ is the characteristic polynomial:ψ = λ[g2]2 − a[g2]λ[g2]− d λ[g2] + d a[g2]− b c[g2] = 0. (A.24)Rearranging gives:d2λ[g2]dg22=λ[g2]d2a[g2]dg22+ bd2c[g2]dg22− dd2a[g2]dg222λ[g2]− d− a[g2] . (A.25)In the original parametersd2a[g2]dg22= −psS1g′′1 [g2] (A.26a)d2c[g2]dg22= sY 1psS1g′′1 [g2], (A.26b)so thatd2c[g2]dg22= γd2a[g2]dg22(A.27)where γ = −sY 1, whether sY 1 = sY 2 or not. For evolutionary stability of thesingular point (9) d2λ[g2]dg22must be negative. This condition can be writtenas:d2a[g2]dg22(λ[g2] + γb− d2λ[g2]− d− a[g2])< 0. (A.28)The part in parentheses is positive if the population growth rate is at orabove the replacement rate (λ[g2] ≥ 1, see equations A.11-A.13). Evolution-ary stability is therefore determined by the sign of d2a[g2]dg22, which is negativewhen g′′1 [g2] > 0 (implying stability).100A.2. Trade-OffsAnnual PlantsHere we demonstrate how trade-offs impact MacArthur’s (1972) model ofgermination rates in an annual system with global migration among patches.A two-environment version of equation (2.2) with a trade-off between ger-mination rates in different environments is given byS[t] = S[0]{p((1−g1[g2])sS1+g1[g2]y1)+(1−p)((1−g2)sS2+g2y2)}t. (A.29)The population growth rate (λ) is therefore given by the term in braces. Thisterm describes the number of seeds resulting from seeds in the previousyear, equivalent to the upper left element of transition matrix TF, a[g2].Therefore, defining a[g2] from equation (A.29), we getdλ[g2]dg2= da[g2]dg2 , andequation (A.6) continues to yield a potential ESS, which now occurs where(sS2 − y2)(sS2 − y2)− (sS1 − y1)g′1[g2]− p = 0. (A.30)Evolutionary stability can again be determined by the sign of d2a[g2]dg22andwill depend on the sign of g′′1 [g2], as above. Therefore, trade-offs can lead tothe evolution of conditional germination or intermediate germination ratesin an annual plant model with purely spatial environmental variation, as inperennials.Incorporating Density DependenceWe can account for any form of density-dependent effects on seedling survivalthrough the transition matrix:TG =(a[g2] bc[g2]comp[A] d). (A.31)Here, we make the assumption that the impact of the adult population sizeis to reduce the survival of all seedlings, regardless of their provenance. Weconsidered the dynamics of a rare mutant with a slightly different germina-tion rate (g2mut = g2res + ) and growth rate (λ[g2mut] = λ[g2res] + ∆λ),where we use the res superscript to denote resident values and mut for themutant. The characteristic polynomial for the invasion of such a mutant is:0 =(λ[g2res] + ∆λ)2 − a[g2res + ](λ[g2res] + ∆λ)− d(λ[g2res] + ∆λ) + d a[g2res + ]− b c[g2res + ]comp[Ares](A.32)101A.2. Trade-OffsWe then conducted a first order Taylor series expansion of  around 0, as-suming that the difference between mutant and resident is small, to obtain0 =(λ[g2res]2 − a[g2res]λ[g2res]− d λ[g2res] + d a[g2res]− b c[g2res]comp[Ares])+ (∆λ(2λ[g2res]− d− a[g2res]) + da[g2res]dg2res(d− λ[g2res])− b comp[Ares]dc[g2res]dg2res)+O[2].(A.33)A mutant with the same trait value as the resident ( = 0) has the samegrowth rate as the resident, which is λ[g2res] = 1 at equilibrium with densitydependence. Therefore, from equation (A.33):1− a[g2res]− d+ d a[g2res]− b c[g2res]comp[Ares] = 0. (A.34)Re-arranging equation (A.33) and ignoring higher order terms in  gives:∆λ =da[g2res]dg2res+ b comp[Ares]dc[g2res]dg2res− d da[g2res]dg2res2− d− a[g2res] . (A.35)The similarity between this equation and (A.21) indicates that density de-pendence does not alter the qualitative results of the model. Indeed, equa-tion (A.35) is the same as equation (A.21) with b now equal to b comp[Ares].Thus, our results are affected by density dependence in a manner akin tohaving birth rates adjusted by the competitive effect of resident adults. Be-cause we have assumed that this competitive effect is the same for typeswith different germination rates, the evolution of these germination rates isqualitatively unaffected. For example, if we assume that sY 1 = sY 2 then, inthe presence of trade-offs, solution (A.6) again represents a potential ESSgiven by equation (2.9).102A.3. Approximating the Cycle MatrixA.3 Approximating the Cycle MatrixThe A and D matrices from equation (2.12) areA =(u11 u12(1− u11) (1− u12))Dτ−1 =(λ1τ−1 00 λ2τ−1)A−1 =( 1−u12u11−u12u12u12−u111−u11u12−u11u11u11−u12 )) (A.36)where λ1 and λ2 are the eigenvalues of T2, u11 and (1−u11) are the elementsof the right eigenvector associated with eigenvalue λ1 and u12 and (1− u12)are the elements of the right eigenvector associated with eigenvalue λ2. Theapproximation we used was to drop the smaller eigenvalue from the normalyear matrix. So that, assuming |λ2| > |λ1|:D˜τ−1 =(0 00 λ2τ−1)(A.37)andT˜2τ−1 =((1−u11)u12λ2τ−1u12−u11u11u12λ2τ−1u11−u12(1−u11)(1−u12)λ2τ−1u12−u11u11(1−u12)λ2τ−1u11−u12), (A.38)in which the ˜ notation is used to indicate that these correspond to theapproximation. Equation (A.38) multiplied on the right by T1 gives theapproximate transition matrix across the entire disturbance cycle as follows,T˜cycle =(sS1u12(1−u11−g1(1−u11(1−sy)))λ2τ−1u12−u11u12(u11sA1−b1(1−u11))λ2τ−1u11−u12sS1(1−u12)(1−u11−g1(1−u11(1−sy)))λ2τ−1u12−u11(1−u12)(u11sA1−b1(1−u11))λ2τ−1u11−u12).