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Evolutionary theory of cooperation and group life Barros Henriques, Gil Jorge 2021

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Evolutionary theory of cooperation and group lifebyGil Jorge Barros HenriquesB.Sc., University of Lisbon, 2013M.Sc., University of Groningen & Uppsala University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Zoology)The University of British Columbia(Vancouver)March 2021© Gil Jorge Barros Henriques, 2021The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the dissertation entitled:Evolutionary theory of cooperation and group lifesubmitted by Gil Jorge Barros Henriques in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Zoology.Examining Committee:Michael Doebeli, Zoology and Mathematics, UBCSupervisorSarah Otto, Zoology, UBCSupervisory Committee MemberJeannette Whitton, Botany, UBCUniversity ExaminerDaniel Coombs, Mathematics, UBCUniversity ExaminerAdditional Supervisory Committee Members:Leticia Avile´s, Zoology, UBCSupervisory Committee MemberChristoph Hauert, Mathematics, UBCSupervisory Committee MemberiiAbstractNatural selection favors behaviors that increase an organism’s survival and reproduction. However,many organisms exhibit traits that benefit others at a cost to themselves, an apparent contradiction thatDarwin called his “special difficulty”. The evolution of cooperation is an important biological questionbecause it underlies group life and the construction of new levels of organization. For example, cellscooperate to make multicellular organisms and social insects as well as humans cooperate to establishlarge-scale societies. In this thesis, I attempt to increase our understanding of the evolution of coop-eration and group life by developing four mathematical models. In Chapter 2, I study a question thatdates back to Darwin: whether multilevel selection can be responsible for intergroup conflicts in humansocieties. Costly conflicts are collective action problems, and it is not clear what mechanisms could ex-plain their prevalence. My model suggests one possible mechanism: the transmission of cultural traitsbetween groups. Chapter 3 focuses on the interplay between the evolution of cooperation and environ-mental change. This model considers how cooperative interactions (public goods games), which evolvein response to changes in group size caused by environmental change, can either promote evolution-ary rescue or, in some cases, lead to evolutionary suicide. In Chapter 4, I investigate the process ofevolutionary branching (the diversification of a population into multiple strains), which can result fromthe evolution of cooperation between individuals. I show that, when multiple phenotypes experienceevolutionary branching, the evolving phenotype distribution of the population can affect the directionof diversification. In the long-term, this may have important consequences for the evolution of divisionof labor. Finally, in Chapter 5, I consider how collectives of cooperating cells—including multicellularorganisms and complex multi-species biofilms—reproduce to create new groups. I develop a multilevelselection model to investigate the consequences of various modes of reproduction, such as the produc-tion of single-cell gametes or vegetative fragmentation. Considered together, these four models expandour understanding of cooperation and group life.iiiLay SummaryGenes that make organisms better are surviving and reproducing get passed on to future generations,in a process called natural selection. But many organisms act in ways that benefit others at a riskto themselves: vampire bats share food with hungry group members, ground squirrels produce alarmcalls that could attract the attention of predators, and many birds take care of each other’s offspring.How can these cooperative behaviors persist in the face of natural selection? In this thesis, I developfour mathematical models that help us understand different aspects of the evolution of cooperation andgroup life. I investigate how cooperation affects adaptation to environmental change, how populationsdiversify into cooperators and non-cooperators, and how groups of cooperative organisms divide tocreate new groups. I also look at our own species, asking why members of human groups risk their livesto engage in conflicts with other groups.ivPrefaceA version of Chapter 2 has been published as: G. J. B. Henriques, B. Simon, Y. Ispolatov, and M.Doebeli (2019). Acculturation drives the evolution of intergroup conflict. Proceedings of the NationalAcademy of Sciences of the USA 116:28, 14089–14097. MD designed the research; GJBH, BS, andMD performed the research; GJBH analysed the data, and GJBH wrote the paper, with manuscript editscontributed by all coauthors.A version of Chapter 3 has been published as: G. J. B. Henriques and M. M. Osmond (2020).Cooperation can promote rescue or lead to evolutionary suicide during environmental change. Evolution74:4, 1255–1273. GJBH designed the research, wrote the simulations, and wrote the article; MMOsupervised the project and contributed substantially to the simulations and to writing. Both authorsparticipated in developing the model.A version of Chapter 4 has been published as: G. J. B. Henriques, K. Ito, C. Hauert, and M. Doebeli(2021). On the importance of evolving phenotype distributions on evolutionary diversification. PLOSComputational Biology 17:2, e1008733. GJBH, CH, and MD conceptualized the project; MD super-vised the project; GJBH and KI performed the analysis; GJBH wrote the paper, with manuscript editscontributed by all coauthors.Chapter 5 is being prepared for submission with coauthors Simon van Vliet and Michael Doebeli.GJBH, SVV, and MD designed the research; MD supervised the project; SVV wrote the simulations; GJBHand SVV analysed the data; GJBH wrote the manuscript, with contributions from SVV and MD.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Darwin’s “special difficulty” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The calculus of selfishness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Golden Rule for the evolution of cooperation . . . . . . . . . . . . . . . . . . . . 51.3.1 Relatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.4 Partner choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.5 Group selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 From evolutionary game theory to adaptive dynamics . . . . . . . . . . . . . . . . . . 91.4.1 Matrix games and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Adaptive dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Goals of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Model 1: Acculturation drives the evolution of intergroup conflict . . . . . . . 131.5.2 Model 2: Cooperation can promote rescue or lead to evolutionary suicide duringenvironmental change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.3 Model 3: On the importance of evolving phenotype distributions on evolution-ary diversification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14vi1.5.4 Model 4: Multilevel selection favors fragmentation modes that maintain coop-erative interactions in multispecies communities . . . . . . . . . . . . . . . . 142 Acculturation drives the evolution of intergroup conflict . . . . . . . . . . . . . . . . . . 162.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Within-group dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Between-group dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Default parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Language and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Population dynamics within groups . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Frequent intergroup conflict promotes warrior production . . . . . . . . . . . . 222.4.3 The evolution of warrior production can lead to whole-population extinction . 232.4.4 Without acculturation, intergroup conflict is not favored . . . . . . . . . . . . 242.4.5 Acculturation drives the evolution of intergroup conflict . . . . . . . . . . . . 252.4.6 Acculturation coevolves with engagement and warrior production . . . . . . . 262.4.7 Overview of model behavior and robustness . . . . . . . . . . . . . . . . . . . 272.