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Adaptive challenges : fitness-valley crossing and evolutionary rescue Osmond, Matthew M 2018

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Adaptive challenges: fitness-valley crossing andevolutionary rescuebyMatthew M OsmondB.Sc. Biology & Mathematics, Queen’s University, 2008M.Sc. Biology, McGill University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Zoology)The University of British Columbia(Vancouver)August 2018c©Matthew M Osmond, 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Adaptive challenges: fitness-valley crossing and evolutionary rescuesubmitted by Matthew M Osmond in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Zoology.Examining Committee:Sarah Otto, ZoologySupervisorAmy AngertSupervisory Committee MemberJonathan DaviesUniversity ExaminerDolph SchluterUniversity ExaminerAdditional Supervisory Committee Members:Michael DoebeliSupervisory Committee MemberMichael WhitlockSupervisory Committee MemberiiAbstractOne of the most striking features of the natural world is the fit of an organism to itssurroundings. Much of this fit, i.e., adaptation, arises from evolution by natural se-lection. Adaptation is thus often thought to be a sure thing; eventually a beneficialallele will arise and/or increase in frequency, ad infinitum. But sometimes adap-tation is more challenging and evolution by natural selection is not a sure thing.This thesis deals with one such type of adaptive challenge, adaptation that requiresthe prolonged persistence of genotypes that are expected to be declining in num-ber. The first example is fitness-valley crossing, where adaptation is the result ofmultiple components that are each selected against when alone but are beneficialin combination. Chapter 2 extends the mathematical framework describing suchsituations to include biased transmission of traits from one generation to the next.The analysis shows that meiotic drive, uniparental inheritance, and cultural inheri-tance can greatly facilitate peak shifts across a valley of low fitness. Chapters 3-5deal with a second example, evolutionary rescue, where declining populations arerescued from extinction by rapid adaptation. Two of the mathematical models ana-lyze how species interactions (predation) and alternative selective surfaces (fitnessfunctions), respectively, affect the ability of a focal population to adapt and persistin a gradually changing world. They find that predators can counterintuitively helpprey persist (e.g., through an ‘evolutionary hydra effect’) and that weakening selec-tion (i.e., antagonistic epistasis) can produce unexpected extinctions (‘evolutionarytipping points’). The final model explores evolutionary rescue following an abruptenvironmental change on a fitness landscape. The analysis shows that rescue canbe more likely by two mutations than one and that the number of mutations thatrescue takes leaves a signature in the distribution of fitness effects among survivors.iiiLay SummaryOne striking feature of the natural world is the fit of an organism to its surround-ings. Much of this fit – adaptation – arises from evolution by natural selection.Sometimes evolution by natural selection is relatively simple; e.g., a beneficialmutation arises and increases in frequency over subsequent generations. But some-times adaptation is more challenging and evolution by natural selection is not as-sured. This thesis deals with one such adaptive challenge, adaptation that requiresthe prolonged persistence of genotypes that are expected to be declining in num-ber. The first example is fitness-valley crossing (Chapter 2), where adaptation isthe result of multiple components that are each deleterious when alone but bene-ficial in combination. The second example is evolutionary rescue (Chapters 3-5),where declining populations are rescued from extinction by rapid adaptation. I usemathematical models to explore when these adaptive challenges are likely to be beovercome – the findings are sometimes counterintuitive.ivPrefaceA version of Chapter 2 has been published [Osmond, M. M., and S. P. Otto. 2015.Fitness-valley crossing with generalized parent–offspring transmission. Theoret-ical Population Biology 105:1–16]. The project was conceived by S.P. Otto andtogether we built and analyzed the mathematical models. I wrote the manuscriptand S.P. Otto contributed to manuscript edits.A version of Chapter 3 has been published [Osmond, M. M., S. P. Otto, andC. A. Klausmeier. 2017. When predators help prey adapt and persist in a changingenvironment. The American Naturalist 190:83–98]. The project was conceivedtogether with C.A. Klausmeier. I built and analyzed the mathematical models withinput from S.P. Otto and C.A. Klausmeier. I wrote the manuscript and S.P. Ottoand C.A. Klausmeier contributed to manuscript edits.A version of Chapter 4 has been published [Osmond, M. M., and C. A. Klaus-meier. 2017. An evolutionary tipping point in a changing environment. Evolution71:2930–2941]. The project was conceived together with C.A. Klausmeier. I builtand analyzed the mathematical models with input from C.A. Klausmeier. I wrotethe manuscript and C.A. Klausmeier contributed to manuscript edits.Chapter 5 is being prepared for submission with coauthors Guillaume Martin,Ophe´lie Ronce, and Sarah Otto. The project was conceived by me and I am thelead investigator. Guillaume Martin, Ophe´lie Ronce, and Sarah Otto have helpedwith concept formation. Guillaume Martin and Sarah Otto have helped with math-ematical analysis. Sarah Otto has contributed to manuscript edits.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Variations on a theme . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The genetic basis of adaptation in mathematical models of evolution 21.2.1 A brief early history . . . . . . . . . . . . . . . . . . . . 21.2.2 One and two locus models . . . . . . . . . . . . . . . . . 31.2.3 Many-locus models of quantitative traits . . . . . . . . . . 41.2.4 The infinite sites model . . . . . . . . . . . . . . . . . . . 71.3 An overarching theme: subcritical adaptation . . . . . . . . . . . 71.3.1 Defining subcritical . . . . . . . . . . . . . . . . . . . . . 71.3.2 Defining adaptation . . . . . . . . . . . . . . . . . . . . . 81.3.3 Two possible types of subcritical adaptation . . . . . . . . 91.3.4 Subcritical intermediates: fitness valleys . . . . . . . . . . 9vi1.3.5 Subcritical progenitors: evolutionary rescue . . . . . . . . 101.4 Goals of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 Chapter 2. Fitness-valley crossing with biased transmission 111.4.2 Chapter 3. Evolutionary rescue: when predators help . . . 121.4.3 Chapter 4. Evolutionary rescue: alternative fitness func-tions cause evolutionary tipping points . . . . . . . . . . . 121.4.4 Chapter 5. Evolutionary rescue: a theory for its genetic basis 131.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Fitness-valley crossing with biased transmission . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Model and Results . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Semi-deterministic analysis . . . . . . . . . . . . . . . . 212.2.2 Stochastic analysis . . . . . . . . . . . . . . . . . . . . . 272.2.3 Three scenarios . . . . . . . . . . . . . . . . . . . . . . . 312.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Evolutionary rescue: when predators help . . . . . . . . . . . . . . . 483.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . 503.2.1 A general model of generalist predation . . . . . . . . . . 503.2.2 Generalist example 1: selective push . . . . . . . . . . . . 573.2.3 Generalist example 2: evolutionary hydra effect . . . . . . 593.2.4 Specialist predation . . . . . . . . . . . . . . . . . . . . . 623.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.1 Selective push . . . . . . . . . . . . . . . . . . . . . . . 683.3.2 Evolutionary hydra effect . . . . . . . . . . . . . . . . . . 683.3.3 Specialist predators and coevolution . . . . . . . . . . . . 703.3.4 Empirical work . . . . . . . . . . . . . . . . . . . . . . . 733.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Evolutionary rescue: alternative fitness functions cause evolutionarytipping points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75vii4.2 Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . 784.2.1 A general model . . . . . . . . . . . . . . . . . . . . . . 784.2.2 The traditional fitness function . . . . . . . . . . . . . . . 794.2.3 An alternative fitness function . . . . . . . . . . . . . . . 804.2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915 Evolutionary rescue: a theory for its genetic basis . . . . . . . . . . 965.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.1 Fisher’s geometric model . . . . . . . . . . . . . . . . . . 995.2.2 Wildtype dynamics . . . . . . . . . . . . . . . . . . . . . 1015.2.3 Mutant lineage dynamics . . . . . . . . . . . . . . . . . . 1025.2.4 Probability of establishment . . . . . . . . . . . . . . . . 1035.2.5 Individual-based simulations . . . . . . . . . . . . . . . . 1045.2.6 Data Availability . . . . . . . . . . . . . . . . . . . . . . 1045.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.1 The probability of 1-step rescue . . . . . . . . . . . . . . 1055.3.2 The DFE following 1-step rescue . . . . . . . . . . . . . 1075.3.3 The probability of 2-step rescue . . . . . . . . . . . . . . 1085.3.4 1-step vs. 2-step rescue . . . . . . . . . . . . . . . . . . . 1125.3.5 The DFE following 2-step rescue . . . . . . . . . . . . . 1125.3.6 k-step rescue . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.7 Inferring the genetic basis of rescue from the DFE . . . . 1165.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.4.1 The number of mutational steps evolutionary rescue takes 1205.4.2 The distribution of fitness effects following rescue . . . . 1235.4.3 Caveats and extensions . . . . . . . . . . . . . . . . . . . 1245.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1266 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2 Chapter-by-chapter conclusions and consequences . . . . . . . . . 128viii6.2.1 Chapter 2. Fitness-valley crossing with biased transmission 1286.2.2 Chapter 3. Evolutionary rescue: when predators help . . . 1296.2.3 Chapter 4. Evolutionary rescue: alternative fitness func-tions cause evolutionary tipping points . . . . . . . . . . . 1316.2.4 Chapter 5. Evolutionary rescue: a theory for its genetic basis1326.3 General conclusions and future directions . . . . . . . . . . . . . 1336.3.1 Subcritical adaptation . . . . . . . . . . . . . . . . . . . . 1336.3.2 The genetic basis of subcritical adaptation . . . . . . . . . 1356.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A Appendices for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 161A.1 Dynamic single mutants . . . . . . . . . . . . . . . . . . . . . . 161A.2 Diffusion approximation . . . . . . . . . . . . . . . . . . . . . . 163A.3 Stochastic crossing times . . . . . . . . . . . . . . . . . . . . . . 168A.4 Stochastic simulations . . . . . . . . . . . . . . . . . . . . . . . 171B Appendices for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 172B.1 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 172C Appendices for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 174C.1 Sufficient conditions for the existence of an evolutionary tippingpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174D Appendices for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 177D.1 Supplementary figures . . . . . . . . . . . . . . . . . . . . . . . 177ixList of TablesTable 2.1 List of parameters . . . . . . . . . . . . . . . . . . . . . . . . 22Table 2.2 Segregation distortion transmission probabilities . . . . . . . . 32Table 2.3 Cytonuclear inheritance transmission probabilities . . . . . . . 37Table 2.4 Cultural inheritance transmission probabilities . . . . . . . . . 41xList of FiguresFigure 2.1 Crossing time with segregation distortion . . . . . . . . . . . 33Figure 2.2 Probability of crossing with segregation distortion . . . . . . 35Figure 2.3 Crossing time with cytonuclear inheritance . . . . . . . . . . 38Figure 2.4 Probability of crossing with cytonuclear inheritance . . . . . . 40Figure 2.5 Crossing time with cultural inheritance . . . . . . . . . . . . 43Figure 2.6 Probability of crossing with cultural inheritance . . . . . . . . 44Figure 3.1 Interaction pathways . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.2 The selective push . . . . . . . . . . . . . . . . . . . . . . . 60Figure 3.3 Selection pressures with a generalist predator . . . . . . . . . 61Figure 3.4 The evolutionary hydra effect . . . . . . . . . . . . . . . . . 63Figure 3.5 Coevolution with a specialist predator . . . . . . . . . . . . . 66Figure 3.6 Generalist vs. specialist . . . . . . . . . . . . . . . . . . . . . 71Figure 4.1 Visual overview of the modelling approach . . . . . . . . . . 82Figure 4.2 Simulation results summary . . . . . . . . . . . . . . . . . . 86Figure 4.3 Generic early-warning signs of tipping points . . . . . . . . . 88Figure 4.4 Evolutionary hysteresis . . . . . . . . . . . . . . . . . . . . . 92Figure 5.1 Genetic paths to evolutionary rescue . . . . . . . . . . . . . . 106Figure 5.2 DFE following 1-step rescue . . . . . . . . . . . . . . . . . . 108Figure 5.3 The probability of rescue by 1 or 2 steps . . . . . . . . . . . . 113Figure 5.4 The probability of 2-step relative to 1-step rescue . . . . . . . 114Figure 5.5 Inferring the genetic path to rescue from the DFE . . . . . . . 117xiFigure A.1 Crossing time approximations . . . . . . . . . . . . . . . . . 164Figure D.1 Mutational distribution in FGM . . . . . . . . . . . . . . . . 177Figure D.2 The probability of 1-step rescue . . . . . . . . . . . . . . . . 178Figure D.3 Comparison with Anciaux et al. (2018) . . . . . . . . . . . . 179Figure D.4 The distribution of mutation size . . . . . . . . . . . . . . . . 180Figure D.5 The probability of 2-step critical rescue . . . . . . . . . . . . 181Figure D.6 Probability of 2-step critical and subcritical rescue . . . . . . 182Figure D.7 DFE following 2-step critical rescue . . . . . . . . . . . . . . 183Figure D.8 DFE following 2-step subcritical rescue . . . . . . . . . . . . 184Figure D.9 Growth rates of potential rescue mutants . . . . . . . . . . . . 184Figure D.10 Growth rates of 1-step rescue mutants in ascending order . . . 185xiiAcknowledgmentsFirst and foremost I would sincerely like to thank my supervisor, Sally Otto, formore than words can say. I feel exceptionally honoured to have had this time towork with you, it has allowed me to grow both academically and personally inways otherwise unreachable. Thank-you!I am also grateful to the Otto lab and all the contributing members of the Bio-diversity Centre. To avoid leaving someone out I will not list names, but you knowwho you are! You make this a wonderful place to work and live and I am happy tohave gotten to know you all. Long live the skit.Finally, I would like to thank my friends and family. I am especially grateful tomy parents for wanting copies of my papers (no matter how incomprehensible!), toJeff Facto for being a constant companion and source of adventure, and to MeganBontrager for everything you have done – and continue to do!I would like to end by thanking my coauthors as well as Amy Angert and MikeWhitlock for helping to refine the contents of this thesis. The mistakes that remainare all mine.Funding has been generously provided by the National Science and Engineer-ing Research Council and the University of British Columbia.xiiiChapter 1Introduction1.1 Variations on a themeFitness-valley crossing, as originally depicted (Wright, 1932), is the transition froma population composed primarily of one genotype to a population composed pri-marily of another when a necessary intermediate between the two genotypes hasthe lowest relative fitness of the three. Evolutionary rescue, in contrast, is the tran-sition from a population composed primarily of genotypes with absolute fitnessesbelow replacement to a population composed primarily of genotypes which can atleast replace themselves (Gomulkiewicz and Holt, 1995). These topics may notat first appear to have much in common. I will argue here, however, that thesetwo processes do in fact share a common theme; both fitness-valley crossing andevolutionary rescue are cases of adaptive evolution that require the prolonged per-sistence of genotypes that are expected to be declining in number. I have calledthis theme “adaptive challenges” in the title but on a more technical level it couldbe called “subcritical adaptation”, as outlined below. There is also a pervasive dif-ference between each chapter that provides variation on the theme – each set ofmathematical models assumes a different genetic basis of adaptation.I will not give a comprehensive review of fitness-valley crossing or evolution-ary rescue here, as these processes are discussed extensively in the chapters thatfollow and have each recently received thorough reviews (e.g., Bell, 2017; Obolskiet al., 2017). Instead, I will first attempt to place the mathematical models used in1this thesis within the broader fields of population and quantitative genetic theory.In particular, I will describe the theoretical foundations of the genetic architecturesassumed to underlie adaptation. I will then clarify just what I mean by subcriticaladaptation and in doing so explain more fully why I think fitness valleys and evo-lutionary rescue are similar challenges for evolution by natural selection. I end bystating the specific goals of each chapter.1.2 The genetic basis of adaptation in mathematicalmodels of evolution1.2.1 A brief early historyTheory describing the genetic basis of adaptation had a turbulent start in evolution-ary biology. Charles Darwin (1859) doubted the efficiency of evolution by naturalselection because of his belief in the theory of blending inheritance; if offspring arean intermediate of their two parents then any new favourable mutation would berapidly lost by blending with the primarily non-mutant population (Fisher, 1930,ch. 1). Mutation rates would then need to be enormous to maintain the variationnecessary for adaptation (Fisher, 1930, ch. 1). This problem was later remediedby particulate Mendelian inheritance, proposed by Gregor Mendel (1866) when hehypothesized that many characters (phenotypes) of pea (Pisum spp.) are coded bya finite number of factors (loci) that are combined but not lost upon mating. Inopposition to blending inheritance, “on this fact a rational theory of natural selec-tion can be based” (Fisher, 1930, p. ix). Mendelianism, however, initially failed toflourish, in part because Mendel focused on characters with a “sharp and certainseparation” instead of those with a “more or less nature” (Mendel, 1866). The gen-erality of Mendel’s findings was therefore challenged by the continuous nature ofmany traits, as emphasized by Francis Galton and the Biometricians (reviewed inProvine, 1971, ch. 5). Fortunately, Fisher (1918) reconciled particulate Mendelianinheritance with continuous traits and their observed relations between relatives byconsidering a character composed of many Mendelian factors. Thus, by 1918, onehundred years ago, the foundations for much of today’s theoretical population andquantitative genetics was in place.21.2.2 One and two locus modelsWhile Mendel (1866) had essentially already presented one, two, and three locusmodels, e.g., giving the genotype frequencies in generation n, he did not considerany other process but selfing. This was later extended to random mating by Hardyand Weinberg, among others, which gave rise to the Hardy-Weinberg principleof genotype frequencies at equilibrium (reviewed in Provine, 1971, ch. 5). Theeffect of selection was then introduced by Punnett and Warren in 1917 before beinggreatly extended by Haldane, Fisher, and Wright (reviewed in Provine, 1971, ch.5).The primary focus of these early mathematical studies was on single loci (Lewon-tin and Kojima, 1960); extensions to multiple loci requires one to grapple withthe additional complexities of epistasis and linkage. Epistasis was a term intro-duced by Bateson in 1908 to mean what evolutionary geneticists now refer to asepistatic dominance (an allele at one locus masks the effect of an allele at another),and this was later generalized (as “epistacy”, without citation) by Fisher (1918) tomean any non-additive interaction between loci (reviewed in Phillips, 1998). Link-age was also observed soon after Mendel’s rediscovery at the turn of the century,when Bateson et al. (1906) noticed colour and pollen shape were often inheritedtogether in sweet pea (Lathyrus odoratus), much like sex and pigmentation in ca-naries (Bateson and Punnett, 1911). They initially termed this “partial coupling”(or “complete repulsion” when linkage was perfect) but had “no probable surmiseto offer”, although they hypothesized that coupling “may be a consequence of thedifference in the geometrical positions of the factors relative to the planes of somecritical division” (Bateson and Punnett, 1911). A surmise was eventually offeredby Morgan who, when working out his theory of a gene, suggested that the cou-pling observed by Bateson and others was due to genes being arranged linearly onchromosomes and thus physically “linked” (Morgan, 1917). Partial coupling washypothesized to be the result of occassional “crossing-over”, where “a piece of onechromosome has been exchanged for the homologous piece of the other” (Morganand Bridges, 1916, p. 7). While theory was soon developed to convert betweenphysical distance and the probability of crossing-over (Haldane, 1919), models ex-amining the effect of linkage on the dynamics of genotype frequencies over time3did not seem to be developed until much later (Geiringer, 1944).The two-locus two-allele model of fitness-valley crossing that I analyze inChapter 2 includes all four of these processes: random mating, selection, epistasis,and linkage. The first models containing all of these ingredients and examiningthe dynamics of genotype frequencies over time appear to be Kimura (1956) andLewontin and Kojima (1960), whose main interests were to find polymorphic equi-libria (the latter also introduced the concept of “linkage disequilibrium”). The firststudies to examine such models in the context of fitness-valley crossing seem to beWright (1959), Crow and Kimura (1965), and Eshel and Feldman (1970).1.2.3 Many-locus models of quantitative traitsAs perceived by Mendel (1866), apparently continuous characters (like flowercolour in the bean Phaseolus nanus) could be reconciled with particulate inheri-tance if the number of factors determining the character was large. This was takento the extreme by Fisher (1918), who considered a character determined by thesum of an effectively infinite number of independent factors (i.e., no epistasis orlinkage) of vanishingly small effect. This is Fisher’s “infinitesimal model” (Turelli,2017). Fisher then invoked the central limit theorem to say that such a character– before selection – will be normally distributed (for a slightly clearer explanationthan Fisher’s see Moran and Smith, 1966).The models used to analyze evolutionary rescue in gradually changing envi-ronments in Chapters 3 and 4 launch from this starting point. In particular, both ofthese chapters assume normally distributed phenotypes coded by many loci. Thiscan be rigorously justified in some instances but is a rough approximation used foranalytical results in others (see ‘Gaussian population’, below). There is also animportant difference between the two chapters. Chapter 3 assumes an effectivelyinfinite number of small effect loci such that we can ignore individual loci andmodel inheritance and genetic variation phenomenologically. Genetic variation isthen maintained at a balance between random mating and segregation, independentof selection (see ‘Gaussian descendants’, below). Meanwhile, Chapter 4 assumesa large but finite number of loci with genetic variation maintained at mutation-selection-drift balance.4Gaussian populationWith an increasing number of additive unlinked loci of vanishingly small effect thecentral limit theorem implies that the distribution of a trait is asymptotically normalbefore selection (Fisher, 1918). Even weak selection generally produces linkagedisequilibrium, however, creating non-independence between loci and thus inval-idating a key assumption of the central limit theorem. The phenotypic distribu-tion thus tends to depart from normality after selection (Turelli and Barton, 1990).There are, though, special cases of selection that do not create linkage disequilib-rium and therefore justify the mathematically convenient assumption that the traitvalues in a population are normally distributed (Turelli and Barton, 1990). A com-mon example where such a “Gaussian population” approximation (Turelli, 2017) isaccurate is under Gaussian stabilizing selection (equivalently quadratic stabilizingselection in continuous time). Further, assuming a constant genotypic variance (seebelow for two alternative ways this can arise), the probability distribution for themean trait value at any time can be calculated from a discrete time “linear Gaussianprocess” (Bu¨rger and Lynch, 1995; Lande, 1976) or a continuous time Ornstein-Uhlenbeck diffusion process (Lynch and Lande, 1993). The probability distributionfor the entire trait distribution at any time is then known once one works out theequilibrium phenotypic variance. The Gaussian population approximation is usedin both Chapters 3 and 4, where forms of selection that do not rigorously maintainnormal trait distributions are sometimes considered but a Gaussian population isassumed anyhow in order to reach simple analytic conclusions (these conclusionsare compared with simulations that assume nothing about the distribution).Gaussian descendantsWhile the Gaussian population approximation is a special case (albeit a very com-mon one), Fisher’s infinitesimal model leads to another, more robust, approxima-tion in many cases (including in the presence of some forms of epistasis): thedistribution of trait values within a family is normally distributed with a meanequal to the average of the two parents and a variance that does not depend onthe parental trait values (see Barton et al., 2017, for a history of this approxima-tion). This “Gaussian descendants” approximation (Turelli, 2017) is very useful5in evolutionary models; because conditioning on parental trait values does not af-fect within-family variance, neither does selection. The dynamics of variance arethen straightforward; with parental variance V , random mating halves the parentalvariance and segregation reintroduces variance α2, leading to an offspring vari-ance of V ′ = V/2+α2. At equilibrium we have V ′ = V giving Vˆ = 2α2 (Bartonet al., 2017). Importantly, this approximation does not guarantee that the popula-tion-wide distribution of trait values is normal, just the within-family distributions.This Gaussian descendants approximation is used in Chapter 3.Mutation-selection-drift balanceWorking out the equilibrium variance of a polygenic trait coded by a finite num-ber of loci turns out to be a much more involved task. Even with a large numberof additive unlinked loci, genotypic variance is reduced under stabilizing selec-tion, as the effects of loci become negatively correlated and linkage disequilibriumbuilds up (Bulmer, 1971). This reduction can be counteracted by mutation; earlyresults for genetic variance under mutation-selection balance with a large but finitenumber of additive loci was worked out in great detail by Latter, Bulmer, Kimura,Lande, and Fleming, as summarized in Turelli (1984). Turelli (1984) went onto promote the house-of-cards approximation (Kingman, 1978), where the varia-tion introduced by mutation each generation overwhelms existing genetic variation.While at face value this might sound absurd, it merely requires that the trait vari-ance associated with new a mutation (relative to the variance associated with ran-dom environmental effects) times the strength of selection is much greater than theper locus mutation rate, m2i V−1s >> µi. A conservative choice of parameters fromTurelli (1984), assuming intermediate strengths of stabilizing selection and mod-erate mutational effects, is m2i = 10−2, Vs = 10, and µi = 10−5. Thus the left-handside of m2i V−1s >> µi is likely to be at least two orders of magnitude greater thanthe right-hand side. This inequality continues to be supported by more recent data(e.g., Huang et al., 2016) and is generally thought to be plausible when the num-ber of contributing loci is large (Johnson and Barton, 2005). The house-of-cardsmodel was later extended to include genetic drift (Bu¨rger, 1989), an approximationused in Chapter 4 along with the purely neutral expectation for genetic variance de-6rived by Lynch and Hill (1986). These models also assume a continuum-of-alleles(Kimura, 1965), that is, they assume the alleles created by mutation are drawn froman infinitely large pool of potential alleles each with a different effect on the trait.1.2.4 The infinite sites modelSo far I’ve discussed traits coded by one, two, n, or an infinite number of loci,each with two or more segregating alleles. An alternative approach is to assume aneffectively infinite number of loci that could have alleles segregating but do not ini-tially, and that any new mutation always appears at a previously unmutated locus.This model was introduced by Kimura (1969), who was inspired by the enormousnumber of nucleotide sites per gene and the incredibly low per site mutation rate(more fully explained in Kimura, 1971). The infinite sites model is often used toask how many sites are expected to be segregating at equilibrium or to differ be-tween two populations. It is therefore valuable for interpreting DNA sequences,where a segregating site is taken to be a single nucleotide polymorphism (SNP),e.g., the infinite sites model gives expectations for the site frequency spectrum un-der neutrality (Nielsen and Slatkin, 2013, p. 53-55). Chapter 5 uses the infinitesites model, with a continuum-of alleles, to analyze evolutionary rescue follow-ing an abrupt environmental change. The model in Chapter 5 further assumes thateach allele affects n traits that are all under selection (i.e., alleles are pleiotropic),as originally envisioned by Fisher (1930).1.3 An overarching theme: subcritical adaptationI will now explain more fully the logic behind my claim that fitness-valley crossingand evolutionary rescue are conceptually tied through what I’ve called subcriticaladaptation.1.3.1 Defining subcriticalI am borrowing the term “subcritical” from the theory of branching processes (fora foundational treatment see Harris, 1964). Branching process theory is a branchof probability theory that has been developing for nearly 200 years (Allen, 2010,p. 159-160). Interestingly, key among the contributors were evolutionary biolo-7gists Francis Galton, Ronald Fisher, and J.B.S. Haldane. In fact, it was a questionposed by Galton, about the probability of surname extinction, that led to the dis-tinction between supercritical, critical, and subcritical. Given an individual leavesk descendants (or surnames) in the next generation with probability pk, regardlessof other individuals and constant across time, the expected number of descendantsleft by an individual in the next generation is m = ∑∞k=0 kpk. Then, given there issome chance that an individual leaves either none or more than one descendants inthe next generation (0< p0 and 0< p0+ p1 < 1), extinction is certain when m≤ 1,while if m > 1 there is some positive probability the lineage never goes extinct(theorem 4.2 in Allen, 2010). A branching process with m > 1 is called “supercrit-ical”. A process with m≤ 1 is further divided; when m= 1 the expected number ofindividuals is constant and we refer to this as a “critical” branching process, whenm < 1 the expected number of individuals decays geometrically and we call this a“subcritical” branching process. Here I use “subcritical” in a more expansive (andless well defined) way to refer to genotypes that deterministically would decline innumber, such that any one subcritical lineage is fated to extinction in the absenceof genetic changes. I allow this to be the result of a process where birth and deathcan depend on time (e.g., environmental change) and on interactions with otherindividuals (e.g., density- or frequency-dependence).1.3.2 Defining adaptationI will consider “adaptation” to be the expected result of evolution by natural selec-tion. Natural selection, and thus the definition of adaptation used here, makes nopredictions about what particular phenotypes will arise (Lewontin, 1978). Insteadthe currency of natural selection is birth and death; evolution by natural selec-tion is the process that results from heritable variation in reproduction and survival(Lewontin, 1970). Individuals that survive and produce more offspring pass on adisproportionate number of genes to the next generation and those genes influencethe birth and death rates of the descendants. Even with a focus only on birth anddeath, one could still consider evolution by natural selection to provide a better“solution” to the “problem” posed by the environment, the problem simply beinghow to better survive and reproduce. This is generally only true, however, if one8considers the environment to consist of both abiotic and biotic components, includ-ing oneself (Lewontin, 1978). The potential feedbacks between the force of naturalselection (the problem) and the populations it creates (the solution) are at least partof what Fisher considered “deterioration of the environment” (Fisher, 1930, p. 41).1.3.3 Two possible types of subcritical adaptationGiven the definitions above, the purging of deleterious variation is adaptation thatinvolves subcritical genotypes, as is the increase in frequency of beneficial variants.However, in neither of these cases does adaptation require the persistence of sub-critical genotypes. For example, genotypes that can replace themselves when ontheir own but not in the presence of certain others only become subcritical oncesuch “beneficial” genotypes exist, in which case the original genotypes are nolonger needed for adaptation. I will restrict my definition of subcritical adaptationto scenarios where subcritical genotypes are required to persist for a sufficientlylong period of time (long enough to lead to non-subcritical genotypes). Then,given evolution by natural selection does not lead, in the long-term, to the fixa-tion of subcritical genotypes (i.e., ignoring “evolutionary suicide”, Ferriere, 2000),there are two options, either (1) the population begins without subcritical genotypesbut adaptation transiently requires them or (2) the population begins with subcriti-cal genotypes and natural selection replaces them. These options are fitness-valleycrossing and evolutionary rescue, respectively.1.3.4 Subcritical intermediates: fitness valleysConsider three genotypes, A, B, and C, and a scenario where natural selection al-ways favours A over B, C over B, and C over A. Genotype B is then subcritical,with a growth rate below replacement, whenever A or C are present. Further, let Bbe a necessary genetic intermediate between A and C (e.g., A is one mutational stepfrom B which is one mutational step from C, but A and C are two mutational stepsapart). We then have a “selective surface” (Wright, 1982) with two peaks, a lowerone at A and a higher one at C. Given sufficient numbers of C, natural selectionwill tend to lead to a population that is composed primarily of C. However, withoutC natural selection cannot move the population off the lower peak at A. In reality9there could be many peaks and “the problem of evolution ... is that of a mechanismby which the species may ... find its way from lower to higher peaks in such a field”(Wright, 1932). We call the transition between peaks “fitness-valley crossing”. Inmy terminology, fitness-valley crossing is subcritical adaptation because it is adap-tation (moving from genotype A to C) that requires the temporary persistence ofsubcritical genotypes (the deleterious intermediate B). The history of fitness-valleycrossing and Sewall Wright’s shifting balance theory (of which valley crossing isthe first of three parts) has been thoroughly discussed (e.g., Coyne et al., 1997,2000; Whitlock and Phillips, 2000; Wright, 1982) and recently reviewed (Obolskiet al., 2017). See Chapter 2 for more information.1.3.5 Subcritical progenitors: evolutionary rescueThe other possibility is that a population begins with subcritical genotypes. Con-sider again genotypes A, B, and C, as above. But this time let natural selectionalways favour B over A and C over B. Then A is subcritical in the presence of B orC and B is subcritical in the presence of C. If none of these genotypes is subcriticalon their own (i.e., their growth rates when alone are above replacement, at leastwhen rare) then no matter what the starting point natural selection will eventuallylead to genotype C. This is adaptation in its most simple form, which differs fromthe topic of this thesis because it does not require the persistence of subcriticaltypes, as discussed above. But consider the case where either A or both A and B aresubcritical regardless of whether any other genotypes are present (i.e., they havegrowth rates below replacement when alone at any density). Then a populationstarting with only subcritical genotypes may go extinct before a non-subcriticalgenotype is created. Adaptation then requires the prolonged persistence of subcrit-ical genotypes. This is “evolutionary rescue” (Gomulkiewicz and Holt, 1995). Thetheory of evolutionary rescue has recently received a thorough review (Alexanderet al., 2014), as has the topic more generally (Bell, 2017). See Chapters 3, 4, and 5for more information.101.4 Goals of the thesisIn this thesis I aim to contribute to our understanding of evolution by focusingon a particularly challenging scenario for adaptation. Specifically, I examine therate, likelihood, and resulting patterns of adaptation that requires the prolongedpersistence of genotypes that are expected to be declining in number. The expecteddeclines are the result of i) contingencies on alleles at other loci (fitness-valleycrossing; Chapter 2) and ii) environmental change (evolutionary rescue; Chapters3-5).1.4.1 Chapter 2. Fitness-valley crossing with biased transmissionDespite nearly 90 years of theory on fitness-valley crossing and the shifting balancetheory, nearly all studies assume Mendel’s “law of segregation”: alleles segregateduring meiosis so that, at each locus, half the gametes carry one allele and halfthe other. This assumption of fair segregation precludes a number of processes wenow know are important. For instance, intragenomic conflict may in fact be thenorm (Rice, 2013), leading to biased transmission through meiotic drive (reviewedin Lindholm et al., 2016). Another common, and more extreme, example of biasedtransmission is uniparental inheritance, as is the case for cytoplasmic elements likemitochondria and chloroplasts in many species (reviewed in Birky, 1995). Further,because some genes required for energy production by cytoplasmic elements arelocated in the nuclear genome and because of the genetic conflicts introduced byuniparental inheritance (reviewed in Rand et al., 2004), cytonuclear inheritancecreates plenty of potential for epistasis (and thus fitness valleys). While biasedtransmission often does impart some selection, it typically cannot be subsumedwithin genic fitnesses, e.g., meiotic drive only has an effect in heterozygotes. Howfitness-valley crossing plays out with biased transmission is unknown.The possibility of cytonuclear interactions causing fitness valleys exposes an-other simplicity of previous models; it is typically assumed that the loci involvedhave equal mutation rates and that the multiple intermediate genotypes have equiv-alently depressed fitness. It is clear that intermediates will not, in general, haveequivalent fitness and it is also widely recognized that cytoplasmic genomes canhave mutation rates that are orders of magnitude above the nuclear