(A.39)Here again, we focus on the case where sY 1 = sY 2 = sY , which does notpermit the evolution of conditional germination in our simple model thatignores temporal variation. The eigenvalues of T˜cycle are then are 0 andλcycle =λ2τ−1u12 − u11(b1(1− u11) + u12(sS1(1− u11)− b1(1− u11)− g1sS1(1− u11(1− sY )))− sA1u11(1− u12)) (A.40)103A.3. Approximating the Cycle MatrixDynamics of Disturbance-Induced Germination Rate, g1The change in long-term growth rate over the entire cycle, λcycle, whendisturbance-induced germination rate (g1) is slightly changed is given by∂λcycle∂g1=sS1u12(1− u11(1− sY ))λ2τ−1u11 − u12 . (A.41)If this quantity is positive then increasing g1 is expected to increase the long-term growth rate across the cycle (λcycle). To evaluate the sign of equation(A.41) we re-write (1− u11(1− sY )) and u11 − u12 as follows:1− u11(1− sY ) = sY (λ2 − sA2)(λ2 − sA2) + g2sS2(1− sY ) (A.42a)u11 − u12 = g2sS2sY (2λ2 − (sA2 + (1− g2)sS2)(λ2 − sS2(1− g2(1− sY )))(g2sS2sY + λ2 − sA2) . (A.42b)Both of which must be positive if we assume that the population is ableto grow in non-disturbance years (λ2 > 1). Thus equation (A.41) is alsopositive and mutants with higher g1 values would have higher long-termgrowth rates.Dynamics of Germination Rate in Non-Disturbance Years, g2To evaluate the effect of a small change in germination rate in non-disturbance years (g2) on λcycle we have to take account of the fact thatu11, u12 and λ2 are all functions of g2 and take a derivative of λcycle withrespect to g2 to get:∂λcycle∂g2=λ2τ−2(u11 − u12)2(xλ2du11dg2− yλ2 du12dg2− z(u11 − u12)(τ − 1)dλ2dg2),(A.43)wherex = b1(u12 − 1)2 + u12 (sA1(u12 − 1) + sS1((1− u12)(1− g1)− g1sY u12)) ,(A.44a)y = b1(u11 − 1)2 + u11 (sA1(u11 − 1) + sS1((1− u11)(1− g1)− g1sY u11)) ,(A.44b)z = b1(u11 − 1)(u12 − 1) + sA1u11(u12 − 1) + sS1u12 ((1− u11)(1− g1)− g1sY u11) ,(A.44c)104A.3. Approximating the Cycle Matrixanddu11dg2=sS2sY (λ2 − sS2)(λ2 − sS2(1− g2))(λ2 − sS2(1− g2(1− sY )))2(2λ2 − sA2 − sS2(1− g2)) , (A.45a)du12dg2=−sS2sY (λ2 − sS2)(λ2 + g2sS2 − sA2)(λ2 + g2sS2sY − sA2)2(2λ2 − sA2 − (1− g2)sS2) , (A.45b)dλ2dg2=(λ2 − sA2)(λ2 − sS2)g2(2λ2 − sA2 − (1− g2)sS2) . (A.45c)We know that, as τ goes to∞ (no disturbances), increasing g2 would increaseλcycle (assuming λ2 > 1). However, for short cycles (low τ),∂λcycle∂g2maybe negative. We denote the critical value of τ at which∂λcycle∂g2transitionsbetween negative and positive as τc. By setting equation (A.43) equal tozero and solving we find:τc =xλ2du11dg2− yλ2 du12dg2 + z(u11 − u12)dλ2dg2z(u11 − u12)dλ2dg2. (A.46)Next we simplify equation (A.43) by assuming that disturbance-inducedgermination rate is as high as possible (g1 = 1, because increasing g1 wasfound to increase λcycle above). We also set the germination rate in non-disturbance years to be high (g2 = 1) to see if decreasing the germina-tion rate from a high value will increase growth rate. Finally, we use thesmallest relevant cycle length (where there is one disturbance and one non-disturbance year, τ = 2) to find when∂λcycle∂g2is negative for very small τ .Written in terms of τc, equation (A.43) then becomes:∂λcycle∂g2∣∣∣∣g1=1,g2=1,τ=2=−(τc − 2)c1c3λ22(sA1sS2c3 + sS1c1c2)sS2c2((τc − 2)c1c2c4 + (c2 + 2c3)c1λ2 + c3s2A2) ,(A.47)which is simplified for presentation using the following positive quantities105A.3. Approximating the Cycle Matrix(assuming λ2 > 1):c1 = λ2 − sA2, (A.48a)c2 = 2λ2 − sA2, (A.48b)c3 = λ2 − sS2, (A.48c)c4 = 2λ2 − sS2. (A.48d)Equation (A.47) is negative when τc > 2 (assuming λ2 > 1). Thus, aslong as τc > 2,∂λcycle∂g2∣∣g1=1,g2=1will start out negative at τ = 2, favouringconditional germination. As the cycle length increases, the sign of∂λcycle∂g2will switch at τc and select against conditional germination at longer cyclelengths.Next, we evaluate whether increasing the life-history parameters from thedisturbance year (equation 11a) will increase or decrease τc by taking thederivative of τc with respect to b1, sA1, sS1 and g1. More general expressionscan be obtained (see supplementary Mathematica file, Wolfram ResearchInc. 2010) but here we present the case where g1 = 1 and g2 = 1 to seeif the region within which conditional germination strategies (g2 < 1) areexpected to evolve (τc) is increased or decreased by these parameters. Forb1 we find:∂τc∂b1∣∣∣∣g1=1,g2=1= −(sY λ22τ (sS1c1c2 + sA1sS2c3)λcycle2c1c22c3), (A.49)which is negative (assuming that λ2 > 1). Therefore, increasing the numberof seeds produced in disturbance years will decrease τc and thus decreasethe parameter space over which we expect conditional germination to befavoured. Similarly, for sA1 we find:∂τc∂sA1∣∣∣∣g1=1,g2=1= −(λ22τ+1(sS1c1c2 + sA1sS2c3)((τc − 2)c2c3 + sA2sS2 + 2c1λ2)sS2c22c3λcycle2(c1c2c3(τc − 2) + s2A2c3 + c1λ2(c2 + 2c3))),(A.50)which is negative (assuming that τc > 2 and λ2 > 1). Thus, increasing theadult survival rates during disturbances also decreases the parameter spaceover which conditional germination is expected to evolve. In contrast, the106A.3. Approximating the Cycle Matrixderivatives of τc with respect to g1 and sS1 are