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Cooperation can promote rescue or lead to evolutionary suicide during environmentalchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.3 Adaptive dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1 Resident dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Evolution of cooperation for a constant lag . . . . . . . . . . . . . . . . . . . 403.3.3 Evolution of cooperation during environmental change . . . . . . . . . . . . . 443.4 Extension to other types of social interactions . . . . . . . . . . . . . . . . . . . . . . 503.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 On the importance of evolving phenotype distributions on evolutionary diversification . 564.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.2 Adaptive dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vii4.2.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.4 G-matrix orientation and direction of branching . . . . . . . . . . . . . . . . . 624.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 Perpendicular branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.2 Changes in the G-matrix orientation underlie the direction of branching . . . . 654.3.3 Effect of the branching direction in more than two dimensions . . . . . . . . . 684.3.4 Effect of the direction of approach for unequal games . . . . . . . . . . . . . . 684.3.5 Eventual fate of the population . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Multilevel selection favors fragmentation modes that maintain cooperative interactionsin multispecies communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2.1 Cell dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.2 Group dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2.3 Evolution of the fragmentation mode . . . . . . . . . . . . . . . . . . . . . . 825.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.1 Multilevel selection can maintain cooperators and avert mutational meltdown . 835.3.2 Complete fragmentation minimizes mutation load . . . . . . . . . . . . . . . . 835.3.3 Multispecies communities are more vulnerable to mutational meltdown . . . . 855.3.4 Size-dependent fragmentation rate and migration prevent mutational meltdownin multispecies communities . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3.5 Strategies that maximize community productivity are evolutionary attractors . . 865.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1 Chapter 2: Acculturation drives the evolution of intergroup conflict . . . . . . . . . . . 916.2 Chapter 3: Cooperation can promote rescue or lead to evolutionary suicide during envi-ronmental change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Chapter 4: On the effect of evolving phenotype distributions on evolutionary diversifi-cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.4 Chapter 5: Multilevel selection favors fragmentation modes that maintain cooperativeinteractions in multispecies communities . . . . . . . . . . . . . . . . . . . . . . . . . 94Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A Supplementary Information for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 111A.1 Model details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A.1.1 Group extinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A.1.2 Fissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111viiiA.1.3 Probability of winning conflict . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.2 Basic model outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.2.1 Within-group equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.2.2 Between-group equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.3 Time-scale of loss of within-group variation . . . . . . . . . . . . . . . . . . . . . . . 114A.4 Selection gradients at the between-group level . . . . . . . . . . . . . . . . . . . . . . 114A.5 Costly acculturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.6 Supplementary figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B Supplementary Information for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 123B.1 Canonical equation modified for non-overlapping generations . . . . . . . . . . . . . . 123B.1.1 Stochastic description of trait substitution sequences . . . . . . . . . . . . . . 123B.1.2 Mean path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B.1.3 Deterministic path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B.2 Equilibrium cooperation for a constant lag . . . . . . . . . . . . . . . . . . . . . . . . 127B.2.1 Proof: there is only one equilibrium value of cooperation below xcrit . . . . . . 127B.2.2 Effect of lag and p on the equilibrium value of cooperation . . . . . . . . . . . 128B.3 Some remarks on post-branching (polymorphic) dynamics . . . . . . . . . . . . . . . 128B.3.1 Polymorphic dynamics for a constant lag . . . . . . . . . . . . . . . . . . . . 129B.3.2 Polymorphic evolution during environmental change . . . . . . . . . . . . . . 132B.4 Alternative cost and benefit functional forms . . . . . . . . . . . . . . . . . . . . . . . 133B.4.1 Nonlinear benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B.4.2 Linear costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135B.5 The evolution of competition during environmental change . . . . . . . . . . . . . . . 140C Supplementary Information for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 144C.1 IBS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.2 OSS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.3 PDE algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145C.4 Branching condition along evolutionary branching line . . . . . . . . . . . . . . . . . 146C.5 Maximum-likelihood invasion path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148C.6 Invasion of large-effect mutants into a two-strain resident . . . . . . . . . . . . . . . . 150D Supplementary Information for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . 154D.1 List of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154D.2 Density regulation based on number of groups . . . . . . . . . . . . . . . . . . . . . . 155D.3 Effect of migration and of size-dependent fragmentation on maximum mutation rate . . 155D.4 Recovering the results of Pichugin et al. (2017) . . . . . . . . . . . . . . . . . . . . . 155ixList of TablesTable 3.1 Possible fitness partitions in the model. . . . . . . . . . . . . . . . . . . . . . . . . 43Table D.1 List of parameters and default values that were using for producing figures. . . . . . 154xList of FiguresFigure 1.1 Prisoner’s Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.2 Natural selection favors defectors . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.3 Many mechanisms for the evolution of cooperation rely on assortment between in-dividuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.4 Matrix games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.1 Evolution of warrior production with constant group-level traits. . . . . . . . . . . 23Figure 2.2 Evolution of warrior production increases the risk of whole-population extinction. . 24Figure 2.3 Coevolution toward peace and coevolution toward conflicts. . . . . . . . . . . . . 26Figure 3.1 The population size depends on distance to the optimum and on the amount ofcooperative investment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.2 Monomorphic evolutionary dynamics for a constant lag when the singular point x?is a stable point and when it is a branching point . . . . . . . . . . . . . . . . . . 41Figure 3.3 Evolution of cooperation with a moving optimum, for a species with fitness partitionD or fitness partition CD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.4 Lag and population size at the dynamical equilibrium, in populations with fitnesspartition D or fitness partition CD, for different velocities of environmental change (v). 47Figure 3.5 Critical velocity (maximal velocity of environmental change for which the pop-ulation is able to track the moving optimum without undergoing extinction) as afunction of the level of cooperative investment (x < xcrit), for fitness partitions Dand CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.6 Evolution of competition with a moving optimum, for a species with viability equalto one and a fecundity that depends on competition (x) and lag (L ). . . . . . . . . 52Figure 4.1 Fitness landscape during approach to the branching point and at the branching point. 