genome (e.g.,11Haag-Liautard et al., 2008). Such asymmetries may have large effects on fitness-valley crossing by recombination, as the intermediates will in general no longer beat equal frequency.Chapter 2 sets out to explore how biased transmission and asymmetries in mu-tation and selection affect the speed and likelihood of fitness-valley crossing. Thecreation of a generalized theory also allows one to explore the possibility of fitnessvalleys created by cultural traits, e.g., a new cultural practice or technology is notadopted in the current system but might be if an additional cultural practice wasmore frequent.1.4.2 Chapter 3. Evolutionary rescue: when predators helpThere has been enormous theoretical progress on the problem of evolutionary res-cue in recent decades (reviewed in Alexander et al., 2014; Bell, 2017). However,the great majority of this effort has been directed towards populations in isolationand the role of species interactions has been largely ignored (e.g., the reviews ofAlexander et al., 2014, Carlson et al., 2014, and Bell, 2017 together mention onlythree relevant studies). This is unfortunate given that species interactions can havestrong demographic and selective effects and thus will surely impact the probabilityof evolutionary rescue in many natural populations. One ubiquitous species inter-action is predation (including herbivory and parasitism). Recent theory (Mellardet al., 2015) and experiments (Tseng and O’Connor, 2015) suggest that predatorscan help their prey adapt, phenotypically, in changing environments, but the effectof predators on the persistence of their prey remains unclear. Chapter 3 thereforesets out to investigate if predators can theoretically promote prey persistence in acontinuously changing environment, and if so, how.1.4.3 Chapter 4. Evolutionary rescue: alternative fitness functionscause evolutionary tipping pointsThere is a strong tradition of using Gaussian functions to describe selection, goingback at least to the discussion of infant mortality by Haldane (1953), as pointed outby Bulmer (1971). The above-mentioned mathematical convenience of workingwith Gaussian selection – a normal phenotypic distribution remains normal in this12special case (Turelli and Barton, 1990) – has surely contributed to its lasting influ-ence (e.g., Lande, 1976, p. 322). Also important is the fact that a Gaussian functionis a good approximation of any smooth selective surface when a population is nearan optimum (Lande, 1976). A growing understanding of environmental change,especially rapid in recent decades, suggests, however, that natural populations maynot always be near optima (e.g., Wilczek et al., 2014). Gaussian selection is then nolonger guaranteed to be a good approximation of reality and the shape of the tailsof the selective surface becomes critical. Despite this, nearly all models investi-gating adaptation and persistence under continuous environmental change – whichinclude the possibility of maladaptation so extreme that it causes extinction – con-tinue to assume Gaussian selection (equivalently quadratic selection in continuoustime; e.g., Bu¨rger and Lynch, 1995; Lynch and Lande, 1993). This assumption ispresumably for historical reasons and to keep analytic approximations as close aspossible to the exact results. An additional concern is that Gaussian selection as-sumes negative epistasis (i.e., a negative curvature in the selective surface on a logscale), but there is no reason to ignore the possibility of positive epistasis (i.e., thebenefit of two mutations that individually move a phenotype closer to the optimumis greater than the sum of the individual benefits, in terms of growth rate). Becauseessentially all of our predictions about adaptation and extinction in a continuouslychanging environment assume Gaussian selection, Chapter 4 sets out to break thisassumption and examine the consequences for population persistence in a changingworld.1.4.4 Chapter 5. Evolutionary rescue: a theory for its genetic basisAs far as I know, all existing theory on evolutionary rescue following an abruptenvironmental change either ignores the genetic basis by which rescue occurs (byworking on the level of highly polygenic phenotypes) or starts by assuming a par-ticular genotype can rescue the population and then asks what the probability ofrescue is given this genetic basis. The latter approach has told us a lot about howrescue depends on the genetic basis of adaptation (e.g., Uecker and Hermisson,2016). In contrast, we know very little about how evolutionary rescue affects thegenetic basis of adaptation observed in surviving populations. Meanwhile, there is13a long history of theory investigating the genetic basis of adaptation in the absenceof demography (reviewed in Orr, 2005). Chapter 5 sets out to remedy this situationby initiating a line of theoretical investigation into the genetic basis of evolutionaryrescue. Specifically, how many mutations is evolutionary rescue likely to take andwhat is the distribution of fitness effects among the survivors?1.5 ConclusionI hope that, with this Introduction, I have helped the reader see (1) where the mathe-matical models that follow come from and how they vary from one another, particu-larly in the genetic basis of adaptation they each assume, and (2) that fitness-valleycrossing and evolutionary rescue share a common theme, namely the reliance ofadaptive evolution on subcritical genotypes. Following the mathematical analysesin Chapters 2-5, I return to these two points in Chapter 6.14Chapter 2Fitness-valley crossing withbiased transmission12.1 IntroductionEpistasis and underdominance create rugged fitness landscapes on which adapta-tion may require a population to acquire multiple, individually-deleterious muta-tions that are collectively advantageous. Using the adaptive landscape metaphor,we say the population faces a fitness “valley” (Wright, 1932). Such valleys ap-pear to be common in nature (Szendro et al., 2013; Weinreich et al., 2005, butsee Carneiro and Hartl, 2010) and affect, among other things, speciation by re-productive isolation, the evolution of sex, the evolvability of populations, and thepredictability of evolution (Szendro et al., 2013). Here we are interested in thespeed and likelihood of fitness-valley crossing, which we determine by examin-ing the first appearance of an individual with the collectively advantageous set ofmutations whose lineage will eventually spread to fixation.Believing epistasis to be ubiquitous, Sewall Wright (1931; 1932) formulatedhis “shifting balance theory”, which describes evolution as a series of fitness-valley1A version of this chapter has been published as Osmond, M. M., and S. P. Otto.2015. Fitness-valley crossing with generalized parent–offspring transmission. Theo-retical Population Biology 105:1–16. A supporting Mathematica file is available at In phase one of the shifting balance process, small, partially-isolatedsubpopulations (demes) descend into fitness valleys by genetic drift. The new mu-tations are selected against when rare, as they will tend to occur alone as singledeleterious alleles. Eventually drift may allow the deleterious mutations to reachappreciable frequencies in at least one deme. Once multiple synergistically-actingmutations arise together, they begin to be locally favoured by selection. In phasetwo, these favoured combinations of mutations sweep to fixation, and those demesascend the new “fitness peak”. Finally, in phase three, the demes that reach the newfitness peak send out migrants whose genes invade and fix in the remaining demes,eventually “pulling” the entire population up to the new fitness peak. Our focushere is in the first appearance of a genotype on the new fitness peak whose lineagewill eventually fix, considering a single isolated deme. This is typically the longeststage of phases one and two (Stephan, 1996) and hence is likely the limiting stepin fitness-valley crossing.Fitness-valley crossing has been investigated in a large number of theoreticalstudies. In the context of multiple loci with reciprocal sign epistasis, the first ap-pearance of the genotype with the best combination of alleles has been the focusof a few studies (Christiansen et al., 1998; Hadany, 2003; Hadany et al., 2004;Phillips, 1996; Weissman et al., 2009, 2010). Many authors have gone on to ex-amine the remainder of phases one and two (Barton and Rouhani, 1987; Crow andKimura, 1965; Eshel and Feldman, 1970; Karlin and McGregor, 1971; Kimura,1985, 1990; Michalakis and Slatkin, 1996; Phillips, 1996; Stephan, 1996; Wein-reich and Chao, 2005; Weissman et al., 2009, 2010), as well as phase three (Bar-ton, 1992; Crow et al., 1990; Gavrilets, 1996; Hadany, 2003; Hadany et al., 2004;Kimura, 1990; Kondrashov, 1992; Phillips, 1993). Similar attention has been givento situations with a single underdominant locus (Barton and Rouhani, 1993; Gille-spie, 1984; Peck et al., 1998; Slatkin, 1981) or a quantitative trait (Barton andRouhani, 1987, 1993; Charlesworth and Rouhani, 1988; Lande, 1985a; Rouhaniand Barton, 1987a,b). The theoretical and empirical support for Wright’s shift-ing balance process has been summarized and debated (Coyne et al., 1997, 2000;Goodnight and Wade, 2000; Goodnight, 2013; Wade and Goodnight, 1998; Whit-lock and Phillips, 2000), and the general consensus appears to be that, unless thevalley is shallow (weakly deleterious intermediates), crossing a fitness valley is16unlikely.Despite the abundance of literature on fitness-valley crossing, the above stud-ies all assume perfect Mendelian inheritance. The question therefore remains: howrobust are our ideas of fitness-valley crossing to deviations from Mendelian inher-itance? Specifically, how does transmission bias (e.g., meiotic drive or uniparentalinheritance) affect the speed and likelihood of valley crossing? Departing fromstrict Mendelian inheritance also allows us to consider the idea of valley crossingin cultures, considering the spread of memes (Dawkins, 1976) rather than genes.This simultaneously adds a level of complexity to current mathematical modelsof cultural transmission, which typically consider only one cultural trait at a time(e.g., Cavalli-Sforza and Feldman, 1981; but see, e.g., Creanza et al., 2012; Iharaand Feldman, 2004).Transmission bias in the form of segregation distortion is likely to have a largeeffect on valley crossing, as distortion represents a second level of selection (Hartl,1970; Sandler and Novitski, 1957). Insight into how segregation distortion af-fects valley crossing comes from models of underdominant chromosomal rear-rangements (mathematically equivalent to models with one diploid biallelic locus),which often find meiotic drive to be a mechanism allowing fixation of a new mu-tant homokaryotype (Bengtsson and Bodmer, 1976; Hedrick, 1981; Walsh, 1982).Populations that have fixed alternate homokaryotypes produce heterokaryotype hy-brids, which have low viability and/or fertility; thus gene flow between these pop-ulations is reduced. Segregation distortion is therefore thought to be a mechanismthat promotes rapid speciation (stasipatric speciation; White, 1978). Although therole of underdominance in chromosomal speciation has recently been questioned(reveiwed in Faria and Navarro, 2010; Hoffmann and Rieseberg, 2008; Kirkpatrick,2010; Rieseberg, 2001), it is hypothesized to be relevant in annual plants (Hoff-mann and Rieseberg, 2008) and appears to play a dominant role in maintainingreproductive isolation in sunflowers (Lai et al., 2005) and monkey flowers (Stathosand Fishman, 2014).Another common form of transmission bias is sex specific, with the extremecase being uniparental inheritance. In genetic transmission, strict uniparental in-heritance is common for organelle genomes, such as the mitochondria, which istypically inherited from the mother. Uniparental inheritance will tend to imply17further asymmetries. For instance, the mutation rate of mitochondrial genes isestimated to be two orders of magnitude larger than the mutation rate of nucleargenes in many animals (e.g., Linnane et al., 1989). Higher mutation rates willlikely facilitate crossing. That said, higher mutation rates in only one gene mayhave limited effect because the production of double mutants by recombinationwill be constrained by the availability of the rarer single mutant. Previous modelsof fitness-valley crossing have tended to ignore asymmetries (but see Appendix Cof Weissman et al., 2010).Transmission bias is an integral characteristic of cultural transmission, whereit is referred to as “cultural selection” (Boyd and Richerson, 1985; Cavalli-Sforzaand Feldman, 1981). However, to the best of our knowledge, no attempts havebeen made to examine the evolution of cultural traits (memes) in the presence ofa “fitness” valley. Boyd (2001) reviews the genetic theory of the shifting balance,and notes that it could be applied to culture, but no explicit cultural models werepresented. Meanwhile, instances such as the so-called “demographic transition” in19th century western Europe, where societies transitioned from less educated, largefamilies to more educated, small families (Borgerhoff Mulder, 1998), suggest thatalternate combinations of cultural traits (e.g., ‘value of education’ and ‘family-sizepreference’) can be stable and that peak shifts may occur in cultural evolution. Infact, alternate stable cultural states may be pervasive (Boyd and Richerson, 2010),as alluded to by the common saying that people are “stuck in their ways.” Paradigmshifts in the history of science (Kuhn, 1962) may provide further examples (Fog,1999). Cultural peak shifts can also be relatively trivial; for instance, changingthe unit of time from seconds, minutes, and hours to a decimal system is onlyadvantageous if we also change units that are based on seconds, such as the jouleand volt (Fog, 1999).Here we focus on a population genetic model with two bi-allelic loci underhaploid selection in a randomly-mating, finite population. This model can easilybe reduced to a single-locus model with two alleles and diploid selection, whichis formally equivalent to a model of chromosomal rearrangements (e.g., a chro-mosome has an inversion or not). Interpreting genes as memes produces a modelof vertically-transmitted cultural evolution. Our model incorporates both trans-mission bias and asymmetries in mutation and initial numbers of single mutants.18We first give a rough semi-deterministic sketch to develop some intuition, thenfollow with a stochastic analysis using a diffusion approximation. Our analysiscorresponds to the stochastic simultaneous fixation regime of Weinreich and Chao(2005), and the neutral stochastic tunnelling and deleterious tunnelling regimes ofWeissman et al. (2010), where the appearance of the new, favourable, and eventu-ally successful “double mutant” occurs before the fixation of the neutral or dele-terious “single mutants”. Finally, we apply our results to the specific cases ofsegregation distortion, uniparental inheritance, and cultural transmission.We derive the expected time until the appearance of a double mutant whoselineage will fix when single mutants are continuously generated by mutation fromresidents (the stochastic model assumes neutral single mutants). We also use thestochastic model to derive the probability that a double mutant appears and fixesgiven an initial stock of deleterious single mutants that is not replenished by muta-tion. Given typical per-locus mutation rates, valley crossing is generally found tobe a slow and unlikely outcome under fair Mendelian transmission, even when sin-gle mutants are selectively neutral. Segregation distortion, in favour of wild-typeor mutant alleles, affects crossing most when recombination and mutation are rare,the scenarios where crossing is otherwise unlikely. Cytonuclear inheritance allowsincreased mutational asymmetries between the two loci; higher mutation rates leadto more single mutants and hence faster valley crossing, but, when holding theaverage mutation rate constant, asymmetries hinder crossing by reducing the prob-ability that the single mutants recombine to produce double mutants. Finally, weshow that, when new cultural ideas or practices are not too poorly transmitted whenarising individually within the previous cultural background, a transmission advan-tage of the new combination greatly facilitates cultural transitions.2.2 Model and ResultsConsider two loci, A and B, with xi j the current frequency of genotype AiB j, wherei, j ∈ {1,2, ..., p} are the alleles carried by the individual. When an AiB j individualmates with an AkBl individual, they produce an AmBn offspring with probabilitybkli j(mn). Summing over all possible offspring types, ∑pm,n=1 bkli j(mn) = 1. We canspecify that the bottom index (here i j) denotes the genotype of the mother, while19the top index (here kl) denotes the genotype of the father. As a consequence, trans-mission biases according to parental sex [bkli j(mn) 6= bi jkl(mn)] are allowed. Whenconsidering sex-biased transmission we assume the frequencies xi j are the same infemales and males (i.e., no sex linkage and no sex-based differences in selection),which is automatically the case in hermaphrodites.Random mating and offspring production is followed by haploid viability se-lection, which occurs immediately before censusing. The population size, N, isconstant and discrete, and generations are non-overlapping. Then the expectedfrequency of AmBn in the next generation, x′mn, solvesV x′mn = wmnp∑i, j,k,l=1xi j xkl bkli j(mn), (2.1)where wmn ≥ 0 is the relative viability of AmBn and V is the sum of the right handside of Equation (2.1) over all genotypes, which keeps the frequencies summed toone.Denote the probability that a mating between an AiB j mother and an AkBl fa-ther produces an AmBn offspring that survives one round of viability selection bybkli j(mn)∗ = wmnbkli j(mn), where the asterisk indicates “after selection”. And let theaverage probability that a mating produces AmBn, regardless of which parent waswhich, be b¯kli j(mn)∗ = 12 wmn[bkli j(mn)+bi jkl(mn)]. Then (as we will see below) selec-tion on AiB j in a population of “residents” (A1B1) is described by si j = 2b¯i j11(i j)∗−1. Letting wi j = 1+di j > 0 describe viability selection and 2b¯i j11(i j) = 1+ ki j de-scribe transmission bias (−1≤ ki j ≤ 1), then si j = (1+di j)(1+ ki j)−1. Here wedefine the relative fitness of genotype AiB j as 1+ si j, which is determined by bothviability and transmission. Thus defined, fitness is a measure of the “transmissi-bility” of a genotype as it includes several processes (e.g., viability, meiotic drive,recombination, mutation) that affect the number of offspring of a given genotypeproduced by a parent of that genotype. We will see that it is transmissibility thatdetermines the dynamics of valley crossing.Without mutation or recombination, fair transmission implies ki j = 0, or b¯i j11(i j)=1/2 ∀ i 6= 1, j 6= 1. In words, with fair transmission we expect half of all offspringfrom matings between A1B1 and AiB j to be of parental type AiB j. Sex-based in-20heritance is expected to arise in the form of bi j11(i j) = 1− b11i j (i j) [e.g., maternalinheritance implies bi j11(i j) = 1 and b11i j (i j) = 0], which does not directly imposeselection as ki j = 0. Segregation distortion can, however, impose selection. For ex-ample, ignoring mutations, if the A2 allele is more likely to be transmitted than theA1 allele (in a B1 background) we would have b¯2111(21)> 1/2, giving k21 > 0. Inter-preting genes as memes, transmission bias ki j determines the strength of “culturalselection” (sensu Cavalli-Sforza and Feldman, 1981) on meme combination AiB j.Previous work on multi-locus peak shifts has assumed that bias does not influenceselection (ki j = 0) and that maternal and paternal types are equally transmitted[bi j11(i j) = b11i j (i j) = 1/2 ∀ i 6= 1, j 6= 1].Here we focus on bi-allelic loci (p = 2). We are specifically interested inthe case where, in a population composed entirely of residents, “single mutants”(A2B1 and A1B2) are selected against while “double mutants” (A2B2) are selectivelyfavoured: s21,s12 < 0 < s22.Given that the population is composed primarily of residents, with no doublemutants as of yet, the population faces a fitness valley. The valley can be created bydifferences in viability alone, or it can be created by differences in transmission, orboth. Here we focus on the limiting step in the peak-shift process, the probabilityand expected time until a double mutant arises whose lineage will eventually fix.Following the lead of Christiansen et al. (1998), we begin by developing a roughsemi-deterministic analysis to gain intuition. A stochastic analysis follows. Table2.1 provides a summary of notation.2.2.1 Semi-deterministic analysisSingle mutant dynamicsSelection against single mutants keeps their frequencies (x21 and x12) small. Letthese frequencies be proportional to some small number ε << 1. Let the probabil-ity that an offspring inherits an allele that neither parent possesses [i.e., mutation;e.g., b1111(21)] be of the same small order ε . Then, for large N, the frequencies of21Table 2.1: Parameters used throughout chapterSymbol Descriptionxi j frequency of AiB j in the current generationx′i j expected frequency of AiB j in the next generationwi j viability of AiB j relative to viability of A1B1V normalizing factorN number of individuals in the populationt time, in units of generationsbkli j(mn) probability AiB j mother and AkBl father produce AmBn offspringb¯kli j(mn)∗ average probability of surviving AmBn offspring from AiB j x AkBl mating,12 wmn[bkli j(mn)+bi jkl(mn)]µkli j (mn)∗ probability of surviving mutant offspring, b¯kli j(mn)∗, m 6∈ {i,k}, n 6∈ { j, l}rkli j (mn)∗ probability of surviving recombinant offspring, b¯kli j(mn)∗, m ∈ {i,k}, n ∈ { j, l},mn 6∈ {i j,kl}b¯kli j(mn)average transmission probability before selection, b¯kli j(mn)∗/wmn[similarly for µkli j (mn) and rkli j (mn)]si j selection on AiB j in a resident population, 2b¯i j11(i j)∗−1T generations until first successful double mutant arisesu22 probability that a double mutant begins a lineage that will fixit number of A2B1 individuals in generation t (similarly for A1B2, jt)X(t) numbers of single mutants in generation t assuming no double mutants, (it , jt)4i change in number of A2B1 individuals, it+1− it (similarly for A1B2,4 j = jt+1− jt)α , β scaling parameters in diffusion processτ scaled unit of time, t/NαY (τ) scaled frequency of A2B1, iτ/Nβ (similarly for A1B2, Z(τ) = jτ/Nβ )µY (y) first moment of4Y = Y (τ+1)−Y (τ) given Y (τ) = y = i/Nβ (similarly for Z)σ2Y (y) second moment of4Y given Y (τ) = y (similarly for Z)κ(y,z) rate diffusion killed by successful double mutants given Y (0) = y,Z(0) = zBkli j(mn) scaled transmission probability, bkli j(mn)NβR1221(22) scaled (rare) recombination from single mutants to double mutants, r1221(22)N1/2Si j scaled selection on AiB j in population of residents, si jNβT˜ (y,z) scaled time until first successful double mutant given Y (0) = y,Z(0) = z, T Nαm index for single mutant types when equivalent (e.g., sm = s21 = s12)cµ mutation rate at locus B relative to locus A, ν/µcn initial number of A1B2 individuals, relative to A2B1, j0/i0ξ scaled frequency of single mutants (y+ z or ci y = z, depending on assumptions)u(y,z) probability no successful double mutant appears given Y (0) = y,Z(0) = zn0 initial number of single mutants, i0+ j022the single mutants in the next generation arex′i j ≈wi jV[b1111(i j)+2b¯i j11(i j)xi j]+O(ε2), (2.2)where i 6= j and O(ε2) captures terms of order ε2 and smaller.We will write µkli j (mn)∗ = b¯kli j(mn)∗ when m 6∈ {i,k} or n 6∈ { j, l} to highlightthe fact that a mutation has occurred. Then, ignoring O(ε2), the frequencies ofsingle mutants, which are initially absent [xi j(0) = 0], in generation t arexi j(t)≈{µ1111 (i j)∗[(1+ si j)t −1]s−1i j : si j 6= 0µ1111 (i j)∗ t : si j = 0(2.3)Viability and transmission are thus coupled together (in si j) throughout our results,and it is primarily the total amount of selection on AiB j in a population of residents(si j) that determines the dynamics. [As a technical aside, this is not true in thefirst generation that mutants appear, via µkli j (mn)∗, but this is simply because of theorder of the life cycle chosen, where these mutants experience viability selection,but not transmission biases, when they first occur.]Equation (2.3) assumes the normalizing factor V remains near 1 over the t gen-erations, which is the case when single mutants are rare, as is generally true whensingle mutants are selected against, si j < 0 ∀ i 6= j. When si j < 0 and there hasbeen a sufficiently long period of selection, t >−1/si j, the single mutant frequen-cies approach mutation-selection balance xi j(t) ≈ −µ1111 (i j)∗/si j. This assumesthe probability of mutation, µ1111 (i j)∗, is small relative to the strength of selection,si j. We next derive a semi-deterministic solution for the crossing time, T , givenmutation-selection balance is reached. In Appendix A.1 we follow Christiansenet al. (1998) to derive the semi-deterministic crossing time when crossing occursbefore mutation-selection balance is reached; this occurs when−si jT << 1, whichcan only be the case if the valley is shallow, −si j << 1.Waiting time for first successful double mutantWe now turn to calculating the waiting time until a double mutant that is able toestablish first arises. Assume the probability two residents mate to produce a dou-23ble mutant (i.e., a double mutation), b1111(22), is very rare, on the order of ε2. Thenthe expected frequency of double mutants in the next generation before selection,assuming single mutant are rare and there are currently no double mutants x22 = 0,isx′22 =[µ1111 (22)+2µ2111 (22)x21+2µ1211 (22)x12+2r1221(22)x21x12]+O(ε3), (2.4)where we write r1221(22) = b¯1221(22) to highlight the fact that a double mutant haseffectively been produced by recombination. The expected frequency of doublemutants (Equation 2.4) is measured before viability selection to avoid artificiallyadjusting the double mutant frequency by its viability difference before it appears.In a truly deterministic model (N→∞) double mutants are present at frequencyx′22 after a single bout of reproduction. However, assuming no double mutants haveyet appeared, we can use x22(t) as a rough approximation for the probability of adouble mutant first arising in generation t (Christiansen et al., 1998). Summing tfrom 0 to t ′ gives the cumulative probability of observing a double mutant in any ofthe t ′ generations. The generation T ′ at which the cumulative probability reaches1/N can be used as an estimate of the time we expect to wait until the first doublemutant has arisen (Christiansen et al., 1998).Here we are more interested in the waiting time until the first successful doublemutant appears (i.e., one whose lineage will eventually fix). We therefore wantto multiply the probability that a double mutant appears at time t, x22(t), by theprobability it will fix before taking the sum over t. Using Kimura’s (1962) approx-imation, the probability a double mutant fixes isu22 =1− e−2s221− e−2Ns22 . (2.5)With a weak double mutant advantage, 0< s22 << 1, in a large population, Ns22 >>1, Equation (2.5) simplifies to the familiar 2s22 (Haldane, 1927).The selection coefficient s22 can be calculated from the number of double mu-tant offspring a newly arisen double mutant is expected to leave in the next gener-ation, given that the mean number of offspring per individual is one, such that thepopulation size is constant. This expectation, 1+ s22, is the probability of mating24with a given type, multiplied by the probability of producing a double mutant off-spring, multiplied by the probability of surviving to the next generation, summedover all possible matings1+ s22 =2∑i, j=12b¯22i j (22)∗xi j, (2.6)where x22 = 0 in the remaining population (i.e., the double mutant does not matewith itself). Without transmission bias, mutation, or recombination, b¯22i j (22) =1/2 ∀ i, j and Equation (2.6) reduces to the familiar s22 = w22− 1. Here we al-low bias, mutation, and recombination, and assume single mutants are sufficientlyrare, giving s22 ≈ 2b¯2211(22)∗−1. This implies that selection on the double mutant(including transmission) is constant over time and that fixation depends only onits dynamics in a population composed almost entirely of residents. With recom-bination and otherwise fair transmission we have b¯2211(22) = (1− r)/2, where r isthe probability of recombination between a double mutant and a resident. Writingw22 = 1+ s and assuming both s and r are small, recovers the well-known first-order approximation s22 = s− r (Crow and Kimura, 1965). This expression high-lights the fact that recombination can reduce the probability of fixation by breakingup favourable gene combinations (Crow and Kimura, 1965).When selection is strong and mutation is rare, relative to the strength of geneticdrift, the time to fixation is dominated by the time to the arrival of a successful mu-tant (Gillespie, 1984; Weinreich and Chao, 2005; Weissman et al., 2010). The wait-ing time until the first successful double mutant, which we derive below, thereforewell approximates the fixation time of a double mutant within a population whendouble mutants are advantageous but rarely produced, x′22 << 1/N < s22.Crossing time given mutation-selection balance When enough time has passed(t > −1/si j) the single-mutant frequencies approach mutation-selection balance(MSB), xi j(t) ≈ −µ1111 (i j)∗/si j. Using these frequencies in Equation (2.4) givesthe expected frequency of double mutants in the next generation, which does notchange until a double mutant arises, i.e., x22(t) = x′22 ∀ t. Summing u22x′22 overTMSB generations, setting equal to 1/N, and solving for TMSB gives an estimate of25the number of generations we expect to wait for a successful double mutant to arisewhen beginning from mutation-selection balanceTMSB ≈ 1u22N[(1+µ1111 (21)∗s21+µ1111 (12)∗s12)2µ1111 (22)−(1+µ1111 (21)∗s21+µ1111 (12)∗s12)(µ1111 (21)∗s21)2µ2111 (22)−(1+µ1111 (21)∗s21+µ1111 (12)∗s12)(µ1111 (12)∗s12)2µ1211 (22)+(µ1111 (21)∗s21)2µ2121 (22)∗+(µ1111 (12)∗s12)2µ1212 (22)+(µ1111 (21)∗s21)(µ1111 (12)∗s12)2r1221(22)]−1−1. (2.7)In our numerical examples, we will track the waiting time until a successfuldouble mutant arises in a population that has recently established and is fixed forthe resident type (e.g., following a bottleneck or a founder event). This time can beapproximated by the time that it takes to reach mutation-selection balance, T0, andthe establishment time once thereT ≈ T0+TMSB. (2.8)Here we use T0 =max{ 1−s21 , 1−s12 }. As the deleterious single mutants approach neu-trality (si j → 0− ∀ i 6= j) the waiting time from mutation-selection balance, TMSB,decreases (because there are more single mutants segregating), but the waiting timeto mutation-selection balance, T0, increases dramatically because it takes longer toproduce the higher segregating frequencies of single mutants. As −si j becomessmall enough such that T < −1/si j the approximation breaks down and we mustuse the non-equilibrium solution derived in Appendix A.1.With symmetric Mendelian assumptions, weak selection on single mutants(δ = 1−wi j ∀ i 6= j), rare mutation (µ), and infrequent recombination [such thatu f ≈ 2(s− r)≈ 2s], the rate of production of successful double mutants from mu-tation selection balance (Equation 2.7) isTMSB−1 ≈ 2sNµ2rδ 2, (2.9)26aligning with equation 4 in Weissman et al. (2010). This result preforms well whenTMSB−1 < δ , or, equivalently, when 3√2sNµ2r < δ .2.2.2 Stochastic analysisMarkov processFitness-valley crossing is naturally a stochastic process. We thus now considerthe Wright-Fisher model, where the next generation is formed by choosing N off-spring, with replacement, from a multinomial distribution with frequency parame-ters x′i j (Equation 2.1). Let the number of A2B1 and A1B2 single mutants in gener-ation t be it and jt , respectively. Given that there are currently no double mutants,we have N− it− jt resident individuals and we let X(t) = (it , jt) describe the stateof the system in generation t. Let the expected frequencies in the t + 1 genera-tion, conditional on X(t) = (i, j), be x′kl(i, j) = x′kl , with x22 = 0. The transitionprobabilities to states without double mutants are thenPkli j =P{X(t+1)= (k, l) |X(t)= (i, j)}=(Nk, l,N− k− l)(x′21)k (x′12)l (x′11)N−k−l.(2.10)Note that summing over all k, l ∈ {0,1, ...,N} gives (1−x′22)N , the probability thatno double mutant is sampled. Equation (2.10) describes a sub-stochastic transitionmatrix for the Markov process.Next, let H be the state with any positive number of double mutants. We thenhave the transition probabilities PHi j = 1−(1−x′22)N ≈Nx′22, where the approxima-tion assumes a small expected frequency of double mutants in the next generation,x′22 << 1. To calculate the waiting time until the first successful double mutant, wereplace PHi j with P˜Hi j = PHi j u22 ≈ Nx′22u22, ignoring the segregation of double mu-tants when lost. H is now the state with any positive number of successful doublemutants. Dividing each x′i j in Equation (2.10) by the probability a double mutantdoes not arise (1− x′22) and multiplying by the probability a double mutant doesnot arise and fix (1− x′22u22) ensures the columns sum to one. To complete thetransition matrix we make H an absorbing state: PHH = 1 and Pi jH = 0.We can describe this process, in part, by the moments for the change in num-27ber of single mutants, conditional on the process not being killed by a successfuldouble mutant. The nth moment for the change in the number of A2B1 individuals,4i = it+1− it , isE[(4i)n|it = i] =N∑k=0(k− i)n(Nk)( x′211− x′22u22)k( x′12+ x′111− x′22u22)N−k. (2.11)Similar equations can be computed for the change in the number of A1B2 individ-uals,4 j = jt+1− jt .To make analytic progress we use the moment equations to approximate theMarkov chain with a diffusion process (Karlin and Taylor, 1981, Ch. 15). Wedo so by taking the large population limit (N → ∞) while finding the appropriatescalings to ensure finite drift and diffusion terms (Appendix A.2).Crossing time with neutral single mutantsThe diffusion process yields a partial differential equation describing the expectedtime until a successful double mutant arises given that we begin with Nβ y individ-uals of type A2B1 and Nβ z individuals of type A1B2 (Christiansen et al., 1998)12σ2Y (y)∂ 2T˜ (y,z)∂y2+12σ2Z(z)∂ 2T˜ (y,z)∂ z2+µY (y)∂ T˜ (y,z)∂y+µZ(z)∂ T˜ (y,z)∂ z−κ(y,z)T˜ (y,z) =−1,(2.12)where T˜ (y,z) refers to time scaled in units of Nβ generations (parameters definedin Table 2.1 and Appendix A.2). In Appendix A.3 we solve Equation (2.12) underthe two scenarios explored in Christiansen et al. (1998): with and without recom-bination from neutral single mutants to double mutants when the population beginswith only residents, but here generalized to allow unequal mutation rates and sex-biased transmission. While the neutrality assumption precludes the existence of afitness valley, it provides a minimum for the expected time to observe a successfuldouble mutant. Previous studies have suggested that fitness valleys will only becrossed if single mutants are nearly neutral (e.g., Walsh, 1982).28Probability of crossing from standing variationThe diffusion process can also be used to describe the production of successful dou-ble mutants from an initial stock of single mutants (i.e., evolution from standingvariation). Specifically, assuming that residents don’t mutate [b1111(12) = b1111(21) =b1111(22) = 0] the process has two absorbing states, fixation of A1B1 and fixation ofA2B2 (a successful double mutant appears and the process is killed). The probabil-ity of fixation of residents is the solution, u(y,z), of (Karlin and Taylor, 1981)12σ2Y (y)∂ 2u(y,z)∂y2+12σ2Z(z)∂ 2u(y,z)∂ z2+µY (y)∂u(y,z)∂y+µZ(z)∂u(y,z)∂ z−κ(y,z)u(y,z) = 0,(2.13)with terms defined in Appendix A.2. The probability that a successful double mu-tant arises is therefore 1−u(y,z). Karlin and Tavare´ (1981) used a similar equationto find the probability of detecting a lethal homozygote in the one locus, diploidcase with Mendelian transmission.Deleterious single mutants without recombination With no recombination fromsingle mutants to double mutants [r1221(22)= 0] we have scaling parameter β = 1/2.Then, with equal selection on single mutants and some mutational symmetry be-tween the two loci, the single mutants are equivalent and we can concern ourselveswith only their sum ξ = y+ z. Equation (2.13) then collapses to12ξd2u(ξ )dξ 2+Smξdu(ξ )dξ−u22w22[Bm11(22)+B11m (22)]ξu(ξ ) = 0, (2.14)where Sm = s21Nβ = s12Nβ is scaled selection on single mutants and Bm11(22)+B11m (22) = [b2111(22) = b1121(22)]Nβ = [b1211(22) = b1112(22)]Nβ is the scaled mutationprobability from single mutants to double mutants.The boundary conditions are u(0) = 1 and u(∞) = 0. Solving the boundaryvalue problem gives the probability of a double mutant appearing when startingwith n0 = i0+ j0 single mutants1−u(n0) = 1− exp[n0[− sm−√s2m+2u222µm11(22)∗]], (2.15)29where sm = s21 = s12 is the total strength of selection on each single mutant type.Setting n0 = 1 gives the probability a newly arisen single mutant will begin a lin-eage which eventually produces a successful double mutant.Interestingly, Equation (2.15) does not depend strongly on population size, N.Without recombination double mutants are primarily produced by mutations fromsingle mutants, which are rare and hence always mate with one of the large numberof residents. In other words, the production of A2 and B2 alleles does not rely onthe number of residents but only on the dynamics of the rare single mutants.Deleterious single mutants with recombination Finally, we examine the probabil-ity of a successful double mutant appearing when there is recombination betweendeleterious single mutants, r1221(22) > 0. With sufficiently strong selection againstsingle mutants the single mutant frequencies scale as cny ≈ z when we begin withinitial frequencies cny(0) = z(0) and both single mutants are under the same selec-tion pressure, S21 = S12. Then, without mutation from residents to single mutants,Equation (2.13) collapses to12(1+ cn)ξd2u(ξ )dξ 2+S21ξdu(ξ )dξ−u22cn2r1221(22)∗ξ 2u(ξ ) = 0, (2.16)where ξ = cn y = z.With boundary conditions u(0) = 1 and u(∞) = 0 the probability of valleycrossing is1−u(i0,N) = 1− exp[− (1+ cn)i0s21]Ai[N(1+cn)2(s21)2+i02u22cn(1+cn)2r1221(22)∗N1/3[2u22cn(1+cn)2r1221(22)∗]2/3 ]Ai[N(1+cn)2(s21)2N1/3[2u22cn(1+cn)2r1221(22)∗]2/3] ,(2.17)where Ai is the Airy function. Equation (2.17) extends the one-locus diploid resultwith Mendelian transmission (equation 28 in Karlin and Tavare´, 1981) by allowingunequal single mutant frequencies (cn 6= 1) while also incorporating transmissionbias, recombination, and double mutant fitness. Equation (2.17) well-approximatesthe Mendelian simulation results of Michalakis and Slatkin (1996).30When (s21)2 and i0 are small, we have the first order approximation1−u(i0,N) = i0[(1+ cn)s21+31/3Γ[2/3]Γ[1/3](2u22cn(1+ cn)2r1221(22)∗N)1/3],(2.18)which is valid only when the term in the large square brackets is positive. Equation(2.18) can be used to show that when holding the initial number of single mutants,(1+ cn)i0, constant, the probability the double mutant fixes is maximized whenthere are equal numbers of single mutants, cn = 1. This is because recombination ismost efficient in creating double mutants when single mutants are equally frequent.2.2.3 Three scenariosWe next apply our results to three different scenarios: segregation distortion, cy-tonuclear inheritance, and cultural transmission.Segregation distortionOne form of segregation distortion, found in the heterothallic fungi Neurosporaintermedia, is autosomal killing (Burt and Trivers, 2006). In heterozygotes, thepresence of a “killer” allele results in the death of a proportion of the spores thatcontain the wild-type (“susceptible”) allele, leading to a (1+k)/2 frequency of thekilling allele at fertilization, 0< k≤ 1. Letting A2 and B2 represent the killing alle-les, and, for the sake of exploration, assuming that cells with one killing allele arefunctionally equivalent to cells with two, the transmission probabilities are shownin Table 