positive:∂τc∂g1∣∣∣∣g1=1,g2=1=sS1λ22τ−1(b1sS2sY (λ2 + sS2 + c1) + λ2(2sA1sS2 + c1(sA1 + sS1))sS2λcycle2c22,(A.51)∂τc∂sS1∣∣∣∣g1=1,g2=1=λ22τ (b1sS2sY c2 + sA1λ2(c1 + sS2))sS2λcycle2c22c3. (A.52)Hence, increasing germination rate and seedling survival in disturbance yearsincreases τc and therefore increases the parameter space (in terms of distur-bance cycle length, τ) over which conditional germination strategies (g2 < 1)have higher growth rates than maximal germination (g2 = 1).Note on the approximated cycle matrix, T˜cycleThe above approximation is most accurate when there is a large differencebetween eigenvalues λ1 and λ2 and when the number of non-disturbanceyears (τ − 1) is large. Care must thus be taken in interpreting the resultswhen the cycle length is short, which is when conditional germination strate-gies tend to be favoured. Therefore, our approximation serves as a guide,but is not quantitatively accurate, in cases with short disturbance cycles.For example, when g1 = 1, b1 = 0 and τ = 2, g2 should have no effect on thelong term growth rate because, in the disturbance year, all seeds germinateand no new seeds are produced, therefore there are no seeds in the seed bankin the subsequent year and g2 cannot affect growth rate. In contrast, usingour approximation, equation (A.43) can be negative at this point (see figureA.1).Figure A.1 shows that∂λcycle∂g2can have a different sign when using theapproximated (equation A.39) vs full transition matrix (equation 11). Forexample, when τ = 4, plants are able to germinate and produce seeds exactlytwice between disturbances so increasing g2 increases the long-term growthrate of the full system. In contrast, when τ is 3 or 5, increasing g2 willreduce the seed bank and increase the number of adults that experience adisturbance before reproducing. Exploring the parameter space numericallyindicates that τc is generally a good indicator of how short disturbance cyclesmust be in order for conditional germination to be favoured, but oscillationssuch as that observed in figure A.1 can cause some values of τ below (above)τc to select against (for) conditional germination.107A.3. Approximating the Cycle MatrixCycle Length, τ∂λcycle∂g2τc2 3 4 5 6 7 8 9−101234Figure A.1: An example of a discrepancy between the approximated transition matrix acrossa disturbance cycle (T˜cycle) and the full matrix. The solid line shows the derivative of λcyclewith respect to g2 taken from equation (A.43) (using the approximation). The points (squaresconnected by a dotted line) show the same derivative where λcycle is calculated from equation(2.12) (unapproximated). Both derivatives are evaluated where g2 = 1 and g1 = 1. The otherparameters used were b1 = 0, b2 = 2, sY = 0.6, sS1 = sS2 = 0.8, sA2 = 0.7 and sA1 = 0.τc is labelled with an arrow. Similar graphs may be explored numerically in the supplementaryMathematica file.108A.3. Approximating the Cycle MatrixAdult survival through disturbance, sA1Seeds produced in a disturbance year, b 10.01 0.1 1.00.010.11.02.0τ = 2τ = 3τ = 4τ = 5τ = 2τ = 3τ = 5Figure A.2: A version of Figure 2.3 that is drawn using the non-approximated transition matrixTcycle. Labelled red lines enclose the parameters for which conditional germination is expectedto evolve for various different cycle lengths (τ = 2, , 3 and 5). This full model includes no regionfor which conditional germination evolves when disturbances occur every four years (no line forτ = 4). Shaded areas represent the parameters for which conditional germination is expected toevolve in the approximated model (equation A.39), as shown in figure 3. Increasingly dark greyareas indicate where conditional germination is expected to evolve for cycle lengths of 2, 3, 4and 5 (lighter regions overlap darker regions). Other parameters are g1 = 1, sY 2 = sY 1 = 0.6,sS1 = sS2 = 0.9, b2 = 2 and sA2 = 0.7.109Appendix BFurther Analysis OfHaploid-Diploid Life CycleEvolutionWe consider four models: two continuous selection models and two discreteselection models with mutations occurring at either meiosis or gamete pro-duction. We allow selfing to occur among gametes at rate σ, following Ottoand Marks (1996). In the main text, we primarily discuss the continuousselection model with mutations at meiosis where σ = 0. We denote thegenotypes MA, Ma, mA and ma by indices 1 to 4, the frequency of these110Appendix B. Further Analysis Of Haploid-Diploid Life Cycle Evolutiongenotypes in the next generation x′1, x′2, x′3 and x′4) are given byx′1 = (1− µ)((1− σ)(x21w11,A + x1x2w12,A + x1x3w13,A + x1x4w14,A − rDw14,A)+σx1w11,A)/W(B.1a)x′2 =((1− σ)(x2x1w12,a + x22w22,a + x2x3w23,a + x2x4w24,a + rDw14,a)+σx2w22,a+µ((1−σ)(x21w11,Aµ + x1x2w12,Aµ + x1x3w13,Aµ + x1x4w14,Aµ − rDw14,Aµ)+σx1w11,Aµ))/W(B.1b)x′3 = (1− µ)((1− σ)(x3x1w13,A + x3x2w23,A + x23w33,A + x3x4w34,A − rDw14,A)+σx3w33,A)/W(B.1c)x′4 =((1− σ)(x4x1w14,a + x4x2w24,a + x4x3w34,a + x24w44,a + rDw14,a)+σx4w44,a+µ((1−σ)(x3x1w13,Aµ + x3x2w23,Aµ + x23w33,Aµ + x3x4w34,Aµ − rDw14,Aµ)+σx3w33,Aµ))/W(B.1d)where D = x1x4 − x2x3 and W is the sum of the numerators. The notationwij,k refers to the fitness of a zygote formed by gametes with