61Figure 4.2 Examples of stochastic replicates of the individual-based and oligomorphic simula-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.3 In IBS, branching tends to occur perpendicularly to the direction of approach, butin OSS, all directions of branching are equally likely. . . . . . . . . . . . . . . . . 65Figure 4.4 In the PDE model, the population branches perpendicularly to the direction of ap-proach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66xiFigure 4.5 Changes in the G-matrix orientation underlie the direction of branching. . . . . . . 67Figure 4.6 Branching in three dimensions also tends to be perpendicular to the direction ofapproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.7 Even when games have different fitness effects, the direction of approach plays arole in determining the direction of branching. . . . . . . . . . . . . . . . . . . . . 70Figure 4.8 Even with different payoff (cost and benefit) functions, the direction of approachplays a role in determining the direction of branching. . . . . . . . . . . . . . . . 71Figure 4.9 The initial direction of approach biases evolution towards one of the two possiblestable states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 5.1 All possible fragmentation strategies in our model can be described using a two-dimensional phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 5.2 In the absence of mutations, complete fragmentation maximizes equilibrium popu-lation size, measured either as total number of groups (G) or as total productivity(Ntot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 5.3 When there are no group events, mutations cause community extinction, but group-level events prevent this fate, as exemplified by the three archetypal modes of frag-mentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.4 When community complexity is high, the productivity peak shifts away from uni-cellular bottlenecks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.5 Migration and size-dependent fragmentation rate allow multispecies communitiesto resist mutational meltdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 5.6 Evolution of fragmentation mode maximizes total number of cells (Ntot, top row)rather than other quantities such as number of groups (G, middle row) or averagegroup size (Ni, bottom row). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure A.1 Fissioning function and probability of winning a conflict for different values of theparameters s and gs, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure A.2 Eq.uilibrium densities of the within-group dynamics for a monomorphic group. . . 113Figure A.3 In the absence of group events, the time-scale at which variation is lost within apolymorphic group is, roughly, linearly proportional to 1/∆p. . . . . . . . . . . . 115Figure A.4 Coevolution toward conflict can occur for different growth rate functional forms,including when nˆ(p) is decelerating for low values of p (which occurs when 2b1 > d).118Figure A.5 Coevolution toward conflict can occur for costly acculturation. . . . . . . . . . . . 119Figure A.6 Sufficiently rare group events lead to a peaceful equilibrium. . . . . . . . . . . . . 120Figure A.7 Coevolution toward conflict occurs even for populations with low numbers of groups.121Figure A.8 Example dynamics when warriors can also reproduce, at a cost c = 0.5. . . . . . . 122Figure B.1 Individual-based simulations showing cooperation values and population size forboth monomorphic and polymorphic dynamics, with static environments. . . . . . 130xiiFigure B.2 Equilibrium strain sizes in a bimorphic population with a constant lag. . . . . . . . 131Figure B.3 As the environment changes, the frequency of cooperators increases which, after aperiod of oscillations, stabilizes the population size and the distance to the optimum. 133Figure B.4 Frequency of cooperators, strain sizes, and distance to the optimum at dynamicalequilibrium, for different velocities of environmental change (v). . . . . . . . . . . 134Figure B.5 Effects of small deviations from linearity with a constant lag, for low and highvalues of p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Figure B.6 Lag and population size at the dynamical equilibrium, in populations with fitnesspartition D or fitness partition CD, for different velocities of environmental change (v).136Figure B.7 Ecological and evolutionary dynamics are qualitatively similar between the nonlin-ear multiplicative cost scenario (main text) and the linear multiplicative cost scenario.137Figure B.8 Evolution of cooperation with a moving optimum in a model with linear multiplica-tive costs, for a species with fitness partition D or fitness partition CD. . . . . . . . 138Figure B.9 Ecological and evolutionary dynamics are qualitatively similar between the nonlin-ear multiplicative cost scenario (main text) and the linear additive cost scenario. . . 139Figure B.10 Evolution of cooperation with a moving optimum in a model with linear additivecosts, for a species with fitness partition CMD or fitness partition CD. . . . . . . . . 141Figure B.11 Competition x decreases the ecological equilibrium of population density nˆ, creatingan eco-evolutionary feedback loop. . . . . . . . . . . . . . . . . . . . . . . . . . 143Figure C.1 In the PDE model, for some initial phenotypic distributions, branching may occurin directions that are different from those predicted by the evolutionary branchingline approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure D.1 When extinction rate grows with number of groups instead of number of cells, themain qualitative model result does not change: increased community complexityshifts the productivity peak away from small bottlenecks and toward binary fission. 155Figure D.2 Size-dependent fragmentation rates allow multispecies communities to resist muta-tional meltdown (more parameters). . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure D.3 Low to intermediate migration rates allow multispecies communities to resist muta-tional meltdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Figure D.4 In the absence of group-level density-dependence, if the cell birth rate increases withgroup size, the complementarity parameter κ determines which strategy maximizesfitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158xiiiAcknowledgmentsI may be the person signing this thesis, but it took numberless people to write it. I want to thank mysupervisor, Michael Doebeli, for always being available to make time for me and guide me when I gotstuck, while also giving me the time and space to become an independent researcher. I must also thankmy coauthors Christoph Hauert, Yaroslav Ispolatov, Koichi Ito, Matt Osmond, Burt Simon, and SimonVan Vleet, as well as my committee members Sarah Otto and Leticia Avile´s. Without their mentorshipand support, none of these papers would have seen the light of day.I am also grateful to every member of the Doebeli, Otto, and Hauert labs, to the Zoology ComputingUnit team (Alistair Blachford, Andy LeBlanc, and Richard Sullivan), and to everyone who contributes tobuild our community at the Biodiversity Research Center. They make this a wonderful place to live andwork. Thank you for organizing, making cookies for, and hosting guest speakers for BRS, BLISS, LET,EDG, and VEG. Thank you for organizing the Huts skit. Thank you for putting together the Christmasparty, for running the coffee co-op, for long chats at lunch time.I’m especially thankful to all the friends that pulled me away from working on this thesis and tookme out for dinner, invited me to their place, or accompanied me on hikes in the mountains around Van-couver. There are far too many people to write down, but I cannot omit Le´onard Dekens, Ailene MacP-hearson, Ba´rbara Neto-Bradley, Ilan Rubin, Szu Shen, Ken Thompson, Mackenzie Urquhart-Cronish,and Brooke Xiang.Huge thanks to the student activists and organizers working at UBC, in groups like Climate JusticeUBC, the Pride Collective, the Social Justice Center, Students Against Bigotry, and of course our Teach-ing Assistant union, CUPE 2278, whose executive and staff fight tirelessly every single day to makesure our rights are respected. The university works because we do! A special thank you to Laura Bulk,Gillian Glass, and David Huxtable.Thank you to all the dogs in the BRC family, including (but not limited to) Bam-boo, Coco, Cosmo,Garbanzo, Harriet, Hasley, Julia, Leyla, Moonie, Pookie, Rush alias Pablo, Sage, and Smudge. Youbring us joy even when we are overwhelmed with work.I acknowledge that the