2.2. The Mendelian case is given by k = 0. When −1 ≤ k < 0 the alleleidentities are reversed: A1 and B1 are killers and A2 and B2 are susceptibles.Since segregation distortion imposes selection on single mutants, we can onlyinvestigate the effect of segregation distortion on valley crossing with the semi-deterministic crossing time estimates allowing selection on single mutants (Equa-tions 2.8 and A.3) and with the crossing probability estimates from standing varia-tion (Equations 2.15 and 2.17).Figure 2.1 shows the crossing time as a function of the probability of recom-bination, and how segregation distortion affects this time. Simulations (X’s) wellmatch the numerical solution (Equation A.1; dots) and the MSB approximation31Table 2.2: Transmission probabilities, bkli j(mn), with segregation distortion (autosomal killing). Recombination occurswith probability r, followed by autosomal killing of strength 0 ≤ k ≤ 1, and mutation with probability µ . Whenk > 0 the killing alleles are the mutant alleles (A2 and B2) and when k < 0 the killing alleles are the resident alleles(A1 and B1).Parents OffspringMother Father A1B1 A2B1 A1B2 A2B2A1B1 A1B1 (1−µ)2 µ(1−µ) µ(1−µ) µ2A1B1 A2B1 1−k2 (1−µ)2 1+k2 (1−µ) 1−k2 µ(1−µ) 1+k2 µ+ 1−k2 µ2A1B1 A1B2 1−k2 (1−µ)2 1+k2 (1−µ) 1−k2 µ(1−µ) 1+k2 µ+ 1−k2 µ2A1B1 A2B2 (1− r)1−k2 (1−µ)2 (1− r)1−k2 µ(1−µ)+ r2 (1− r)1−k2 µ(1−µ)+ r2 (1− r)1+k2 +(1− r)1−k2 µ2A2B1 A1B1 1−k2 (1−µ)2 1+k2 (1−µ) 1−k2 µ(1−µ) 1+k2 µ+ 1−k2 µ2A2B1 A2B1 0 1−µ 0 µA2B1 A1B2 r 1−k21−r2 (1−µ) 1−r2 (1−µ) r 1+k2A2B1 A2B2 01−µ2 01+µ2A1B2 A1B1 1−k2 (1−µ)2 1+k2 (1−µ) 1−k2 µ(1−µ) 1+k2 µ+ 1−k2 µ2A1B2 A2B1 r 1−k21−r2 (1−µ) 1−r2 (1−µ) r 1+k2A1B2 A1B2 0 0 1−µ µA1B2 A2B2 0 01−µ21+µ2A2B2 A1B1 (1− r)1−k2 (1−µ)2 (1− r)1−k2 µ(1−µ)+ r2 (1− r)1−k2 µ(1−µ)+ r2 (1− r)1+k2 +(1− r)1−k2 µ2A2B2 A2B1 01−µ2 01+µ2A2B2 A1B2 0 01−µ21+µ2A2B2 A2B2 1 0 0 032(Equation 2.8; solid curves in top panel) over the range of parameters tested. Whenvalley crossing occurs before reaching MSB (bottom panel) a transition occurs be-tween when mutation drives crossing (dashed line; Equation A.2) and when re-combination does (solid curves; Equation A.3), here approximately r ≈ 10−4. Thelargest effect of segregation distortion occurs when the crossing time is long, thescenario in which single-mutants must persist the longest before a successful dou-ble mutant appears. In addition, observe that as the probability of recombination,r, increases above a critical value such that s22 < 0, the double mutant is brokenapart faster than its selective advantage and valley crossing takes longer [equationsB21 and B25 in Weissman et al., 2010 approximate the crossing times with nosegregation distortion (k = 0) when s22 < 0; see also Altland et al., 2011; Lynch,2010].Figure 2.2 shows the probability of crossing from standing variation. Again,segregation distortion has a large effect when mutation (top panel) and recombina-tion (bottom panel) are rare, the conditions under which single mutants must per-sist the longest before a successful double mutant is formed. When crossing occursby recombination our analytical approximation (Equation 2.17) overestimates theFigure 2.1: Expected number of generations until a double mutant begins tofix, T , as a function of the probability of recombination, r. The dotsshow the full semi-deterministic solution (numerical solution to Equa-tion A.1, including higher order terms, allowing both recombinationand mutation to generate double mutants). The solid curves show thesemi-deterministic results when (top) mutation-selection balance is firstreached (Equation 2.8) and (bottom) mutation-selection balance is notreached and crossing can occur by recombination (Equation A.3). Thedashed line gives the crossing time when crossing occurs by mutationonly, before mutation selection balance is reached, and single mutantsare selectively neutral (Equation A.2). The X’s are mean simulation re-sults (Appendix A.4). The grayscale corresponds to (dark) distortionfavouring single mutants, k = 10−4; (medium) the Mendelian case, k =0; and (light) distortion favouring wild-type, k = −10−4. Parameters:µ = 5× 10−7, N = 106, w22 = 1.05, and (top) w21 = w12 = 1− 10−3and (bottom) w21 = w12 = 1−10−5.33x x xxxxx x xxxxx xxxx010 00020 00030 00040 000Tx x xxxxx x xxxxx x xxxxk = - 110000k = 0k = 11000010-6 10-5 10-4 0.001 0.01 0.1r0200040006000800010 00012 000probability of crossing (bottom panel), especially when the initial number of singlemutants is small and therefore subject to strong stochasticity (results not shown).This occurs because the assumption that the ratio of single mutant frequencies inthese simulations remains roughly cn = 1 is violated, reducing the probability thatdouble mutants are formed by recombination.34Cytonuclear inheritanceWe next explore how fitness-valley crossing is affected by uniparental inheritanceof one of the traits. This might occur if, for example, there was reciprocal sign epis-tasis between cytoplasmic and nuclear loci. Without loss of generality we assumethat the B trait is always inherited from the mother. For simplicity we assume indi-viduals are hermaphroditic. Here we can use only those results that allow recombi-nation (Equations 2.8, A.3, A.20, and 2.17), as cytoplasmic and nuclear elementsare expected to be inherited independently (i.e., r1221(22)+ r2112(22) = 1/2).One likely implication of uniparental transmission is asymmetric mutation prob-abilities. For instance, in animals the mitochondrial mutation rate is two orders ofmagnitude larger than typical nuclear rates (Linnane et al., 1989). Let µ be themutation probability in the biparentally inherited A trait and ν be the mutationprobability in the uniparentally inherited B trait, with cµ = ν/µ the ratio of uni-parental to biparental mutation probabilities. The transmission probabilities areshown in Table 2.3.The top panel of Figure 2.3 shows the crossing time as a function of the muta-tion probability in the B locus, ν . Increasing ν increases the rate at which sin-gle and double mutants are created, aiding valley crossing. The bottom panelof Figure 2.3 shows the crossing time as a function of the ratio of the mutationprobabilities at the two loci, cµ , while holding the average mutation probability,(µ+ν)/2 = µ(1+cµ)/2, constant. When holding the average mutation probabil-ity constant the time to fixation is minimized when ν = µ because single mutanttypes are then equally frequent, increasing the chances they mate with one anotherto produce a double mutant by recombination. As cµ departs from one, the muta-tion rate at one of the loci becomes small, causing those single mutants to becomeFigure 2.2: The probability, P = 1− u, of crossing the valley given an ini-tial stock of single mutants (with no further mutations from resident-resident matings) as a function of the rate at which single mutants pro-duce double mutants (top) without recombination (Equation 2.15) and(bottom) by recombination only (Equation 2.17). The X’s are simula-tion results (Appendix A.4). Parameters and grayscale as in Figure 2.1(bottom) with i0 = j0 = 1000.35xxxxxxxxxxk = 110000k = 0k = - 11000010-8 10-7 10-6 10-5Μ0. 10-4 0.001 0.01 0.1r0.0050.0100.0500.1000.5001.000rare. The highly stochastic nature of the rare single mutant frequencies causesour semi-deterministic (Equation A.3) and stochastic (Equation A.20) approxima-tions to underestimate the crossing time and, instead, the single mutants first reach36Table 2.3: Transmission probabilities, bkli j(mn), with cytonuclear inheritance.The A locus is biparentally inherited with µ the mutation probabilityfrom A1 to A2. The B locus is uniparentally inherited with ν the mutationprobability from B1 to B2.Parents OffspringMother Father A1B1 A2B1 A1B2 A2B2A1B1 A1B1 (1−µ)(1−ν) µ(1−ν) (1−µ)ν µνA1B1 A2B11−µ2 (1−ν) 1+µ2 (1−ν) 1−µ2 ν 1+µ2 νA1B1 A1B2 (1−µ)(1−ν) µ(1−ν) (1−µ)ν µνA1B1 A2B21−µ2 (1−ν) 1+µ2 (1−ν) 1−µ2 ν 1+µ2 νA2B1 A1B11−µ2 (1−ν) 1+µ2 (1−ν) 1−µ2 ν 1+µ2 νA2B1 A2B1 0 1−ν 0 νA2B1 A1B21−µ2 (1−ν) 1+µ2 (1−ν) 1−µ2 ν 1+µ2 νA2B1 A2B2 0 1−ν 0 νA1B2 A1B1 0 0 1−µ µA1B2 A2B1 0 01−µ21+µ2A1B2 A1B2 0 0 1−µ µA1B2 A2B2 0 01−µ21+µ2A2B2 A1B1 0 01−µ21+µ2A2B2 A2B1 0 0 0 1A2B2 A1B2 0 01−µ21+µ2A2B2 A2B2 0 0 0 1mutation-selection balance (Equation 2.8; dashed gray curve).Given that crossing occurs by recombination from standing variation (Equation2.17), asymmetric mutation rates have little effect given a particular starting pop-ulation (i0, j0). However, standing variation will also tend to vary in proportion tomutation rates, implying that uniparental inheritance will cause differences in theinitial numbers of the two single mutants, which can have a large effect. Let cµ nowalso determine the ratio of the initial numbers of single mutants, cµ = cn = j0/i0.Figure 2.4 shows the probability of crossing from standing variation as a func-tion of cµ . When we hold i0 and µ constant and increase j0 and ν (grey curve),the probability of crossing increases with cµ as there are then more single mu-tants segregating. When we instead hold the total initial number of single mutants(i0 + j0) and the average mutation probability [(µ+ν)/2] constant (black curve),37the probability of crossing is maximized at cµ = 1 because the single mutants arethen equally frequent and hence more likely to mate with one another and producea double mutant through recombination.Cultural inheritanceFinally, we remove Darwinian selection, such that transmission bias alone deter-mines the dynamics, and interpret the model in a cultural context. For the sake ofexposition we consider only one simplified case of cultural transmission. Let traitcombinations with only one new trait (A2B1 and A1B2) be inherited relative to theprevious combination (A1B1) with probability q. Let the new combination of cul-tural traits (A2B2) be inherited relative to the previous combination with probabilityp. We are most interested in the case of a “transmission valley”, where the previouscombination of traits is transmitted more effectively than mixed combinations ofnew and old (q < 1/2), but the all-new combination spreads even more effectivelythan the previous combination (p > 1/2). We assume that parental trait combina-tions can be broken up with probability r and mutation occurs with probability µ .The transmission probabilities are shown in Table 2.4.Figure 2.3: Expected number of generations until a double mutant begins tofix, T , as a function of (top) the mutation probability in locus B, ν ,and (bottom) the relative mutability of the two loci, cµ = ν/µ . Thetop panel holds the mutation probability in locus A (µ = 5× 10−7)constant while the bottom panel holds the average mutation probability[(µ+ν)/2= µ(1+cµ)/2= 5×10−7] constant. The solid curves showthe stochastic crossing time by recombination with neutral single mu-tants (Equation A.20). The dashed curves show the semi-deterministicresults when crossing occurs (black) before (Equation A.3) and (gray)after (Equation 2.8) mutation-selection balance is first reached. The dotsshow the full semi-deterministic solution (numerical solution to Equa-tion A.1, including higher order terms, allowing both mutation and re-combination to generate double mutants). The X’s are mean simulationresults (Appendix A.4). Parameters as in Figure 2.1 (bottom), exceptw22 = 2.01, which ensures s22 ≥ 0 ∀ cµ .38xxxxxxxxx10-11 10-9 10-7 10-5 0.001 0.1Ν1101001000104105TxxxxxMutation-selection balance HMSBLBefore MSBStochastic10-4 0.01 1 100 104cΜ101001000104105T39xxx x xxx x xnuclear mutation & i0 constanttotal mutation & i0+ j0 constant1 10 100 1000 104cn0. 2.4: The probability, P = 1− u, of crossing the valley given an ini-tial stock of single mutants (with no further mutations from resident-resident matings) as a function of the ratio of the initial numbers of sin-gle mutants and mutation probabilities, cµ = ν/µ = cn = j0/i0 (Equa-tion 2.17). The gray curve holds the initial number of A2B1 (i0 = 100)and the mutation probability in the A locus (µ = 5× 10−7) constantand varies the initial number of A1B2 ( j0) and the mutation probabil-ity in the B locus (ν). The black curve holds the initial number ofsingle mutants (n0 = i0 + j0 = 200) and average mutation probability[(µ + ν)/2 = 5× 10−7] constant. The X’s are simulation results (Ap-pendix A.4). Other parameters as in Figure 2.3.40Table 2.4: Transmission probabilities, bkli j(mn), with cultural inheritance. Parental trait combinations are broken upwith probability r, followed by biased transmission (A2B1 and A1B2 are passed down over A1B1 with probabilityq, A2B2 is passed down over A1B1 with probability p), and mutation with probability µ .Parents OffspringMother Father A1B1 A2B1 A1B2 A2B2A1B1 A1B1 (1−µ)2 µ(1−µ) µ(1−µ) µ2A1B1 A2B1 (1−q)(1−µ)2 (1−q)µ(1−µ)+q(1−µ) (1−q)µ(1−µ) (1−q)µ2+qµA1B1 A1B2 (1−q)(1−µ)2 (1−q)µ(1−µ) (1−q)µ(1−µ)+q(1−µ) (1−q)µ2+qµA1B1 A2B2 (1− r)(1− p)(1−µ)2 (1− r)(1− p)µ(1−µ)+ r2(1−µ) (1− r)(1− p)µ(1−µ)+ r2(1−µ) (1− r)[(1− p)µ2+ p]+ rµA2B1 A1B1 (1−q)(1−µ)2 (1−q)µ(1−µ)+q(1−µ) (1−q)µ(1−µ) (1−q)µ2+qµA2B1 A2B1 0 1−µ 0 µA2B1 A1B2 r(1− p)(1−µ)2 1−r2 (1−µ)+ r(1− p)µ(1−µ) 1−r2 (1−µ)+ r(1− p)µ(1−µ) (1− r)µ+ r[p+(1− p)µ2]A2B1 A2B2 0 (12 − p+q)(1−µ) 0 (12 − p+q)µ+ 12 − p+qA1B2 A1B1 (1−q)(1−µ)2 (1−q)µ(1−µ) (1−q)µ(1−µ)+q(1−µ) (1−q)µ2+qµA1B2 A2B1 r(1− p)(1−µ)2 1−r2 (1−µ)+ r(1− p)µ(1−µ) 1−r2 (1−µ)+ r(1− p)µ(1−µ) (1− r)µ+ r[p+(1− p)µ2]A1B2 A1B2 0 0 1−µ µA1B2 A2B2 0 0 (12 − p+q)(1−µ) (12 − p+q)µ+ 12 − p+qA2B2 A1B1 (1− r)(1− p)(1−µ)2 (1− r)(1− p)µ(1−µ)+ r2(1−µ) (1− r)(1− p)µ(1−µ)+ r2(1−µ) (1− r)[(1− p)µ2+ p]+ rµA2B2 A2B1 0 (12 − p+q)(1−µ) 0 (12 − p+q)µ+ 12 − p+qA2B2 A1B2 0 0 (12 − p+q)(1−µ) (12 − p+q)µ+ 12 − p+qA2B2 A2B2 0 0 0 141Figure 2.5 shows that the crossing time is substantially faster when the newcombination of traits has a stronger transmission advantage (T decreases with p;compare thick curve with thin). Nevertheless, even combinations that are transmit-ted very effectively (thick curve) spread very slowly when their component traitsare passed on poorly in the previous cultural background (q<< 1/2). In particular,the crossing time increases most quickly as q decreases from 1/2, demonstratingthat slight biases in the transmission of the new traits when arising within the previ-ous cultural background have a strong influence on the spread of new combinationsof cultural traits, effectively preventing establishment if q << 1/2.Figure 2.6 shows the probability of crossing from standing variation. In thiscase, with such a large mutation rate, crossing can be more likely by mutation(Equation 2.15) than by recombination (Equation 2.17). Recombination has theadded effect of breaking up the new combination of traits, reducing the probabil-ity of crossing. With a lower mutation rate crossing is most likely with moder-ate amounts of recombination (e.g., Figure 2.1). Figure 2.6 again shows that thetransmission advantage of the new combination of traits (p; compare thick linesto thin) and slight biases in the transmission of new traits in the previous culturalbackground (q ≈ 1/2) greatly influence the probability that a new combination ofcultural traits successfully spreads.2.3 DiscussionOur results support the general consensus that, given reasonable population sizesand per locus per generation mutation rates, crossing a particular fitness valleyby genetic drift is typically a slow and unlikely event (Bengtsson and Bodmer,1976; Coyne et al., 1997; Crow and Kimura, 1965; Hedrick, 1981; Lande, 1979,1985b; Michalakis and Slatkin, 1996; Phillips, 1996; Walsh, 1982). For example,with a per locus per generation mutation probability of µ = 10−8, a double mutantviability of w22 = 1.01, recombination between the two loci with probability r =0.01, and a population size of N = 104, the waiting time for a successful doublemutant, in the best case scenario where single mutants are selectively neutral, ison the order of 107 generations (Equation A.20). As this is the typical age forliving animal genera (Lande, 1979; Van Valen, 1973), we should not expect to see42x x xxxxxxxxWeak A2B2 transmission advantageStrong A2B2 transmission advantage0.0 0.1 0.2 0.3 0.4 0.5 q1101001000104105TFigure 2.5: Expected number of generations until the new combination ofcultural traits (A2B2) begins to fix, T , as a function of the probabilityof inheritance, q, of the new traits singly (A2B1, A1B2) over the previ-ous combination (A1B1). The curves show the estimate given mutation-selection balance is first reached (which assumes A2B1 and A1B2 aredisfavoured, q < 0.5; Equation 2.8). The dots show the full semi-deterministic solution (numerical solution to Equation A.1, includinghigher order terms, allowing both recombination and mutation to gen-erate double mutants). The X’s are mean simulation results (AppendixA.4). The transmission advantage for the new combination of culturaltraits is either weak (thin curves, small dots: p = 0.51) or strong (thickcurves, large dots: p = 0.6). Parameters: N = 103, µ = 10−3, r = 0.01,and w21 = w12 = w22 = 1.this fitness valley forded. Of course, with many potential fitness valleys across thegenome, the chance that one of them is forded can become substantial.By broadening previous treatments to allow for non-Mendelian inheritance,we have shown that a small amount of segregation distortion can greatly impactthe chances of fitness-valley crossing. Of course, segregation distortion has a largeimpact because it provides a second level of selection (Sandler and Novitski, 1957),43x xxxxxx xxxxxNo recombination È r = 0Recombination È r = 0.010.45 0.46 0.47 0.48 0.49 0.50 q10-510-40.0010.010.11PFigure 2.6: The probability, P = 1− u, that the new combination of culturaltraits (A2B2) fixes given an initial number of A2B1 and A1B2 (with nofurther mutations from resident-resident matings) as a function of theprobability of inheritance, q, of the new traits singly (A2B1, A1B2) overthe previous combination (A1B1). The grey curves show the probabil-ity of crossing in the absence of recombination (r = 0; Equation 2.15),with a strong (thick curves: p = 0.6) or weak (thin curves: p = 0.51)transmission advantage for the new combination of cultural traits. Theblack curves show the probability of crossing by recombination only(r = 0.01; Equation 2.17). The X’s are simulation results (AppendixA.4). With such a large mutation probability, crossing can be morelikely without recombination, which has the added effect of breakingapart the new combination. Parameters as in Figure 2.5 with i0+ j0 = 20and c = 1.often acting like gametic selection (but see Hartl, 1970, 1977). When the A2 andB2 alleles are more likely to be passed down than the A1 and B1 alleles, respec-tively, in matings between single mutants and residents, the depth of the valleyis effectively reduced and hence crossing is much more likely. For example, whensingle mutants have a relative viability of wm = 0.95, the mutation rate is µ = 10−8,44double mutants are weakly favoured (w22 = 1.01), and we begin with one singlemutant (n0 = 1) in a population of size N = 104, in the absence of recombination[r1221(22) = 0] and segregation distortion (ki j = 0), the probability of crossing is onthe order of 10−9 (Equation 2.15). With a 5% distortion in favour of A2 and B2 alle-les (k21 = k12 = 0.05) the single mutants are effectively neutral and the probabilityincreases seven orders of magnitude to 10−2. And with a 10% distortion the singlemutants are selectively favoured and the double mutant fixes with probability 0.25.Segregation distortion, in the form of meiotic drive, has often been impli-cated as a force that could help fix underdominant chromosomal rearrangements(Bengtsson and Bodmer, 1976; Faria and Navarro, 2010; Hedrick, 1981; Sandlerand Novitski, 1957; Walsh, 1982). Chromosomal rearrangements, such as translo-cations and inversions, are often fixed in alternate forms in closely related species(Coyne, 1989; Faria and Navarro, 2010; White, 1978). Because heterokaryotypestypically have severely reduced fertility (Lande, 1979; Sandler and Novitski, 1957),such rearrangements are thought to promote rapid speciation (stasipatric specia-tion; White, 1978, but see Faria and Navarro, 2010; Kirkpatrick, 2010). The trou-ble is explaining how such rearrangements originally increase in frequency whenthey are so strongly selected against when rare (Kirkpatrick, 2010; Navarro andBarton, 2003). Meiotic drive provides one possible answer. Our results can beused to investigate valley crossing with chromosomal rearrangements by assumingA and B are homologous chromosomes, with A2 and B2 being the novel chromo-somes, and A2B1 and A1B2 interchangeable. For example, with free recombina-tion [r1221(22) = 1/4, r2211(22) = 1/4], a 5% viability reduction in heterokaryotypes(wm = 0.95), no meiotic drive [b¯m11(m) = 1/2], a very beneficial mutant homokary-otype (w22 = 2.5), and a spontaneous chromosome mutation rate of µ = 10−3(Lande, 1979), when starting with one copy of each mutant chromosome (i0 = 1,cn = 1) in a population of size N = 104, Equation (2.17) gives a 0.4% chanceof fixing the mutant homokaryotype. When the mutated chromosome has a 70%chance of being passed down in matings with residents, a relatively weak amountof drive (Sandler and Novitski, 1957), the chance of crossing increases two ordersof magnitude, to nearly 75%.Here we have shown that, for a given number of single mutants, the chance ofcrossing a valley by recombination is best when the two single mutant types are45at equal frequencies. This is an important factor when the mutation rates in A andB are highly asymmetric. One instance where this asymmetry is likely is whenone locus (say B) is in the mitochondrial genome, and is passed down maternally,while the other (say A) is in the nuclear genome, and is passed down biparentally.Mutation rates in the mitochondria can be orders of magnitude higher than in thenucleus (Linnane et al., 1989). With r1221(22) = r/2 = 1/4, N = 104, w22 = 2,b1111(21) = µ = 10−6, and neutral single mutants, when the mutation rates in A andB are equal [b1111(12)/b1111(21) = cµ = 1] the crossing time is 40,000 generations.When the mutation rate in B is two orders of magnitude larger (cµ = 100) thewaiting time is reduced to 2,500 generations. But when the average mutation rate(1+cµ)b1111(21)/2 is held constant, the asymmetrical mutation rates instead hindercrossing, increasing the crossing time to nearly 120,000 generations.By expanding a mathematical model of fitness-valley crossing beyond symmet-rical Mendelian inheritance we gain insight into transitions between alternate stablestates in non-genetic systems, such as culture. As mentioned in the introduction,culture may often exhibit alternate stable states; here valley crossing correspondsto a shift between alternate combinations of cultural ideas or practices (e.g., thedemographic transition; Borgerhoff Mulder, 1998). The valley is a “transmissionvalley”, created by new cultural traits that are transmitted effectively in concertbut poorly when arising individually within the previous cultural background. Inthis case our simplified example above demonstrates that, given that the compo-nent pieces are not passed on too poorly in the previous cultural background, theprobability that a new set of practices or ideas becomes pervasive in society isgreatly improved by its transmission advantage over the previous set. Valley cross-ing might also be relevant in the context of gene-culture coevolution, where onetrait is cultural and the other genetic. For instance, the ability to absorb lactoseas an adult is largely genetically determined and is positively correlated with thecultural practice of dairy farming, reaching frequencies over 90% in cultures withdairy farming but typically remaining less than 20% in cultures without (Feldmanand Laland, 1996). If, as seems reasonable, the ability to absorb lactose as anadult has a cost in the absence of dairy farming and the cultural practice of dairyfarming has a cost when adults are unable to absorb lactose, then the transitionfrom non-pastoralist non-absorbers to pastoralist absorbers may represent another46example of fitness-valley crossing outside the purely genetic arena. We have usedour generalized model to begin to explore cultural transitions, but it should be em-phasized that we neglect oblique and horizontal transmission, common featuresof cultural evolution (Cavalli-Sforza and Feldman, 1981) and likely componentsof the demographic transition (Ihara and Feldman, 2004). Generalizing models offitness-valley crossing further to include oblique and horizontal transmission wouldimprove insight into cultural transitions.We have incorporated transmission bias in a model of multi-locus fitness-valleycrossing. This allows us to investigate fitness-valley crossing in new scenarios,such as in genetic systems with segregation distortion and/or uniparental inheri-tance. Segregation distortion acts as a second level of selection and therefore cangreatly help or hinder fitness-valley crossing, especially when crossing is other-wise unlikely. Uniparental inheritance will often imply asymmetric mutation rates,which in turn lead to unequal frequencies of single mutants, and therefore, all elsebeing equal, a lower probability of fitness-valley crossing by recombination. How-ever, uniparental-inherited cytoplasmic elements tend to have increased mutationrates, which helps crossing. Generalizing transmission also allows us to begin toextend the theory of valley crossing to non-genetic systems, such as culture. De-spite component traits being passed on poorly in the previous cultural background,we find that small advantages in the transmission of the new set of cultural traitswill greatly facilitate a cultural transition. While crossing a deep fitness valley isdifficult under Mendelian inheritance, it can be easier when Mendel is left behind.47Chapter 3Evolutionary rescue: whenpredators help13.1 IntroductionPopulations are increasingly facing directional environmental shifts caused by cli-mate change and other anthropologic impacts (reviewed in Davis et al. 2005; Hoff-mann and Sgro` 2011; Lavergne et al. 2010; Parmesan 2006; Parmesan and Yohe2003; Visser 2008). For example, migrating birds face earlier springs (Møller et al.2008), diapausing insects face later winter onsets (e.g., Bradshaw and Holzapfel2001), and ectotherms face warming temperatures (e.g., Huey et al. 2009; Thomaset al. 2012). Dispersal and plasticity can help maintain adaptedness and prolongpersistence (Gienapp et al. 2008; Holt 1990), but long-term survival ultimately re-quires genetic adaptation (Gienapp et al. 2013; Visser 2008).The ability of a population to adapt to environmental change commonly de-pends on interactions with other species (Gilman et al. 2010; Harrington et al. 1999;Holt 1990; Lavergne et al. 2010; Lawrence et al. 2012; Tylianakis et al. 2008; Vander Putten et al. 2010). One ubiquitous interaction is predation, which can impose1A version of this chapter has been published as Osmond, M. M., S. P. Otto, and C. A. Klaus-meier. 2017. When predators help prey adapt and persist in a changing environment. The Ameri-can Naturalist 190:83–98. A supporting Mathematica file and simulation scripts are available at demographic and selective effects (Abrams 2000; Dawkins and Krebs 1979;Reznick et al. 2008). Because predators tend to reduce prey population sizes (Holtet al. 2008; Salo et al. 2010), thus imposing ‘demographic costs’, they can increasethe probability of prey extinction following environmental change (Schoener et al.2001). However, predators can also induce adaptive phenotypic evolution. For ex-ample, Tseng and O’Connor (2015) have shown that Daphnia pulex, which tendto have smaller body sizes in warm water, evolve to smaller sizes and achievehigher population growth rates when reared in warm water with predators thanwithout. This suggests that predators can sometimes help prey adapt phenotypi-cally in changing environments. However, increased phenotypic adaptation doesnot necessarily result in increased persistence. If predators are to help maintainbiodiversity in the face of environmental change it must be shown that predator-induced adaptive phenotypic evolution can outweigh predation’s demographic costand allow prey to persist in environments in which they otherwise could not. Theresults have obvious implications for the fate and conservation priority of popula-tions with and without predators (e.g., continental vs. island, endemic vs. invasive)as well as the conservation practice of removing the predators of threatened popu-lations (Reynolds and Tapper 1996; Smith et al. 2010).There is a small but growing number of theoretical studies on adaptation togradual, directional environmental change that incorporate interactions betweenspecies (de Mazancourt et al. 2008; Johansson 2008; Jones 2008; Mellard et al.2015; Norberg et al. 2012). Only two of these studies assessed predator-prey in-teractions. Jones (2008) used simulations to study, among other things, a predator-prey system where the predator trait evolves to match the prey trait and the preytrait evolves both to match the changing environmental optimum and escape thepredator. His main finding was that predation can sometimes increase the per-sistence time of the prey, presumably when the predator removes prey that aremaladapted to the abiotic environment. Mellard et al. (2015) studied an explicitresource-plant-herbivore trophic chain using a primarily numerical approach to askhow the addition of an herbivore affects plant persistence in a warming environ-ment. They analytically clarified the conditions required for the predator to hastenprey evolution but found in simulations that, regardless of the effect on evolution-ary rates, herbivory almost never aided plant persistence. In addition, Northfield49and Ives (2013), who did not explicitly model a gradually changing environment,analytically demonstrated that ‘conflicting’ interactions, such as predation, tendedto dampen the direct effect of small environmental changes on the equilibriumdensities of interacting species, potentially facilitating persistence. In their model,coevolution facilitates persistence through negative feedback loops. For example,as the predator nears extinction the prey invests less in defence, making it eas-ier for the predator to consume prey and rebound in abundance. Thus, up to thispoint, while it is clear that predators can theoretically help prey adapt in changingenvironments, the effect of predators on prey persistence remains unresolved.Here we clarify the effect of predators on the ability of prey to persist in a grad-ually changing environment within a well established quantitative-genetic frame-work (Bu¨rger and Lynch 1995, 1997; Chevin et al. 2010a; Lande and Shannon1996; Lynch et al. 1991; Lynch and Lande 1993; Pease et al. 1989), which canbe applied to both laboratory (Willi and Hoffmann 2009) and wild (Gienapp et al.2013) populations. The framework allows analytical predictions of population den-sities and mean trait values, as well as the critical rate of environmental change,defined as the minimum rate of environmental change that causes an equilibriumpopulation size of zero. Under this framework, predators affect prey persistence ifthey alter the prey’s critical rate of environmental change.While there are many ways for predators to hinder prey persistence (most obvi-ously by reducing prey population size), we find that predators can help prey persistwhen 1) predators prefer maladapted prey and thus push the mean prey trait closerto its moving optimum (‘selective push’) and when 2) the potentially non-selectivemortality induced by predators increases prey birth rates and thus increases the rateof selective events, accelerating prey evolution (‘evolutionary hydra effect’).3.2 Methods and Results3.2.1 A general model of generalist predationLet prey individuals with trait value z, in a prey population of density N, be born ata per capita rate of b(z,N) and die with predator-independent per capita mortalityrate m(z,N). Additionally, let a population of predators kill prey individuals at a per50prey rate of k f (z,N), where k is the overall predation intensity and f includes thepotential dependence on prey trait value, which describes direct selective effects(e.g., increased consumption of slow, large, or maladapted individuals), as well asthe potential dependence of predation on prey density, which allows, for example,a type-II or type-III functional response. We do not explicitly model predator den-sity and trait values here (but see the “Specialist predation” section). Instead, weassume the predator is a generalist that consumes a sufficient number of other preyspecies such that its density and trait values are relatively constant and the predationrate depends only on the density and trait values of the focal prey. Alternatively,constant predator trait values and density could result from the predator having amuch longer generation time than the prey or from predator migration from nearbyhabitats. For example, with a type-II functional response with handling time h andattack rate a(z) we have k f (z,N) = a(z)[1+ ha(z)N]−1P, where predator densityP is approximately constant over the timescale of interest. Although we wish tointerpret k f (z,N) as generalist predation here, it could arise from any source ofmortality whose per capita rate does not vary too much in time with factors otherthan prey trait value and population density. Under these conditions the rate ofchange in prey density is approximatelydNdt= N∫ ∞−∞[b(z,N)−m(z,N)− k f (z,N)] p(z)dz= N[b¯(z¯,N)− m¯(z¯,N)− k f¯ (z¯,N)] , (3.1)where p(z) is the probability distribution of trait values in the prey population, b¯,m¯, and k f¯ are the population mean per capita birth, non-predator mortality, andpredation rates, respectively, and z¯ is the population mean trait value.When phenotypic variance is small the rate of evolution in the mean trait valueis roughly (Lande 1976)dz¯dt=VA∂∂ z¯(1NdNdt)∝∂ b¯∂ z¯− ∂ m¯∂ z¯− k∂ f¯∂ z¯,(3.2)where VA > 0 is the additive genetic variance, which determines the strength ofthe response to selection. Predators could potentially affect prey adaptation by in-51fluencing prey variance, VA, but here we simplify the dynamics and assume thevariance is constant and independent of predation. In the examples that follow,a constant variance results from assuming that offspring trait values are near themean of their parents. This causes an equilibrium to be reached between the vari-ance introduced by segregation and recombination and the variance lost throughrandom mating.We now consider prey evolution in a changing environment. Let there be some‘optimal’ trait value, θ , that maximizes prey growth rate at a given density in theabsence of predators (i.e., z = θ maximizes b−m at a given N). We assume thatthe environmental change causes this optimum to increase linearly at rate δ . Ifthe prey persists the mean prey trait will evolve to track the optimum, eventuallysettling into a steady-state lag (sensu Lynch et al. 1991) with the mean trait valuetypically trailing the optimum by a constant amount. A steady-state is achievedwhen prey density is constant in time (Equation 3.1 is zero) and the rate of evo-lution (Equation 3.2) is equal to the rate of change in the environment, δ . As-suming a stable steady-state exists we can solve for equilibrium population density(Nˆ) and the corresponding steady-state lag [Lˆ = limt→∞L = limt→∞(θ − z¯)]. Theminimum rate of change in the optimal trait value that causes extinction at thissteady-state lag (Nˆ = 0) gives the critical rate of environmental change, δc (sensuLynch et al. 1991). This is the slowest rate of environmental change that determin-istically causes the prey to go extinct (unless the steady-state first loses stability,see discussion in Generalist Example 2).What we ultimately want to know is how increased predation intensity affectsprey persistence, dδc/dk. This will depend on how predators affect prey adaptation,which we turn to first. We begin with a very general description and follow thiswith concrete examples, where we choose specific functional forms for the termsin Equation (3.1).Effect of predators on prey adaptationWe begin with the reasonable assumption that increasing predation intensity de-creases prey density at a given mean lag, ∂N/∂k < 0 (see Abrams 2009 for ex-ceptions). Without loss of generality, we specify the direction in which the trait52is measured such that the mean prey trait value lags behind the optimum. Then asmall increase in the mean trait value would increase the mean prey growth ratein the absence of the predator, ∂ (b¯− m¯)/∂ z¯ > 0. We say that predation selects inthe same direction as the environment when increasing mean prey trait value de-creases predation as well as increases the mean prey growth rate in the absence ofpredators, that is, when ∂ f¯/∂ z¯ and ∂ (b¯− m¯)/∂ z¯ have opposite signs.Despite its negative effect on prey density, predation could help the prey persistif it helps the prey adapt, that is, if increasing the predation intensity increases therate at which prey evolve. Mathematically, predators help prey adapt when0 <ddkdz¯dt0 <∂∂N(∂ b¯∂ z¯)∂N∂k− ∂∂N(∂ m¯∂ z¯)∂N∂k− k ∂∂N(∂ f¯∂ z¯)∂N∂k− ∂ f¯∂ z¯.