indices i andj that produces a haploid of type k without mutation, wij,kµ is similarbut where the k haploid produced by meiosis mutates. These fitnesses forthe discrete and continuous selection models are given in table B.1. Whenmutations occur at gamete production, mutation does not affect fitness andwij,Aµ = wij,A. The fitness values where mutations occur at meiosis aregiven in table B.2.We then calculate the frequency of the a allele (qˆa) when the modifierlocus is fixed for a resident allele, M , which is given byqˆa =µw11,Aµw11,A − (1− σ)w12,a − σw22,a , (B.2)where we ignore terms on the order of µ2. For the continuous selectionmodel with mutations at meiosis and σ = 0, this is equivalent to equation111Appendix B. Further Analysis Of Haploid-Diploid Life Cycle EvolutionTable B.1: Fitnesses in discrete and continuous selection modelsFitness Continuous selection Discrete selectionw11,A wAA(tMM )wA(tMM ) wAAdMM + wA(1− dMM )w12,A wAa(tMM )wA(tMM ) wAadMM + wA(1− dMM )w12,a wAa(tMM )wa(tMM ) wAadMM + wa(1− dMM )w13,A wAA(tMm)wA(tMm) wAAdMm + wA(1− dMm)w14,A = w23,A wAa(tMm)wA(tMm) wAadMm + wA(1− dMm)w14,a = w23,a wAa(tMm)wa(tMm) wAadMm + wa(1− dMm)w22,a waa(tMM )wa(tMM ) waadMM + wa(1− dMM )w24,a waa(tMm)wa(tMm) waadMm + wa(1− dMm)w33,A wAA(tmm)wA(tmm) wAAdmm + wA(1− dmm)w34,A wAa(tmm)wA(tmm) wAadmm + wA(1− dmm)w34,a wAa(tmm)wa(tmm) wAadmm + wa(1− dmm)w44,a waa(tmm)wa(tmm) waadmm + wa(1− dmm)Table B.2: Fitnesses of mutated types when mutations occur at meiosisFitness Continuous selection Discrete selectionw11,Aµ wAA(tMM )wa(tMM ) wAAdMM + wa(1− dMM )w12,Aµ wAa(tMM )wa(tMM ) wAadMM + wa(1− dMM )w13,Aµ wAA(tMm)wa(tMm) wAAdMm + wa(1− dMm)w14,Aµ = w23,Aµ wAa(tMm)wa(tMm) wAadMm + wa(1− dMm)w33,Aµ wAA(tmm)wa(tmm) wAAdmm + wa(1− dmm)w34,Aµ wAa(tmm)wa(tmm) wAadmm + wa(1− dmm)(3.1). As in the main text, we then evaluate the spread of a rare modifierusing the leading eigenvalue (λl) of the system described by equations B.1cand B.1d. Full expressions of λl for each of the life cycles considered can befound in the supplementary Mathematica notebook.In the models in which mutations occur at gamete production, and as-suming that the fitnesses of A haploids and AA diploids are equal (such thatw11,A = w13,A = w33,A = 1), invasion occurs (λl > 1) if0 <σ(w22,a − w44,a)(w12,A − w14,A(1− r))+ r(1− σ)(w12,Aw14,a + w14,A(w12,a − 2w14,a)+ (w12,A − w14,A)(1− w14,a(1− σ)− w22,aσ).(B.3)Increased selfing can either increase or decrease the parameter range over112Appendix B. Further Analysis Of Haploid-Diploid Life Cycle Evolutionwhich this inequality is satisfied unless it is further assumed that the fitnessof a haploids and aa diploids are equal (such that w22,a = w44,a and the firstterm in B.3 is 0).When the fitnesses of haploids and homozygous diploids are equal andmutations occur at gamete production, Otto and Marks (1996) showed thathaploidy is always favoured over a larger parameter space when selfing ishigher in the discrete selection model. Similarly, in the continuous selectionmodel, where we also assume that modifiers have a small effect, tMm−tMM =δtMm is of order µ, modifiers that increase the length of the haploid phase(δtMm > 0) invade ifh(wAA(tMM )wA(tMM )− (1− σ)wAa(tMM )wa(tMM )− σwaa(tMM )wa(tMM ))> r(1− σ)(1− 2h)wa(tMM )wAA(tMM ).(B.4)This condition is always met when h > 1/2 and is always satisfied for agreater parameter range with higher selfing rates (higher σ) if h < 1/2.In the continuous selection model with mutations at meiosis, however,the impact of selfing is not so simple. Even when we assume the fitnesses ofhaploids and homozygous diploids is equal (sh = sd and σd = σh = 0) andmodifiers have a small effect (tmm−tMM = δtmm and tMm−tMM = hmδtmm,where δtmm is of order µ and terms of O(µ2) are discarded) and make thefurther assumption that recombination is free (r = 1/2), haploidy is favouredwhenh >1− (1− hm)(1− σ)(1 + σwa(tMM )wAa(tMM )/K1)2hm, (B.5)where K1 = wAA(tMM )wA(tMM ) − σwaa(tMM )wa(tMM ). For dominantmodifiers (hm = 1), this condition is satisfied if and only if h > 1/2, such thatselfing has no effect on whether haploidy or diploidy is favoured. When 0 <hm < 1, increased selfing increases the right hand side of inequality (B.5).Therefore, increased selfing decreases, rather than increases, the parameterrange under which haploidy is favoured. Although selfing can facilitate theevolution of haploidy when r < 1/2 (presumably because the impact ofdisequilibrium is greater), our overall finding is that when mutations occurat meiosis, selfing does not uniformly favour haploidy even when we assumethat the fitness of haploids and homozygous diploids are equal.In addition, the convergence properties of discrete and continuous se-lection models differ. For example, Hall (2000) found that, without selfing113Appendix B. Further Analysis Of Haploid-Diploid Life Cycle Evolutionor intrinsic fitness differences, haploid-diploid life cycles can evolve in thediscrete selection model where mutations occur at meiosis. However, in themain text we show that haploid-diploid life cycles do not evolve in the con-tinuous selection model where mutations occur at meiosis