beautiful land where I lived and worked for the past six years is the ancestral,unceded, traditional and occupied territory of the xwm@Tkw@y´@m, Skwxu´7mesh, and s@l´ilw@taPì peoples.I’m immensely grateful for being welcomed as a visitor on the most gorgeous place I have ever lived.I would like to end by thanking my family: my parents, who every weekend encouraged me tocontinue (even though they would prefer if I were closer to home); my brother, who has always beenmy role model and inspiration; and, more than anyone, Linne´a Sandell—your support and your love arethe reason I reached the end of this journey.xivDedicationThis thesis is dedicated to the essential and front line workers who risked their lives to keep our shelvesstocked, our hospitals, nursing homes and schools open, our public transport circulating, our streets andbuildings clean, our mail delivered, our food growing, and our society running during the long monthsof lockdown while I wrote the final chapters.xvChapter 1IntroductionAs soon as we study animals—not in laboratories and museums only, but in the forest and theprairie, in the steppe and the mountains—we at once perceive that though there is an immenseamount of warfare . . . amidst various classes of animals, there is, at the same time, as much, orperhaps even more, of mutual support, mutual aid, and mutual defence.— Pyotr Kropotkin1.1 Darwin’s “special difficulty”In a 1988 article published in Natural History, palaeontologist and historian of science Stephen JayGould recounts the story of Leo Tolstoy’s final journey. Aged eighty-two, having contracted pneumoniaduring a long train ride during the cold Russian winter, Tolstoy passed away in the station master’s homeat a railroad stop. As he died, too weak to write, he dictated a last letter to his son and daughter (citedin Gould 1988):“The views you have acquired about Darwinism, evolution, and the struggle for existencewon’t explain to you the meaning of your life and won’t give you guidance in your actions,and a life without an explanation of its meaning and importance, and without the unfailingguidance that stems from it is a pitiful existence. Think about it. I say it, probably on theeve of my death, because I love you.”Tolstoy’s exhortation to his children echoes a prevalent perception among 19th century intellectuals:that natural history in general, and Darwinian thought in particular, posed a threat to the very foundationsof Christian morality. For example, in a famous stanza from In Memoriam A. H. H., published sixty yearsprior to Tolstoy’s death, Lord Tennyson wrote about the predicament of “Man”, “Who trusted God waslove indeed / And love Creation’s final law– / Tho’ Nature, red in tooth and claw / With ravine, shriek’dagainst his creed” (Tennyson 1850).It is not hard to understand the source of this accusation. After all, Charles Darwin contended thatsuccess in the evolutionary arena was all about a vicious “struggle for existence”: a bloody battle ofall against all in which only the fittest survive (Darwin 1859). Darwin’s contemporary interpreters in1the popular press were even more cantankerous. For Thomas Henry Huxley, the natural world wascharacterized by “intense and unceasing competition” of a “gladiatorial” nature (Huxley 1895), andfor Herbert Spencer (who coined the phrase “survival of the fittest”), Darwinism justified mercilesscompetition and provided a biological apology for laissez faire (Hofstadter 1955). In the popular cultureof the time, Darwinism (just like its intellectual ancestor, Malthusianism) came to be understood as ascientific vindication of the Hobbesian conception of the state of nature: bellum omnium contra omnes.This bleak view of nature was then a fairly novel development in Western thought. It was contraryto the hitherto dominant Aristotelian perspective that considered humans, together with bees, wasps,and ants, to be “social animals” (angelaia) that lived together in harmony (Arnhart 1994). However,from a Darwinian point of view, the altruistic interactions that characterize the societies of bees, wasps,ants, and humans are—or at least appear to be—mysterious. After all, natural selection favors behaviorsthat increase one’s reproductive success. But cooperation—i.e., giving up reproductive potential for thebenefit of others, as social insects do—runs counter to this principle (as further detailed in section 1.2).Darwin himself was aware that this posed a “special difficulty” to his theory (Herbers 2009).The list of organisms engaging in cooperative behaviour has only grown since Darwin’s time (Dugatkin2002). Naked mole rats and the Synalpheus genus of shrimps joined the ranks of the eusocial animals(those who, like Aristotle’s bees, wasps, and ants, exhibit a division of labor into reproductive and non-reproductive groups), see Wilson and Ho¨lldobler (2005). Meerkats (Clutton-Brock et al. 1999), groundsquirrels (Dunford 1977), and pied flycatchers (Krams et al. 2010) give alarm calls, which alert theirgroup-mates to the presence of predators. Rats and vampire bats donate food to conspecifics (in thecase of bats, by regurgitating blood, Wilkinson 1984). And many species, such as cichlids (Bergmu¨lleret al. 2005), azure-winged magpies (Komeda et al. 1987), and tamarins (Cronin et al. 2005), engage incooperative breeding.Cooperation is prevalent among microbial species as well. Microbes engage in many collective be-haviors, such as biofilm formation, quorum sensing, and dispersal (Tarnita 2017; West et al. 2006). Forinstance, during food shortages, individual slime mold cells of the species Dictyostelium discoideumband together, forming a fruiting body for dispersal purposes. Those cells that form the stalk of thefruiting body sacrifice their own reproductive success for the benefit of the other cells (Strassmann et al.2000). Another form of cooperation that is commonplace in microbial populations is the productionof costly extracellular compounds or functions that provide a collective benefit (public goods, Morris2015). Some examples include siderophores (for acquiring iron), antibiotics (for microbial competi-tion), exopolysaccharides (which provide structure in biofilms), or enzymes such as invertase (for sugardigestion).How can such behaviors be maintained in the face of natural selection? After all, yeast cells thatdo not produce invertase still reap the benefits of sugar digestion by their more cooperative neighbours,without paying any of the metabolic costs. Meerkats that give alarm calls risk being seen and eatenby predators, while their less altruistic partners run to safety and survive. The evolutionary success ofa given phenotype is traditionally measured by fitness—a quantitative representation of reproductivesuccess and contribution to the next generation (Ariew and Lewontin 2004; Krimbas 2004; Orr 2009).2On the face of it, it would seem that cooperators like the invertase-producing yeast or the alarm-callingmeerkat should have very low fitness, especially when compared with their less cooperative counter-parts.1.2 The calculus of selfishnessWhen faced with conundrums like these, evolutionary biologists turn to simple mathematical models.Such models (much like controlled laboratory experiments) provide simplified abstractions of some bi-ological problem. They keep only those variables that are believed to be essential to the question at hand(Fisher 1930). In doing so, they make the problem tractable and (hopefully) allow us to draw helpfullessons that we might not be able to learn from studying the question in all of its natural complexity.Though some of these models may say little about any particular biological species, they tell us some-thing important about the possible relations and patterns that we should be on the lookout for (Caswell1988). And, while they cannot teach us what is true, they can set limits to what could be true:“Theoretical population biology is the science of the possible; only direct observation canyield a knowledge of the actual.” (Lewontin 1968)Let us then consider a simple mathematical model of cooperation. Two individuals interact; each ofthem may act as a cooperator or as a defector. A cooperator pays a fitness cost c to provide some fitnessbenefit b to their neighbour. A defector, on the other hand, pays no cost and provides no benefit. If theinteraction partner decides to cooperate, a focal individual will maximize their own fitness by defecting;the same is true if