(3.3)Thus, to know if increasing predation intensity could speed up the rate of preyevolution we need to know the likely signs of the terms on the right hand side ofEquation (3.3).Most intuitively, the last term in Equation (3.3) shows that when predators se-lect in the same direction as the environment (here ∂ f¯/∂ z¯ < 0) – what we call a‘selective push’ – increased predation intensity will help the prey adapt directlythrough selection regardless of whether the predation rate is density-dependent(Figure 3.1). Of course, selection from the predator that opposes selection fromthe environment (a ‘selective pull’, here ∂ f¯/∂ z¯ > 0) will hinder prey adaptation.Even without direct selection via predation (∂ f¯/∂ z¯ = 0), predators can facili-tate adaptation by altering prey demography. Negative density-dependence causesbirth rates to decline with population density, ∂ b¯/∂N < 0. Thus we may expectthat the mean per capita birth rate will increase less with mean trait value at higherdensities, ∂ (∂ b¯/∂ z¯)/∂N < 0. Therefore, with negative density-dependent pop-ulation growth and our initial assumption that predation decreases prey density(∂N/∂k < 0), increasing predation intensity can help prey adapt when selectionand density-dependence act on births (first term in Equation 3.3). Essentially,the predator (or any other form of non-selective mortality) depresses prey den-sity, which elevates prey birth rates, increasing the expected number of selective53kpredationintensityN−m¯mortalityb¯birthf¯predation−+ ∂b¯∂z¯− ∂m¯∂z¯−k ∂ f¯∂z¯+dz¯dtrate of evolution++evolutionaryhydra effect+ selective push(− ∂ f¯∂z¯ > 0)Figure 3.1: Pathways by which predation intensity, k, can affect the rate ofprey evolution, dz¯/dt. Bold lines show our two examples: the evolu-tionary hydra effect (top) and the selective push (bottom). Positive andnegative symbols give the sign of the partial derivative of the right vari-able with respect to the left variable in our examples (e.g., the negativesymbol between k and N indicates that prey density, N, declines withincreasing predation intensity, ∂N/∂k < 0). Increasing predation inten-sity increases the rate of prey evolution (towards larger trait values) viaa specific pathway when the product of the signs along that pathway in a given time and thus accelerating the rate of evolution in a changing en-vironment (Figure 3.1). We call this phenomenon the ‘evolutionary hydra effect’(after the non-evolutionary ‘hydra effect’ described by Abrams 2009), referring tothe ability of the mythical monster to regrow two heads for every one severed byHeracles. In our case, due to selection, the heads that grow back tend to be betteradapted than the one that was severed. The evolutionary hydra effect can occurwhenever generation times can be decreased, allowing more opportunity for selec-tion per unit time. This applies to univoltine species with overlapping generations(e.g., many temperate bird species), whenever the average age of one’s parent isdecreased by mortality. Generation times cannot, however, be decreased in univol-tine species with non-overlapping generations, where generation time is fixed (e.g.,annual plants).54The opposite is true for per capita death rates; with negative density-dependence,increasing density increases death rates, ∂ m¯/∂N > 0, making it likely that deathrates decrease more with increases in trait value at high densities, ∂ (∂ m¯/∂ z¯)/∂N <0. That is, the opportunity for selection via natural mortality is likely to be strongerat larger population sizes (in the absence of predators) than at smaller ones (withpredators). Thus, selection and negative density-dependence acting on prey mor-tality (along with dN/dk) means that predation likely reduces the ability of preyto adapt (the second term in Equation 3.3 is likely negative). However, with posi-tive density-dependence (e.g., when the population size is very small and there areAllee effects) the situation is reversed; increased predation rates could then helpthe prey adapt when selection and density-dependence both act on death, ratherthan birth.Assuming negative density-dependence again, predation can also help the preyadapt if −∂ (∂ f¯/∂ z¯)/∂N < 0 (third term in Equation 3.3). This is most likelywhen per capita predation rates weaken as the population size grows (∂ f¯/∂N < 0)so that predation is more effective – and more selective – when predation has de-pressed prey densities. Whether this occurs depends on the nature of the func-tional response. A type-I functional response implies that density has no effect onper capita predation rates (∂ f¯/∂N = 0), a type-II functional response implies thatdensity has a negative effect on per capita predation rates (∂ f¯/∂N < 0), while atype-III functional response implies that density has a positive effect on per capitapredation rates at low densities (∂ f¯/∂N > 0) and a negative effect on per capitapredation rates at high densities (∂ f¯/∂N < 0). It is therefore possible for increas-ing predation intensity to aid adaptation indirectly through changes in density (asin the evolutionary hydra effect described above) even if prey birth and naturalmortality are not selective (first two terms in Equation 3.3 are zero) but preda-tion is (∂ f¯/∂ z¯ 6= 0). This would occur, for example, with a type-II functionalresponse and predation that selects in the same direction as the environment. Inthis case, the decreased prey density resulting from increased predation intensitycauses an increase in the per capita rate of predation, and since predation selectsin the same direction as the environment it accelerates the evolution of the meanprey trait value towards the optimum (i.e., the evolutionary hydra effect enhancesthe selective push).55Effect of predators on prey persistenceWhile considering the form of Equation (3.3) provides insight into when predationcan help prey adapt (decreased phenotypic lag), satisfying this condition does notnecessarily mean that predation will help prey persist (maintain positive populationdensity) in a more rapidly changing environment. Because predation reduces preydensity, it can lower the critical rate of environmental change, δc, even if the preyare more adapted.From Equation (3.2), we can write the critical rate of environmental change asδc =VA∂g(z¯,k)∂ z¯∣∣∣z¯=z¯c(k), (3.4)where g(z¯,k) = b¯(0, z¯)− m¯(0, z¯)− k f¯ (0, z¯) is the per capita growth rate when rare(invasion fitness) and z¯c(k) is the mean trait value that causes equilibrium popula-tion size to be zero (critical trait value). Letting h(z¯c(k),k) = ∂g(z¯,k)/∂ z¯∣∣z¯=z¯c(k)bethe selection gradient at the critical rate, we can take the derivative with respect topredation intensity on both sides, showing that predation increases the critical ratewhen0 <dδcdk=VA[∂h∂k+∂h∂ z¯c∂ z¯c∂k]0 <VA[∂ 2g∂ z¯∂k+∂ 2g∂ z¯2∂ z¯c∂k]z¯=z¯c(k),(3.5)Restricting ourselves to cases where extinction occurs because equilibrium popula-tion density goes to zero (as opposed to an unstable steady-state), predation helpspersistence when it helps adaptation by bringing the trait closer to the optimum(Equation 3.3 is satisfied) and increases the critical rate of environmental change(Equation 3.5 is satisfied).The first term inside the brackets of Equation (3.5) is a mixed derivative thatdescribes the impact of selective predation on persistence. It is negative when thereis a selective pull, positive when there is a selective push, and zero otherwise. Thesecond term is comprised of two derivatives. One of these is a second derivativethat describes how the shape of invasion fitness changes with mean trait value. Itis zero when invasion fitness depends linearly on trait value, negative when inva-sion fitness increases slower with trait value at trait values closer to the optimum56(concave down), and positive when invasion fitness increases faster with trait valuecloser to the optimum (concave up). The second part of the second term describeshow the critical trait value depends on predation intensity. Since predation is as-sumed to lower per capita growth rate, the critical trait value will increase withincreases in predation intensity, and thus this derivative is positive (i.e., a preypopulation on the brink of extinction needs to become better adapted to the abi-otic environment if it is to persist with more predation). We can therefore reasonthat in the absence of selective predation the evolutionary hydra effect will helppersistence if and only if invasion fitness is concave up at the critical trait value.For example, if selection occurs only through births, the evolutionary hydra effectcannot help persistence if birth rate is a quadratic function of phenotype (concavedown for all trait values), but it can if birth rate is a Gaussian function (concaveup for critical trait values that are sufficiently far from the optimum). In otherwords, small decreases in lag at the persistence boundary must cause large enoughincreases in invasion fitness for the evolutionary hydra effect to help persistence.Recall too that Equation (3.3) must also be satisfied, so that if selection is actingthrough births, so too must density-dependence.We now explore two specific examples, comparing prey populations with andwithout a generalist predator, to demonstrate that, when the environment begins tochange, predation can indeed increase prey persistence by facilitating prey evolu-tion through 1) a selective push and 2) the evolutionary hydra effect. We do sowith analytical expressions for steady-state mean phenotypic lag and equilibriumdensity as well as numerical and simulated temporal trajectories beginning fromdemographic equilibrium in an originally constant environment. These specificexamples confirm the generic statements drawn from Equations (3.3) and (3.5).3.2.2 Generalist example 1: selective pushHere we explore an example where a selective push from the predator can increaseprey persistence. To eliminate the possibility of an evolutionary hydra effect weassume that per capita prey birth rates are independent of trait value and negativelydensity-dependent, b(z,N) = b(N) = bmax(Rtot −N). The per capita birth rate,equal for all individuals, reaches its maximum, bmax > 0, at low prey density, N, and57declines linearly as prey density increases, consistent with decreasing resources.We assume that initial prey density is not greater than the total amount of resourcesin the system, 0≤ N(0)≤ Rtot , which ensures 0≤ N(t)≤ Rtot and a non-negativebirth rate, b(z,N)≥ 0, for all time t ≥ 0. We measure population density in units ofresource content, such that if R∗tot is the density of some limiting nutrient and eachunit of prey density requires density q of this nutrient for life then Rtot = R∗tot/q.We assume the per capita prey death rate is density-independent with stabiliz-ing selection on trait values around an optimum, m(z,N) = m(z) = mmin + γ(θ −z)2. Individuals with the optimum trait value, z = θ , achieve the minimum percapita death rate, mmin > 0. Per capita death rate increases with the square of thedeviation of the trait value from the optimum, with quadratic selection strengthγ > 0. Thus, in the absence of predators there is a single selective pressure, whichdrives the mean prey trait value, z¯, towards the optimum, θ .Finally, the selective push is modelled by assuming that the per capita predationrate on prey with trait value z is k f (z,N) = k f (z) = k[1+ γk(θ − z)2], describing asystem where maladapted prey also suffer more predation. Prey with the optimumtrait value are predated upon least, at per capita rate k≥ 0, and the rate of predationincreases quadratically as prey traits vary from θ . Predation thus produces a secondselective pressure, which magnifies the strength of selection and drives the meanprey phenotype towards the optimum. We have also analyzed a model where theselective push originates from predators preferentially consuming prey with thesmallest trait values (results not shown; the results do not differ qualitatively).With the above assumptions, trait and density-dependence never affect thesame demographic rate, so the first three terms in Equation (3.3) must be zero. Onlythe final term remains, describing the selective push of predation (−∂ f¯/∂ z¯ > 0when z¯ < θ ).We next derive the rate of change in prey density assuming prey trait valuesare normally distributed with phenotypic variance Vz. Following De´barre and Otto(2016) we derive approximations for the rate of evolution and the equilibrium ad-ditive genetic variance in a large population assuming that mating is random andthat offspring inherit the average value of the two parental trait values plus a smallnormally distributed random effect with mean zero and variance α2 (which mimicssegregation and recombination). We then use these approximations to calculate the58equilibrium density, steady-state lag, and critical rate of environmental change.Figure 3.2A shows how the mean prey trait value with a predator (black) is‘pushed’ closer to the optimum than the mean prey trait value without a predator(gray). Even though predation decreases prey density for a given trait value (seeFigure 3.2B at time 0), predation indirectly increases prey density in the long-runthrough the above-mentioned selective push (Figure 3.2B). Throughout, pheno-typic variance remains roughly constant near its predicted equilibrium value (Fig-ure 3.2C). Figure 3.3A shows the various selective pressures experienced by theprey population with (black) and without (gray) predation at the steady-state shownin Figure 3.2A-C.Figure 3.2D-F shows the steady-state lag and equilibrium density and pheno-typic variance as functions of the rate of environmental change for a prey popula-tion with (black) and without (gray) predation. Here we can see that prey popula-tions with a predator persist at higher rates of environmental change than thosewithout predators, demonstrating that a selective push from predators can helpprey persist (Nˆ > 0) in more extreme environments. Finally, it should be notedthat higher rates of predation can depress prey density to the point that the de-mographic cost of predation outweighs the selective push and predation no longerhelps prey persist (i.e., Equation 3.5 is not satisfied because the negative secondterm dominates)3.2.3 Generalist example 2: evolutionary hydra effectHere we show that, even in the absence of a direct selective effect, predation (or anyother form of non-selective mortality) can increase prey persistence when selectionand negative density-dependence act on prey per capita birth rate. This occurs byincreasing the rate of selective events via the evolutionary hydra effect.Let the per capita birth rate have the same form of linear negative density-dependence as in the previous example but now also depend on trait value. Weassume that, at a given density, only individuals with trait value z = θ achieve themaximum per capita, per available resource birth rate, bmax > 0, and that birth ratedeclines to zero as trait values deviate in either direction from θ . Using a Gaussianform of decline, the per capita birth rate is b(z,N)= bmaxExp[−(γ/2)(θ − z)2](Rtot −N),59Optimum phenotype HΘ LMean phenotype HzLSteady-state expectation0 200 400 600 800 100005101520APhenotypeWith predationæææææææææææææææ0 0.01 0.02 ∆c ∆c0. 0.01 0.02 0.03 0.04 predationDensity HNLEquilibrium density HN`L0 200 400 600 800 1000020040060080010001200BDensityæææææææ ææ æææææææææ æ0 0.01 0.02 ∆c ∆c0200400600800100012000.00 0.01 0.02 0.03 0.04 0.05020040060080010001200EEquilibriumdensityVariance HVzLEquilibrium variance HV`zL0 200 400 600 800 10000.0000.0010.0020.0030.0040.0050.0060.007TimeCPhenotypicvarianceææ ææ ææ ææ æ æ æ æ0 0.01 0.02 ∆c ∆c0.0000.0010.0020.0030.0040.0050.0060.00 0.01 0.02 0.03 0.04 0.050.0000.0010.0020.0030.0040.0050.006FEquilibriumvarianceRate of environmental changeFigure 3.2: The selective push. (A,B,C) Temporal dynamics of mean preyphenotype, population density, and phenotypic variance when δ = 0.02.Dashed curves are analytical solutions for equilibrium values. Thickcurves are numerical solutions. Thin curves are results from one simu-lation (obscured by thick curves in A; Appendix B.1). (D,E,F) Equilib-rium dynamics of steady-state phenotypic lag, population density, andphenotypic variance as functions of δ . Curves give analytical results.The dots give simulation results (mean ± 1.96 SE of 10 replicates; Ap-pendix B.1). Parameters: bmax = 0.01, Rtot = 1000, mmin = 0.1, γ = 1,γk = 1, α2 = 0.05, k = 0 (gray), k = 2 (black).where γ > 0 determines how quickly the birth rate declines with trait deviationsfrom θ . We assume the per capita death and predation rates are constants that donot depend on trait values or density: m(z,N) = m and k f (z,N) = k. Thus, with or60Density, NHzLBirth, bHz,NLDeath, mHz,NLPredation, kf Hz,NLz Θ02004006008001000 8 9 10 11APreydensityPercapitarateWith predation02468zWithout predation With predationz Θ02004006008001000 3.5 4.0 4.5 5.0 5.5 6.0PhenotypeBPreydensityPercapitarate0. 3.3: Graphical depiction of selective pressures in our two generalistpredator examples with (black) and without (gray) predation. (A) Theselective push at the equilibrium shown in Figure 3.2A-C. (B) The evo-lutionary hydra effect at the equilibrium shown in Figure 3.4A-C. In Athe per capita birth rate is higher in the absence of predation (grey) be-cause the population is further from the optimum and hence experiencesmore deaths, freeing up resources that increase the birth rate.without predators the prey experience only one selection pressure (through births),although the strength of this selection pressure now varies with available resources,Rtot−N. Because predation affects prey density it also changes the amount of avail-able resources in the system, altering the rate of selective events. Using the same61assumptions as the previous example, we summarize the results below.Figure 3.4A shows that non-selective predation can help prey adapt in a chang-ing environment and Figure 3.4B shows that being better adapted can translate intohigher population density at equilibrium. Figure 3.3B shows the selective pressuresexperienced by the prey population with (black) and without (gray) predation at thesteady-state shown in Figure 3.4A-C. Figure 3.4D then illustrates how the higherpopulation densities with predation allow persistence at faster rates of environmen-tal change. Thus even non-selective predators can help prey persist by increasingthe rate of births and hence the rate of selective events (the evolutionary hydra ef-fect). As in the selective push example, higher rates of predation can depress preydensity to the point that they overwhelm the benefit provided by the evolutionaryhydra effect, hindering prey persistence (the critical lag becomes smaller, causingthe second term in Equation 3.5 to become negative).In this example, not only does the rate of change in prey density directly dependon mean trait value, but the rate of change in the mean trait value directly dependson prey density (the latter was not true in the selective push example). There isthus potential for strong eco-evolutionary feedbacks. In particular, large additivegenetic variances strengthen feedbacks that can cause eco-evolutionary cycles toemerge through a Hopf bifurcation. These cycles can grow and become unstable,causing prey extinction at rates of change smaller than the critical, δc. Such cyclesrequire that evolutionary and ecological changes occur on similar time frames anddo not emerge when genetic variation in the trait is small or predation intensity isstrong.3.2.4 Specialist predationSpecialist predators – those that primarily consume a single prey species – re-quire a sufficient amount of biomass production by their prey to persist. Thus, ifa specialist did not affect prey lag the predator would go extinct at a lower rate ofenvironmental change than its prey (at the point where prey biomass productiondropped below the threshold required for predator persistence), and therefore spe-cialists would not affect prey persistence. However, specialist predators can affectprey lag in the same ways that generalists can. It can therefore be reasoned that, if62Optimum phenotype HΘ LMean phenotype HzLSteady-state expectation0 5000 10 000 15 000 20 000024681012APhenotypeWith predationæææææææææææææ0 0.002 0.004∆c ∆c0. 0.001 0.002 0.003 0.004 0.005 0.0060. predationDensity HNLEquilibrium density HN`L0 5000 10 000 15 000 20 000020040060080010001200BDensityææææææææææ æ æææææ0 0.002 0.004∆c ∆c0200400600800100012000.000 0.001 0.002 0.003 0.004 0.005 0.006020040060080010001200EEquilibriumdensityVariance HVzLEquilibrium variance HV`zL0 5000 10 000 15 000 20 0000.0000.0010.0020.0030.0040.0050.0060.007TimeCPhenotypicvarianceææææææææ æ æ æ æ æ0 0.002 0.004∆c ∆c0.0000.0010.0020.0030.0040.0050.0060.0070.000 0.001 0.002 0.003 0.004 0.005 0.0060.0000.0010.0020.0030.0040.0050.0060.007FEquilibriumvarianceRate of environmental changeFigure 3.4: The evolutionary hydra effect. (A,B,C) Temporal dynamics ofmean prey phenotype, population density, and phenotypic variancewhen δ = 0.0006. Dashed curves are analytical solutions for equilib-rium values. Thick curves are numerical solutions. Thin curves areresults from one simulation (obscured by thick curves in A and B;Appendix B.1). (D,E,F) Equilibrium dynamics of steady-state pheno-typic lag, population density, and phenotypic variance as functions of δ .Curves give analytical results. The dots give simulation results (mean± 1.96 SE of 10 replicates; Appendix B.1). Parameters: bmax = 0.01,m = 0.1, Rtot = 1000, γ = 1, α2 = 0.05, k = 0 (gray), k = 1 (black).a specialist predator reduces prey lag enough to maintain sufficient prey biomassproduction at rates of environmental change greater than that which would allowthe prey to persist on its own, the predator could sustain itself and, by necessity,63help its prey persist.We demonstrate the above logic by explicitly modeling specialist predator den-sity, P, and trait values, zP. In general, per capita rates of prey birth (b), death (m),and predation (k f ) could now depend on the traits and densities of both species.However, our goal here is only to show that the selective push and evolutionaryhydra effect can also help prey persist when the predator is a specialist. In the maintext we present an example where a selective push arises from trait-matching co-evolution. Coevolution is not necessary, however, and we can show that specialistswith a constant trait value, zP, can also promote persistence in a manner similar tothe generalist examples above (via a selective push and/or the evolutionary hydraeffect; results not shown). In all cases the predator promotes prey persistence byincreasing prey adaptedness.To model coevolution with a specialist we let the predation rate depend onboth prey and predator trait values as well prey and predator densities. In themain text we assume trait-matching, such that a predator is best able to catch preywith the same trait value as itself. This could be the case, for example, when zand zP describe spatial locations or times of peak activity, or, alternatively, preda-tor trait values could be defined on this basis (i.e., the trait value of a predator isthe trait value of prey they best consume). The predator’s mean trait value thenevolves toward the mean prey trait value, while selection from predation drivesthe prey’s mean trait value away from it. Specifically, we assume the per preyrate of predation on prey with trait value z by predators with trait value zP isk f (z,zP,P) = k×max[0,1− (γk/2)(z− zP)2]P. At given densities, predation oc-curs at the maximum per prey per predator rate k > 0 when the trait values match.This rate falls off quadratically at rate γk > 0 as the trait values diverge from oneanother. In our analytic approximations we assume that γk is small enough thatf (z,zP,P) remains positive for the vast majority of prey and predators.We assume predators are born at per capita rate ek f (z,zP,P)N, where e de-scribes how efficiently consumed prey are turned into new predators, and die atconstant per capita rate mP. Thus the predator is not directly affected by the chang-ing abiotic optimum, θ , but is indirectly influenced through its effect on the prey.Like the prey, predators are assumed to randomly mate, with their offspring inherit-ing the mean trait value of the two parents plus a random normal segregation effect64with mean 0 and variance α2P. We assume predator trait values remain normallydistributed with mean z¯P and variance Vz,P.The prey’s per capita birth rate is assumed to be b(N,P) = bmax(Rtot −N−P),where we subtract P from Rtot−N to account for resources bound up in predators.The prey’s per capita mortality rate is m(z) = mmin + (γ/2)(θ − z)2, describingstabilizing selection around z = θ . Prey growth rate in this model is thereforesimilar to what it was in the selective push example given above, except that thepredation rate now depends on how close prey trait values are to predator traitvalues (instead of how close they are to θ ) and on predator density. We present abrief summary of our analytical results here.None of the prey’s per capita rates depend on both trait value and prey den-sity, and therefore there can be no evolutionary hydra effect (the first three termsin Equation 3.3 are zero). Instead we find, as expected, that direct selection fromthe coevolving specialist predator (∂ f¯/∂ z¯) can have dramatic effects on prey adap-tation. At low to intermediate rates of environmental change, demographic andcoevolutionary cycles occur (explaining the deviation from equilibrium in Figure3.5A and the large variance in the simulation results in Figure 3.5D at low to in-termediate δ ; evolutionary stability analysis results not shown). At fast rates ofchange, the cycles dissipate and the predator can ‘push’ the prey mean trait valueto evolve in advance of its changing optimum (Figure 3.5A-C). Finally, at yet fasterrates of change, the mean prey trait value falls back towards the optimum (Figure3.5D). The decrease in absolute prey lag at sufficiently fast environmental changesis caused by the selective push of predation (compare gray to black in Figure 3.5D).This push maintains both prey and predator populations at rates of change thatwould cause prey extinction in the absence of the specialist predator (Figure 3.5E).Thus, when the specialist promotes prey persistence, both prey and predator mustgo extinct at the same rate of environmental change (since the prey cannot persistat sufficiently high rates of environmental change without the predator). Interest-ingly, in this example, the critical rate of environmental change occurs not wherethe prey and predator densities simultaneously pass through zero but where theysimultaneously become complex (a catastrophic saddle-node bifurcation). Theseresults imply that small increases in the rate of environmental change can causeseemingly sustainable communities to suddenly go extinct.65Abiotic optimum HΘ LMean phenotypes Hz, zPLSteady-state expectations0 5000 10 000 15 000 20 0000102030405060 APhenotypePredator Prey with predatorææ æææ æ æææ æ æææææ0 0.001 0.003 0.005∆c *-505101520250.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007-50510152025DSteady-statelagPrey without predatorDensities HN, PLEquilibrium densities HN`, P`L0 5000 10 000 15 000 20 000020040060080010001200BDensityææ æ æ æ ææææ æ æ æ æ æ æææææææ æ æ0 0.001 0.003 0.005∆c *0200400600800100012000.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007020040060080010001200EEquilibriumdensityVariances HVz,Vz,PLEquilibrium variances HV`z, V`z,PL0 5000 10 000 15 000 20 0000.0000.0010.0020.0030.0040.0050.0060.007TimeCPhenotypicvarianceæ æ æ æ æ æ ææ ææ0 0.001 0.003 0.005∆c *0.0000.0010.0020.0030.0040.0050.0060.0070.000 0.001 0.002 0.003 0.004 0.005 0.006 0.0070.0000.0010.0020.0030.0040.0050.0060.007FEquilibriumvarianceRate of environmental changeFigure 3.5: Coevolution between the prey (black) and a specialist predator(red). (A,B,C) Temporal dynamics of mean prey and predator pheno-types, population densities, and phenotypic variances when δ = 0.003.Dashed curves are analytical solutions for equilibrium values. Thickcurves are numerical solutions. Thin curves are results from one sim-ulation (gray obscured by thick curve in A; Appendix B.1). (D,E,F)Equilibrium dynamics of steady-state phenotypic lags, population den-sities, and phenotypic variances as functions of δ . Curves give analyt-ical results. The dots give simulation results (mean ± 1.96 SE of 10replicates; Appendix B.1). Asterisk (*) denotes rate of environmentalchange where coevolving prey and predator equilibrium densities be-come complex values. Parameters: b = 0.01, mmin = 0.1, Rtot = 1000,γ = 0.03, γk = 0.5, e = 1, mP = 4, α2 = 0.05, α2P = 0.05, k = 0 (gray),k = 0.01 (black and red).66Even if specialist predators do not directly exert selection on the prey, they canalso allow prey to persist at more extreme rates of environmental change via theevolutionary hydra effect (if they increase prey birth rates and if fitter prey individ-uals are more likely to give birth; results not shown). Again, the prey populationwith predators persists at higher rates of environmental change than it could with-stand without predators because of the increased rate of selective events (reducedgeneration time). When this occurs both prey and predator go extinct at the exactsame rate of environmental change.3.3 DiscussionWe have shown two ways that predators can help prey persist in a gradually chang-ing environment. The first is termed the ‘selective push’ and is the more intuitivephenomenon. Here predation acts to increase the strength of directional selec-tion towards the optimum, and it thus speeds adaptation. Importantly, we haveshown that this increase in the speed of adaptation can help prey persist in moreextreme environments, despite the mortality imposed by predation. Similar ideashave been described to explain the potential effects of both predators (Jones 2008)and competitors (Osmond and de Mazancourt 2013) on a focal population’s abil-ity to adapt and persist in a changing environment. Surprisingly, the second waywe find predators to help prey persist does not require the predator to exert a di-rect selective pressure on the prey. Instead, decreases in prey density feedback toincrease rates of prey evolution. We term the phenomenon of increased prey mor-tality increasing rates of prey evolution the ‘evolutionary hydra effect’ and demon-strate with specific examples that this can indeed lead to increased prey persistence.While our study has been motivated by recent empirical work on predation (Tsengand O’Connor, 2015), the selective push and evolutionary hydra effect can equallyarise from other sources of mortality, for example, from parasitism, competition,additional environmental stressors, or potentially even cannibalism. In particular,Equations (3.3) and (3.5) continue to apply to other sources of mortality whosestrength is also proportional to k.673.3.1 Selective pushThe selective push acts much like an increase in the strength of selection (e.g.,total strength of selection in the example in the main text equals γ + kγk). Pre-vious models of sexually reproducing populations evolving alone in continuouslychanging environments have also shown that increasing the strength of selectioncan help persistence. For example, Lynch and Lande (1993) and Bu¨rger and Lynch(1995, 1997) found the critical rate of environmental change to be maximized atintermediate strengths of selection, as we find here (at intermediate rates of pre-dation, k). In contrast, more recent treatments (Jones, 2008; Mellard et al., 2015)have found intermediate strengths of selection to minimize prey persistence. Inthe case of Mellard et al. (2015) the discrepancy is the result of differing modelassumptions. They assume the prey is monomorphic (except for the occasional ap-pearance and loss or fixation of mutations), and hence there is no decline in meanprey growth rate due to trait variance around the mean (genetic load; Lande andShannon, 1996), which decreases the cost of strong selection. They also assumethat the rate of evolution is proportional to the population-scaled mutation rate, Nµ ,instead of additive genetic variance. Since weakening selection increases equilib-rium population sizes, N, it also increases the response to selection (in contrast, inour model the additive genetic variance reaches an equilibrium that is independentof selection or N). Together, these assumptions allow prey persistence to be maxi-mized at the extremes of selection. The discrepancy with the simulations of Jones(2008) is due to the range of selection strengths he examined. The parameter valuesin Jones (2008) were borrowed from Bu¨rger and Lynch (1995). The latter went onto show, however, that persistence is often maximized at intermediate values whenexamining a wider range of selection strengths. Thus, our results are consistentwith previous quantitative genetic models while demonstrating that predation canbe the cause of the increased selection strength.3.3.2 Evolutionary hydra effectThe evolutionary hydra effect – increasing mortality causing faster evolution – isrelated to previous ideas in the literature. For example, Zeineddine and Jansen(2009) have shown how increasing adult mortality can theoretically increase rates68of evolution and therefore explain why an annual life-history may be favouredover a perennial one in changing environments. Similarly, Kuparinen et al. (2010,and references within) discuss how adaptation to longer growing seasons is ham-pered by low adult mortality in long-lived tree species and use simulations to showthat increasing mortality enhances adaptation. Our study extends this previouswork by including the possibility that predators could be the cause of increasedmortality and, more importantly, by analyzing when the increase in adaptednessbrought about by the evolutionary hydra effect outweighs the mortality cost andhelps species persist.Increased mortality has been shown to increase persistence in models of evo-lutionary rescue. In these models mortality facilitates persistence by decreasingcompetition (Uecker et al. 2014) and maladaptive epistasis (Uecker and Hermisson2016). This increases the probability of adaptation by increasing the probability ofestablishment of beneficial genotypes. These population genetic models assumenon-overlapping generations with fixed generation times, where mortality affectsadaptation by altering survival probabilities. The evolutionary hydra effect de-scribes another way for mortality to increase rates of evolution and aid persistence:by decreasing generation times.Generation times can be reduced in populations with overlapping generations(including, e.g., univoltine insects with resting stages and annual plants with seedbanks) and in populations with non-overlapping generations where the time be-tween generations can diminish (e.g., biennial plants becoming annuals). How-ever, the evolutionary hydra effect cannot act in populations with non-overlappinggenerations where the time between breeding seasons is fixed (e.g., in univoltineinsects and annual plants with no resting stages or seed banks). Non-selectivedeath could then still help prey persist, but not by reducing generation time andonly under arguably more complex scenarios. For example, if decreased popula-tion density disproportionally increased the birth rate of well adapted individualsthen increased random mortality could accelerate evolution and aid persistence.693.3.3 Specialist predators and coevolutionWe have found that specialist predators, like generalists, can help prey adapt andpersist. The main difference between specialist and generalist predators is the pres-ence or absence of demographic feedbacks (as well as evolutionary feedbacks inthe case of a coevolving specialist), where declining prey numbers sustain fewerspecialist predators and thus reduce deaths from predation. These demographicfeedbacks are a double-edged sword. At low predation intensities, the specialistpredator becomes rare and loses its ability to help the prey adapt at high rates ofenvironmental change. In this case, a generalist (i.e., with constant predator den-sity) could nevertheless help the prey adapt and persist. On the other hand, at highpredation intensities, prey can persist in the presence of a specialist predator athigher rates of environmental change than they can with a generalist, because thespecialist becomes rare as the prey becomes less well-adapted and does not inducetoo high a mortality cost. As a consequence, whether a specialist or a generalistpredator better allows prey to persist in rapidly changing environments depends onthe exact model conditions and particularly on the intensity of predation (Figure3.6).There are at least three other theoretical studies suggesting that coevolutionwith specialist predators can, at least temporarily, facilitate prey persistence inchanging environments. The earliest of these is Jones (2008), who uses simulationsto show that coevolution between a specialist predator and its prey can often in-crease the prey population’s persistence time, given that all finite populations havesome chance of going extinct each generation (Bu¨rger and Lynch 1995). While themodel in Jones (2008) differs from ours in a number of ways (e.g., non-overlappinggenerations, density-independent predation), both studies show that predator-preycoevolution can help prey persist when they otherwise would not. Here we ana-lytically clarify why coevolution can help: a selective push from coevolution canreduce prey lags enough to allow prey growth at rates of environmental change thatwould otherwise cause extinction.In another study, Northfield and Ives (2013) show that coevolution betweena specialist predator and its prey can reduce the effect of small environmentalchanges on equilibrium densities of both prey and predator through negative feed-70Prey onlyPrey with specialistPrey with generalist0.000 0.005 0.010 0.015 0.020 0.02502004006008001000Low predation intensity Hk=0.01LAEquilibriumdensity*Prey onlyPrey with specialistPrey with generalist0.000 0.005 0.010 0.015 0.020 0.02502004006008001000High predation intensity Hk=0.05LC*Only generalist helpsSpecialist and generalist help0.000 0.005 0.010 0.015 0.020 0.0250. of environmental changeBAbioticselectionstrength Only specialist helpsSpecialist and generalist help0.000 0.005 0.010 0.015 0.020 0.0250. of environmental changeDFigure 3.6: Prey persistence with a specialist vs. a generalist predator. Shownhere is the case of a non-coevolving specialist predator that causes a se-lective push by predating maladapted prey [replacing f (z,zP,P) in coe-volving case with f (z,P) = k[1+ γk(θ − z)2]P]. (A,B) With low pre-dation intensity the specialist predator becomes rare and does not suf-ficiently push the prey at higher rates of environmental change. (C,D)With high predation intensity the generalist predator exerts too high ademographic cost, while the specialist becomes rare when the prey arerare and can help the prey persist. (A,C) Equilibrium prey densitiesacross rates of environmental change with γ = 0.1. (B,D) Parameterregions where predators help prey persist (dashed line corresponds tothe γ used in A and C). Asterisks (*) indicate the rate of environmen-tal change where the specialist predator and prey equilibrium popula-tion sizes simultaneously become complex values. Parameter values:bmax = 0.01, mmin = 0.1, Rtot = 1000, e = 0.25, mP = 1, γk = 0.1,α = 0.05.backs. We also observe negative feedbacks in our coevolutionary specialist preda-tor model. This is particularly apparent in a constant environment, where theeco-evolutionary system, as parameterized, is predicted to cycle. Here, the meanprey trait departs from the abiotic optimum, decreasing prey growth rate, whichdecreases predator density. As predator density declines, the predator’s selective71pressure on the prey dissipates, allowing the prey to evolve back towards the abi-otic optimum, increasing prey growth rates, and hence predator density. Despitethe similar feedback, direct comparison with Northfield and Ives (2013) is difficultsince they model one step changes in the environment, allowing them to comparemodels with and without evolution. With a continuous environmental change, theabsence of evolution at any nonzero rate of environmental change would cause themean prey trait lag to grow indefinitely, bringing both populations to extinction.We instead compare models with and without a specialist predator (Figure 3.5), acomparison Northfield and Ives (2013) do not make, allowing us to ask whetherthe presence of a predator (not evolution) helps the prey adapt and persist.In a third theoretical study, Mellard et al. (2015) find that a specialist predatorcan increase prey persistence times under “strict conditions”. The strict conditionis that coevolution causes the mean prey trait to evolve in the direction of the en-vironmental change before the environment begins to change, giving the prey ahead-start. This transient effect, not observed in our steady-state results, does notaffect the critical rate of environmental change and hence the ability of large pop-ulations to persist indefinitely in changing environments. That is, Mellard et al.