without intrinsicfitness differences. For the purposes of this study, one important distinctionbetween models is whether haploid-diploid life cycles evolve for recessivedeleterious mutations with selfing and loose linkage (σ > 0, r = 1/2). Infigure B.1, we show a numerical example of life cycle evolution with selfing,loose linkage, and sd = sh. For these parameters, haploid-diploid life cyclesevolve for low h in the discrete selection model but not in the continuous se-lection model (where mutations occur at gamete production in both cases).Thus in both the case considered by Hall (2000) (mutations at meiosis withno selfing) and in figure B.1 (mutations at gamete production with selfing),life-cycle models in which selection occurs continously (figure 3.1b) favourhaploid-diploid life cycles less often than discrete life cycle models (figure3.1a)Finally, we clarify how selfing affects the disequilibrium between the Mand A loci, which was discussed in Otto and Marks (1996). Using the samemodel and assumptions as Otto and Marks (1996), where wAA = wA = 1,wAa = 1 − hs, and wa = waa = 1 − s we find that the disequilibrium,D = x1x4 − x2x3 during invasion of a modifier is given byD =(dMm − dmm)(1− h)µ(1− σ)K5(1− dMM (1− h)(1− σ)) (B.6)where K5 = r(1− σ) + s(1− dMm)(1− h)(1− r) + hs(1− r)(1− σ) + σs isstrictly positive. Thus, disequilibrium has the same sign as (dMm − dMM )and is positive for modifiers that increase the the diploid phase (modifiersassociated with the less fit allele) and negative for modifiers that increasethe haploid phase, as found by Otto and Marks (1996). However, the mag-nitude of this disequilibrium decreases with increasing selfing, contrary tothe result stated in Otto and Marks (1996). In the supplementary Mathe-matica file we show that the magnitude of the disequilibrium increases withincreasing selfing if qˆa is held constant but because selfing also helps purgingand reduces qˆa, the net effect on disequilibrium is opposite.114Appendix B. Further Analysis Of Haploid-Diploid Life Cycle Evolution-0.1 -0.05 0.05 0.1ÈSd È0.250.50.751.1.25h-0.1 -0.05 0.05 0.1ÈSd È0.250.50.751.1.25hFigure B.1: Here we plot whether haplont, diplont, or haploid-diploid life cycles are favouredwhen there is selfing among gametes as a function of the intrinsic fitness of diploids (Sd) for(a) the discrete selection model with mutations at gamete production and (b) the continuousselection model with mutations at gamete production. To evaluate expected life cycle evolutionwe evaluated the stability of pure haplont (dMM = 0, tMM = 1) or diplont (dMM = 1, tMM = 0)strategies using equation (3.5) with the full expression of λl where terms on the order of µ2 arediscarded, which can be found in the supplementary Mathematica file. In both plots σ = 0.4,r = 1/2, sd = sh = −0.3, U = 0.1, L = 1000, Sh = 0, and modifiers have a small and dominanteffect (tmm = tMm, |tMm − tMM | = 1/10, 000, dmm = dMm, |dMm − dMM | = 1/10, 000).115Appendix CEvolution of RecombinationRate on Sex ChromosomesC.1 Recursion EquationsIn each generation we census the genotype frequencies in male and femalehaploids before haploid selection, e.g., sperm/pollen and eggs/ovules. Be-fore haploid selection, the frequency of X-bearing male and female haploidsare given by Xmi and Xfi and the frequency of Y-bearing haploids is givenby Y mi where the index i specifies genotypes MA, Ma, mA, and ma. Se-lection then occurs among male haploids according to the A locus allele,k, carried by individuals with genotype i. Assuming that the fraction ofX-bearing haploids produced by males is f , the genotype frequencies afterhaploid selection are Xm,si = fwkXmi /w¯H and Ym,si = (1 − f)wkY mi /w¯H ,where w¯H =∑4i=1 fwkXmi + (1− f)wkY mi is the mean fitness of male hap-loids. Random mating then occurs between gametes to produce diploidfemales with genotype ij at frequency xij = Xfi Xm,sj and diploid malesat frequency yij = Xfi Ym,sj . In females, individuals with genotype ijare equivalent to those with genotype ji. For simplicity we denote thefrequency of genotype ij in females to the average of these frequencies,xij = (Xfi Xm,sj + Xfj Xm,si )/2. Note that the sex ratio before diploid se-lection depends both on the production of X-bearing haploids by fathers(f) and on haploid selection (wk). However, f does not enter into any re-sults, indicating that the main force driving recombination evolution is notto balance the current sex ratio.Table C.1: Fitness of different genotypes.Genotype A a AA Aa aaFitness in males wA wa wmAA wmAa wmaaFitness in females 1 1 wfAA wfAa wfaa116C.1. Recursion EquationsTable C.2: Marginal fitnesses of YA and Xa haplotypesw¯mYA = (wA(pXfwfAA + (1− pXf )wfAa))w¯mat,mXa = pY mwAwmAa + (1− pY m)wawmaaw¯pat,fXa = pXfwawfAa + (1− pXf )wawfaaw¯mat,fXa = pXmwAwfAa + (1− pXm)wawfaaSelection among diploids then occurs according to the diploid geno-type at the A locus, k, for an individual of type ij (see Table C.1).The diploid frequencies after selection are given by xsij = wfkxij/w¯f in fe-males and ysij = wmk yij/w¯m in males, where w¯f =∑4i=1∑4j=1wfkxij andw¯m =∑4i=1∑4j=1wmk yij are the mean fitnesses