the interaction partner decides to cooperate (Fig. 1.1). This particular model is aninstance of a Prisoner’s Dilemma: a “game” in which mutual defection is the only result such thateach “player” could only do worse by unilaterally changing strategy. Here, the word “game” refers tointeractions between individuals. The application of game theory to evolutionary biology (the “calculusof selfishness”, in the words of Sigmund 2010) was pioneered by Robert Trivers, John Maynard Smith,and George R. Price (Maynard Smith 1982; Maynard Smith and Price 1973; Trivers 1971).Clearly, if cooperation is in fact similar to the Prisoner’s Dilemma game, it seems unlikely that itwould evolve by natural selection. Let us see if this is true. Denote by x the frequency of cooperatorsand by y = 1− x the frequency of defectors. Given these frequencies, and the payoff matrix from Fig.1.1, the average fitness for a cooperator is wC(x) = (b− c)x+(−c)y = bx− c. The average fitness for adefector is wD(x) = bx+0y = bx. Hence, defectors always have a higher fitness than cooperators.We can also keep track of the evolutionary dynamics of the population (Taylor and Jonker 1978):x˙ = x(wC(x)−w)y˙ = y(wD(x)−w), (1.1)where the overdot represents differentiation with respect to time and w = xwC(x)+ ywD(x) = (b− c)x3       Figure 1.1: The Prisoner’s Dilemma is a game in which mutual defection is the only result such that each playercould only do worse by unilaterally changing strategy. If both players cooperate, they both receive the rewardR for cooperating. If both players defect, they both receive the punishment payoff P. If the focal player defectswhile the partner cooperates, then the focal player receives the temptation payoff T , while the partner receives the“sucker’s” payoff, S. In a Prisoner’s Dilemma, T > R > P > S. In the example given in the text, b is the fitnessbenefit and c is the fitness cost of cooperation: this example obeys the aforementioned condition.is the mean population fitness. Since x+ y = 1, we can write the system above asx˙ = x(1− x)(wC(x)−wD(x)), (1.2)which for this particular game equals −cx(1− x), meaning that the number of cooperators decreaseslogistically over time toward extinction (Fig. 1.2).Instead of pairwise interactions, we can also imagine a group of N individuals who live together. Anumber k of them are cooperators (just like in the Prisoner’s Dilemma above, they pay a fitness cost c toprovide some fitness benefit b to their neighbour). The other N− k are defectors, who neither providebenefits nor pay costs. Then, the fitness of each cooperator is given by wC = b(k− 1)/(N − 1)− c,because they receive a share (1/(N− 1)) of the benefit produced by the k− 1 other cooperators, whilealso paying the cost. The fitness of each defector, on the other hand, is wD = bk/(N−1): they receivea share of the benefit produced by all cooperators in the group and pay no cost. The average fitnessis w = (b− c)k/N. This example is a particular instance of what are sometimes called “public goodsgames” (reviewed in Archetti and Scheuring 2012). Just like in the pairwise game, defectors alwayshave higher fitness than cooperators: they would reproduce at a higher rate and, over time, drive thecooperators extinct (Fig. 1.2).Note that, in our models, fitness is not a context-independent quantity. Cooperators who belongto a cooperator-rich group may have higher fitness than defectors who are themselves surrounded bydefectors (in situations where the success of a given phenotype depends on the phenotypes of the otherindividuals in the population, fitness is said to be frequency-dependent). In fact, a population consistingonly of cooperators has the highest possible average fitness. But remarkably, in these simple models,natural selection drives the average fitness of the population to lower and lower values. This flies inthe face of intuition. We tend to picture fitness as a landscape full of hills and valleys (Wright 1932).Any given point in the landscape corresponds to a phenotype value, and the higher the landscape atthat point, the more fit that phenotype is. Natural selection, in this metaphor, is a process that drivesevolving populations uphill until they reach the nearest peak. In fact, if this fitness landscape were static,4C CCCC  C CD CCCD CDDDDDDDFigure 1.2: In the absence of any mechanisms that benefit cooperation, natural selection favors defectors. Thefigure shows the appearance by mutation of a defector, D, in a resident population of cooperators, C. Becausedefectors have a higher fitness than cooperators, natural selection reduces the abundance of cooperators. Thiscauses a decline in the mean population fitness. Figure adapted from Nowak (2006).the population’s mean fitness should never decrease (Fisher 1930). Frequency-dependence, however,implies that as the population’s trait values change, so does the fitness landscape. The population alwaysmoves uphill, only to find the landscape shifting beneath its feet. This is the great tragedy of publicgoods games: those individuals who do what is best for themselves get rewarded by natural selection,yet this process leads to ever decreasing benefits for everyone. As the Nobel laureate Joseph Stiglitzquipped, Adam Smith’s invisible hand—the idea that competition leads to efficiency as if guided byunseen forces—“is invisible, at least in part, because it is not there” (Stiglitz 2002).What processes, then, account for the evolution and maintenance of cooperation?1.3 The Golden Rule for the evolution of cooperationThe Gospel of Matthew describes the Sermon on the Mount, which introduces the Christian version ofa moral principle found in many religions and cultures—the Golden Rule:“Therefore all things whatsoever ye would that men should do to you:do ye even so to them: for this is the law and the prophets” (Matthew 7:12)The Golden Rule is a general prescriptive principle for altruism between people. As it turns out, avery similar descriptive principle underlies most explanations for the evolution of cooperation in nature.We have seen that the more cooperators a group contains, the higher the average fitness of thatgroup. That is because even though all cooperators pay a cost, they also benefit from each others’contributions. The fitness of a cooperator surrounded by other cooperators is higher than the fitness ofa defector surrounded by other defectors. Therefore, cooperation can be favored by natural selection,provided that cooperators interact preferentially with other cooperators. In other words, assortmentbetween cooperators is the engine of cooperation. This general principle is a common ingredient tomost models of social evolution: we may consider it as a Golden Rule for the evolution of cooperation.Although they are very different from each other, many modeling traditions within evolutionary biologyrely on assortment (under different names) as an explanation for cooperation (Fletcher and Doebeli2009). In this section, we briefly describe some of them: relatedness, reciprocity, structure, partnerchoice, and group selection (Fig. 1.3).5  Figure 1.3: Many mechanisms for the evolution of cooperation rely on assortment between individuals. As-sortment allows cooperation to evolve by ensuring that the benefits of cooperation flow preferentially towardcooperators (in blue) rather than defectors (in red). Related individuals are likely to share similar genotypes.Reciprocity occurs when cooperation is a behavioral response to cooperation. Spatial or social structure allowsclusters of cooperators to share benefits with each other. Partner choice occurs when individuals actively seekothers based on their phenotype. In group selection, groups with more cooperators grow faster than groups withfewer cooperators. Figure inspired by Nowak (2006).1.3.1 RelatednessPerhaps the earliest and most celebrated mechanism incorporating this Golden Rule was suggested byJ. B. S. Haldane (1932, 1955) and formalized by William D. Hamilton (1963, 1964a,b). The ideais that natural selection will benefit cooperation if the donor and the recipient of an altruistic act arerelatives. Specifically, cooperation will increase in frequency if rb > c, where b and c are the fitnessbenefit and cost of cooperation, respectively, and r is the genetic relatedness between the donor andthe recipient (Hamilton’s rule). More concretely, r is the regression coefficient of recipient genotypeon donor genotype (Michod and Hamilton 1980). In other words, relatedness measures whether therecipient of cooperation is more likely to also be a cooperator than a random individual in the population.Relatedness is high when both individuals belong to the same family, in which case they are likelyto share identical alleles because they inherited them from a common ancestor. For example, a diploidindividual’s relatedness to their sibling is 1/2, to their uncle or aunt is 1/4, and to their cousin is 1/8.Thus, when asked whether he would jump into a river to save a brother, Haldane, anticipating Hamilton’swork, is rumored to have quipped that he would not, but would do so for two brothers or eight cousins.