(2015) show that the predator can increase prey persistence time, but not whetherit can increase long-term prey persistence. We extend this result by showing thatthe selective push from predator-prey coevolution can also affect long-term preypersistence.Finally, there is one additional piece of theory that, despite assuming a con-stant environment and no evolution, aligns surprisingly closely with our results.Holt and Barfield (2009) discuss how predation affects the geographic range limitsof prey when assuming that prey become increasingly maladapted away from thecentre of their range. This situation turns out to be roughly analogous to ours whenone replaces time with space. Despite the fact that prey traits are fixed in theirmodels (i.e., there is no evolution), Holt and Barfield (2009) reach many of thesame conclusions. First, they find that generalist predators can expand prey rangeswhen mortality increases prey density (the non-evolutionary hydra effect Abrams2009, a scenario we don’t consider). Second, they find that specialist predatorsmake demographic stability more likely nearer prey range edges (here nearer thecritical rate of environmental change; stability analysis not shown) and maximize72prey density away from its range centre (here away from a constant environment;Figure 3.5E). Finally, Holt and Barfield (2009) find that specialist predators are lesslikely to determine prey range edges than generalist predators but may do so underspecial circumstances. One of these circumstances is sufficient predator dispersal,which, when replacing space with time, uncouples predator survival from localprey densities (acting more like our generalist model). Another special circum-stance is when the specialist predator causes prey to flee and colonize new patchesin a metapopulation, expanding the prey’s range, which is roughly analogous tocausing the prey to evolve faster in our model. Combining the two models to lookat the effect of predators on the evolution of range edges could produce interestingpredictions.3.3.4 Empirical workFew empirical studies examine the effect of predators on adaptation to environ-mental change. A notable exception is Tseng and O’Connor (2015), who raisedDaphnia pulex at a range of temperatures with or without predatory Chaoborus lar-vae. They found that Daphnia populations reared with predators evolved smallerbody sizes than those populations without, potentially increasing the adaptednessof this ectotherm in warmer waters (Atkinson and Sibly 1997). Consistent withthis adaptive hypothesis is the fact that, in warmer waters, Daphnia had higherintrinsic growth rates with predators than without. However, Daphnia populationsizes tended to be smaller with predators than without, at all temperatures, and atthe highest temperature assayed, Daphnia populations with predators went extinctbefore those without, suggesting that Chaoborus hinders Daphnia persistence. In-terpreted in the light of our analysis, the benefits of greater adaptation do not out-weigh the demographic costs (Equation 3.3 is satisfied but Equation 3.5 is not).More experiments in this area are needed to see if the potential selective ben-efits of predation ever outweigh its demographic costs. It would be especially in-teresting to examine the dynamics of prey near their extinction threshold (δc) andsee how this threshold is affected by the presence of predators. This could be doneby adding a predator treatment in experiments with gradual environmental change(e.g., Willi and Hoffmann, 2009) or, perhaps more easily, by adding a predator73treatment in evolutionary rescue experiments (e.g., Bell and Gonzalez 2009). Al-ternatively, predation could be imposed by the experimenter themselves. For ex-ample, a selective push would result from the occasional removal of individualsthat are maladapted. Or, in a system where intraspecific competition is reducingprey birth rate and better adapted prey produce more offspring, the evolutionaryhydra effect could be induced by diluting the population at a higher rate.3.4 ConclusionWe are in need of theory that explains and predicts how communities respond to en-vironmental change (Low-De´carie et al. 2015). With communities comes speciesinteractions, which affect how – and if – populations within communities adapt(Lavergne et al. 2010). Trophic interactions are ubiquitous, and it has been shown,both theoretically (Jones 2008; Mellard et al. 2015; Northfield and Ives 2013) andempirically (Tseng and O’Connor 2015), that predators can help prey adapt. How-ever, whether predators can ever help prey persist has remained unclear. Here wehave theoretically demonstrated that predators can indeed help prey persist, despitethe mortality cost that predation imposes, and have highlighted two ways that thiscan occur. Predators can help prey adapt through a ‘selective push’ or throughthe newly defined ‘evolutionary hydra effect’, whereby increasing prey death ratesaccelerate prey evolution by decreasing generation times. We have thus shownthat predators can catalyze adaptation and allow prey to persist in more extremeenvironments, but it remains an open question how often this occurs in nature.74Chapter 4Evolutionary rescue: alternativefitness functions causeevolutionary tipping points14.1 IntroductionMany populations currently face gradual directional changes in their environment(reviewed in Davis et al., 2005; Hoffmann and Sgro`, 2011; Lavergne et al., 2010;Parmesan, 2006; Visser, 2008). Those populations with limited dispersal and plas-ticity can persist only if they evolve fast enough (Lynch and Lande, 1993). Themaximum rate of environmental change a population can adaptively track – anddemographically tolerate – has recently received considerable theoretical attention(reviewed in Alexander et al., 2014; Kopp and Matuszewski, 2013; Walters et al.,2012).Typically these studies follow a quantitative genetic approach (for alternativessee Bertram et al., 2016; Johansson, 2008). They first assume some unimodal map-ping from phenotype to absolute fitness (the ‘fitness function’). Then, for a given1A version of this chapter has been published as Osmond, M. M., and C. A. Klaus-meier. 2017. An evolutionary tipping point in a changing environment. Evolution71:2930–2941. A supporting Mathematica file and simulation scripts are available at of change in the trait value that maximizes fitness (the ‘environmental opti-mum’), the fitness function is used to derive the rate of evolution in the mean traitvalue and the expected difference between the mean trait value and the optimumat equilibrium (the ‘steady-state lag’). The rate of environmental change that pro-duces a steady-state lag resulting in a population mean growth rate (when rare)of zero is dubbed the ‘critical rate of environmental change’ (Lynch et al., 1991).Critical rates of environmental change are now being estimated and used to predictwhether particular species will survive or go extinct in the face of global climatechange (Aitken et al., 2008; Gienapp et al., 2013; Vedder et al., 2013; Willi andHoffmann, 2009).To the best of our knowledge, all quantitative genetic theory developed so farimplicitly assumes that the maximum rate of environmental change to which apopulation can adapt is determined by demography (i.e., ‘selective load’ sensuLynch and Lande 1993, or ‘demographic constraint’ sensu Gomulkiewicz andHoule 2009). This assumption results from the shape of the specific fitness func-tions used. In particular, Gaussian fitness functions, W (z), are used in models withnon-overlapping generations in discrete time (Bu¨rger, 1999; Bu¨rger and Lynch,1995, 1997; Charlesworth, 1993; Chevin et al., 2010a; Gomulkiewicz and Houle,2009; Marshall et al., 2016; Matuszewski et al., 2015), while quadratic fitness func-tions, r(z), are used in models with overlapping generations in continuous time(Aguile´e et al., 2016; Lynch et al., 1991; Lynch and Lande, 1993; Pease et al.,1989; Polechova´ et al., 2009). These are equivalent given log(W ) = r (Crow andKimura, 1970, Chapter 1) and have presumably been chosen for mathematical con-venience (e.g., they maintain a normal trait distribution) as well as their ability toapproximate – when near the optimum – any smooth fitness function imposingstabilizing selection (Lande, 1976). This particular fitness function is therefore arelatively mild assumption under the historical paradigm of weak selection, but itbecomes a strong yet biologically arbitrary assumption when environments changequickly enough that populations find themselves considerably maladapted.The rate of evolution in a mean trait value can be approximated by the prod-uct of additive genetic variance and the selection gradient (Lande, 1976). Withoverlapping generations in continuous time, the selection gradient is the deriva-tive of mean fitness with respect to mean trait value (Lande, 1982, equation 11),76while with non-overlapping generations in discrete time, it is the derivative of thelogarithm of mean fitness (Lande, 1976, equation 7). Thus the strength of selec-tion becomes a linear function of mean phenotypic lag in all models listed above.This implies that the strength of selection has no limit and therefore that, givena large enough steady-state lag, evolution can proceed arbitrarily fast (as long asadditive genetic variance remains non-zero). Population persistence is then onlydetermined by the population mean growth rate at the steady-state lag that causesevolution to proceed as fast as the environment changes, i.e., there is a critical rateof environmental change at which populations cease to grow.Here we show that the existence of a critical rate of environmental change de-pends on the choice of fitness function. Moreover, decreases in the strength ofselection (the slope of the fitness function) with increasing maladaptation causelocal maxima in the rate of evolution. These local maxima can create an ‘evolu-tionary tipping point’, where rates of environmental change less than the tippingpoint result in stable steady-state lags and population persistence while rates ofenvironmental change greater than the tipping point lead to an apparent existen-tial crisis: the population ceases to adapt as the selective pressure relaxes, causingthe steady-state lag to rapidly increase and the population to go extinct. This ex-istential crisis is brought about by what is known as a saddle-node bifurcation.Many dynamical systems are thought to experience saddle-node bifurcations, fromglobal finance to climate, and there is a substantial literature devoted to developinggeneric early-warning signs to detect impending bifurcations (reviewed in Schef-fer et al., 2009). Two common early-warning signs are increased variance andlag-1 autocorrelation, both of which are caused by slow recovery from perturba-tion, or a ‘critical slowing down’, and have been detected in climate and ecolog-ical data (Lenton, 2011; Scheffer et al., 2009). We therefore use simulations tosee if generic early-warning signs have the potential to detect evolutionary tippingpoints, granted one has extensive time series of difficult-to-measure parameterssuch as mean phenotypic lag. Finally, we show how the existence of an evolution-ary tipping point induces ‘evolutionary hysteresis’, which can create an extinctiondebt: transitory increases in mean phenotypic lags (e.g., due to sudden environ-mental changes) can initiate the above mentioned existential crisis, with extinctionoccurring many generations later even if the rate of environmental change returns77to moderate levels. Overall, our results demonstrate that our current understandingof evolutionary rescue in directionally changing environments is highly sensitiveto the – relatively unknown – shape of fitness functions as populations becomeincreasingly maladapted.4.2 Methods and Results4.2.1 A general modelFollowing Lynch and Lande (1993), we consider a well-mixed and randomly mat-ing population of short-lived, hermaphroditic individuals with overlapping gener-ations in continuous time. Individuals are characterized by a quantitative trait, z,which is the sum of genetic and environmental effects, z= g+e. The genetic effectis determined by a large number of equivalent, additive, and freely-recombiningdiploid loci. The environmental effect is an independent random normal variablewith mean 0 and variance σ2e . The population mean trait value is then the meangenetic effect, z¯ = g¯, while the phenotypic variance is the sum of additive geneticand environmental variance, σ2z = σ2g +σ2e .Ignoring frequency-dependence for simplicity, let r(z) be the per capita growthrate when rare (hereafter fitness) of individuals with quantitative trait z. Let density-dependence affect all individuals equally. The expected rate of change in the cur-rent mean trait value due to natural selection on standing genetic variation is thenapproximately the product of standing genetic variance and the selection gradient,E[dz¯/dt] = dg¯/dt ≈ σ2g ∂ r¯/∂ g¯, where r¯ is the population mean growth rate. Weassume additive genetic variance remains constant at some equilibrium (which weestimate in specific examples below and compare to simulations).Now assume there is some trait value, θ , that maximizes fitness, r(z), andlet this value increase linearly in time at rate k, such that its value at time t isθ(t)= kt. A quasi-steady-state is then achieved when the expected rate of evolutionmatches the rate of change in the environment, dg¯/dt = k. If at this steady-state theexpected population mean growth rate is positive, r¯> 0, the population will persist.If instead the growth rate is negative, r¯ < 0, the rate of environmental change is toofast and the population goes extinct. The rate of environmental change that causes78an expected growth rate of zero, r¯ = 0, at steady-state is termed the critical rate ofenvironmental change, kc.However, there is also the – yet to be discussed – possibility that such a steady-state does not exist. In particular, a steady-state does not exist if the rate of evolu-tion has some maximum and the rate of environmental change is beyond this. Moreimportantly, if, over the range of phenotypic lags that allow population persistence,the rate of evolution is maximal at some intermediate lag, then population growthrate at steady-state will not decline continuously towards zero as the rate of envi-ronmental change increases (see Appendix C.1 for a more technical discussion).Instead, the long-run population growth rate will jump from a potentially largepositive number to a potentially very negative number as the rate of environmen-tal change increases through the maximum rate of evolution. Technically, this isdue to an inflection point in the fitness function causing a saddle-node bifurcation.When this bifurcation causes extinction we refer to the maximum rate of evolutionas an ‘evolutionary tipping point’. When an evolutionary tipping point exists it isthe meaningful predictor of persistence (disregarding stochastic factors), and thereis no critical rate of environmental change as defined by Lynch and Lande (1993).To demonstrate the effect of changes in the shape of the commonly assumedfitness function more concretely, we will next compare results arising from the‘traditional’ fitness function to those arising from an alternative fitness functionthat imposes a limit on the rate of evolution. In doing so we do not mean to implythat our alternative fitness function is necessarily always more biologically relevantthan the traditional. Our alternative fitness function is used only to demonstrate thatsubtle changes in the shape of the fitness function may have dramatic effects on ourpredictions for adaptation and persistence in a rapidly changing world.4.2.2 The traditional fitness functionWe begin with the traditional fitness function in continuous time, r(z) = rm− (θ −z)2/(2σ2w) (Lynch and Lande, 1993, equation 1), where rm is the maximum percapita growth rate and σ2w determines the strength of stabilizing selection (strongerif smaller) around θ . Averaging over the phenotypic distribution, we find that pop-ulation mean growth rate, r¯, is reduced by the magnitude of the mean phenotypic79lag, l¯ = θ − z¯, and by standing genetic variance (Lande and Shannon, 1996), e.g.,when the mean trait value matches the optimum, l¯ = 0, the mean growth rate isr¯m = rm−σ2z /(2σ2w). Furthermore, this function implies that as mean trait valuedeparts from the optimum population growth rate declines ever more rapidly, andthere is no bound on how negative it can become (gray curve in Figure 4.1A).The expected rate of evolution given the current mean genotypic value is dg¯/dt =σ2g (θ− g¯)/σ2w (Lynch and Lande, 1993, equation 5). This is a linear function of theexpected mean phenotypic lag, E[l¯] = θ − g¯, and therefore as lag increases so toodoes the rate of evolution, without bound (gray curve in Figure 4.1B). Thus, there isalways a solution to the quasi-steady-state equation dg¯/dt = k, i.e., there is alwayssome expected mean lag, E[l¯] = lˆ, that produces the required rate of evolution.In this particular case the steady-state lag is lˆ = kσ2w/σ2g (gray curve in Figure4.1C). Evaluating the population mean growth rate at this lag gives the expectedlong-run population growth rate for an infinitely large population in a deterministicenvironment. Increasing rates of environmental change cause a smooth decline inthis long-run growth rate (gray curve in Figure 4.1D). We can therefore solve forthe rate of environmental change, k, that makes the long-run growth rate 0, givingthe critical rate of environmental change, kc = σ2g√2r¯m/σ2w (Lynch and Lande,1993, equation 11).4.2.3 An alternative fitness functionHere we alter the assumption that fitness declines increasingly fast as trait valuesdepart from the optimum. Instead, we depict a scenario where, far from the opti-mum, small departures from the optimum have smaller and smaller fitness conse-quences. This could result from selection becoming weaker with increasing mal-adaptation (for which there is some evidence; Agrawal and Whitlock, 2010), whichin turn could be caused by a lower bound on fitness (i.e., there is some maximumrate at which a population can decline). For example, when selection acts onlythrough birth rate, which cannot be negative, while death rate (m > 0) is fixed,Malthusian fitness is bounded below by −m. However, we would like to empha-size that growth rates do not have to be bounded below for evolutionary tippingpoints to exist – all that is required is an inflection point.80Consider an alternative fitness function r(z)= rm−d[1− exp(−(θ − z)2/(2σ2w))].This is a Gaussian fitness function (in continuous time) with maximum growth raterm at z= θ and minimum growth rate rm−d, achieved as lags tend to plus or minusinfinity. For comparison, the alternative fitness function has been constructed suchthat when d = 1 it is equivalent to the traditional fitness function, to second order,when trait values are near the optimum. Averaging over the phenotypic distribu-tion, we find that the population mean growth rate (black curve in Figure 4.1A) hasan inflection point at mean lag E[l¯] =V 1/2, where V = σ2w+σ2z +σ2θ .The expected rate of evolution is dg¯/dt = dσ2gσwE[l¯]exp[−E[l¯]2/(2V )]/V 3/2(black curve in Figure 4.1B). The rate of evolution is no longer a linear functionof expected mean lag, E[l¯]. Instead, there is a maximum rate of evolution, ktip =dσ2gσw exp(2)/V , at the inflection point, E[l¯] =V 1/2.When the rate of environmental change is less than this maximum rate of evolu-tion, k< ktip, the steady-state lag is lˆ =(V wk)1/2 (solid black curve in Figure 4.1C),where wk is the solution to wkewk = (kV )2/(dσ2gσw)2 (i.e., wk(x) is the Lambert Wfunction, and here x = (kV )2/(dσ2gσw)2; Lehtonen, 2016). If this lag remains bi-ologically valid (real) at the point where the expected long-run population growthrate becomes zero, there is a critical rate of environmental change, kc, that deter-mines persistence. If, on the other hand, there is no valid steady-state lag that givesa population growth rate of zero then there is no ‘critical rate of environmentalchange’, as typically defined (Lynch and Lande, 1993). Instead, it is the maxi-mum rate of evolution that determines persistence (with weak selection this occurswhen d[1− exp(−1/2)] < rm < d; the upper bound is required to ensure the pop-ulation goes extinct as lag tends to infinity), and the maximum rate of evolution isan ‘evolutionary tipping point’ (black curves in Figure 4.1D).When the rate of environmental change is less than the maximum rate of evolu-tion the population mean growth rate is r¯= rm−[1−σw exp(−lˆ/(2(σ2w+σz)2))]>0. This can be substantially positive right up to the tipping point (where lˆ =V 1/2)when the maximum growth rate is large, rm≈ d. However, as soon as the rate of en-vironmental change increases above the maximum rate of evolution, the mean lagincreases quickly without bound, leading to a population growth rate of r¯≈ rm−d,and therefore rapid extinction when the maximum population growth rate is small,rm << d. In any case, at the evolutionary tipping point, long-run population growth81rates go from positive values to negative values without ever crossing zero, caus-ing what may appear to be highly sustainable populations to rapidly begin to goextinct.TraditionalAlternative2 4 6 8 10-0.4- 0.02 0.04 0.06 0.08 0.100246810CSteady-statelagEvolutionary tipping point0 2 4 6 8 phenotypic lagBSelectiongradientkΣg20.02 0.04 0.06 0.08 0.10-0.4- of environmental changeCritical rate of environmental changeFigure 4.1: Visual overview of the modelling approach. Population meangrowth rates, r¯, shown in A, are derived by integrating the traditional(gray) and alternative (black) fitness functions, r(z), over the phenotypicdistribution, p(z). Taking the derivative of mean population growth ratewith respect to the mean trait value, dr¯/dz¯, gives the selection gradientsshown in B. Setting the the rate of evolution equal to the rate of changein the environmental optimum (σ2g dr¯/dz¯ = k; where the dashed line in-tersects the solid curves in B) gives the steady-state lags, lˆ, shown in C.With the traditional fitness function all steady-state lags are stable (filledcircles in B and solid lines in C), while those that are on the decreasingportion of the selection gradient with the alternative fitness function areunstable (open circle in B and dashed lines in C). Evaluating populationmean growth rate at a stable steady-state lag gives the long-run popula-tion growth rates shown in D. The rate of change that causes a long-rungrowth rate of zero is the critical rate of environmental change. Becausethe long-run population growth rate with an alternative fitness functionswitches sign without crossing zero at the bifurcation point in C, we callthis rate of environmental change an evolutionary tipping point. Param-eters: rm = log(2), σ2w = 9, σ2e = 1, σ2g ≈ 0.18, and d = 1.824.2.4 SimulationsWe next use stochastic, individual-based simulations to (i) compare the dynamicsarising from the traditional and alternative fitness functions, (ii) examine genericearly-warning signs of approaching tipping points, and (iii) demonstrate the con-sequences of evolutionary hysteresis.Simulation methodsWe use discrete time simulations with non-overlapping generations (as describedin Bu¨rger and Lynch, 1995), which allows us to compare our results to previousstudies and provides us with analytical predictions for the additive genetic vari-ance (equations 14 and 15 in Bu¨rger and Lynch, 1995) as well as empirically jus-tified parameters (Bu¨rger and Lynch, 1995). To convert our continuous time mod-els into discrete time, we set the expected number of offspring per parent to B =exp(rm) and the probability of survival to adulthood W (z) = exp[r(z)− rm], suchthat growth rates in the absence of density-dependence are equivalent, BW (z) =exp[r(z)] (Crow and Kimura, 1970, Chapter 1). All scripts written in Python(Python Software Foundation, version 3.5; ).Briefly, each individual’s trait is determined by n additive, freely-recombiningdiploid loci plus a random normal environmental effect with mean 0 and variance1. All simulations are initiated as in Bu¨rger and Lynch (1995); we create 5 ancestralalleles at each locus, their effects being random normal variables with mean 0 andvariance (0.1α)2. The first generation of individuals are then created by randomlychoosing two ancestral alleles for each locus, with replacement. Simple densitydependence then acts by randomly choosing K individuals if there are more thanK. These pair at random (potentially leaving one individual out) and each pairproduces 2B offspring by fair Mendelian transmission. Each gamete mutates withprobability nµ . If it does mutate one locus is chosen at random and a randomnormal effect, with mean 0 and variance α2, is added. Viability selection then acts,with survival probability W (z). A maximum K surviving offspring become theparents of the next generation. The first 1,000 generations are used as a burn-inwith k = 0. Simulations continue until the population goes extinct or the maximumnumber of generations is reached (11,000 in Figures 4.2 and 4.4; 201,000 in Figure834.3).Comparing the dynamics arising from traditional and alternative fitnessfunctionsFigure 4.2 shows the effect of the rate of environmental change on evolution andpersistence with the traditional (A-E) and alternative fitness functions (F-J). PanelsA-C and F-H show that our analytical predictions (broken curves; discrete timeanalysis not shown) for steady-state lag, equilibrium additive genetic variance, andpopulation mean growth rate perform fairly well for those populations that persist(black circles). In particular, the simulation results are intermediate between ourpredictions using the neutral (dotted) and stochastic-house-of-cards (dashed) ap-proximations for the genetic variance (equations 14 and 15 in Bu¨rger and Lynch,1995), which therefore provide reasonable bounds. Comparing Figure 4.2B to thecircles in figure 6 in Bu¨rger and Lynch (1995) further suggests that our simulationmethod is accurate.With the traditional fitness function, population growth rates decline as the rateof environmental change increases (Figure 4.2C), as expected from the analyticaltheory. However, in contrast to analytical expectations, the growth rates of sur-viving populations do not reach values close to zero. Thus, even with a traditionalfitness function we see a dynamic similar to that expected from an evolutionary tip-ping point: a small increase in the rate of environmental change causes populationswith a relatively large growth rate to suddenly begin to go extinct. This dynamic islikely caused by a negative feedback between genetic variance, which is constantin the analytical theory, and mean lag (as described in Bu¨rger and Lynch, 1995).When genetic variance declines, the population evolves slower and the mean lagincreases. Vice versa, when the mean lag increases, selection becomes strongerand genetic variance declines. Since large lags cause low growth rates, this feed-back can spiral to extinction. The extinction spiral can be initiated by either about of reduced genetic variance caused by random genetic drift or a period ofincreased mean lag because beneficial genotypes fail to arise by chance (given seg-regation and mutation are random events). This spiral is therefore reminiscent of“mutational meltdown” (Lynch and Gabriel, 1990), where genetic drift increasesthe probability of fixing deleterious alleles. The extinction spiral observed here, in84a changing environment, additionally involves the loss of genetic variance due togenetic drift (including a reduced probability of maintaining beneficial alleles) anda deterministic decrease in the rate of beneficial mutations (∼ nµNB per generationwhen the lag is sufficiently large).With the alternative fitness function, population growth rates of surviving pop-ulations also fail to reach values near zero as the rate of environmental changeincreases (Figure 4.2H), this time as expected form the analytical theory. Instead,growth rates suddenly drop from well above zero to the minimum, negative growthrate (dot-dashed line). In addition, the rate of environmental change that causesthis sudden drop in growth rate is roughly what we expect the evolutionary tippingpoint to be given that genetic variance is intermediate between the two analyticalpredictions.Panels D-E and I-J further show how the transition from persistence to extinc-tion is fairly abrupt for both fitness functions. Although, with these parameters, thetransition might be slightly more abrupt in the presence of an evolutionary tippingpoint, the traditional fitness function exhibits similarly sharp transitions as carryingcapacity, and thus effective population size, is increased (results for K = 1024 notshown). An increase in the sharpness of the transition from persistence to extinc-tion with larger population size is also demonstrated in figure 2B in Bu¨rger andLynch (1995).Early-warning signs of evolutionary tipping pointsTwo common, generic early-warning signs of saddle-node bifurcations are increasesin lag-1 autocorrelation and in temporal variation (Lenton, 2011; Scheffer et al.,2009). If these metrics can reliably predict a nearby evolutionary tipping pointthey may be useful in pinpointing at-risk populations whose population growthrates do not advertise the possibility of imminent extinction (Figure 4.2H).Generic early-warning signs are only predicted to work when a gradual changein a parameter brings the system closer to a saddle-node bifurcation (Boettiger andHastings, 2012). We therefore ran simulations where the rate of environmentalchange, k, increased from 0 by a small amount each generation. Panels A andB of Figure 4.3 show how mean phenotypic lags (black) increase and population85growth rates (gray) decrease as the rate of environmental change speeds up overtime, for both the traditional fitness function and the alternative fitness function(10 replicates for each). Panels B-F show the changes in the early-warning signs:temporal variation and lag-1 autocorrelation (calculated within each replicate us-ing non-overlapping windows of 3000 generations, each data point 100 genera-tions apart). As measured by Kendall rank correlation coefficients (Dakos et al.,2008), temporal variance increases in all cases (all τ > 0; panel G) and the in-crease in variance is not more consistent when approaching the evolutionary tip-ping point than it is when approaching the critical rate of environmental change[two-sided t-test comparing Kendall’s τ in variance between traditional and alter-native fitness functions: T =−0.48, p= 0.64, df= 12.41 (mean lag) and T = 1.75,p= 0.11, df = 10.84 (population growth)]. An increase in temporal variance there-fore does not provide a reliable signal of nearby evolutionary tipping points. Onthe other hand, the Kendall rank correlation coefficients in lag-1 autocorrelation areFigure 4.2: Discrete-time, individual-based simulation results with tradi-tional (A-E) and alternative (F-J) fitness functions. In discrete time thetraditional fitness function is W (z) = exp[−(θ − z)2/(2σ2w)] (Bu¨rgerand Lynch, 1995, equation 1) and the alternative fitness function isW ∗(z) = exp [d(W (z)−1)]. “Population growth rate” is the number ofoffspring surviving viability selection (before density-dependence) di-vided by the number of parents, minus one. “Fraction extinct” is thenumber of replicates that go extinct before the end of the simulation(generation 11,000). In A-C and F-H, circles give mean values over thelast 10 generations for each replicate simulation, or over all generationssince the burn-in if populations went extinct in less than 10 generationsfollowing the burn-in. Gray circles are replicates that went extinct be-fore the end of the simulation. Ten replicates are shown for each rateof environmental change. Broken curves in A-C, and F-H give analyticresults using the stochastic-house-of-cards (dashed) and neutral (dot-ted) approximations for genetic variance (equations 14 and 15 in Bu¨rgerand Lynch, 1995). The dot-dashed curve in H is the minimum growthrate, approached as mean lag goes to infinity. Parameters as in Bu¨rgerand Lynch (1995): B = 2, σ2w = 9, σ2e = 1, K = 512, µ = 2× 10−4,α2 = 0.05, n = 50, and d = 1.86ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 0.402468 AMeanphenotypiclagTraditionalëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 BGeneticvarianceëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 0.4-1.0-èèèèèèèèèèèèèèèèèèèèèèèèèè0.0 0.1 0.2 0.3 DFractionextinctëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 0.40200040006000800010000Rate of environmental changeEGenerationextinctÈextinctëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 0.402468101214 FAlternativeëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 Gëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 0.4-1.0-èèèèèèèèèèèèèèèèèèèèèèèèèè0.0 0.1 0.2 0.3 Iëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë0.0 0.1 0.2 0.3 0.40200040006000800010000Rate of environmental changeJ87generally greater when approaching the evolutionary tipping point than when ap-proach the critical rate of environmental change [panel H; two-sided t-test compar-ing Kendall’s τ in lag-1 autocorrelation between traditional and alternative fitnessfunctions: T =−3.18, p = 0.01, df = 18 (mean lag) and T =−2.89, p = 0.01, df= 13.05 (population growth)]. However, the majority of the τ’s for lag-1 autocor-relation are negative except those for population growth rate with the alternativefitness function, but the mean of this distribution is not significantly different fromzero (two-sided t-test: T = 1.31, p = 0.22, df = 9). Thus, a consistent increasein the lag-1 autocorrelation of growth rate may provide a hint that a populationis approaching an evolutionary tipping point, but the absence of this pattern sayslittle.Figure 4.3: Generic early-warning signs of tipping points. Here the rate ofenvironmental change, k, gradually increases from 0 by 10−6 pheno-typic units every generation, eventually causing extinction. With thetraditional fitness function (A-C) there is no saddle-node bifurcationand extinction occurs as the rate of environmental change approachesk = 0.175, as in Figure 4.2. On the other hand, with the alternative fit-ness function (D-F) there is a saddle-node bifurcation and extinction iscaused by an evolutionary tipping point near k= 0.125, as in Figure 4.2.Nevertheless, in both cases the temporal variance in mean phenotypiclag (black) and population growth rate (gray) tend to increase (B,E) andKendall rank correlation coefficients, τ , do not differ significantly be-tween the two fitness functions (G; details in text). C and F show thedynamics of lag-1 autocorrelation in mean phenotypic lag and popula-tion growth rate for both fitness functions, and the Kendall rank cor-relation coefficients (H) indicate that a consistent increase in the lag-1autocorrelation of population growth rate may be the best predictor of anapproaching evolutionary tipping point for this set of parameters (detailsin text). Shown are ten replicate simulations for each fitness function,with parameters as in Figure 4.2. Variance and lag-1 autocorrelation aremeasured for each replicate separately, using non-intersecting windowsof 30 consecutively recorded time points, each 100 generations apart.880 50 000 100 000 150 000 200 00002468AMeanphenotypiclagTraditional-1.0- Populationgrowthrate0 50 000 100 000 150 000 200 00002468DMeanphenotypiclagAlternative-1.0- Populationgrowthrate0 50 000 100 000 150 000 200 0000. 0 50 000 100 000 150 000 200 0000. 50 000 100 000 150 000 200 000-1.0- 50 000 100 000 150 000 200 000-1.0-æMean lagTraditional AlternativePopulation growth rateTraditional Alternative0.780.800.820.840.860.88 GKendall'sΤTemporal varianceMean lagTraditional AlternativePopulation growth rateTraditional Alternative-0.3-0.2- HKendall'sΤLag-1 autocorrelationEvolutionary hysteresisIn the presence of an evolutionary tipping point, a population experiencing a slowlychanging environment, k < ktip, is expected to attain a sustainable steady-state lag.Deterministically, it will maintain increasing yet sustainable steady-state lags as89the rate of environmental change increases, until the rate of environmental changeincreases beyond the tipping point, ktip < k. Weakening selection then causes thesteady-state lag to make a discontinuous jump (or be lost entirely), and the pop-ulation begins to go extinct. However, even if we ignore demographics and ex-tinction, the dynamics as we decrease the rate of environmental change throughthe tipping point are not the same. For example, with the alternative fitness func-tion used here, when the rate of environmental change is beyond the tipping point,ktip < k, the mean lag quickly increases towards infinity as selection becomes van-ishingly weak. Decreasing the rate of environmental change below the tippingpoint, k < ktip, then only results in a stable steady-state lag if the current mean laghas remained small enough to produce a rate of evolution greater than the currentrate of environmental change. Otherwise the mean lag falls outside the basin ofattraction of the stable steady-state lag, where selection is too weak to allow it tocatch-up. Since the dynamics of the system passing through the tipping point in onedirection are not the same when passing through in the opposite direction, we cansay that the state of the system depends on its history, which is called hysteresis.Because in this case hysteresis involves an evolving trait, we call the phenomenonof the attainment of a steady-state lag depending on the past history of environ-mental change ‘evolutionary hysteresis’. Hysteresis has been described in otherevolutionary contexts, which differ from ours by involving feedbacks with demog-raphy; temporary reductions in the size of habitat patches can cause permanentlosses of genetic polymorphism (Kisdi and Geritz, 1999) and temporary increasesin the rate of migration between habitat patches can cause permanent reductions inpopulation size (Ronce and Kirkpatrick, 2001).Now considering demographics in our case, note that a short period of fast en-vironmental change, ktip < k, can cause eventual extinction, even after the rate ofenvironmental change has been reduced below the tipping point, k < ktip. In otherwords, evolutionary hysteresis produces an extinction debt. Extinction debts havealso been predicted in non-evolving communities of competitors exposed to habi-tat destruction (Tilman et al., 1994) and in evolving communities of competitorsexposed to gradually changing environments (Norberg et al., 2012), but neither ofthese debts are caused by evolutionary hysteresis and both are only predicted tooccur when the environment remains in its changed state.90Evolutionary hysteresis can also be induced by a sufficiently large jump in theoptimum or mean trait value, as either of these can displace the mean lag fromthe basin of attraction of a sustainable steady-state lag. Figure 4.4 shows howa large jump in the optimum trait value can result in evolutionary rescue in theabsence of evolutionary tipping points (panels A-D) but evolutionary hysteresisand an extinction debt in their presence (panels E-F). In this example, the optimumtrait value increases by a small amount each generation (k = k1 < ktip) for the first5000 generations. The optimum then makes a much larger jump at generation5000, and from there continues to increase at the original rate (k = k1). Regardlessof whether there is a tipping point, the large jump in the optimum trait value atgeneration 5000 causes mean lags to increase so much that populations begin todecline. However, in the absence of a tipping point, the increase in mean lag alsocauses the strength of selection, and hence the rate of evolution, to increase, whichrescues half of the replicates from extinction. In sharp contrast, the evolutionarytipping point causes selection to become weaker when the mean lag is increasedat generation 5000. The rate of evolution thus slows and the mean lag increasesdramatically, causing 9/10 replicates go extinct (the mean lag of one lucky replicatedoes not escape the basin of attraction; dotted line in panel E). For these parametervalues, extinction tends to occur ≈ 300 generations after the jump in the optimum,meaning that short term environmental perturbations can lead to extinctions far intothe future (i.e., an extinction debt).4.3 DiscussionAdaptive evolution requires population persistence, heritable variation, and selec-tion. Previous authors have shown how persistence (e.g., Bu¨rger and Lynch, 1995;Lynch and Lande, 1993) and variation (Gomulkiewicz and Houle, 2009) can con-strain evolution. However, because of the specific fitness functions commonlyassumed in theoretical quantitative genetics for the sake of mathematical conve-nience, the idea that selection can also constrain evolution has, up till now, largelybeen overlooked. In particular, we have shown that when the strength of selectiondoes not uniformly increase with maladaptation, selection itself can be the limitingfactor determining the ability of a population to evolve and persist in the face of91directional environmental change. With limiting selection, a qualitatively differ-ent persistence threshold arises, a difficult to detect evolutionary tipping point thatgives rise to an extinction debt. This is particularly worrying given that all currentquantitative genetic predictions effectively use the same specific fitness function,which assumes selection is never limiting.One obvious question that follows from our work is what fitness functionslook like in nature. Much of our knowledge about the shape of empirical fitnessfunctions comes from four main sources: selection gradient analysis (Lande andFigure 4.4: Evolutionary hysteresis prevents evolutionary rescue and createsan extinction debt. Here the optimum trait value increases gradually(k = 0.1), experiences a sudden jump (5 phenotypic units) at generation5000, and from there continues to increase at the gradual rate (k = 0.1).With the traditional fitness function (A-D), the sudden increase in meanlag at generation 5000 causes an increase in the strength of selectionand hence in the rate of evolution, rescuing populations from extinc-tion. With the alternative fitness function (E-F), the mean lag increasesto