of females and males, re-spectively. Finally, these diploids undergo meiosis to produce the next gen-eration. The haplotype frequencies in the next generation of eggs/ovules isgiven by:Xf′MA = 4∑j=1xs1j−Rf (xs14 − xs23) (C.1a)Xf′Ma = 4∑j=1xs2j+Rf (xs14 − xs23) (C.1b)Xf′mA = 4∑j=1xs3j+Rf (xs14 − xs23) (C.1c)Xf′ma = 4∑j=1xs4j−Rf (xs14 − xs23) (C.1d)which only involve the recombination rate between the A locus and theM locus in females (Rf ). In males, recombination between the SDR andthe A locus or the M also affects the frequencies of haplotypes produced.The frequency of haplotypes among X-bearing sperm/pollen (before haploid117C.1. Recursion Equationsselection) in the next generation are given byXm′MA = 4∑j=1ys1j− rMM (ys12 − ys21)− (Rm + rMm − 2χ)(ys13 − ys31)− (Rm + rMm − χ)ys14+ (rMm − χ)ys41 + χ ys23 + (rMm − χ)ys32(C.2a)Xm′Ma = 4∑j=1ys2j− rMM (ys21 − ys12)− (Rm + rMm − 2χ)(ys24 − ys42)− (Rm + rMm − χ)ys23+ (rMm − χ)ys32 + χ ys14 + (rMm − χ)ys41(C.2b)Xm′mA = 4∑j=1ys3j− rmm(ys34 − ys43)− (Rm + rMm − 2χ)(ys31 − ys13)− (Rm + rMm − χ)ys32+ (rMm − χ)ys23 + χ ys41 + (rMm − χ)ys14(C.2c)Xm′ma = 4∑j=1ys4j− rmm(ys43 − ys34)− (Rm + rMm − 2χ)(ys42 − ys24)− (Rm + rMm − χ)ys41+ (rMm − χ)ys14 + χ ys32 + (rMm − χ)ys23(C.2d)and the frequencies of Y-bearing sperm/pollen haplotypes (before haploidselection) are given byY m′MA = 4∑j=1ys1j− rMM (ys21 − ys12)− (Rm + rMm − 2χ)(ys31 − ys13)− (Rm + rMm − χ)ys41+ (rMm − χ)ys14 + χ ys32 + (rMm − χ)ys23(C.3a)118C.2. Invasion of Recombination ModifiersY m′Ma = 4∑j=1ys2j− rMM (ys12 − ys21)− (Rm + rMm − 2χ)(ys42 − ys24)− (Rm + rMm − χ)ys32+ (rMm − χ)ys23 + χ ys41 + (rMm − χ)ys14(C.3b)Y m′mA = 4∑j=1ys3j− rmm(ys43 − ys34)− (Rm + rMm − 2χ)(ys13 − ys31)− (Rm + rMm − χ)ys23+ (rMm − χ)ys32 + χ ys14 + (rMm − χ)ys41(C.3c)Y m′ma = 4∑j=1ys4j− rmm(ys34 − ys43)− (Rm + rMm − 2χ)(ys24 − ys42)− (Rm + rMm − χ)ys14+ (rMm − χ)ys41 + χ ys23 + (rMm − χ)ys32(C.3d)C.2 Invasion of Recombination ModifiersInvasion of modifiers that create a strong linkage between the X and a alleleis determined by the largest solution to the characteristic polynomialλXa2 − λXaw¯mat,fXa /w¯f − (w¯pat,fXa /w¯f )(w¯mat,mXa /w¯m) = 0. (C.4)This can be solved for λXa if we assume that the selected locus is initiallyloosely linked to the SDR (rMM ) and that there are no sex differences inselection (wmij = wfij = wij). The equilibrium frequency of the A allele whenmaintained at a polymorphic equilibrium by selection is thenpˆXm = pˆY m = pˆXf =2wawaa − wAa(wA + wa)2 (wA(wAA − wAa) + wa(waa − wAa)) . (C.5)119C.2. Invasion of Recombination ModifiersThis equilibrium is valid and stable whenwAa(wA + wa) > 2wAwAA andwAa(wA + wa) > 2wawaa.(C.6)Therefore, a polymorphism can be maintained either if there is heterozygoteadvantage in diploids (wAa > waa and wAa > wAA) or if there is antagonisticselection between haploids and diploids (e.g., wA > wa and waa > wAa >wAA) or a combination of both (Immler et al. 2012).After this equilibrium is reached, the invasion of a modifier that bringsthe A allele into linkage with the Y is given byλY A = 1 +(wA − wa)wAa(wA + wa)(wAa(wA + wa)− 2wAAwA)(wA + wa)(wAa2(wA + wa)2 − 4wAwAAwawaa) , (C.7)where λY A > 1 indicates that the modifier increases in frequency. Giventhat a polymorphism at the A locus is initially stable (conditions C.6 aremet) the sign of λY A− 1 depends on the sign of wA−wa. That is, modifiersthat bring the allele favoured in haploids (e.g., A when wA > wa) into tightlinkage with the Y will spread.Similarly, condition (4.2) for the invasion of modifiers that bring the aallele into tight linkage with the X chromosome is satisfied if(wA − wa)wAa(wA + wa)(wAa(wA + wa)− 2wAAwA)2(wA + wa)(wAa(wA + wa)− wAwAA − wawaa) > 0, (C.8)which requires wA > wa, given that conditions (C.6) are met. These resultsindicate that recombination modifiers invade if they bring the X into tightlinkage with the allele that is less fit during haploid selection, even withoutthe weak selection assumptions in equation (4.4) and without sex differencesin selection in the diploid phase.In the main text and above, we consider the invasion of recombina-tion suppressors that bring the a allele into tight linkage with the X whenthe A locus is initially loosely linked to the SDR (rMM = 1/2) such thatpˆXm = pˆY m. Here, we consider cases where rMM < 1/2 and define the differ-ence in the frequency of the A allele between X- and Y-bearing pollen/spermas δXY = pˆY m − pˆXm. We assume that selection is weak relative to recom-bination such that δ, δXY , and δH are all small (of order 2). Invasion is120C.3. Invasion of Modifiers that Increase Recombination from an Initially Low Levelthen given byλ′Xa = λXa(1− (1− 2 rMM )(3 + 2wfAa/w¯f ))+wfAaδXY3w¯f(C.9)Under the conditions where λXa > 1, we would expect that the a alleleis associated with the X such that δXY < 0. Thus, (C.9) indicates thatselection in favour of modifiers that suppress recombination is less strongwhen rMM < 1/2 (λ′Xa < λXa), in which case intralocus conflicts are initiallypartially resolved by reduced recombination.C.3 Invasion of Modifiers that IncreaseRecombination from an Initially Low LevelWe consider a population