(This particular version of the story, retold from McElreath and Boyd 2007, is probably apocryphal; buta very similar if less witty argument can be found in Haldane 1955.) If interactions occur frequentlybetween individuals of the same family, that is a form of assortment that can promote the evolution ofcooperation—a process that Maynard Smith (1964) famously described as “kin selection”.61.3.2 ReciprocityAre there no conditions under which natural selection would favor altruism between unrelated individu-als? Trivers (1971) addressed this question using the framework of game theory. He depicted encountersbetween individuals as a Prisoner’s Dilemma game (Fig. 1.1). As we saw in section 1.2, this scenariousually leads to the proliferation of defectors (Fig. 1.2). But what if individuals encounter each otherrepeatedly during their lifetimes? Then, when cooperative individuals encounter a cooperative partner,they will get enormous benefits as the number of interactions increases. And should a cooperator en-counter a defecting partner, they can always switch to defection in future rounds and minimize theirlosses.This idea was vindicated when Axelrod and Hamilton (1981), in a series of computer tourna-ments, showed that a strategy named “tit-for-tat” (TFT) performs remarkably well in iterated Prisoner’sDilemma games. The strategy is simple: start as a cooperator, then copy your partner’s previous move.The payoff of a player who always defects (ALLD) against a TFT-player is b in the first round, and zerofor each additional round. In contrast, the payoff of TFT against another TFT is b− c per round. If,at the end of every round, there is a probability p that the players initiate a new round, then the to-tal payoff becomes b− c+ p(b− c)+ p2(b− c)+ · · · = (b− c)/(1− p). Using these payoffs, we candetermine under what condition a TFT resident population can resist the invasion of an ALLD mutant:(b− c)/(1− p) > b. This simplifies to pb > c, which looks just like Hamilton’s rule. This similaritymakes a lot of sense, since the parameter p expresses the stability of reciprocation. By acting towardyour partner as they have acted toward you, cooperation gets rewarded over the long run. Reciprocatorsbecome engaged in long, virtuous chains of cooperative interactions: cooperation begets cooperation.The strategy of reciprocity ensures that, in the long run, the recipients of cooperation will be thoseindividuals most likely to cooperate back—in other words, it is a form of assortment (Fletcher and Doe-beli 2009). But interactions need not always occur between the same two individuals. Encounters areoften fleeting, between individuals who do not anticipate seeing each other again. In these cases, indi-viduals may build a reputation as their actions are observed by others in the population, and reciprocitycan still take place indirectly (Nowak and Sigmund 1998; Nowak and Sigmund 2005).1.3.3 StructureAnother form of assortment occurs when populations are not well-mixed. The most straightforwardexample of this are viscous populations, i.e., populations in which limited dispersal leads to spatialstructure. For example, if individuals occupy specific patches from which they do not move—e.g., inisland or stepping-stone models— they can only interact with their neighbors. The simplest such caseoccurs when each individual occupies a patch in a two-dimensional, square lattice (Nowak and May1992). In each round of the game, each individual interacts with its immediate neighbors. Cooperatorscan then survive by forming large, compact clusters. In these clusters, most individual cooperators onlyinteract with other cooperators, thus restricting exploitation by defectors to the cluster edges.A generalization of the same idea places individuals on the vertices of a graph, connected to theirinteraction partners by the graph’s edges (De´barre et al. 2014b; Ohtsuki et al. 2006). This network7could represent spatial or social population structure. Just like in the square lattice case, cooperators canpersist by forming network clusters.Interestingly, there are some circumstances that severely limit the beneficial effect of viscosity oncooperation. Imagine a spatially structured population where individuals inhabit separate demes, con-nected by migration. If density regulation occurs within each deme (after social interactions and repro-duction but prior to migration), then social behavior has no effect on the number of emigrants (this issometimes called soft selection or local regulation). In other words, even though cooperators may helpone another, they must later compete with one another (Hamilton 1971). This problem disappears ifdensity regulation occurs after dispersal and at a global scale (hard selection).1.3.4 Partner choiceAnother way to generate assortment is when individuals actively chose their interaction partners. Oneexample is self-referential phenotype-matching, the process of individuals comparing their own pheno-typic characteristics with those of others and interacting preferentially with those with whom they sharethe same phenotype (Lacy and Sherman 1983). A similar phenomenon occurs when there is a linkagedisequilibrium between the gene encoding an outwardly recognizable phenotype and the gene encodingcooperation (Hamilton 1964b). This has been called the greenbeard effect (Dawkins 1976), after theexample of a gene that causes a conspicuous feature (e.g., a green beard) and is linked to the gene forcooperation, allowing the bearer of this feature to recognize it in other individuals and behave preferen-tially toward them (Jansen and Van Baalen 2006). Individuals may also chose their partners based ontheir reputation (Fu et al. 2008). Partner choice mechanisms are susceptible to deception by individualswho mimic the phenotype or otherwise circumvent the recognition mechanism.A form of partner choice can also be combined with the Axelrod–Hamilton model of iterated gamesby making players cease a relationship after a number of interactions. When a relationship ceases, bothplayers switch partners. Perhaps counter-intuitively, this type of model actually restricts the evolution ofcooperation, because defectors can move rapidly through a population of cooperators in search of newpartners to exploit (Enquist and Leimar 1993). Critically, however, when the dissolution of relationshipsis contingent on partner strategy (“walk-away” strategies), cooperation is favored (Aktipis 2004).1.3.5 Group selectionNatural selection can operate in any population of replicating entities, provided that (Lewontin 1970):• Entities vary in their properties (phenotypic variation);• Sampling (reproduction) of entities is biased with respect to those properties (differential fitness);• These properties are correlated between parents and offspring (fitness is heritable).These principles describe the process by which one phenotype may increase its proportional representa-tion in the population relative to other phenotypes. We usually think of the replicating entities as being8individuals; however, the same three principles could apply to other entities, such as genes, groups ofindividuals, species, or even pre-biotic molecules (Lewontin 1970).Whether or not natural selection at levels other than the individual plays a significant role on theevolution of life on Earth is a very controversial topic (West et al. 2007; Wilson 1983, 2008). The firstverbal version of the idea was put forward by Darwin (1871), who argued that selection operating at thelevel of the group could drive the evolution of helping behavior in humans. Groups containing manycooperators, he argued, would grow faster than groups containing many defectors. Here, we see alreadythe seeds of the idea that came to be known as group selection (or multilevel