values that are often just beyond the basin of attraction of the steady-state lag at k = 0.1 (dotted line in E, using the neutral approximation forgenetic variance; k= 0.1 is beyond the tipping point with the stochastic-house-of-cards approximation for genetic variance). (F) The rate of evo-lution then declines (except in one lucky replicate that does not escapethe basin of attraction), causing further increases in the mean lag, whichfurther decreases the rate of evolution, and so on, leading to an apparentexistential crisis. Broken lines show the maximum rate of evolution us-ing the neutral (dotted) and stochastic-house-of-cards (dashed) approx-imations for genetic variance. (G) The ever increasing mean lag lowersthe population mean growth rate, eventually reaching values below re-placement (horizontal line). (H) This drop in population growth ratesultimately, some∼300 generations later, results in extinction. The hori-zontal line is the maximum number of parents, K. Here the fitness func-tions (see Figure 4.2 caption) are multiplied by (1−d′), the probabilitythat an optimally adapted individual survives viability selection. Thisgeneralization gives more flexibility in minimum growth rate withoutaffecting the strength of selection. Parameters as in Figure 4.2, exceptB = 3 and d′ = 0.1.924600 4800 5000 5200 540001020304050AMeanlagTraditional4600 4800 5000 5200 5400- 4800 5000 5200 5400-1.0- 4800 5000 5200 5400020040060080010001200GenerationDNumberofsurvivingoffspring4600 4800 5000 5200 540001020304050EAlternative4600 4800 5000 5200 5400- 4800 5000 5200 5400-1.0- 4800 5000 5200 5400020040060080010001200GenerationHArnold, 1983), cubic spline analysis (Schluter, 1988), aster analysis (Shaw andGeyer, 2010; Shaw et al., 2008), and mutation accumulation/reverse genetics (re-viewed in de Visser and Krug, 2014). Selection gradient analysis is a linear orquadratic regression of fitness on trait value. Thus, even if fitness was measuredas growth rate or lifetime fitness (r or W ≈ exp(r), respectively) it would not bepossible to detect potential tipping points (inflection points in r or in log(W )), and93hence is of little value here. Cublic spline analysis removes the parametric con-straint, and thus could suggest the presence of inflection points if one measuredlifetime fitness (e.g., Re´ale et al., 2003; Wilson et al., 2005). However, most cu-bic spline analyses relate only one aspect of absolute fitness (e.g., survival) to traitvalue (e.g., Figure 4 in Reimchen and Nosil, 2002). Conflicting selection at otherlife-stages (e.g., Robinson et al., 2006) could therefore drastically change the shapeof this function. Meanwhile, aster analysis is designed to calculate lifetime fitnessand can simultaneously estimate fitness functions (e.g., Figure A2 in Shaw et al.,2008). However, aster analysis fits a quadratic as the fitness function (parametricbootstrap on a scaled measure of fitness; Shaw et al., 2008) and therefore may alsomiss inflection points. Thus, combining lifetime fitness estimates from aster withnonparametric cubic spline analysis – along with experimentally-induced environ-mental change (e.g., Weis et al., 2014) or phenotypic manipulation (e.g., Simons,2009; Sinervo et al., 1992) to probe the tails of fitness functions – is one promisingway to identify potential evolutionary tipping points. Finally, mutation accumula-tion and reverse genetics can be used to construct mutant genotypes and evaluatetheir fitness, producing incredibly detailed fitness landscapes of microbial popu-lations in the lab (e.g., Figure 2 in Bank et al., 2016). Beginning from near theoptimal genotype and with fitness measured as population growth rate, a patternof antagonistic (positive) epistasis between deleterious mutations (i.e., each addi-tional mutation adds a smaller detrimental effect to r = log(W )) would indicatethat selection gets weaker with maladaptation and therefore that an evolutionarytipping point might exist. It has been suggested that antagonistic epistasis is morelikely in organisms with simpler genomes, where there is less genetic robustness(Sanjua´n and Elena, 2006) – suggesting such organisms might be more likely toexperience evolutionary tipping points – but it is unclear if this result will hold upto more data (Agrawal and Whitlock, 2010). It is worth noting that sterilizing orlethal mutations (in particular those that cause W (z) = 0 or r(z)→−∞), which aredifficult to detect in studies that do not construct mutants (e.g., mutation accumu-lation), create strong synergistic (negative) epistasis (e.g., Lalic´ and Elena, 2012)and hence reduce the possibility of tipping points induced by limiting selection. Atthe same time, these mutations impose their own kind of tipping point by puttingan irreversible end to all lineages that acquire them.94In the process of illustrating how limiting selection can cause evolutionary tip-ping points, we unexpectedly discovered a sudden transition from relatively largepositive growth rates to extinction with small changes in the rate of environmen-tal change in simulations of the ‘traditional’ quantitative genetics model (Figure4.2C). This transition is caused by a negative feedback between genetic varianceand maladaptation (Bu¨rger and Lynch, 1995), a process akin to mutational melt-down (Lynch and Gabriel, 1990) but with a stronger dependence on the mainte-nance of genetic variance and the acquisition of beneficial mutations, both of whichare necessary for populations to persist in changing environments. The extinctionspiral observed here therefore differs from evolutionary tipping points, which arecaused by negative feedbacks between maladaptation and the strength of selection(opposite to above), and which are expected to occur even in infinitely large popu-lations and when genetic variance is constant. The expected effects also differ, as isexemplified in Figure 4.4, where it is shown that only the evolutionary tipping pointstrongly diminishes the probability of evolutionary rescue for these parameter val-ues. While it has been noticed that simulated populations tend to go extinct at ratesof change less than the critical in the traditional model, and the reasons for it havebeen discussed (Bu¨rger and Lynch, 1995), the implications for detecting popula-tions near extinction thresholds has not been appreciated. Just as predicted near anevolutionary tipping point, small changes in the rate of environmental change in thetraditional model can cause populations with relatively large positive growth ratesto suddenly go extinct, giving little information on how to, for example, triagepopulations of conservation concern. Thus, while critical rates of environmentalchange estimated from simple analytical models may give us rough estimates ofthe conditions under which extinction or persistence will occur, the added com-plexities of a dynamic genetic variance and limiting selection add caution to theirinterpretation and use.95Chapter 5Evolutionary rescue: a theory forits genetic basis15.1 IntroductionOur understanding of the genetic basis of adaptation is rapidly improving due to thewidespread use of experimental evolution and genetic sequencing (see examples inBell, 2009; Schlo¨tterer et al., 2015; Stapley et al., 2010). These empirical advanceshave been developed in parallel with a growing body of population genetic theory(reviewed in Orr, 2005). One key observation arising from both the theory anddata follows the Pareto principle: the majority of fitness effects are due to substi-tutions at a minority of loci (Orr, 2005). This is not in contradiction with the factthat much of the genome may underlie phenotypic variation (Boyle et al., 2017);the genetic basis of standing genetic variation may differ from the genetic basis ofadaptation (Bell, 2009), for instance due to pleiotropy or opposing selection con-straining allele frequency changes at many loci (Hoffmann and Willi, 2008). ThePareto principle appears to be especially true with rapid evolution driven by strongselection (Bell, 2009), suggesting that we may be able to build a predictive theoryof how populations respond to novel environments (Bell, 2009).Under sufficiently challenging conditions populations will decline in size. Rel-1A version of this chapter is being prepared for publication with coauthors Guillaume Martin,Ophe´lie Ronce, and Sarah Otto96atively rapid adaptation may then be required for population persistence, a phe-nomenon termed “evolutionary rescue” (Bell and Gonzalez, 2009; Gomulkiewiczand Holt, 1995). It is at this very upper limit of challenge that much of our knowl-edge about the genetic basis of adaptation is derived, because the genotypes thatgrow in the face of such challenges (hereafter “rescue genotypes”) can be easilyisolated from non-mutants and because of the applied need to prevent drug resis-tance. For example, the genetic basis of adaptation is well known for bacterial drugresistance (reviewed in MacLean et al., 2010), anti-fungal resistance (reviewed inRobbins et al., 2017), and anti-viral resistance (reviewed in Yilmaz et al., 2016).Potential rescue genotypes, unfiltered by their establishment probabilities whenrare, can be isolated using fluctuation tests (Luria and Delbru¨ck, 1943) and theirfitness subsequently assayed (reviewed in Bataillon and Bailey, 2014). Assayingfitness in the challenging environment used to isolate mutants (i.e., in the selectivemedia) then provides the distribution of fitness effects of potential rescue genotypes(e.g., Kassen and Bataillon, 2006; MacLean and Buckling, 2009).While there is a large amount of data on the genetic basis of evolutionary res-cue and the fitness effects of potential rescue genotypes can be readily measured,theory has so far focused almost solely on determining whether or not rescue willoccur (reviewed in Alexander et al., 2014). Extending theory on evolutionary res-cue to predict its genetic basis and resulting fitness effects could be helpful for anumber of reasons. For instance, such a theory might be able to answer fundamen-tal questions, such as how predictable and parallel evolutionary rescue is. Further,we may be able to use this theory to infer whether evolutionary rescue has recentlyoccurred, providing a tool to assess the frequency with which evolutionary rescueoccurs in nature. Such an extended theory on evolutionary rescue might also leadto new perspectives on how to prevent drug resistance and how to measure theeffectiveness of drugs (i.e., whether rescue was likely or not).While most models of evolutionary rescue ask what the probability of rescueis given a fixed genetic basis, a few vary the genetic basis enough to make someinference about how evolutionary rescue is likely to happen at the genetic level.In the context of pathogen emergence, Alexander and Day (2010) showed that theprobability of getting the m mutations required for rescue (when the first m− 1have no effect) depends on how likely it is to gain multiple mutations simultane-97ously; populations in the most challenging environments (lowest starting R0) aremuch more likely to rescue by multiple simultaneous mutations (e.g., mutations ata few loci) than by many single steps (e.g., mutations at many loci), while the dif-ference is negligible in less challenging environments. The two-locus, two-allelemodel of Uecker and Hermisson (2016) also allows some flexibility in how res-cue can occur; for instance, rescue by a mutation at each locus can be more likelythan rescue by mutation at a single locus with standing genetic variation, frequentrecombination, and a near lethal wildtype. Wilson et al. (2017) do not allow thegenetic basis of evolutionary rescue to vary in their one-locus two-allele model,but they show that multiple rescue events can occur simultaneously (i.e., a soft se-lective sweep) during mild environmental challenges, a genetic signature that hasbeen used to assess the efficiency of HIV drug treatment (Feder et al., 2016). Mod-els of (potentially multidimensional) quantitative traits tracking gradually movingoptima have predicted a great deal about the genetic basis of adaptation to envi-ronmental change (Kopp and Hermisson, 2007, 2009a,b; Matuszewski et al., 2014,2015), e.g., alleles of intermediate effect are most likely to fix first, in contrast tolarge effect alleles under sudden environmental change (Orr, 1998), because smalleffect alleles take a longer time to fix while large effect alleles must wait longer tobecome beneficial. One major difference between these moving-optimum modelsand evolutionary rescue following sudden environmental change is that the formerassume constant population sizes and thus their predictions lack demographic in-fluence. Finally, Anciaux et al. (2018) has recently modelled evolutionary rescuefollowing an abrupt environmental change with Fisher’s geometric model (Fisher,1930). This is the same model that has been used to describe the genetic basis ofadaptation in the absence of demography (reviewed in Orr, 2005). Anciaux et al.(2018) concentrated on the probability of rescue by one mutational step but alsoidentified a characteristic level of initial maladaptation below which rescue wasmore likely to occur by a hard selective sweep rather than soft one.Here we follow the lead of Anciaux et al. (2018) in using Fisher’s geomet-ric model to describe adaptation following an abrupt environmental change thatinstigates population decline. The key difference between these and earlier mod-els using Fisher’s geometric model (e.g., Orr, 1998) is that we assume variationin absolute, rather than relative, fitness. Variation in absolute fitness implies that98population sizes are not constant, allowing a feedback between demography andevolution, which could strongly impact the resulting genetic basis of adaptation.In contrast to Anciaux et al. (2018), our focus here is on the genetic basis of evo-lutionary rescue. In particular, we ask: (1) How many mutations is evolutionaryrescue likely to take given an initial level of maladaptation? and (2) What is theexpected distribution of fitness effects (and growth rates) among the survivors?5.2 Methods5.2.1 Fisher’s geometric modelWe model the genotype to phenotype to fitness map using Fisher’s geometric model,originally introduced by Fisher (1930, p. 38-41) and reviewed by Tenaillon (2014).Each genotype is characterized by a point in n-dimensional phenotypic space, ~z.We assume selection is stabilizing in each dimension, phenotype ~o having maxi-mum fitness. When the population is far from ~o, as would be the case followingenvironmental change, selection is effectively directional. The n phenotypic axesare chosen and scaled such that fitness can be described by a multivariate nor-mal function derived from a normal distribution with variance 1 in each dimensionand no covariance, but height Wmax. Thus the fitness of phenotype ~z is W (~z) =Wmax exp(−||~z−~o||2/2), where ||~z−~o||=√(z1−o1)2+(z2−o2)2+ ...+(zn−on)2is the Euclidean distance of~z from the optimum. Here we are interested in absolutefitness; we take ln[W (~z)] = m(~z) = mmax− ||~z−~o||2/2 to be the continuous timegrowth rate of phenotype~z (m is for Malthusian fitness, we reserve r for extensionswith recombination). We ignore density- and frequency-dependence in m(~z) forsimplicity. The fitness effect, i.e., selection coefficient, of phenotype z′ relative to zin a continuous time model is exactly s= log[W (z′)/W (z)] =m(z′)−m(z) (Martinand Lenormand, 2015). This is approximately equal to the selection coefficient indiscrete time (W (z′)/W (z)−1) when selection is weak (W (z′)−W (z)<< 1).To make analytical progress we use the isotropic version of Fisher’s geometricmodel, where mutations (in addition to selection) are assumed to be uncorrelatedacross traits. Universal pleiotropy is also assumed, so that each mutation affectsall phenotypes. In particular we use the “classic” Fisher’s geometric model (Har-99mand et al., 2017), where the probability density function of a mutant phenotypeis multivariate normal, centred on the current phenotype, with variance λ in eachdimension and no covariance. This implies a continuum-of-alleles (Kimura, 1965).We further assume that each mutation arises at a new locus, i.e., that there are in-finite sites (Kimura, 1969). This implies that multiple mutants can in principle beidentified (e.g., double mutants will have mutations at two loci, barring hitchhik-ing) and allows extensions to recombination (e.g., lineages with different mutationscan recombine to form double mutants). Mutations are assumed to be additive inphenotype, which causes negative epistasis in fitness as the fitness function hasnegative curvature (d2m/d~z2 = −1). An obvious and important extension wouldbe to relax the simplifying assumptions of isotropy and universal pleiotropy, butwe leave this for future work. Note that with a constant population size the non-isotropic case corresponds closely to an isotropic model with a lower number ofphenotypic dimensions (Martin and Lenormand, 2006); whether a similar equiva-lence applies under evolutionary rescue is unknown.Given this genotype to fitness mapping and distribution of mutation effects onphenotype, the distribution of fitness effects and growth rates of new mutations canbe predicted exactly. Let m′ be the growth rate of a genotype and m the growth rateof a mutant derived from it. Then let so = mmax−m′ be the selective effect of amutant with the optimum genotype and s=m−m′ the selective effect of the actualmutant. The probability density function of the selective effects of new mutations,s, is then given by equation 3 in Martin and Lenormand (2015). Converting togrowth rate, the probability density function for mutant growth rate m from anancestor with growth rate ma is (cf. equation 2 in Anciaux et al., 2018)f (m|ma = m′) = 2λ fχ2n(2(mmax−m)λ,2(mmax−m′)λ)(5.1)where χ2n (c) is a non-central chi-square deviate with n degrees of freedom andnoncentrality c > 0 and fχ2n (x,c) is its corresponding probability density functionfor positive real x (equation 26.4.25 in Abramowitz and Stegun, 1972). We willuse the shorthand f (m|ma = m′) = f (m|m′) from now on.While equation 5.1 is exact, it can be unwieldy for integration. In what followswe will often use the displaced gamma approximation (Martin and Lenormand,1002006)f˜ (m|m′) =[mmax−mα(m′)]β (m′) exp[−(mmax−m)/α(m′)](mmax−m)Γ(β (m′)) , (5.2)whereα(m′) = λ1+2ε(m′)1+ ε(m′), β (m′) =n2[1+ ε(m′)]21+2ε(m′)(5.3)with ε(m′) = (mmax−m′)/s¯ where s¯ = nλ/2 is the expected value of |s| for a newmutation from the optimum (Martin and Lenormand, 2006). Figure D.1 shows theaccuracy of this approximation against simulated mutants.5.2.2 Wildtype dynamicsWe are envisioning a scenario where N0 clonally reproducing wildtype individ-uals experience a sudden environmental change, causing a negative growth ratemwt < 0. With non-overlapping generations, the expected number of wildtype in-dividuals in generation t is Nt = N0W ( ~zwt)t = N0 exp(mwtt). Summing over all t,the total number of wildtype individuals expected before extinction is ∑∞t=0 Nt =N0/(1− exp(mwt)). We will treat the wildtype dynamics deterministically, whichis reasonable for large enough N0 (Martin et al., 2013). Note that we have ignoredmutations when calculating the cumulative number of wildtypes; this is valid aslong as mutations are rare and/or small.Each individual mutates with probability U (i.e., when U << 1 it is the pergenome per generation mutation rate in fitness in the given environment). Let pbe the probability a mutant lineage is either directly or indirectly (through furthermutations) “successful”, that is, produces a lineage with positive growth rate thatescapes random loss when rare. The expected number of successful lineages isthen N0(1− exp(m))−1U p; we assume this is small such that the realized numberis roughly Poisson distributed. The probability of rescue is then one minus theprobability that no mutant lineage is successful, which is approximatelyP = 1− exp(− N01− exp(mwt)U p). (5.4)What remains is to find p, the probability a mutant lineage successfully rescues thepopulation.1015.2.3 Mutant lineage dynamicsHere we follow the lead of Weissman et al. (2010) and Uecker and Hermisson(2016) in approximating our discrete time process with a continuous time branch-ing process (see chapter 6 in Allen, 2010). Consider a birth-death process, whereindividuals give birth at rate b and die at rate d. One can then obtain the probabilitygenerating function for the number of individuals at a given time, n(t), given theinitial number, n(0). We are primarily interested in new mutant lineages, n(0) = 1.The generating function then allows us to calculate the probability a lineage per-sists at least until time t and the distribution of n(t) given it does (see below).To convert between birth and death rates and our Malthusian parameter, m, weneed to specify a model of reproduction. We will assume a Poisson number of off-spring per generation, with expectation exp(m). Following Uecker and Hermisson(2016), we then convert between models by equally distributing the growth rate mbetween birth and death, b = (1+m)/2 and d = (1−m)/2, such that m = b− dand the continuous time process exhibits the same amount of drift as the discretetime process (and matches discrete-time simulations well; Uecker et al., 2014). Wecan now report the necessary results in terms of m.Denoting the extinction time as T , the probability a mutant with growth rate mpersists until time t is approximately (Weissman et al., 2010)P(T > t)≈2/t t << |1/m|−2mexp(mt) t >>−1/m > 0 (5.5)As pointed out in Weissman et al. (2010) (whose equation A2 differs from equa-tion 5.5 by a factor 2 because they assume binary fission instead of birth-death),the distribution of persistence times has a long tail (like 1/t) until being cut off(declining exponentially) at t =−1/m.Given a lineage persists until t, the distribution of n(t) is roughly (Weissmanet al., 2010)P(n(t) = n|n(t)> 0)≈2(1/t)(1+2/t)−n t << |1/m|−2m(1+m)n−1 t >>−1/m > 0 (5.6)102As pointed out in Weissman et al. (2010) (whose equation A3 only differs fromequation 5.6 by constants), the distribution of n(t) is approximately geometric forsmall or large t, implying n(t) is very unlikely to be greater than the minimum of tand −1/m.In what follows we will use two key results: while t < |1/m|, (1) the probabilityof persisting to time 2t is roughly 1/t and (2) the expected number of individuals ina lineage at time 2t, given persistence, is nearly ∑n P(n(t) = n|n(t)> 0)n≈ t. Notethat if a mutant lineage has growth rate m = 0 these key results hold for any t. Inbranching process theory, a lineage that has a growth rate exactly at replacement,m = 0, is called “critical” (Allen, 2010, p. 167). Lineages that are expected togrow (m > 0) are “supercritical” and those that are expected to decline (m < 0)are “subcritical” (Allen, 2010). The expected dynamics of a critical lineage is thesame as the expectation for a neutral lineage in a population of constant size (cf.Weissman et al., 2009). Likewise, one can refer to lineages that are effectivelycritical, while t < |1/m|, just as deleterious or beneficial lineages in populations ofconstant size are effectively neutral while rare (e.g., Weissman et al., 2009). We saythat lineages “behave critically” while t < |1/m| and that a lineage is “sufficientlycritical” if it behaves critically until some predefined time T (we will derive this Tbelow). Lineages that are not sufficiently critical are either sufficiently subcritical(if m < 0) or sufficiently supercritical (if m > 0); for simplicity we refer to thesesimply as subcritical and supercritical.5.2.4 Probability of establishmentHere we follow Martin et al. (2013) in approximating our discrete time (Poisson)process with a Feller diffusion to get the probability a lineage with positive growthrate m << 1 establishes:pest(m) = 1− exp(−2m). (5.7)This reduces to the 2(s+mwt) result in Otto and Whitlock (1997) when m= s+mwtis small, which further reduces to the famous 2s in a population of constant size,mwt = 0 (Haldane, 1927). Using p = pest(m) ≈ 2(s+mwt) in equation 5.4 werecover the probability of rescue in a one-locus two-allele model (equation 3 in Orr103and Unckless, 2008). When m≤ 0 the mutant cannot establish, pest(m) = Individual-based simulationsSimulations are initialized with N0 wildtype individuals, each of which have phe-notype ~zwt and Malthusian fitness mwt = m(zwt). Each generation, an individualwith phenotype~z produces a Poisson number of offspring (e.g., by mitosis), withmean W (~z), and dies. Each offspring mutates with probability U , with mutationsdistributed as described above (see Fisher’s geometric model). The simulationcontinues until extinction or until there are ≥100 individuals with positive growthrate, at which point the population is considered rescued. We record the percentof replicates that are rescued and the unique growth rates among the survivors toqualitatively compare with our analytical and numerical results below.5.2.6 Data AvailabilityAnalytical and numerical results were derived in Mathematica (version 9.0;Wolfram Research Inc., 2012). Simulations were written in Python (version 3.5;Python Software Foundation, ResultsIn this model rescue can occur in an infinite number of ways. The most directpath to rescue is when the immediate descendant of a wildtype has a mutation thatendows it with a positive growth rate and this lineage establishes without need offurther mutation; we call this 1-step rescue. It is also possible for a mutant lineageto fail to establish without further mutation (either due to a negative growth rateor stochastic loss when rare) but persist long enough to produce a second mutationthat, when combined with the first mutation, endows a positive growth rate andsuccessful establishment; we call this 2-step rescue. This can be extended to k-step rescue more generally. 1-step rescue in Fisher’s geometric model has recentlybeen analyzed by Anciaux et al. (2018). We give a brief alternative derivationof 1-step rescue, which allows us to compare 1-step vs. 2-step rescue and derivethe distribution of fitness effects in rescued lineages (which was not a focus ofAnciaux et al., 2018). Figure 5.1 illustrates the 1-step and 2-step pathways; as104we will soon see, the 2-step path will require us to differentiate between caseswhere the first mutant is sufficiently critical (i.e., has a growth rate near enoughto replacement, m ∼ 0) or subcritical (i.e., has a growth rate far enough belowreplacement, m << 0). We end by demonstrating how the probability of k-steprescue rapidly falls off with increasing k (provided U < 1).5.3.1 The probability of 1-step rescueThe probability a mutant from the wildtype has a growth rate in the small range[m,m+dm) and rescues the population is f (m|mwt)pest(m)dm. Integrating over allpossible growth rates, the per mutant probability of 1-step rescue is approximatelyp1 =∫ ∞−∞f (m|mwt)pest(m)dm. (5.8)This can be integrated numerically. Alternatively, we can approximate f (m|mwt)with the displaced gamma distribution, f˜ (m|mwt) (equation 5.2). Using f˜ (m|mwt)we can evaluate equation 5.8, giving a closed form approximation for the per mu-tant probability of 1-step rescue:p˜1 =1−[e−2mmax(1−2αwt)−βwt[Γ(βwt)−Γ(βwt ,mmax(1−2αwt)αwt)]+Γ(βwt ,mmaxαwt)]/Γ(βwt),(5.9)where αwt = α(mwt) and βwt = β (mwt) and Γ(a,x) is the incomplete gamma func-tion (equation 6.5.3 in Abramowitz and Stegun, 1972). Figure D.2 compares thisapproximation to individual-based simulations. Figure D.3 compares our approx-imation to the alternative derivation of Anciaux et al. (2018), who approximateequation 5.8 by assuming small mutational effects and by using the Laplace methodof approximating integrals. For a discussion of the implications of equations 5.8and 5.9 we refer readers to Anciaux et al. (2018).105m > 0m < 01-step2-step critical2-step subcriticalFigure 5.1: Genetic paths to evolutionary rescue. Here we show an n = 2dimensional phenotypic landscape. Continuous time growth rate (m)declines quadratically from the centre, becoming negative outside thegray “persistence zone”. Red, orange, and blue circles show wildtypephenotypes with increasingly negative growth rates (i.e., more severeenvironmental challenges). Light gray circles show first step mutationsalong 2-step paths. Dark gray circles show rescue genotypes. The 2-step path is split into cases where the first mutant is sufficiently critical(i.e., near replacement, m ∼ 0) or subcritical (i.e., declining quickly,m << 0).1065.3.2 The DFE following 1-step rescueWe can now use our results to predict the distribution of mutant growth rates inrescued populations. From Bayes theorem, the distribution of mutant growth rates,m, given 1-step rescue isg(m|1-step rescue) = f (m|mwt)pest(m)∫ ∞−∞ f (m|mwt)pest(m)dm≈ f˜ (m|mwt)pest(m)/ p˜1.(5.10)This approximation is verified against simulated mutants in Figure 5.2, where wehave translated from growth rate to fitness effect, using s = m−mwt , to emphasizethe difference with the constant population size case.Some insight can be gained from our approximation in equation 5.10 and itsrepresentation in Figure 5.2. Any mutant with fitness effect less than s = mwt can-not establish (because then m< 0, making pest(m) = 0) while the maximum fitnesseffect is primarily determined by mutational variance (there is also a hard boundat s = mmax−mwt). Therefore, as the wildtype becomes more maladapted withmore severe environmental challenges, the distribution of fitness effects becomescharacterized by a larger mean and smaller variance. This increasing concentra-tion of greater fitness effects can be thought of as an extension to the conclusionsreached by Fisher (1930) and Kimura (1983). Fisher showed that the phenotypi-cally smallest mutants are most likely to be beneficial and Kimura then pointed outthat the smallest of these will be lost by drift and thus mutations of intermediatesize are most likely to be observed; we have shown that rescue increases the meanmutational size further by imposing a minimum for growth (Figure D.4). The anal-ogous arguments in terms of s are shown in Figure 5.2: in the case of a constantpopulation size (mwt = 0), the probability density function of beneficial fitness ef-fects is shown by the dashed line while the probability density function of fitnesseffects of mutants that establish is shown in gray. Adding the extra condition thata mutation must be large enough to rescue a declining population increases the ex-pected fitness effect further (red and yellow). This latter observation aligns withthe prevailing idea that rapid adaptation, often driven by strong selection in novelenvironments, is typically achieved by mutations of relatively large effect (Bell,1072009).mwt = -0.2mwt = -0.1mwt = 00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350510152025fitness effect, s = m - mwtfrequencyFigure 5.2: The distribution of fitness effects, s = m−mwt , following 1-steprescue. Shown are histograms of simulated mutants that establish ac-cording to pest(m) (equation 5.7). Also shown is the analytic approxi-mation for the probability density function given in equation 5.10, con-verted from growth rates to fitness effects. The special case of mwt = 0(constant population size) is shown for comparison. The dashed curvegives the probability density function of beneficial fitness effects whenmwt = 0. Parameters: n = 4, s¯ = 0.01, mmax = The probability of 2-step rescueWe now turn to the probability of 2-step rescue, where a mutant lineage arisingfrom the wildtype fails to establish until gaining a second mutation. Consider amutant lineage with growth rate m1 that has just arisen from the wildtype. Theprobability that it does not establish without further mutation is 1− pest(m1). Theprobability that one individual in this lineage produces a lineage with an additionalmutation that goes on to establish is U pm1 , where U is the probability of mutationand pm1 =∫ ∞−∞ f (m|m1)pest(m)dm is the probability the double mutant establishes.108What remains is to find out how many individuals will arise in the first mutant lin-eage given m1. To do this, we first condition on the lineage persisting for a givennumber of generations, and we will then examine the probability of such a persis-tence time. As pointed out above (see Mutant lineage dynamics), the distributionof persistence times and lineage sizes depends on a mutant’s growth rate, m1. Wefirst examine the case where a mutant lineage with growth rate m1 goes extinct atsome time less than |1/m1|. Such a lineage “behaves critically” over its entire tra-jectory (just as a deleterious mutation is “effectively neutral” while rare in constantpopulation size models; Weissman et al., 2009). This then tells us what range offirst step mutant growth rates can lead to 2-step rescue while the first mutant lin-eage behaves critically. The remaining range of growth rates, where the first step issubcritical, will then be treated differently (first step mutants that are supercritical,m >> 0, are ignored as they tend to arise only when 1-step rescue is already verylikely).2-step critical A mutant lineage arising from the wildtype persists for time 2T <<|1/m1| with a probability of roughly 1/T (equation 5.5). The expected number ofindividuals at time 2t ≤ 2T is then nearly t (from equation 5.6). We now follow theheuristic approach that Weissman et al. (2009) took for fitness-valley crossing in atwo-locus two-allele model. Summing over the 2T generations of persistence, theexpected cumulative number of individuals in the lineage is roughly ∑2Tt=1 t/2≈ T 2for reasonably long-lived lineages, T >> 1. Each of these individuals will pro-duce a double mutant lineage with probability U . The probability that a mutantlineage that persists for 2T generations leads to 2-step rescue is then T 2U pm1 , untilthis approaches 1 at large T . Because the probability of rescue given persistenceto 2T increases as T 2 while the probability of persistence to 2T only declines as1/T , rescue will be dominated by rare, long-lived lineages (as in valley-crossing;Weissman et al., 2009). In particular, because T 2U pm1 reaches 1 when T becomesT ∗ = 1/√U pm1 , any lineage that persists for that long while behaving criticallywill almost surely lead to rescue. Considering only these lucky long-lived lin-eages, the probability of 2-step rescue is the probability of a lineage surviving toT ∗, which from equation 5.5 is roughly 2/T ∗ = 2√U pm1 given a first step mutantwith growth rate satisfying |m1|< 1/T ∗ =√U pm1 arises. First step mutants with109growth rates satisfying m1 < −√U pm1 are too subcritical to reach lineage sizeslarge enough to rescue reliably and must be treated separately (see next section).To calculate the rate at which first step mutants lead to 2-step rescue whilebehaving critically we need to integrate the probability that a mutant lineage hasgrowth rate m1, does not itself establish, but produces a double mutant that does,f (m1|mwt)[1− pest(m1)]2√U pm1 , over the m1 for which |m1| <√U pm1 . Lettingm∗+ be the positive solution to |m1| =√U pm1 and m∗− the negative solution, theprobability a mutant lineage leads to 2-step rescue through a sufficiently criticalfirst step is then roughlyp(c)2 =∫ m∗+m∗−f (m1|mwt)[1− pest(m1)]2√U pm1dm1. (5.11)Our next task is to derive an analytic approximation for equation 5.11. Theprobability a mutant lineage deriving from the first mutant lineage will rescue ispm1 , which, importantly, depends on m1. However, when the mutation rate is nottoo large the initial mutant lineage must persist for a long time in order to reli-ably lead to 2-step rescue while behaving critically, meaning that these first stepmutants must be very close to critical, m1 ∼ 0. We can therefore approximatepm1 with pm1=0 =∫ ∞−∞ f (m|0)pest(m)dm, which does not depend on m1, and setm˜∗ =√U pm1=0 as the boundary of sufficiently critical. If we additionally use thegamma approximation for the distribution of mutations (equation 5.2), the proba-bility a mutant lineage leads to 2-step rescue through a sufficiently critical first stepis roughlyp˜(c)2 =∫ m˜∗−m˜∗f˜ (m1|mwt)[1− pest(m1)]2√U pm1=0dm1. (5.12)This can be integrated, giving a large but analytic solution, which is compared withnumerical evaluation of equation 5.11 in Figure D.5. This approximation tends tooverestimate the probability of rescue through sufficiently critical first step muta-tions as mutation rates increase. With a higher mutation rate the first step does nothave to persist as long to almost surely give rise to a successful double mutant,which allows mutants with more negative growth rates to contribute to 2-step crit-ical rescue. However, as the first step becomes more subcritical, the probability110that a second mutation is large enough to rescue the population declines, but ourapproximation (pm1 ≈ pm1=0) does not take this into account (it assumes the muta-tions produced by a sufficiently critical mutant are the same as those produced bya perfectly critical mutant). Fortunately, this overestimation will often occur whenrescue is already nearly assured, making equation 5.12 a reasonable approximationacross most U .2-step subcritical When the first step mutation is subcritical, such that m1 <−√U pm1 , it cannot reach a cumulative size large enough to rescue reliably. In-stead, such a lineage is unlikely to persist longer than −1/m1 generations. Givensuch a lineage does persist until −1/m1 it produces a cumulative ∼ 1/(2m21) in-dividuals, leading to rescue with a probability near [1/(2m21)]U pm1 . Consideringonly these most long-lived lineages, which persist for −1/m1 generations withprobability near −2m1, the probability a subcritical first step mutant lineage pro-duces a double mutant that rescues the population is roughly (−1/m1)U pm1 . Theprobability a mutant lineage leads to 2-step rescue through a subcritical first step isthen approximatelyp(s)2 =∫ m∗−−∞f (m1|mwt) 1−m1U pm1dm1. (5.13)It is more difficult to find an adequate analytic approximation for the integral in thiscase (as opposed to equation 5.11) as we are integrating over a much wider rangeof mutant growth rates here than in the critical case. We therefore numericallyintegrate equation 5.13 when needed.2-step critical vs. subcritical Figure D.6 compares the probability of 2-step crit-ical and subcritical rescue. The subcritical pathway dominates at low mutationrates, when a mutation would have to be extremely critical to initiate a lineage thatreaches a large enough size to almost surely rescue. The subcritical pathway alsodominates during severe environmental challenges (low wildtype growth rates),when the probability of a mutation large enough to be near critical is very unlikely.1115.3.4 1-step vs. 2-step rescueThe probability of 2-step rescue is the probability of either 2-step critical or 2-step subcritical rescue, p2 = p(c)2 + p(s)2 . This is compared against individual-basedsimulations in Figure 5.3 and against the probability of 1-step rescue in Figure5.4. As we can see in Figure 5.3, 1- and 2-step rescue are the predominate pathsto rescue under these parameter values (allowing individuals to carry mutations at10 loci does not increase the probability of rescue). Figure 5.4 shows that 1-steprescue dominates when mutation rates are small and the environmental challengeis mild (more slowly declining wildtypes). Since 2-step rescue increases fasterthan linearly in U , it therefore comes to dominate at high mutation rates (if theprobability of 1-step rescue has not already saturated at 1). 