in which linkage is tight between the A locusand the SDR (rMM is of order , where the M allele is initially fixed).Recombination has no effect if the A locus is fixed for one allele, we thereforefocus on the five equilibria that maintain both A and a alleles, of which fourare given to leading order by:(A) pˆY m = 0, pˆXf =αα+ β, pˆXm =wmAaαwmAaα+ wmaaβ(A′) pˆY m = 1, pˆXf = 1− α′α′ + β′, pˆXm = 1− wmAaα′wmAaα′ + wmaaβ′(B) pˆY m = 0, pˆXf = 1, pˆXm = 1(B′) pˆY m = 1, pˆXf = 0, pˆXm = 0α =wfAa(wmaawa + wmAawA)− 2wfaawmaawaα′ =wfAa(wmAAwA + wmAawa)− 2wfAAwmAAwAβ =wfAa(wmaawa + wmAawA)− 2wfAAwmAawAβ′ =wfAa(wmAAwA + wmAawa)− 2wfaawmAawaA fifth equilibrium (C) also exists where A is present at an intermediatefrequency on the Y chromosome (0 < pˆY < 1). However, equilibrium (C) isnever locally stable when rMM ≈ 0 and is therefore not considered further.Thus, the Y can either be fixed for the a allele (equilibria A and B) or theA allele (equilibria A′ and B′). The X chromosome can then either be poly-morphic (equilibria A and A′) or fixed for the alternative allele (equilibria121C.3. Invasion of Modifiers that Increase Recombination from an Initially Low LevelB and B′). Since equilibria (A) and (B) are equivalent to equilibria (A′)and (B′) with the labelling of A and a alleles interchanged, we discuss onlyequilibria (A′) and (B′), in which the YA haplotype is favoured (as in theprevious section), without loss of generality.We next calculate when (A′) and (B′) are locally stable for rMM = 0.According to the ‘small parameter theory’ (Karlin and McGregor 1972a;b),these stability properties are unaffected by small amounts of recombinationbetween the SDR and A locus, although equilibrium frequencies may beslightly altered. For the A allele to be stably fixed on the Y requires thatw¯mYA > w¯mY a, where the marginal fitnesses of YA and Ya haplotypes are w¯mYA(as above) and w¯mY a = wmAapXf + wmaa(1 − pXf ), respectively. SubstitutingpˆXf from above, fixation of the A allele on the Y requires that γi > 0 whereγ(A′) = wA(wmAaα′ + wmAAβ′) − wa(wmaaα′ + wmAaβ′) for equilibrium (A′) andγ(B′) = wmAawA−wmaawa for equilibrium (B′). Stability of a polymorphism onthe X chromosome (equilibrium A′) further requires that α′ > 0 and β′ > 0.Fixation of the a allele on the X (equilibrium B′) is mutually exclusive with(A′) and requires that β′ < 0. We will assume that these conditions are metsuch that population has reached a stable equilibrium at the A locus whenconsidering evolution at the modifier locus.To consider recombination rate evolution, we evaluate whether a mutantallele, m, can invade if it modifies the recombination rate between A andthe SDR by a small amount (|rmm−rMM | and |rMm−rMM | are of order ).As above, we use the leading eigenvalue, λ, from a local stability analysisto evaluate the spread of a rare mutant modifier, where now λi determinesinvasion into a population at equilibrium i. Firstly, because stability ofequilibrium (A′) requires that α′ > 0 and β′ > 0 and all fitnesses must benon-negative, we can define the following series of κ terms, which must bepositive when (A′) is locally stable.122C.3. Invasion of Modifiers that Increase Recombination from an Initially Low Levelκ1 = wfaaα′ + wfAaβ′κ2 = wfAaα′ + wfAAβ′κ3 = wmAaα′ + wmAAβ′κ4 = wfaaα′ + wfAAβ′κ5 = wmAawa + wmAAwAκ6 = wmAawawmAAwAκ7 = wfaawmAawaα′ + wfAAwmAAwAβ′κ8 = wmaaα′α′ + 2wmAaα′β′ + wmAAβ′β′κ9 = wmAawaα′ + wmAAwAβ′κ10 = wfAaκ9 + 2κ6κ4/κ5These are useful in determining the magnitude of λ(A′), which determinesinvasion of modifiers and is given byλ(A′) = 1 + (rMm − rMM )wmAaα′K1waRm(wmaaα′ + wmAaβ′)K2(C.10)where we neglect terms of order 2 and higher and K2 is strictly positive,K2 =Rf2wfAaκ3κ5(α′ + β′)κ10 +RfRmwmAawmAA2wawAK3κ3κ4/κ5+RmwmAawmAA(1− 2Rf )(waβ′κ1(2wmAAwAκ2 + κ10) + wAα′κ2(2wmAawaκ1 + κ10))such that λ(A′) > 1 if and only if (rMm − rMM )K1 > 0, whereK1 =− (1− 2Rf )Rmγ(A′)κ1κ2κ6 −RmRfγ(A′)κ4κ6(κ7/κ5 + wfAa(α′ + β′)/2)−Rfγ(A′)wfAawaκ1κ3κ5+RfwfAawmAa(γ(A′)α′ +Rmwaκ8)((wmAa − wmAA)wawAκ4 + (wA − wa)wfAaκ5(α′ + β′)/2)Modifiers that increase recombination (rMm − rMM > 0) therefore onlyspread if K1 > 0. Only the last term of K1 can be positive, and this termcan only be positive if either wmAa > wmAA or wA > wa. Thus, for increasedrecombination to be favoured by selection (K1 > 0), heterozygous malesmust be more fit that males homozygous for the allele fixed on the Y and/orthe allele fixed on the Y must be favoured during haploid selection. Since theA allele is fixed on the Y, wmAa > wmAA implies that X chromosomes bearing123C.3. Invasion of Modifiers that Increase Recombination from an Initially Low Levelthe a allele are favoured during selection in males. If a polymorphism ismaintained on the X (equilibrium A′), counter-selection must favour the Aallele during haploid selection and/or selection in females when wmAa > wmAA.In addition, when linkage between the modifier locus and the selected locusis tight (at least in females, Rf = 0), K1 is always negative and increasedrecombination is never favoured.We next consider the invasion of a recombination modifier into a popu-lation at equilibrium (B′). Local stability of this equilibrium requires that(−β′) > 0 and γ(B′) > 0. Ignoring terms of order 2 and higher,λ(B′) = 1+(rMm − rMM )K44(γ(B′) +Rmwmaawa)((−β′) + wfAa(RfwmAawa +RmwmAAwA(1−Rf ))whereK4 =− 2γ(B′)(−β′)− (2Rf +Rm(1−Rf ))wfAawmAAwAγ(B′)−Rm(−β′)wmaawa+Rf (wA − wa)wfAawmAa(2γ(B′) +Rmwmaawa)+RfRm(wmAa − wmAA)wfAawmAawawATherefore λ(B′) > 1 if and only if (rMm − rMM )K4 > 0. The only terms inK4 that can be positive again involve the factors (wA−wa) and (wmAa−wmAA),such that either wmAa > wmAA or wa > wA are again necessary (but not suffi-cient) conditions for the invasion of modifiers that increase recombination.Finally, we re-write the condition K4 > 0 to obtainwfaa <wfAa(1− γ(B′)Rf (2−Rm)Rm)− γ(B′)(wmAa − wmAA)K5 + (wA − wa)K6)/K7(C.11)where the following terms are positiveK5 =(1−Rf )(2γ(B′)(1−Rm) +RmwmAawa)/wmAaK6 =(RfRmwAwmAa2 + (wmAA(1−Rf ) +RfwmAa)(2γ(B′)(1−Rm) + wmAawARm))K7 =4γ(B′) + 2wmaawaRmThus, if haploid selection favours the A allele, then condition (C.11) can bemet whether selection among diploid females favours allele A or a (wfaa <124C.3. Invasion of Modifiers that Increase Recombination from an Initially Low LevelwfAa or wfaa > wfAa). However, if haploid selection favours the a allele (wa >wA), the evolution of increased recombination requires that wmAa > wmAA (seeabove), and equation (C.11) shows that selection must favour the A alleleduring selection in females (wfaa < wfAa). Thus, increased recombination isonly favoured if the A allele is favoured during selection in females (wfaa <wfAa) and/or the A allele is favoured during haploid selection (wA > wa).Only under these conditions is it possible for recombination between the XAand Ya to produce XA gametes that are favoured over the short term (indaughters and/or gametes/gametophytes, respectively).One might not expect selection to favour XA haplotypes because an Aallele on an average X background should either have the same fitness asan a allele (when a polymorphism is maintained, equilibrium A′) or lowerfitness (when A is fixed, equilibrium B′). However, an XA haplotype createdby recombination in males is found in a male haploid (pollen or sperm),not on an average X background (which is weighted across X-bearing malesperm/pollen and female eggs/ovules). Increased recombination does notevolve if Rf and Rm are small because the modifier remains linked to thehaplotypes it creates, which will eventually be found on all backgrounds.However, when Rf and Rm are sufficiently large, modifiers that increaserecombination can gain a transient fitness advantage. XA pollen/spermhaplotypes can gain a transient fitness advantage during haploid selectionand/or selection in females. The evolution of increased recombination isonly consistent with this form of selection.125C.3. Invasion of Modifiers that Increase Recombination from an Initially Low Level0. 0.5 1. 1.50.0.51.1.5waawAAwa=10. 0.5 1. 1.50.0.51.1.5waawa=0.750. 0.5 1. 1.50.0.51.1.5waawa=0.50. 0.5 1. 1.50.0.51.1.5waamwAAmwa=10. 0.5 1. 1.50.0.51.1.5waamwa=0.950. 0.5 1. 1.50.0.51.1.5waamwa=0.90. 0.5 1. 1.50.0.51.1.5waamwAAmwa=10. 0.5 1. 1.50.0.51.1.5waamwa=0.90. 0.5 1. 1.50.0.51.1.5waamwa=0.8126C.3. Invasion of Modifiers that Increase Recombination from an Initially Low LevelFigure C.1 (preceding page): Selection can favour increased recombination between the sex-determining region (SDR) and a selected locus that is closely linked to the SDR (rij ≈ 0), evenwhen selection in males is not overdominant. The grey regions show where one or more of thepolymorphic equilibria are stable and thus recombination modifiers can affect fitness. Colouredregions show where increased recombination is favoured in a population at equilibrium (A) in blue,(B) in green, (A′) in red, and (B′) in orange. Since this model is symmetrical, red/orange regionscan be exchanged with blue/green regions if the labelling of A and a alleles is switched. Acrosscolumns we vary the fitness of a-bearing haploids relative to the A-bearing haploids (wA = 1).Grey lines show the fitness of heterozygous diploids wkij = 1. In the first row, there are no dif-ferences in selection between male and female diploids (wfij = wmij = wij), where waa and wAAare varied along the x and y axes, respectively. As haploid selection becomes stronger, increasedrecombination can evolve with weaker overdominance in diploids and also with ploidally antag-onistic selection (waa > 1 > wAA). In the second and third rows, we consider sex differencesin selection, where wmaa and wmAA are varied along the x and y axes (wmAa = 1). In the secondrow, where selection in females is overdominant (wfAA = 0.75, wfAa = 1, wfaa = 0.75), increasedrecombination can be favoured when selection is directional (or underdominant) in males andhaploid selection is moderately strong. In the third row, selection favours the A allele in females(wfAA = 1.05, wfAa = 1, wfaa = 0.75) and increased recombination can also be favoured withsexually antagonistic selection (wmAA < 1 < wmaa). For this plot, we assume that the modifier ofrecombination is unlinked (Rf = Rm = 1/2).127

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