selection): that allelesencoding traits that are detrimental within a given group may still increase in frequency if those traitsconfer an advantage to the group to which that individual belongs.Although there have been different approaches to modeling group selection, a general frameworkconsiders both individual-level events (such as births or deaths of individuals) and group-level events(such as group fission, extinction, or fusion). Then, the dynamics of both groups and individuals can betracked over time (Simon et al. 2013). This framework can be applied to the question of cooperation. Forexample, we can consider a population consisting of different groups (whenever individuals reproduce,their offspring are added to the same group). Inside each group, cooperators always perform worsethan defectors. Hence, if there were no group events, cooperators would eventually go extinct. Nowimagine that, occasionally, groups split into two, and that bigger groups split more often. Then, groupswith many cooperators (which grow fast) will split more often than groups with many defectors (whichgrow slow). Therefore, selection within groups favors defectors, but selection between groups favorscooperators (Simon 2010; Simon et al. 2013; Simon and Nielsen 2012; Traulsen and Nowak 2006).1.4 From evolutionary game theory to adaptive dynamicsThe field of evolutionary game theory goes much beyond the Prisoner’s Dilemma–style games we havediscussed so far. Because it provides a natural mathematical language to study frequency-dependentselection, it has been applied to questions from mate choice to animal contests to breeding strategies(McGill and Brown 2007).1.4.1 Matrix games and beyondCooperative interactions are not always well-described by the parameter values in the payoff matrixfrom Fig. 1.1 (T > R > S > P). Different parameter values result in other evolutionary dynamics (Fig.1.4). For example, we can envision a task whose completion requires a total investment c and providesthe same benefit b to each individual. If both individuals cooperate, the cost is split between the two;but if only one individual cooperates, the cooperator will pay the full cost. Then, some of the entries inthe payoff matrix will look different from Fig. 1.1. Specifically, R = b− c/2 and S = b− c. With thesepayoffs, S > P and T > R, which means that the strategy that increases fitness for the focal player is theopposite of whichever strategy the interaction partner plays. This arrangement of payoffs is called theHawk–Dove game or the Snowdrift game, and it can be a good descriptor of social interactions in whichboth individuals benefit from the completion of a given task, but a single cooperator is sufficient to90.2 0.4 0.6 0.8 1.0-0.10.10.2 0.4 0.6 0.8 1.00.10.2 0.4 0.6 0.8 1.00.10.2 0.4 0.6 0.8 1.00.1-0.1-0.1 -0.1    Figure 1.4: There are four possible symmetric binary-choice matrix games, depending on the order of the payoffelements R,S,T, and P. The figure shows the phase lines (Eq. 1.2) for each of the games. Increases in the numberof cooperators are depicted with blue arrows; increases in the number of defectors are depicted with red arrows.perform that task (Maynard Smith 1982; Maynard Smith and Price 1973). When interactions are well-described by a Snowdrift game, the evolutionary equilibrium comprises both cooperators and defectors(top-right panel in Fig. 1.4).The Prisoner’s Dilemma, Snowdrift game, and the other games in Fig. 1.4 are all examples of binary-choice games, where only two strategies are possible. They are also pairwise games, because theydescribe interactions between two individuals at a time (we already encountered non-pairwise gameswhen we briefly discussed a multiplayer public-goods game in section 1.2). Finally, they are symmetric,since the payoff matrix is equal for all players involved in the game. All of these assumptions canbe violated, resulting in a rich language capable of describing many types of social interactions. Forexample, ternary-choice games—represented by 3× 3 matrices—could describe rock-paper-scissors–style games, which result in oscillatory temporal dynamics (Maynard Smith 1982), as observed in somenatural systems (Buss and Jackson 1979; Kerr et al. 2002; Sinervo and Lively 1996). In the case ofsocial evolution, we might consider a third option beyond cooperation and defection, such as refrainingfrom participating in the game (Hauert et al. 2002).Another extension of evolutionary game theory does away with payoff matrices altogether, andinstead considers continuous-trait values, such as body size, date of flowering, or allocation of resourcesto the reproductive tissue (McGill and Brown 2007). In the case of cooperation, a continuous traitvalue could be, for instance, the amount of investment into producing a given public good (Doebeliet al. 2004; Killingback and Doebeli 2002). This is in contrast to the discrete strategies (either produceor do not produce a public good) that we have considered so far. The challenge with this approachis that a population consists of a cloud of points in phenotype space, rather than two (or a few) well-defined discrete phenotypes. Hence, calculating the fitness of any given (focal) phenotype is analytically10unwieldy. The problem vanishes, however, if the population has very little standing genetic variation, inwhich case all non-focal strategies can be approximated by the average strategy of the whole population.This consideration opened the way to the development of adaptive dynamics.1.4.2 Adaptive dynamicsBuilding on evolutionary game theory, the mathematical language of adaptive dynamics (Dieckmannand Law 1996; Geritz et al. 1998; Metz et al. 1992, 1996) was developed in order to study the long-termevolution of continuous-valued traits. The foundation of adaptive dynamics is the back-and-forth inter-action between ecological and evolutionary dynamics, which generalizes frequency-dependent selectionfrom evolutionary game theory.Adaptive dynamics arises from individual-based dynamics as the limit at which mutations are rareand small and the population is large (Champagnat 2006; Champagnat et al. 2008, 2006; Dieckmannand Law 1996). Because mutations are rare, we can assume that the population achieves population-dynamical (ecological) equilibrium by the time a new mutant type arrives. This amounts to a separationof time-scales between the slower evolutionary time-scale and the faster ecological time-scale. Separat-ing time scales is mathematically convenient. On the one hand, it ensures that the resident population ismonomorphic, which will make the fitness of the next mutant type easier to compute. On the other hand,it means that we can calculate the population dynamics of the resident in between mutant introductionevents. This provides a key linkage between population dynamics and evolutionary dynamics.Invasion fitnessLet us represent the evolving trait value by x. Once the resident achieves equilibrium, we can then ask:what would be the per capita growth rate of an initially rare mutant, x′, introduced into this residentpopulation? This quantity, w(x′,x) is called the invasion fitness, and it is the central concept of adaptivedynamics (Metz et al. 1992). For each resident trait value x, the invasion fitness can be thought ofas the fitness landscape experienced by initially rare mutants. Because the resident is at equilibrium(w(x,x) = 0), it follows that, if the mutant’s invasion fitness is positive (w(x′,x) > 0), the mutant willgrow in frequency and supplant the resident (this is called an invasion). Thanks to the separation oftime-scales, it will then reach a new equilibrium before the appearance of a new mutant. Critically,at every successful invasion event, the fitness landscape changes, since its shape is determined by theresident’s trait value.The slope of the invasion fitness at the resident trait value, dw(x′,x)/dx′, is called the selectiongradient. If the selection gradient is positive (negative), mutants with slightly higher (lower) trait valuescan successfully invade. Therefore, thanks to the small mutation assumption, the selection gradientdetermines the direction of evolutionary change:x˙ ∝dw(x′,x)dx′. (1.3)The above is a simplified version of the canonical equation of adaptive dynamics (Dieckmann and Law111996).Convergent stability and evolutionary stabilityArmed with the concept of the selection gradient, we can now calculate the evolutionary equilibrium x?:the