2-step rescue alsodominates during severe environmental challenges, when it is very unlikely that asingle mutation is large enough to rescue the population.5.3.5 The DFE following 2-step rescueAs we did in the case of 1-step rescue, we can now ask what the distribution ofgrowth rates and fitness effects will be among the surviving mutants in populationsrescued in 2-steps. We focus here on the fitness of rescue genotypes, rather thanthe fitness of the individual mutations that compose them, as the fitness of the en-tire genotype can be measured (e.g., isolated by fluctuation tests and measured asexponential growth rate in the selective media, e.g., Kassen and Bataillon, 2006)without constructing mutants. One can also predict the distribution of fitness ef-fects for the individual mutations that comprise the rescue genotype (results notshown).1121-step 2-step 1- or 2-step11122210101011122210101011221010mwt=-0.1mwt=-0.2mwt=-0.3Simulations10 max 10-step2 max 2-step1 max 1-step10-610-510-4 0.001 0.01 0.1 110-610-510-40.0010.010.11Mutation probability, UProbabilityofrescueFigure 5.3: The probability of rescue by 1 or 2 steps. The thin solid curvesgive the probability of 1-step rescue, using equation 5.9 as p in equation5.4. The dashed curves give the probability of 2-step rescue, using theanalytic solution of equation 5.12 plus the numeric solution of equation5.13 as p in equation 5.4. The thick solid curve gives the probability of1-step or 2-step rescue. Individual-based simulation results (104 repli-cates for each point) prevent individuals with 1, 2, or 10 mutations frommutating further. Parameters: n = 4, s¯ = 0.01, mmax = 0.5.113mwt=-0.3mwt=-0.2mwt=-0.110-610-510-4 0.001 0.01 0.1 110-40.011100Mutation probability, UProbability2-step:Probability1-stepFigure 5.4: The probability of 2-step relative to 1-step rescue. Otherwiseidentical to Figure 5.3.2-step critical Using the logic developed above, the distribution of growth ratesamong the survivors of 2-step critical rescue should be roughlyg(m|2-step critical rescue)=∫ m∗+m∗−f (m1|mwt)[1− pest(m1)]2√U f (m|m1)pest(m)dm1∫ ∞−∞∫ m∗+m∗−f (m1|mwt)[1− pest(m1)]2√U f (m|m1)pest(m)dm1dm≈ f (m|0)pest(m)∫ ∞−∞ f (m|0)pest(m)dm≈ f˜ (m|0)pest(m)pm1=0.(5.14)In words, the growth rates of these rescue genotypes are very nearly those that es-tablish after arising from first step mutants with growth rate zero (at rate f (m|0)pest(m)),at least while mutation rates are low (so that critical is almost perfectly critical).Note that this is the same rate that 1-step rescue genotypes are created from a wild-114type with growth rate mwt (mwt = 0 in the numerator of equation 5.10). In terms offitness effects relative to the wildtype, s=m−mwt , equation 5.14 then implies thatfollowing 2-step critical rescue, the distribution of s will be nearly the same as thedistribution of establishing first step mutants in the constant population size case,but shifted to the right by −mwt (the minimum needed for growth). This impliesthat, conditioning on 2-step critical rescue, the mean fitness effect of rescue geno-types is increased but there is essentially no effect on the variance (Figure D.7), asopposed to the 1-step case (Figure 5.2).2-step subcritical When the first step is not near critical the distribution of growthrates and fitness effects amongst the survivors of 2-step rescue is more complicated.In this case the rescue genotypes arise from subcritical lineages, whose growthrates (and hence mutational distributions) depend on the wildtype and in principlecan be quite varied. Regardless, given the logic developed above, the distributionof growth rates among the survivors of 2-step subcritical rescue should be nearlyg(m|2-step subcritical rescue)=∫ m∗−−∞ f (m1|mwt)(−1/m1)U f (m|m1)pest(m)dm1∫ ∞−∞∫ m∗−−∞ f (m1|mwt)(−1/m1)U f (m|m1)pest(m)dm1dm≈∫ m˜∗−∞ f˜ (m1|mwt)(−1/m1) f˜ (m|m1)pest(m)dm1∫ ∞−∞∫ m˜∗−∞ f˜ (m1|mwt)(−1/m1) f˜ (m|m1)pest(m)dm1dm(5.15)Figure D.8 shows the accuracy of this (numerically integrated) approximation interms of fitness effects, s = m−mwt . As in the 2-step critical case, the distributionof fitness effects is shifted to the right as the severity of the environmental challengeincreases, but in this case we also see a reduction in variance (although not asextreme a reduction as in the 1-step case, Figure 5.2). Variance is reduced withinitial maladaptation because, in 2-step subcritical rescue, when the wildtype isfurther from the optimum the subcritical mutants it creates will also tend to befurther from the optimum. These first step mutants will therefore be less likely toproduce double mutants near the optimum, and instead the rescuing double mutantswill have growth rates that are clustered at small positive values, analogous to whatwe saw under 1-step rescue (Figure 5.2).1155.3.6 k-step rescueIn principle we can extend the logic developed in the 2-step case to an arbitrary k≥2 steps. This is complicated somewhat by the possibility that a subcritical lineagebegets a nearly critical lineage that begets a subcritical lineage that begets a rescuegenotype. However, if we ignore these unlikely paths which take ”backwards”steps from nearly critical to subcritical genotypes then the number of routes torescue is considerably reduced. In particular, in a k-step rescue event, all of thefirst k−2 genotypes must have growth rates that are subcritical, while the (k−1)thgenotype may be either subcritical or sufficiently critical.Let pk be the probability of k-step rescue given a mutation arises from thewildtype. Letting p(mk−1|mk−2) be the probability the (k− 1)th mutant leads tok-step rescue (conditioned on one arriving from the (k−2)th mutant), we havepk ≈∫ m∗−−∞...∫ m∗−−∞[Πk−2j=1 f (m j|m j−1)(−1/m j)U]p(mk−1|mk−2)dm1...dmk−2,(5.16)where m0 = mwt and p(mk−1|mk−2) = p(c)2 + p(s)2 (equations 5.11 and 5.13) withmwt = mk−2 and m1 = mk−1. As the mutation rate goes to zero so too does m∗−and p(c)2 , making the probability of k-step rescue proportional to Uk. Increasing themutation rate only makes the dependence on U stronger (as shown for the 2-stepcase, Figure D.6), and thus the probability of k-step rescue declines very quicklywith k when U << 1 and very quickly with U < 1 when k >> 1. We thereforefocus on 1- and 2-step rescue for the remainder of this paper.5.3.7 Inferring the genetic basis of rescue from the DFEWe can now use the predicted distribution of fitness effects for each of the 1- and2-step genetic pathways developed above to infer how rescue has occurred across anumber of replicate experiments (demonstrated with individual-based simulationshere). As we can see from equations 5.10, 5.14, and 5.15, inferring the geneticpathway of evolutionary rescue from the fitness effects of rescue genotypes re-quires we have a reliable estimate of the distribution of growth rates of new muta-tions ( f (m|m′)), the wildtype growth rate (mwt), and the per genome per generation116mutation rate in traits affecting fitness (U , which is needed for the integration limitm˜∗− in the case of 2-step subcritical rescue).Figure 5.5 illustrates the approach for three different combinations of wildtypegrowth rates and mutation probabilities. In panel A, where the environmental chal-lenge is relatively mild, the observed distribution of fitness effects suggests thatrescue occurs almost solely by one mutational step. This aligns with our expecta-tions from the probability of rescue under these parameter values (Figure 5.3) andwith the distribution of mutant growth rates from such a wildtype (Figure D.1). Themoderate environmental challenge and higher mutation rate in panel B appears tocause a mix of 1 and 2-step rescue events, as we see a concentration of fitness ef-fects under the predicted 1-step distribution (solid), but with a much heavier tail. Infact, based on the probability of rescue, 1-step and 2-step rescue are equally likelyunder these conditions (Figure 5.4). Further, the probability of 2-step critical andsubcritical rescue are also nearly equivalent (Figure D.5). Contributions from the2-step critical path (dotted) could explain the right tail. Finally, in panel C, wherethe environmental challenge is yet more severe, the distribution of fitness effectssuggests few rescue events happen by 1-step. As in panel B, rescue appears tooccur through both 2-step critical and subcritical pathways, with perhaps a greaterpercentage of subcritical contributions, as expected from the probability of rescueunder these parameter values (Figure D.5).Figure 5.5: Inferring the genetic pathway of evolutionary rescue from the dis-tribution of fitness effects among survivors. (A) A relatively mild en-vironmental challenge and low mutation rate. (B) A more moderateenvironmental challenge and a higher mutation rate. (C) A more diffi-cult environmental challenge and with the higher mutation rate. Solidcurves show equation 5.10, dotted curves the last line of equation 5.14,and dashed curves numerical integration of the last line of equation 5.15.In gray is the distribution of mutations expected to establish in a pop-ulation of constant size (mwt = 0), for comparison. Histograms showunique selection coefficients in each rescued replicate, across (A) 104,(B) 1.1× 105, and (C) 5× 105 individual-based simulations. Parame-ters: n = 4, s¯ = 0.01, mmax = 0.5.117mwt = -0.11-step2-step subcritical2-step critical0.0 0.1 0.2 0.3 0.4 0.50510152025frequencyU = 0.0002Amwt = -0.20.0 0.1 0.2 0.3 0.4 0.50510152025frequencyU = 0.002Bmwt = -0.30.0 0.1 0.2 0.3 0.4 0.50510152025fitness effect, s = m - mwtfrequencyU = 0.002C1185.4 DiscussionHere we have used Fisher’s geometric model with absolute fitness to explore thegenetic basis of evolutionary rescue. In particular, we deduce the number of mu-tational steps evolutionary rescue is likely to take and the expected distribution offitness effects (and growth rates) among survivors. With mild environmental chal-lenges and low mutation rates we find that evolutionary rescue tends to occur by asingle mutational step, while under more severe environmental challenges and highmutation rates evolutionary rescue is more likely to take multiple mutational steps(Figure 5.4). In general, the probability of rescue by k-steps is expected to decreasefaster than Uk as mutation rates decline (where U is the per genome per genera-tion mutation rate for fitness; equation 5.16). The distribution of fitness effectsamong survivors provides a signature of the number of mutational steps evolution-ary rescue has taken, providing an avenue by which we might infer the genetics ofevolutionary rescue through measures of fitness alone. Following 1-step rescue thedistribution of fitness effects has a smaller variance than the distribution expectedin a population of constant size, and the variance shrinks further with more severeenvironmental challenges (compare solid curves across panels in Figure 5.5). Thedistribution of fitness effects following rescue by multiple steps is also expected tohave reduced variance relative to the constant population size case, but its varianceis less severely affected by the severity of the environmental challenge (comparebroken curves across panels in Figure 5.5) as rescue genotypes can be producedby lower order mutants that are close to the “persistence zone” (m > 0). This willdepend on the relative probability of the (k− 1)th step of k-step rescue having agrowth rate near critical (m = 0); the closer the (k− 1)th mutant growth rate is tom= 0 the more impervious the shape of the distribution of fitness effects will be tothe severity of the environmental challenge (compare dotted curves across panels in5.5). Following this logic, the more mutational steps that evolutionary rescue takesthe greater the expected variance in the distribution of fitness effects among thesurviving genotypes, and the less effect the severity of the environmental challengehas on this variance.1195.4.1 The number of mutational steps evolutionary rescue takesHow do our predictions about the number of mutational steps that evolutionaryrescue is likely to take align with the available data? Below we summarize fourwell-studied empirical examples.Azole resistance in yeast While much is known about the genetic basis of resis-tance to the anti-fungal fluconazole in both Saccharomyces cerevisiae and Candidaalbicans, evolutionary experiments in this system have typically been designed toavoid extinction (reviewed in Cowen et al., 2002). One exception is provided byAnderson et al. (2009), who deleted six nonessential genes whose expression wasconsistently elevated in resistant strains of S. cerevisiae. When grown at varyinglevels of fluconazole, some of these deletion strains went extinct while all othersexhibited the characteristic “U”-shaped population size trajectories of evolution-ary rescue (see figure 4 in Anderson et al., 2009). The genetic basis of rescue inthe deletion strains was not reported, but this shows that rescue is possible in thissystem and that there are multiple potential rescue genotypes. Whether multiplemutations would arise in the same genotype during rescue remains to be seen.A second exception is provided by Hill et al. (2013), who initiated an evo-lutionary rescue experiment with both S. cerevisiae and C. albicans strains thatwere resistant to azoles when two molecular chaperones (Hsp90 and calcineurin)were functional, but not when either was inhibited. The experiment was conductedin the presence of azoles plus Hsp90 or calcineurin inhibitors, causing > 95% ofthe lineages to go extinct. Each of the 14 surviving lineages appeared to have asingle adaptive mutation at the only drug concentration used in the experiment.Because these strains already carried resistance mutations to azoles, these resultssuggests that double mutants may allow evolutionary rescue to the two-drug com-bination used here. Supporting this idea, double mutants resistant to the 2 drugcombination have been observed in a lineage of C. albicans infecting a human host(Cowen and Lindquist, 2005). It is not clear whether a single pleiotropic muta-tion could have the same result as the double mutant; if it cannot than a modelwith restricted pleiotropy (Chevin et al., 2010b) may be more appropriate than theuniversal pleiotropy assumed here.120Quinolone resistance in Pseudomonas fluorescens Strains of Pseudomonas flu-orescens with resistance to the quinolone antibiotic nalidixic acid were isolatedby fluctuation tests (Kassen and Bataillon, 2006) and later sequenced (Bataillonet al., 2011). Of the 18 most resistant strains there were at least 5 double mu-tants (∼28%), which tended to be dramatically more resistant than single mutants.Thus there is the potential for 2-step rescue in this system at high drug concentra-tions. Further support comes from the fact that quinolone-resistant clinical strainsof Pseudomonas aeruginosa often contain multiple mutations; patients with acuteinfections may have higher drug concentrations and also harbour strains with moreresistance mutations than cystic-fibrosis patients (reviewed in Wong and Kassen,2011).Rifampicin resistance in Pseudomonas aeruginosa and Escherichia coli Rifampicin-resistant strains of P. aeruginosa isolated from fluctuation tests contained a singlemutation that arose in one of two genes (MacLean and Buckling, 2009), suggestinglittle potential/need for 2-step rescue at this drug concetration. Consistent with this,surviving strains of Escherichia coli isolated after sudden exposure to rifampicinalso each contained only a single mutation (Lindsey et al., 2013). Meanwhile,surviving strains from populations exposed to less sudden changes in drug concen-tration typically contained multiple mutations. This appears to be in opposition toour prediction that more severe environmental challenges are more likely to lead to2-step rescue (when one equates “severe” with more sudden). The discrepancy islikely because of historical contingency, which relies on sign epistasis in both geno-type and environment; mild challenges select for mutations that are not favouredwhen the challenge is more severe, but together with a more severe challenge theseinitial mutations make previously deleterious mutations beneficial (Lindsey et al.,2013). Sign epistasis in genotype (e.g., single mutants are favourable but the dou-ble mutant overshoots the optimum) and sign epistasis in environment (e.g., largemutations are disfavoured at mild challenges if they overshoot the optimum but be-come favourable during severe challenges) can occur in Fisher’s geometric model.However, we have here assumed that small mutations are most likely (i.e., littlechance of overshooting) and that any mutation favoured in the face of a mild en-vironmental challenge is also favoured when the challenge is more severe. We121discuss the limitations of these assumptions further below (see Caveats and exten-sions).Copper and nystatin resistance in Saccharomyces cerevisiae Strains of S. cere-visiae capable of growing in the anti-fungal nystatin tend to contain a single mu-tation (Gerstein et al., 2012), while strains capable of growing at elevated copperconcentrations tend to contain multiple (Gerstein et al., 2015). In addition, resis-tance took longer to arise in copper (Gerstein et al., 2015), suggesting that copperimposed a more severe environmental challenge. In fact, Gerstein et al. (2015)discussed the possibility that adaptation to copper may have been facilitated byfirst step mutants that could not grow sufficiently but allowed a lineage to persistlong enough and reach lineage sizes large enough to gain a second mutation. How-ever, they also note that an alternative hypothesis is that sufficient growth by thedouble mutant might have been the result of positive epistasis (double mutant fit-ness greater than the sum of single mutant fitnesses). Fisher’s geometric modelinstead exhibits negative epistasis, consistent with the growth of yeast in nystatin(Ono et al., 2017). Our model thus illustrates that positive epistasis is certainly notrequired for multi-step rescue, and that the “buying time” hypothesis is plausibleeven when double mutants are less fit than expected from single mutant fitnesses.It is also possible that due to the finite nature of real organisms and genomes, thenumber of mutations that affect growth in copper (i.e., its mutational target size) islarger than the number that affect growth in nystatin (see Caveats and extensionsbelow for further discussion of mutation assumptions).These experiments also introduce a complexity that can be captured by Fisher’sgeometric model but is not explored in the analyses above (where mutations withsmall fitness effects are always available); 1-step rescue might be more likely than2-step rescue during mild environmental challenges if double mutants tend to over-shoot the optimum so much that they have lower fitness than either single mutant(reciprocal sign epistasis). More severe challenges should then alleviate such ex-treme epistasis and eventually allow only the double mutant to rescue, as has beenshown by constructing double mutants and growing them in nystatin (Ono et al.,2017). Altering the assumption of mutation as a normal distribution about the an-cestor could cause overshooting and therefore could be an interesting alternative122to the model we’ve presented here (see Caveats and extensions below for furtherdiscussion).5.4.2 The distribution of fitness effects following rescueDistributions of potential rescue genotypes Fluctuation tests (Luria and Delbru¨ck,1943) provide us with the distribution of fitness effects of potential rescue geno-types when fitness is measured in the same selective media that was used to isolatemutants. In this case a period of relaxed selection under exponential growth allowsfor the rapid build-up of standing genetic variation, allowing even weakly resistantmutants a chance to grow when plated on the selective media (i.e., relaxed selectionreduces the likelihood weak resistance mutants are lost because they are plated inlarge numbers). Site-directed mutagenesis or mutation accumulation experimentscan also be used to generate a collection of mutants in the near absence of selection(reviewed in Bataillon and Bailey, 2014), and their fitness effects can be tested ina selective media of choice.In our model, when the wildtype declines relatively quickly in the selectivemedia (mwt << 0) the first step mutations that can grow will be chosen from a dis-tribution with a maximum at zero growth rate, m = 0, that monotonically declineswith m (i.e., the positive range of the right tails in Figure D.1 rescaled to integrateto 1, as shown in Figure D.9). More specifically, because growth rate is truncatedat mmax, as the wildtype becomes more maladapted this distribution falls into theWeibull domain of attraction (see discussion of domains in Beisel et al., 2007),as long as phenotypic dimension, n, is finite (Martin and Lenormand, 2008). Asreviewed by Bataillon and Bailey (2014), the data are quite mixed (see their table1); Weibull distributions have been suggested, but so have others. In some casesthe fitness effects of potential rescue genotypes even had a maximum frequency atan intermediate growth rate (Kassen and Bataillon, 2006; MacLean and Buckling,2009). This suggests that our model of mutation might not always be appropriate;it might sometimes be better to model mutations as draws from a distribution that isnot centred on the current phenotype. Harmand et al. (2017) discuss further exten-sions to Fisher’s geometric model, specifically a relaxation of universal pleiotropy,based on their fluctuation tests with E. coli in nalidixic acid (see Caveats and ex-123tensions below for further discussion). Alternatively, these experiments may onlybe sampling the mutations which provide the most resistance, which would be thefirst to show visible growth following plating on selective media.Distributions of realized rescue genotypes When an experiment is initiated byplacing an ancestral strain in an environment where it is unable to grow (mwt < 0),any mutant showing substantial growth is a rescue genotype. Isolating these mu-tants and measuring their growth rates gives the realized distribution of fitness ef-fects of rescue genotypes. In this case their appearance depends on both arising andestablishing when rare, as in our model. Lindsey et al. (2013) provide one example,showing that the mean growth rate of E. coli surviving rifampicin may be slightlygreater when the severity of the environmental challenge increases suddenly ratherthan gradually. Only in the gradual treatment did resistant E. coli strains acquiremultiple mutations. This finding thus runs counter to our predictions here; we havepredicted a higher mean growth rate following 2-step rescue (compare broken withsolid curves in Figure 5.5). However, our model has not been designed with grad-ual environmental change in mind and does not include the historical contingenciesobserved in this system (as discussed above). Gerstein et al. (2012) provides an-other example of a realized distribution of rescue genotype growth rates, showingthat nystatin-resistant mutations in S. cerevisiae show a discrete jump in their ef-fect on resistance (their figure 3) that translates into a distribution of growth rates(their figure 4C) that is similar to what we have predicted here following a mildenvironmental challenge (Figure D.10). A similar result was later found in strainsresistant to high levels of copper (figure 3A in Gerstein et al., 2015).5.4.3 Caveats and extensionsIsotropy and universally pleiotropy One unrealistic assumption of our model isisotropy; for simplicity and analytical results both fitness and mutation are assumedto be uncorrelated across the same set of scaled phenotypes. Relaxing this assump-tion in future work would greatly extend the generality of this theory. Another as-sumption made for simplicity and analytical progress is universal pleiotropy, whereeach mutation affects all phenotypes. Relaxing this assumption in future work may124help account for the extensive genotype-by-environment interactions sometimesobserved (Harmand et al., 2017).An infinite number of primarily small effect mutations Based on the data summa-rized here, another key alteration of the model might be to relax the assumptionof a continuous distribution of mutational effects (which implies an infinite num-ber of possible mutations) that is phenotypically centred on the phenotype thatmutates. For example, one could assume there are a finite number of possiblemutations, each with a different effect (e.g., the small and large mutations in the 1-dimensional moving-optima model of Kopp and Hermisson, 2007). Choosing theset of possible mutations would depend on knowledge of the system however, butmight be deduced, for example, from biophysical properties of resistance mech-anisms (MacLean and Buckling, 2009). Alternatively, one could still assume acontinuous distribution of mutations in phenotypic space but shift the mode. Ei-ther of these approaches could allow more sign epistasis in genotype (overshootingoptima, e.g., Ono et al., 2017) and a distribution of growth rates among potentialrescue genotypes with a positive mode (e.g., Kassen and Bataillon, 2006). Whilethe above modifications appear essential for a theory on the evolution of drug re-sistance in microbes, where there seems to be a limited number of large effect loci,the simplistic model we have explored here may perform better when rescue is theresult of adaptation in a more polygenic trait, although even in that case adaptationcan result from the fixation of a few of the largest effect loci (de Vladar and Barton,2014).Constant fitness Another unrealistic assumption of our model is the independencebetween individuals invoked by the branching process method. This means there isno density- or frequency-dependence in our model. Such an assumption might be areasonable first pass given that evolutionary rescue is often taking place at low den-sities, where interactions between individuals is less strong, but a relaxation of thisassumption could produce interesting results. For example, density-dependencecan make rescue more likely following a severe environmental challenge, whichreleases beneficial mutants from competition (Uecker et al., 2014), and frequency-125dependent competition can exert additional selection pressures that can help orhinder rescue (Osmond and de Mazancourt, 2013).Asexuality Here we’ve assumed an asexual population initiated with a singlegenotype. This is in part for simplicity but also to align with common experimen-tal design, e.g., when investigating the evolution of drug resistance in microbes.Sexual reproduction would have a number of interesting impacts; it would allowtwo single mutant lineages to recombine into a double mutant, which could poten-tially rescue the population, and it will also break down favourable combinations ofmutations, e.g., double mutants recombining back with the wildtype (Uecker andHermisson, 2016). It is not clear if or when recombination would be the most likelyavenue to rescue on the fitness landscape assumed here, or what its resulting distri-bution of fitness effects would be. Recombination would also reduce the extent ofhitchhiking (Maynard Smith and Haigh, 1974) and therefore potentially improveinferences of the genetic basis of rescue based on the predicted and observed dis-tribution of fitness effects (if mutations that affect growth rate in the challengingenvironment are hitchhiking).Clonal interference An interesting aspect of asexuality is clonal interference,where multiple beneficial mutations compete amongst one another and therebyslow or prevent each others spread (Gerrish and Lenski, 1998). By using simplebirth-death process that assumes independence among lineages we have not con-sidered such an effect. Clonal interference tends to increase the mean fitness effectof fixed mutants (Rozen et al., 2002); it is therefore another way (in additionalto alternative mutational distributions and the sampling of only the most resistantgenotypes, discussed above) to explain a relatively large mode in the distributionof rescue genotype growth rates.5.4.4 ConclusionThe past century has seen substantial theoretical and empirical progress on thegenetic basis of adaptation in the absence of demography (reviewed in Orr, 2005).Over this same period, a pressing need to prevent the evolution of drug resistance126has uncovered much about the potential genetic basis of resistance (e.g., reviewedin MacLean et al., 2010). However, we have little theory to predict and generalizehow resistance might be realized, which will often have to occur during a period ofpopulation decline, i.e., as evolutionary rescue (reviewed in Alexander et al., 2014).Here we have used Fisher’s geometric model with absolute fitness to begin a lineof theoretical inquiry into the genetic basis of evolutionary rescue. In particular,we predict how many mutational steps evolutionary rescue is likely to take andthe distribution of fitness effects (and growth rates) among survivors. There is stillrelatively little data available to test our predictions, likely due in large part to thedifficulties of finding the conditions under which extinction occurs but rescue isstill occasionally observed (i.e., the “characteristic stress window” predicted byAnciaux et al., 2018). What data we have reviewed still seems equivocal, butprevious empirical studies suggests future theoretical progress may lie in modelsthat can explain a relatively large mode in the growth rate of rescue genotypes(e.g., some large effect loci or clonal interference), at least when applied to theevolution of drug resistance. We hope that the work presented here adds to a risingawareness of the importance of demography in the evolution and genetic basis ofdrug resistance, inspiring new theory and experiments alike.127Chapter 6Conclusions6.1 OutlineHere I recap the major conclusions of the preceding four chapters and highlightthe paths these research questions have led me down. I then briefly discuss moregenerally what I’ve learned about subcritical adaptation and its genetic basis. I endwith a few closing remarks on the synergy between biology and mathematics.6.2 Chapter-by-chapter conclusions and consequences6.2.1 Chapter 2. Fitness-valley crossing with biased transmissionIn Chapter 2, I modelled the evolution of a complex trait, where each component ofthe trait is selected against when alone but is favoured in combination (i.e., fitness-valley crossing). I was particularly interested in the effects of biased transmission– meiotic drive, uniparental inheritance, and cultural transmission – on the speedand likelihood of the complex trait arising. I therefore generalized existing modelsto allow for any form of between-generation inheritance (with Mendelian inheri-tance a special case). Analysis of this model showed that meiotic drive and biasedcultural transmission can greatly increase or decrease the speed at which com-plex adaptations arise, depending on the direction in which they bias transmission.Meanwhile, uniparental inheritance itself has little effect, but large differences in128mutation rates (say, the greater than 10-fold difference between the mitochondrialgenome and the nuclear genome) can severely slow the speed at which complextraits arise when holding the total mutation rate constant. However, if the totalmutation rate is not held constant, fitness-valley crossing is promoted by interac-tions with more mutable genes. Finally, despite component traits being passed onpoorly in the previous cultural background, small advantages in the transmissionof a new combination of cultural traits can greatly facilitate a cultural transition. Itherefore conclude that, while peak shifts are unlikely under many of the commonassumptions of population genetic theory, relaxing some of these assumptions canpromote fitness-valley crossing and therefore allow the evolution of complex traits.This chapter has had a number of lasting influences on me. On a technicaland somewhat personal level, it was my first model with explicit genetics andrecombination, my first model of finite populations, and my first use of diffu-sion approximations; useful skills for an aspiring evolutionary theoretician. Ona more biological level, this project introduced me to the ideas of cultural evo-lution (Cavalli-Sforza and Feldman, 1981) and genetic conflict (Burt and Trivers,2006). The former idea, cultural evolution, has led to many interesting thoughtsand discussions, attendance at a “complexity” summer school, and has greatly ex-panded my view of where mathematical models are possible and useful. The latteridea, genetic conflict, has similarly opened my mind and has led to a very excit-ing and enjoyable collaboration on the effect of meiotic drive (among other things)on sex chromosome evolution (Scott et al., 2018). I continue to think about newpossibilities in this area, e.g., the effect of selective sweeps and genetic variationin maternally-inherited cytoplasmic elements on the stability of male vs. femaleheterogamety.6.2.2 Chapter 3. Evolutionary rescue: when predators helpIn Chapter 3, I analyzed mathematical models describing the demography and evo-lution of a focal population in the face of a changing environment, with and withouta predator, to ask if predators could ever help their prey persist. In the end I foundtwo mechanisms by which predators can help. The first is when predators kill moreprey of a particular type, for example the sick and weak or perhaps the large and129tasty, and this drives the prey to evolve in the same direction that is favourable inthe new environment. That is, predators can impose a “selective push” that helpsthe prey keep up with the environmental change (as I have also shown with com-petitors; Osmond and de Mazancourt, 2013). In the other circumstance, predatorshelped even when they did not induce selection on the prey but simply ate themat random. By eating more prey, predators freed up resources that allowed preyto reproduce faster, reducing the time between being born and giving birth. Theseshorter generation times led to faster evolution, allowing prey to persist in a rapidlychanging environment. I termed this mechanism the “evolutionary hydra effect”.The possibility of predators helping their prey persist raises concern for the conser-vation practice of removing predators for threatened populations; while this mayhelp the prey survive in the short run, it may pose a longer term danger by hinderingtheir ability to evolve to a changing world.One of the main things I learned from this project, on a more general level,is that modelling birth and death separately can sometimes give deeper insightthan modelling compound parameters like growth rate (birth minus death). For in-stance, the evolutionary hydra effect in Chapter 3 requires density-dependence inbirth rate; the effect depends on shortened generation times, which can only hap-pen when there is selection and density-dependence in birth. Such an effect wasnot noticed in my previous modelling experience where I used compound param-eters (Osmond and de Mazancourt, 2013). Another, more specific, insight fromthis model is that the evolutionary hydra effect might depend on the genetic ba-sis and mode of reproduction (Chapter 3 assumes an infinite number of unlinkedloci, sexual reproduction, and random mating). I am currently exploring this pos-sibility (with Dr. Vincent Calvez, ENS Lyon); it appears that in some limits (e.g.,many small effect loci) the evolutionary hydra effect may not occur in asexuallyreproducing populations. This likely has to do with the different amounts of ge-netic variation maintained in sexual vs. asexual populations as well as the relatedfact that sex and random mating cause the best adapted traits to be broken up byrecombination. Finally, this work has also instigated great cross-talk with an em-piricist (Dr. Michelle Tseng, UBC) whose previous work (Tseng and O’Connor,2015) helped inspire the chapter. Together we have been delving into just why thepredator helps the prey adapt in her experiments with midge larvae and Daphnia130under warming; in particular, might this be a selective push or the evolutionaryhydra effect?6.2.3 Chapter 4. Evolutionary rescue: alternative fitness functionscause evolutionary tipping pointsIn Chapter 4, I relaxed the pervasive assumption of Gaussian selection in modelsof adaptation to environmental change to see whether our predictions about extinc-tion are robust. Strikingly, I found that relatively minor changes to the Gaussianassumption (in particular weakening selection in the tails of the fitness function,i.e., antagonistic epistasis) can create what I call “evolutionary tipping points”,where seemingly sustainable populations suddenly begin to go extinct. Further, Ifound that tipping points cause extinction debts, meaning that temporary increasesin the rate of environmental change can lead to extinction many generations later.The results therefore call for caution when using current theory to predict whetherpopulations will adapt or persist in changing environments and highlight the im-portance of characterizing how selection changes with maladaptation in naturalpopulations.One clear lesson deriving from this work is to challenge current assumptions,however common. Many common habits (in science and beyond) are taken up dueto historical precedence or convenience; this constrains our view of the world andthus our ability to make predictions. In addition, the dramatic consequences stem-ming from this simple adjustment of fitness functions has greatly increased my cu-riosity about the shape of fitness functions in nature. As we gather more data frommaladapted individuals, e.g., due to rapid environmental change or experimentaltransplants, it may be possible to characterize more than just the peak of stabilizingfitness functions. To this end I have begun to develop a method which fits splines tofitness measures across a range of trait values in an attempt to find potential evolu-tionary tipping points (inflection points). It is my goal to publish this method alongwith an R package that allows users to readily characterize non-parametric fitnessfunctions and look for evolutionary tipping points in their own data (a beta versionis currently available at Chapter 5. Evolutionary rescue: a theory for its genetic basisIn Chapter 5, I extended theory on the genetic basis of adaptation in the absenceof demography to the case of evolutionary rescue. This led to predictions on thegenetic pathways (i.e., number and size of mutations) by which rescue will occurand the distribution of fitness effects (DFE) of rescued populations, looking at onlythose adaptive walks that end in persistence. I found that rescue by two mutationsmay be more likely than rescue by one relatively large mutation in cases of se-vere environmental change and high mutation rates. Further, I found that the DFEfollowing rescue by one mutation has an increased mean and decreased variancerelative to the DFE of mutations that establish in populations of constant size andthat this variance is further reduced with increased maladaptation of the wildtype.The variance in the DFE following rescue by more mutations is less sensitive toinitial maladaptation, because rescue then tends to occur by mutations from mu-tant lineages that are less maladapted, reducing the dependence on the wildtype.This suggests that information about the genetic pathways by which evolutionaryrescue occurs may be provided in measures of growth rate alone. By summarizinghow these predictions align with existing observations, in particular the evolutionof drug resistance in microbes, I found that, while some patterns are qualitativelyconsistent, future theory in this area may need to contend with the finiteness of theset of genotypes that can confer such rapid adaptation.This project has really made me wonder if we might one day be able to de-tect past instances of evolutionary rescue, and eco-evolutionary dynamics moregenerally, in data from natural extant populations. While this study focused on in-ferences based on measures of growth rate, I am curious if patterns at the geneticsequence level might also be informative. For example, perhaps we can look forstrong selective sweeps at specific loci that are temporally coincident with past pop-ulation bottlenecks (the characteristic “U”-shape of evolutionary rescue) inferredfrom genome-wide averages of heterozygosity or long-range linkage disequilib-rium. Looking past this naive coincidence, perhaps the feedback between demog-raphy and selection (i.e., the sweep has caused the recovery in population size)leaves some more robust signature. These ideas have inspired me to learn moreabout genomics and methods of inference from genetic data and to better connect132the theory I do with data that is being collected; I am excited to be following thisline of inquiry in my future post-doc.6.3 General conclusions and future directions6.3.1 Subcritical adaptationIn the Introduction, I attempted to conceptually link fitness-valley crossing andevolutionary rescue through the idea of subcritical adaptation; both are scenarioswhere adaptation requires sufficiently long persistence of genotypes that are ex-pected to be declining in number. The previous four chapters show, however, thatthere is no general conclusion for when subcritical adaptation is likely. For exam-ple, fitness-valley crossing can be much more likely with small population sizes,where there is more stochasticity in the dynamics of deleterious intermediates (ge-netic drift). In contrast, evolutionary rescue is nearly always found to be morelikely with higher initial population sizes (assuming density-dependence is neg-ligible; Uecker et al., 2014), which provide more mutational opportunities. Thisstrong contrast may in part be the result of the relatively simplistic genetic archi-tectures typically assumed in models of evolutionary rescue. Only one of the mostrecent theoretical extensions has considered epistasis and therefore the potential forfitness valleys during evolutionary rescue (Uecker and Hermisson, 2016). Fromthe other direction, empirical work on the evolution of drug resistance has beenstrongly focused on epistasis but completely ignores demography when inferringpotential evolutionary paths (e.g., Kouyos et al., 2012; Ogbunugafor et al., 2016;Schenk et al., 2013; Weinreich et al., 2006; but see Lindsey et al., 2013). Only oncewe begin to consider epistasis at the same time as absolute fitness and demogra-phy will the historically separated topics of fitness-valleys and evolutionary rescuebecome entwined and any synthetic predictions for subcritical adaptation arise.Combining epistasis and absolute fitness would create an enormous number ofpossibilities to model. For instance, in a haploid with n loci each with k allelesthere are kn possible genotypes. Even restricting ourselves to constant fitnesses,there are kn! ways to rank relative fitnesses (ignoring equivalences) and kn + 1ways to choose whether individuals will have an absolute fitness above or below133replacement (either 0, 1, 2, ..., kn genotypes can be above replacement). This gives(kn + 1)! combinations, which quickly becomes greater than the number of elec-trons and protons in the universe (Wright, 1932). Letting one allele mutate to anyother gives n(k− 1) mutational neighbours, but there are only two intermediatesbetween any two genotypes separated by two mutational steps. Considering twogenotypes separated by two mutational steps, there are therefore only four ways tohave a fitness valley with one peak below replacement (the genotype at either peakcan be below replacement and there are two ways to rank the fitness of the interme-diates in the valley). Each of the kn genotypes has n(k−1)(n(k−1)−1)/2 two-stepneighbours, giving knn(k−1)(n(k−1)−1)/4 two-step pairs (we divide by 2 to pre-vent counting pairs twice). There are therefore knn(k−1)(n(k−1)−1) ≈ kn+2n2possible fitness valleys with one peak below replacement. Thus, as the numberof loci and alleles grows the potential for interactions between fitness valleys andevolutionary rescue skyrockets.Yet the thought experiment above also shows that the number of ways to havesuch an interaction is quickly dwarfed by the number of alternative scenarios. Whatare these other scenarios that arise when epistasis and absolute fitness combine?One scenario is when the genotypes in a fitness valley have absolute fitnesses be-low replacement but the genotypes at the peaks can replace themselves. Such ascenario would break the common assumption of constant population size typi-cally invoked in models of fitness-valley crossing and may lead to surprising newdynamics. For instance, variation maintained at mutation-selection balance maythen depress population size (Agrawal and Whitlock, 2012); to what extent wouldthe increased strength of genetic drift ease fitness-valley crossing? It may then beinteresting to consider local fitness peaks, above replacement, with some of theirn(k−1) neighbours above replacement and others below. Which pathways off thepeak are most likely? Alternatively, it may only be the highest peak that has afitness above replacement. This may help the beneficial genotype spread betweendemes once it has arisen (Peck and Welch, 2004), but could hamper its arrival byforcing the population to be rescued by evolution. Under what scenarios (e.g., ofenvironmental change, environmental gradients) would allow the beneficial geno-type to arise but also spread? Such questions might be hard to answer analyti-cally, but simulations of evolution and demography on empirical fitness landscapes134(De Visser and Krug, 2014) could help. The results have important consequencesfor preventing drug resistance and for the repeatability and predictability of evolu-tion. In sum, considering sign epistasis in models with variation in absolute fitnesswill create new types of subcritical adaptation that may lead to many interestinginsights, for both the evolution of drug resistance and evolutionary thinking moregenerally.6.3.2 The genetic basis of subcritical adaptationSo what is the genetic basis of subcritical adaptation? What modelling assumptionsbest align with the data? The answer, of course, depends on the organism and theenvironment to which it is adapting.As I have alluded to above and in Chapter 5, the genetic basis of microbesadapting to drugs is often relatively simple on the one hand, in the sense that itinvolves few loci of large effect (Bell, 2009), but also incredibly complex on theother, in the sense that there is pervasive sign epistasis. The best tractable modelsare thus likely those with a few loci each with a handful of alleles, as hypothesizedby Mendel (1866) and used in Chapter 2. Meanwhile, larger data sets and im-proved sequencing and statistical techniques are showing us that standing variationin many traits, like human height, is coded by an enormous number of small-effectloci (Boyle et al., 2017). Surely such traits are involved in subcritical adaptation,e.g., evolutionary rescue, sometimes? Over a large number of loci the expectedepistatic effect is negligible and the best models are then those that follow fromFisher’s infinitesimal limit (Fisher, 1918), as used in Chapters 3 and 4.This seems to give rise to a paradox; many traits are composed of many lociof small effect but subcritical adaptation is often the result of few loci with largeeffect. In other words, traits are omnigenic (Boyle et al., 2017) but subcriticaladaptation is oligogenic (Bell, 2009). This is not unlike the Mendelian vs. Biome-trician battle over 100 years ago (reviewed in ch. 5 of Provine, 1971). Part of theresolution this time likely lies in the fact that the genetic basis of adaptation doesnot have to be the same as the genetic basis of standing variation (Bell, 2009). Inparticular, the allele frequencies at many of the loci underlying variation may beconstrained by pleiotropy or epistasis (as in Fisher’s geometric model, Chapter 5).135Further, when the contribution of each locus varies (i.e., they harbour alleles ofunequal effect), as expected (Orr, 1998), rapid adaptation in a highly polygenictrait following environmental change can occur by selective sweeps in a few lociof large effect (de Vladar and Barton, 2014). Another part of the resolution likelylies in our inability to disentangle traits coded by an intermediate number of lociof intermediate effect; the effects at each locus will then not be large enough toisolate single mutants (either statistically or experimentally), but they will also notbe small enough to ignore epistasis and linkage. Such a scenario is a difficult chal-lenge both empirically and theoretically. Machine-learning and simulation willhelp, especially as computing power improves, but I believe that simple, analyt-ically tractable mathematical models will always be needed for intuition. As wecontinue to work out the genetic basis of subcritical adaptation, and adaptationmore generally, it will therefore be essential to maintain and expand upon the di-versity of tractable theoretical approaches we have been using to make sense ofevolution since the pioneering works of Mendel and Fisher.6.4 Closing remarksThe models presented here are abstractions of a complex world. While I have triedin many places to connect the theory with data this is not always possible on aquantitative level. The goal instead has been to use math to deduce the qualitativeconsequences of a set of biological assumptions in a strictly logical fashion. Someof these consequences may not yet have been observed in nature, and more besidesmay never be realized. This is not as grave a problem as it sounds; part of the beautyof mathematical models is that they can expand our notion of what is possible. In anideal world this expansion of thought inspires us to collect new data, which in turnmotivates the refinement of old models and the creation of new ones. Vice versa,surprising biological findings spur on new models, giving insight into potentialmechanisms or consequences that can then be sought out experimentally. Thisfeedback between biology and mathematics is an engine for the train of knowledge,as pointed out long ago (Fisher, 1930, p. ix):“The most serious difficulty to intellectual co-operation [between bi-ologists and mathematicians] would seem to be removed if it were136clearly and universally recognized that the essential difference lies,not in intellectual methods, and still less in intellectual ability, butin an enormous and specialized extension of the imaginative faculty,which each has experienced in relation to the needs of [their] specialsubject. I can imagine no more beneficial change in scientific educa-tion than that which would allow each to appreciate something of theimaginative grandeur of the realms of thought explored by the other.”137BibliographyAbramowitz, M., and I. A. Stegun, eds. 1972. Handbook of mathematicalfunctions with formulas, graphs, and mathematical tables. United StatesDepartment of Commerce, Washington, DC, USA. → pages 100, 105Abrams, P. A. 2000. 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While single mutants arefar from mutation-selection balance, |si jt|< 1 ∀ i 6= j, we can write x22 as a functionof t by replacing xi j in Equation (2.4) with the appropriate xi j in Equation (2.3).With rare mutation, rare single mutants, and weak selection on single mutants, theexpected number of generations until a successful double mutant appears, T , solves1u22N= µ1111 (22)+T[µ1111 (22)[1−µ1111 (21)∗−µ1111 (12)∗]+µ1111 (21)∗µ2111 (22)+µ1111 (12)∗µ1211 (22)+13µ1111 (21)∗s21µ2111 (22)+13µ1111 (12)∗s12µ1211 (22)+13µ1111 (21)∗µ1111 (12)∗r1221(22)]+T 2[µ1111 (21)∗µ2111 (22)+µ1111 (12)∗µ1211 (22)+µ1111 (21)∗µ1111 (12)∗r1221(22)−µ1111 (22)[µ1111 (21)∗+µ1111 (12)∗]]+T 3[23µ1111 (21)∗µ1111 (12)∗r1221(22)+13µ1111 (21)∗s21µ2111 (22)+13µ1111 (12)∗s12µ1211 (22)]+O(ε5). (A.1)161The O(ε5) terms disappear and the equation is exact when single mutants are neu-tral, si j = 0. Otherwise, with selection against single mutants, the higher orderterms can only be ignored as long as the crossing time, T , is much smaller than theinverse of the selection coefficients, s21 and s12. Because Equation (A.1) is a cubicin T , its solution is cumbersome. Here we examine two scenarios which give moreinterpretable approximations for T .Without selection on single mutants (s21 = s12 = 0) and without recombinationfrom single mutants to double mutants [r1221(22) = 0] the T3 term in Equation (A.1)vanishes. In addition, if the crossing time T is long, the dominant term is the oneproportional to T 2. Solving for T from this term alone givesT ≈[u22N[µ1111 (21)∗µ2111 (22)+µ1111 (12)∗µ1211 (22)]]−1/2, (A.2)where we have ignored double mutants arising instantaneously [µ1111 (22) = 0].Equation (A.2) shows that the crossing time without selection on or recombinationamong single mutants is roughly proportional to N−1/2 generations. The cross-ing time decreases with N because increasing N increases the per generation inputof mutations. Holding mutation input θ kli j (mn) = Nµkli j (mn) constant, the cross-ing time becomes proportional to N1/2. When the single-mutation transmissionprobabilities are equal [µ1111 (21)∗ = µ1111 (12)∗ = 2µ2111 (22) = 2µ1211 (22) = µ] and wecalculate the first appearance of any double mutant (successful or not; u22 = 1),the expected time until the first double mutant appears simplifies to the neutral ge-netic case without recombination, T ≈ 1/√µ2N (equation 8 in Christiansen et al.,1998). Equation (A.2) clarifies the role of the various, potentially different, mu-tation probabilities µ i j11(kl) on the time until the first double mutant, while alsoallowing us to ignore double mutants that are lost.When there is recombination between single mutants to produce double mu-tants [r1221(22) > 0] and the crossing time, T , is smaller than the inverse of theselection coefficients, s12 and s21, the dominant term in Equation (A.1) is pro-portional to T 3. This term is positive when recombination is frequent relative toselection against single mutants. Again, if the time T is long we can use this term162alone to approximate T , which givesT ≈31/3[u22N[µ1111 (21)∗µ1111 (12)∗r1221(22)+ s21µ1111 (21)∗µ2111 (22)+ s12µ1111 (12)∗µ1211 (22)]]−1/3,(A.3)where we have once again ignored the instantaneous production of double mutants.Notice that, for a given mutation input θ kli j (mn), when there is recombination be-tween single mutants, the crossing time is roughly proportional to N1/3 generations(rather than N1/2 generations without recombination), implying that recombinationbetween single mutants tends to shorten the expected time until the first (success-ful or unsuccessful) double mutant arises. However, because recombination canalso occur between residents and double mutants (reducing u22) Equation (A.3)shows that the waiting time until the first successful double mutant is minimized atintermediate levels of recombination.Equation (A.3) reduces to T ≈ 1/ 3√Nrµ2/3 (equation 9 in Christiansen et al.,1998) when we ignore the weak selection against single mutants (si j = 0), there isequal mutation probability at each locus [µ1111 (21)∗ = µ1111 (12)∗ = µ], and we waituntil the first double mutant appears, successful or not (u22 = 1). Once again ouranalysis clarifies the role of the various, potentially different, mutation probabilitiesµkli j (mn) on the waiting time until the first successful double mutant. Equation(A.3) also allows (weak) selection on single mutants and incorporates transmissionbias, which we explore more fully in the main text.Figure A.1 compares the approximations derived here (Equations A.2 and A.3)with that derived in the text assuming mutation-selection balance is reached beforecrossing (Equation 2.8). The approximations given by Equations (A.2) and (A.3)break down as the depth of the valley (δ = −s21 = s12) increases such that thecrossing time becomes long, T > 1/δ .A.2 Diffusion approximationHere we take the large population limit (N → ∞), scale time such that one unit oftime in the scaled diffusion process (τ ∈ Z≥0) is Nα generations in the unscaledMarkov process (4t = τNα ) and define new frequency parameters Y (τ) = iτ/Nβ163and Z(τ) = jτ/Nβ , with 0 < α,β < 1.We are concerned with three quantities for each variable4Y and4Z. The firstis the infinitesimal meanµY (y) = limN→∞E[4Y |Y (τ) = y = i/Nβ ] = limN→∞NαNβE[4i|it = i]. (A.4)The second quantity is the infinitesimal varianceσ2Y (y) = limN→∞E[(4Y )2|Y (τ) = y = i/Nβ ] = limN→∞NαN2βE[(4i)2|it = i]. (A.5)And the third quantity of interest is a higher (n > 2) infinitesimal momentlimN→∞E[(4Y )n|Y (τ) = y = i/Nβ ] = limN→∞NαNnβE[(4i)n|it = i]. (A.6)We can similarly calculate µZ(z), σ2Z(z), and a higher moment in4Z.The final quantity of interest is the scaled “killing rate”κ(y,z) = limN→∞Nα P˜Hi j ≈ limN→∞NαNx′22u22, (A.7)where the approximation assumes x′22u22 << 1.For the Markov chain to converge to a diffusion process as N→ ∞ we require:1) µY (y) and µZ(z) to be finite; 2) σ2Y (y), σ2Z(z), and κ(y,z) to be positive and finite;Figure A.1: Expected number of generations until a double mutant beginsto fix, T , as a function of recombination, r, given (gray) mutation-selection balance is first reached (Equation 2.8) or mutation-selectionbalance is not reached and (black, solid) crossing can occur by recom-bination (Equation A.3) or (black, dashed) crossing occurs by mutationonly and −s21 = −s12 = δ = 0 (Equation A.2). The dots show thefull semi-deterministic solution (numerical solution to Equation A.1,including higher order terms, allowing both recombination and muta-tion to generate double mutants). The mutation-selection balance esti-mate (gray) performs better than the dynamic estimates (black) whenδ T > 1, and vice-versa. Parameters: symmetrical, Mendelian inheri-tance with N = 105, s22 = 0.05, and µ = 5×10−7.164020 00040 00060 00080 000100 000T∆ =1100000010 00020 00030 00040 00050 00060 000∆ =11000010-6 10-5 10-4 0.001 0.01 0.1r0100 000200 000300 000400 000∆ =11000and 3) some higher moment (in both4Y and4Z) to be equal to zero (Karlin andTaylor, 1981). We first take a hint from the genetic case (Christiansen et al., 1998)165and scale transmission probabilities asbkli j(mn) =Bkli j(mn) : m ∈ {i,k},n ∈ { j, l}Bkli j (mn)N2 +O(1/N3) : m 6∈ {i,k},n 6∈ { j, l}Bkli j (mn)N +O(1/N2) : otherwise(A.8)In the genetic case this can be interpreted as making the probability of mutationproportional to the inverse of population size µ = B/N. Then, as N→ ∞ mutationprobability decreases (µ → 0), such that mutation input B = Nµ is constant. Thisprevents the process from taking large jumps in frequency space, which violate thediffusion process (Karlin and Taylor, 1981).In order for the transmission parameters to satisfy the logical constraint∑2m,n=1 bkli j(mn) = 1 the diffusion also requires, as N→ ∞, thatBi ji j(i j) = 1+O(1/Nβ ) (A.9)andBkli j(i j)+Bkli j(kl) = 1+O(1/Nβ ) (A.10)when either {i 6= k, j = l} or {i = k, j 6= l}. In words, the sum total mutationprobability for parents AiB j and AkBl must be relatively small, at most on the orderof 1/Nβ .Finally, our approximation requires weak selection, relative to w11 = 1. Inparticular, total selection on single mutants must be weak, on the order of 1/Nβ ,wi j[bi j11(11)+b11i j (i j)] = 1+Si j/Nβ +O(1/N2β ) (A.11)for i 6= j, where Si j is the scaled selection strength. And selection on double mu-tants must also be weak, such thats22 = S22/N+O(1/N2). (A.12)With the above assumptions (Equations A.8-A.12) the Markov chain converges166to a diffusion process as N→ ∞ whenα = β ={1/2 : r2112(22)≤ O(1/N1/2)1/3 : otherwise(A.13)This scaling implies that if recombination between single mutants to make doublemutants r2112(22) is less likely that mutation (which is on the order of N−1/2; Equa-tion A.9), then the time until the process is killed scales with N1/2. Meanwhile, ifrecombination is more likely than mutation the killing time scales with N1/3. Theseresults align with our semi-deterministic analysis (Equations A.2 and A.3).When α = β the infinitesimal variances are σ2Y (y) = y and σ2Z(z) = z. Theinfinitesimal means and the killing term depend on the probability of recombina-tion. When recombination is rare the single mutants are expected to reach higherfrequencies and therefore have a greater influence on the dynamics. To simplify,when recombination is rare [r1221(22) ≤ O(1/N1/2)] we assume weak transmissionbias for residents mating with single mutants [bi j11(11)+ b11i j (11) = 1+O(1/Nβ )]and for single mutants mating with each other [bkli j(i j)+ bi jkl(i j) = 1+O(1/Nβ )].We further assume weak viability selection on single mutants, wi j = 1−O(1/Nβ ),regardless of recombination. The infinitesimal mean is then alwaysµY (y) = B1111(21)− y S21 (A.14)and similarly for µZ(z). The first term, B1111(21)≈ b1111(21)N, describes mutation tosingle mutants in resident-resident matings and the second term, with S21 ≈ s21Nβ ,describes the removal of single mutants by selection (both through transmissionbias when mating with the resident and survival).The killing terms areκ(y,z)=u22w22[y[B2111(22)+B1121(22)]+ z[B1211(22) +B1112(22)]+ y z R1221(22)]: r1221(22)≤ O(1/N1/2)u22 y z r1221(22)∗ : otherwise(A.15)where R1221(22)≈ r1221(22)N1/2 describes a (low) probability of recombination. Thefirst line shows that the process can be killed by mutations in single mutants that167mate with residents [Bi j11(22) + B11i j (22) ≈(bi j11(22) + b11i j (22))N] or by rare re-combination between single mutants to produce double mutants R1221(22). Whenrecombination is more likely than N−1/2 the process is essentially always killed byrecombination r1221(22) (second line in Equation A.15).A.3 Stochastic crossing timesNeutral single mutants without recombination With no chance of recombinationfrom single mutants to double mutants [r1221(22) = 0] we have scaling parameterβ = 1/2. Then, without selection on single mutants (s21 = s12 = 0) and withsome mutational symmetry between the two loci [b¯2111(22) = b¯1211(22) = b¯m11(22)],the single mutants are equivalent and we can concern ourselves with only theirsum, ξ = y+ z. Letting m be either single mutant type (m = 21 or 12), Equation(2.12) reduces to12ξd2T˜ (ξ )dξ 2+[B1111(21)+B1111(12)]dT˜ (ξ )dξ−u22ξ[Bm11(22)+B11m (22)]w22 T˜ (ξ ) =−1(A.16)where Bkli j(mn) = bkli j(mn)N.When there are an infinite number of single mutants a successful double mu-tant is produced immediately, giving one boundary condition limξ→∞ T˜ (ξ ) = 0.The second boundary condition is dT˜ (0)/dξ = −[B1111(21) + B1111(12)]−1, whichcan be derived directly from Equation (A.16) by setting ξ = 0 (see appendix A inChristiansen et al., 1998, for a more complete derivation).The solution to the boundary value problem, evaluated at ξ = 0, correspondingto the expected number of generations until a successful double mutant arises whenbeginning with only residents, T = N1/2T˜ (0), is thenT =N1/2Γ[1/2]Γ[B1111(21)+B1111(12)]Γ[1+B1111(21)+B1111(12)]√u222[Bm11(22)+B11m (22)]w22, (A.17)where Γ[·] is the gamma function. Setting mutation probabilities equal [B1111(21) =168B1111(12) = 2Bm11(22) = 2B11m (22) = θ =Nµ] reduces Equation (A.17) to the neutralgenetic case (equation 27 in Christiansen et al., 1998) divided by√u22w22 becausewe census after selection and consider double mutant fixation. By separating thevarious mutational terms our analysis clarifies that, while the crossing time is in-versely proportional to the mutation probability from residents to single mutants,it is inversely proportional to the square root of mutation probabilities from singlemutants to double mutants. The crossing time is therefore increased much more bya reduction in mutations from residents to single mutants than it is by a reductionin mutations from single mutants to double mutants.When mutations from residents to single mutants [b1111(21) and b1111(12)] arerare, an approximation for the crossing time, in terms of our unscaled parameters,isT ≈ 1N[b1111(21)+b1111(12)]√u224µm11(22)∗. (A.18)Increasing the mutational supply of single mutants [N(b1111(21)+ b1111(12))] or theprobability of mutation from single mutants to successful double mutants [u22µm11(m)∗]decreases the amount of time we expect to wait before a successful double mutantarises. Holding mutation input, θ , constant, Equation (A.18) shows that the cross-ing time without recombination is roughly proportional to N1/2 generations, align-ing with the semi-deterministic analysis (Equation A.2) and indicating that, for agiven mutational input, genetic drift increases the speed at which fitness valleys arecrossed.Neutral single mutants with recombination With recombination the scaling pa-rameter is β = 1/3. We can reduce and solve Equation (2.12) with recombinationwhen the frequencies of single mutants remain proportional to one another, suchthat we need follow only cµ y = z = ξ , where cµ is a constant. This requires mu-tation input [Nb1111(21), Nb1111(12)] to be large enough to make the dynamics of yand z relatively deterministic. We further assume no selection on single mutants(s21 = s12 = 0). We then have cµ y = z for all time, t, when the ratio of mutationprobabilities is cµ [i.e., cµb1111(21) = b1111(12)] and we begin with cµy(0) = z(0).169Equation (2.12) then collapses toξ2(1+ cµ)d2T˜ (ξ )dξ 2+B1111(21)dT˜ (ξ )dξ−u22cµr1221(22)∗ ξ 2 T˜ (ξ ) =−1. (A.19)The boundary conditions are limξ→∞ T˜ (ξ ) = 0 and dT˜ (0)/dξ = −B1111(21)−1.The solution to the boundary-value problem, evaluated at ξ = 0, in units of gener-ations, T = N1/3T˜ (0), isT =25/3pi311/6Γ[2/3] N1/3(1+ cµ)Γ[2(1+ cµ)B1111(21)/3]Γ[2[1+(1+ cµ)B1111(21)]/3]3√u22cµ(1+ cµ)r1221(22)∗. (A.20)Letting cµ = 1, B1111(21) = θ = Nµ , and r1221(22) = r/2 reduces Equation (A.20)to the neutral genetic case (equation 30 in Christiansen et al., 1998) divided by3√u22w22 because we census after selection and consider double mutant fixation.Our result extends the insight of Christiansen et al. (1998) by allowing the fre-quencies of single mutants to differ, cµ 6= 1. Holding average mutation input[(1+ cµ)Nb1111(21)/2] constant, Equation (A.20) shows that the crossing time isminimized when there are equal numbers of the two single mutants (cµ = 1) andincreases as the asymmetry grows. This occurs because recombination is mosteffective in creating double mutants when the single mutants are equally frequent.Converting the full solution back in terms of our unscaled parameters and let-ting the mutation probability b1111(21) be small, we have the approximationT ≈ 22/3pi35/6Γ[2/3]2 1N2/3b1111(21)3√u22cµ(1+ cµ)r1221(22)∗. (A.21)Holding mutation input [Nb1111(21)] constant, Equation (A.21) shows that the cross-ing time is roughly proportional to N1/3 generations, aligning with the semi-deterministicanalysis (Equation A.3).170A.4 Stochastic simulationsWe performed stochastic simulations to verify our analytical and numerical results.Briefly, we performed random multinomial sampling of genotypes with frequencyparameters given by Equation (2.1) and transition probabilities defined in Tables2.2-2.4. Crossing time simulations ended on double mutant fixation and the gener-ation in which this occurred was recorded as the crossing time. Crossing time wasaveraged over all trials (103 trials in Figure 2.1, 102 trials in Figure 2.3 and 2.5).Crossing probability simulations ended on resident or double mutant fixation andthe genotype which fixed in each trial was recorded. The crossing probability wascalculated as the fraction of trials in which the double mutant fixed (103 trials inFigure 2.2 and 2.4, 105 trials in Figure 2.6).171Appendix BAppendices for Chapter 3B.1 Simulation MethodsIndividual-based simulations followed a Gillespie algorithm (Gillespie, 1977). Pop-ulations were represented by a vector of trait values (one entry for each individual),which was initiated by choosing n entries (corresponding to demographic equilib-rium in the absence of selection) from a random normal distribution with mean 0and variance 2α2 (the predicted equilibrium phenotypic variance).For each iteration (1) the propensities for each type of reaction (birth, death,predation) were calculated for each individual, (2) the propensities were summedto give the total propensity, a0, (3) time was increased by τ =−log(r1)/a0, wherer1 is a random uniform number in [0,1), (4) a particular reaction was chosen bygiving a reaction with propensity ai a probability of occurrence ai/a0 and choosinga random uniform number in [0,1), and (5) the reaction occurred and abundanceswere updated accordingly. To speed up the coevolutionary simulations we madethe propensity for a prey to be predated a function of the difference between itstrait value and the mean predator trait value (instead of a separate value for eachpredator individual) and the propensity for a predator to capture a prey a functionof the difference between its trait value and the mean prey trait value (instead of aseparate value for each prey individual). When an individual was chosen to repro-duce, a second individual of the same population was chosen randomly as the mate(possibly the same individual), and the offspring inherited the average of the two172parental trait values plus a random normal effect with mean 0 and variance α2.The abundances and the mean and variance in the trait distributions were recordedevery 104 iterations (events). Simulations ended when the prey went extinct or thefinal time was reached, t ≥ 104 (or t ≥ 2∗104 in the evolutionary hydra example).Means of the final 10 recorded values were used to estimate steady-state lags, equi-librium population densities, and equilibrium genetic variances across replicates.The exception is in the coevolution example, where we used the last 98 recordedvalues in order to average over the cycling behaviour observed at low rates of en-vironmental change.All simulations were implemented in Python (version 2.7.10; CAppendices for Chapter 4C.1 Sufficient conditions for the existence of anevolutionary tipping pointHere we sketch out in more detail what is required for an evolutionary tipping pointto exist for any continuous, real, thrice differentiable fitness function, r(z), whichis monotonically declining from a sufficiently positive local maxima at z = θ toa negative number as the lag between the local maxima and trait value increases,l = θ − z→ ∞.Let the position of the local maxima, θ , at time t be kt. Expected populationmean fitness then monotonically declines from, E[r¯|l¯ = 0] = r¯m > 0, as the expectedpopulation mean lag, E[l¯] = E[θ − z¯] = kt− g¯, increases from 0. Let l¯c be the meanlag that causes an expected population growth rate of zero, E[r¯|l¯ = l¯c] = 0. We thenhave E[r¯|l¯]> 0 ∀ l¯ ∈ [0, l¯c) and E[r¯|l¯]< 0 ∀ l¯ ∈ (l¯c,∞).As described in the main text, the expected rate of evolution given mean addi-tive genetic value g¯ is approximately E[dg¯dt∣∣g¯]≈σ2g dr¯dg¯ , where σ2g > 0 is the additivegenetic variance (a constant that is independent of k) and r¯ is population mean fit-ness. A quasi-steady-state is reached when the expected rate of evolution equalsthe expected rate of change in the optimum, or equivalently, dr¯dl¯ = −k/σ2g . Onethen wants to solve this equation for the steady-state lag, lˆ, which is the mean lagat which mean fitness declines with mean lag at rate k/σ2g .Given that fitness, r, and thus mean growth rate, r¯, has a local maxima at θ ,174in a constant environment, k = 0, a quasi-steady-state is achieved when the meanlag is zero, l¯ = 0. Since the expected growth rate at this lag is positive, E[r¯|l¯ =0] = r¯m > 0, the population can persist at this steady-state. We assume this isthe starting point of the population. Because mean growth rate, r¯, is continuousand monotonically declining as mean lag, l¯, increases from zero, i.e., dr¯dl¯ < 0 forall l¯ > 0, we are guaranteed that near l¯ = 0 the steady-state lag increases withk. This is because near the local maxima, θ , the mean growth rate is necessarilyconcave down, d2 r¯dl¯2 < 0, i.e., the strength of selection, and thus the rate of evolutionwith constant additive genetic variance, increases with increasing mean lag nearl¯ = 0. However, as we depart from l¯ = 0 the monotonicity of r¯ is not enough todetermine the sign of d2 r¯dl¯2 . Thus, the expected rate of evolution, σ2gdr¯dl¯ , can increaseor decrease as mean lag increases. In particular, inflection points in the fitnessfunction, which cause inflection points in mean growth rate, d2 r¯dl¯2 = 0, create localminima and maxima in the expected rate of evolution as a function of mean lag.Let L = {l¯1, l¯2, ..., l¯n} be the ordered set of mean lags at which there are lo-cal minima and maxima in the expected rate of evolution (i.e., at which there areinflection points in the fitness function, d2rdl¯2 ) and let M = {m1,m2, ...,mn} be thecorresponding expected rates of evolution, i.e., E[dg¯dt |l¯i] = mi. Due to the mono-tonicity of mean growth rate, r¯, the first extrema, at l¯ = l¯1, must be a maximum. Ifthe lag that causes this first maxima in the rate of evolution is greater than the lagthat causes a mean growth rate of zero, l¯1 > l¯c, then the expected rate of evolutionis monotonically increasing as the mean lag increases from 0 to l¯c, and thereforethe expected rate of evolution at l¯ = l¯c is the critical rate of environmental change(i.e., the k that causes r¯ = 0). If, however, l¯1 < l¯c, then the steady-state lag con-tinuously increases as the rate of environmental change, k, increases from 0 to m1,where the population can persist (given l¯1 < l¯c), after which the steady-state lagmakes a discontinuous increase. Technically, there is a saddle-node bifurcation atk = m1. The size of the discontinuous increase in the steady-state lag as the rateof environmental change, k, increases through the first maxima in the rate of evo-lution, m1, and the consequences for population persistence, depends on the otherlags that cause extrema, L, and their respective rates of evolution, M. In particular,if the first local maxima is the global maxima, m1 > mi ∀ i > 1, then there is noquasi-steady-state solution when the rate of environmental change is greater than175it, k > m1, and the mean lag will increases towards infinity. Thus the populationwill go extinct for any k > m1 and m1 is an evolutionary tipping point. This is thesituation discussed in the main text, as our alternative fitness function only createsone extrema in the rate of evolution as a function of mean lag. However, if thereis a maxima that is greater than the first, m1 < mi for some i > 1, then as the rateof environmental change, k, increases through m1 the steady-state lag increasesto the next largest mean lag that produces an expected rate of evolution slightlylarger than m1. If this next largest mean lag is greater than the lag that causes amean growth rate of zero, l¯c, the population is still expected to go extinct for anyk > m1, and m1 is still an evolutionary tipping point. But if the next largest meanlag that produces an expected rate of evolution slightly larger than m1 is less thanl¯c, then m1 is not an evolutionary tipping point and the arguments above for l¯1 canbe repeated for l¯3 (the next maxima). I.e., if l¯3 > l¯c then the critical rate of changedetermines persistence, while if l¯3 < l¯c the other lags and respective evolutionaryextrema determine whether l¯3 is an evolutionary tipping point or not.This argument can be generalized by letting l¯ j be the mean lag in [0, l¯c] thatproduces the maximum expected rate of evolution. If l¯ j < l¯c it must cause a localmaximum in the rate of evolution and thus be in L (with j odd). Extinction thenoccurs whenever k > m j. We then call the height of the largest local maxima inthe expected rate of evolution within the persistence zone, m j, an evolutionarytipping point, as a saddle-node bifurcation occurs as k increases through m j. Thisbifurcation causes long-run population growth rates to go from E[r¯|l¯ = l¯ j]> 0 to anegative value without ever crossing zero.176Appendix DAppendices for Chapter 5D.1 Supplementary figuresmwt = -0.3mwt = -0.2mwt = -0.1-0.6 -0.4 -0.2 0.0 0.2012345growth rate, mfrequencyFigure D.1: The distribution of mutant growth rates before selection. Shownare histograms of 106 mutant growth rates (drawn from the mutationalprocess described in Fisher’s geometric model) and the gamma approx-imation f˜ (m|mwt) for their probability density function (Equation 5.2)for three different wildtype (ancestor) growth rates, mwt . Parameters:n = 4, s¯ = 0.01, mmax = 0.5.17711111111mwt = -0.1mwt = -0.2mwt = -0.310-510-4 0.001 0.01 0.1 110-610-510-40.0010.010.11Mutation probability, UProbabilityof1-steprescueFigure D.2: The probability of 1-step rescue. Shown is the analytic ap-proximation (equation 5.4 with p given by equation 5.9) and resultsof individual-based simulations (”1”s) for three wildtype growth rates(104 replicates for each point). The simulations enforced 1-step rescueby not allowing individuals with a mutation to mutate further. Parame-ters: n = 4, s¯ = 0.01, mmax = 0.5.178Osmond et al.Anciaux et al.Numeric0.05 0.10 0.15 0.20 0.25-mwt0. D.3: Comparison with Anciaux et al. (2018). Probability of 1-steprescue as a function of wildtype decline rate. The red curves plot theirequation 7a. The blue curves plot our equation 5.9 as p in equation5.4. The black curves are obtained by numeric integration of equation5.8 as p in equation 5.4. Parameters as in figure 2A of Anciaux et al.(2018): mmax = 0.5, n = 4, N0 = 105, Es = 0.01, U = 2× 10−5 (left)and U = 2×10−4 (right).179y = beneficial HFisher 1930Ly = fix HKimura 1983Ly = rescue0 1 2 3 mutation size, xfrequencyÈyFigure D.4: The distribution of mutation size. Here size is the scaled Eu-clidean distance from the ancestor in phenotypic space, x = r√n/d,with d/2 the phenotypic distance of the ancestor from the optimumand r the true phenotypic size of the mutation. Fisher (1930) showedthat the smallest mutations are most likely to fix, Kimura (1983) thenpointed out that the smallest of these will be lost to drift (multiplyingFisher’s result by 2x and normalizing); we point out that there is a min-imum size required for rescue (multiplying Fisher’s result by 2(x− xd)and normalizing, where xd is the scaled minimum distance of the wild-type from a phenotype with growth rate 0).180full approximatemwt = -0.1mwt = -0.2mwt = -0.310-510-4 0.001 0.01 0.1 110-910-710-50.0010.1Mutation probability, UProbabilityof2-stepcriticalrescueFigure D.5: The probability of 2-step critical rescue. The solid curves usethe numerical solution to equation 5.11 and the dashed use the analyticsolution to equation 5.12, respectively, as p in equation 5.4. Parameters:n = 4, s¯ = 0.01, mmax = 0.5.1812-step critical 2-step subcritical 2-stepmwt=-0.1mwt=-0.2mwt=-0.310-610-510-4 0.001 0.01 0.1 110-610-510-40.0010.010.11Mutation probability, UProbabilityofrescueFigure D.6: Comparing the probability of 2-step critical and subcritical res-cue. The thin solid curves give the probability of 2-step critical rescueusing the analytic solution to equation 5.12 as p in equation 5.4. Thedashed solid curves give the probability of 2-step subcritical rescue us-ing the numerical solution to equation 5.13 as p in equation 5.4. Thethick solid curves give the probability either 2-step pathway rescues thepopulation. Parameters: n = 4, s¯ = 0.01, mmax = 0.5.182mwt = -0.2mwt = -0.10.0 0.1 0.2 0.3 0.402468101214fitness effect, s = m - mwtfrequencyFigure D.7: The distribution of fitness effects following 2-step critical rescue.Shown are histograms of fitness effects of simulated mutants that rescuethe population according to the first line of equation 5.14. Also shownis the analytic approximation for the probability density function givenin the last line of equation 5.14, converted from growth rates to fitnesseffects. The distribution for first step mutants that establish in a constantpopulation (gray) is shown for comparison. Parameters: n = 4, s¯ =0.01, mmax = 0.5.183mwt = -0.2mwt = -0.10.0 0.1 0.2 0.3 0.402468101214fitness effect, s = m - mwtfrequencyFigure D.8: The distribution of fitness effects following 2-step subcritical res-cue. Shown are histograms of fitness effects of simulated mutants thatrescue the population according to the first line of equation 5.15. Alsoshown is the numerical evaluation for the probability density functionof s given in the last line of equation 5.15, converted from growth ratesto fitness effects. The distribution for first step mutants that establishin a constant population (gray) is shown for comparison. Parameters:n = 4, s¯ = 0.01, mmax = 0.5.mwt = -0.3mwt = -0.2mwt = -0.10.00 0.02 0.04 0.06 0.08 0.100102030405060growth rate, mfrequencyFigure D.9: The distribution of growth rates of potential rescue genotypes.Dashed curves are the right tails of the distributions shown in FigureD.1, renormalized to sum to one over m > 0. Solid curves are renor-malized right tails of the exact distributions (equation 5.1).184mwt = -0.2mwt = -0.10 100 200 300 4000. mutant indexgrowthrate,mFigure D.10: Growth rates of 1-step rescue genotypes in ascending order. res-cue genotypes were sampled as in Figure 5.2. The data is presented inthis alternative way to more readily compare with plots like figure 4Cin Gerstein et al. (2012) and figure 3A in Gerstein et al. (2015).185


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