value of x at which the selection gradient vanishes (i.e., is equal to zero).Furthermore, if the slope of the selection gradient is negative at the equilibrium, then evolution willdrive the population toward this point, meaning that it is an attractor of the evolutionary dynamics. Thisis often called a convergent stable equilibrium.What will happen if the population reaches a convergent stable equilibrium? Intuitively, since theselection gradient vanishes, evolution should come to a halt. After all, throughout the evolutionaryprocess the population has constantly moved uphill on the fitness landscape, climbing upwards in thedirection determined by the selection gradient (Eq. 1.3), until it reached a point at which the fitnesslandscape is locally flat (i.e., the equilibrium). Surely, then, that must correspond to a fitness peak: anendpoint of evolution, at which no mutant has positive invasion fitness. Remarkably, this is not alwaysthe case. Recall that the fitness landscape changes with every successful invasion. Climbing uphillis no guarantee that the population will reach a peak if the landscape keeps changing at every step.Furthermore, while fitness peaks are indeed locally flat, so are fitness minima.If the fitness landscape at the convergent stable equilibrium is a fitness maximum, d2w(x′,x)/dx′2|x′=x=x? < 0, then evolution comes to a rest: this is called an evolutionarily stable equilibrium (aterm borrowed from evolutionary game theory). Alternatively, the fitness landscape at the convergentstable equilibrium may be a fitness minimum, d2w(x′,x)/dx′2|x′=x=x? > 0. Then, selection is disruptiveand mutants on either side of the equilibrium can invade. If the population reproduces asexually, it willthen eventually branch into two strains and become dimorphic: a process called evolutionary branching.Evolutionary branching occurs frequently in models of social interactions (Doebeli 2011; Doebeliand Dieckmann 2000), including in models describing the evolutionary dynamics of cooperation. As anexample, Doebeli et al. (2004) use adaptive dynamics to describe the change, over evolutionary time,in the amount of investment x into a public good game. Individuals engage in pairwise interactionswhere cooperation incurs costs to the actor and accrues benefits to both the actor and the recipient: acontinuous-valued analogue of the Snowdrift game. The benefits and costs of cooperation, B(x) andC(x), respectively, are both smooth, increasing functions satisfying B(0) =C(0) = 0. Then, an individ-ual with strategy x′ interacting with an individual with strategy x has payoff P(x′,x) = B(x′+x)−C(x′).It turns out that in this model, when the curvature of the benefit function is larger than the curvatureof the cost function, the evolutionary equilibrium is also a fitness minimum. In that case, evolutionarybranching occurs, and the initially monomorphic population diversifies into two strains: a cooperativestrain that makes large investments, and a defector strain that makes no investments whatsoever.1.5 Goals of this thesisIn this thesis, I aim to add to our growing understanding about the evolution of cooperation and grouplife. I do so by developing four different models, each of which explores a different as-yet poorly under-12stood facet of social evolution theory. The models cover a broad range of topics—from human culture tomicrobial evolution—and methods—from adaptive dynamics to multi-level stochastic simulation. De-spite their differences, they are bound together by the common goal of understanding the origin andconsequences of social evolution and group life in biological systems. I finish this introduction bybriefly introducing each of the models that will be discussed at length in the following chapters.1.5.1 Model 1: Acculturation drives the evolution of intergroup conflictConflict between groups of individuals is a prevalent feature in human societies. A common theoreticalexplanation for intergroup conflict is that it provides benefits to individuals within groups in the formof reproduction-enhancing resources, such as food, territory, or mates. However, it is not always thecase that conflict results from resource scarcity. In Chapter 2, I show that intergroup conflict can evolvedespite not providing any benefits to individuals or their groups. The mechanism underlying this processis acculturation: the adoption, through coercion or imitation, of the victor’s cultural traits. Acculturationacts as a cultural driver (in analogy to meiotic drivers) favoring the transmission of conflict despite apotential cost to both the host group as a whole and to individuals in that group. I illustrate this processwith a two-level model incorporating state-dependent event rates and evolving traits for both individualsand groups. Individuals can become ‘warriors’ who specialize in intergroup conflicts but who are costlyotherwise. Additionally, groups are characterized by cultural traits, such as their tendency to engage inconflict with other groups and their tendency for acculturation. I show that, if groups engage in conflicts,group selection will favor the production of warriors. Then, I show that group engagement can evolveif it is associated with acculturation. Finally, I study the coevolution of engagement and acculturation.The model shows that horizontal transmission of culture between interacting groups can act as a culturaldriver and lead to the maintenance of costly behaviors by both individuals and groups.1.5.2 Model 2: Cooperation can promote rescue or lead to evolutionary suicide duringenvironmental change.The adaptation of populations to changing conditions may be affected by interactions between individ-uals. For example, when cooperative interactions increase fecundity, they may allow populations tomaintain high densities and thus keep track of moving environmental optima. Simultaneously, changesin population density alter the marginal benefits of cooperative investments, creating a feedback loop be-tween population dynamics and the evolution of cooperation. In Chapter 3, I model how the evolutionof cooperation interacts with adaptation to changing environments. I hypothesize that environmentalchange lowers population size and thus promotes the evolution of cooperation, and that this, in turn,helps the population keep up with the moving optimum. However, I find that the evolution of coop-eration can have qualitatively different effects, depending on which fitness component is reduced bythe costs of cooperation. If the costs decrease fecundity, cooperation indeed speeds adaptation by in-creasing population density; if, in contrast, the costs decrease viability, cooperation may instead slowadaptation by lowering the effective population size, leading to evolutionary suicide. Thus, cooperationcan either promote or—counter-intuitively—hinder adaptation to a changing environment. Finally, I13show that this model can also be generalized to other social interactions by discussing the evolution ofcompetition during environmental change.1.5.3 Model 3: On the importance of evolving phenotype distributions on evolutionarydiversification.Evolutionary branching occurs when a population with a unimodal phenotype distribution diversifiesinto a multimodally distributed population consisting of two or more strains. Branching results fromfrequency-dependent selection, which is caused by interactions between individuals. For example, apopulation performing a social task may diversify into a cooperator strain and a defector strain. Branch-ing can also occur in multi-dimensional phenotype spaces, such as when two tasks are performed si-multaneously. In such cases, the strains may diverge in different directions: possible outcomes includedivision of labor (with each population performing one of the tasks) or the diversification into a strainthat performs both tasks and another that performs neither. In Chapter 4, I show that the shape of thepopulation’s phenotypic distribution plays a role in determining the direction of branching. Further-more, I show that the shape of the distribution is, in turn, contingent on the direction of approach to theevolutionary branching point. This results in a distribution–selection feedback that is not captured inanalytical models of evolutionary branching, which assume monomorphic populations. Finally, I showthat this feedback can influence long-term evolutionary dynamics and promote the evolution of divisionof labor.1.5.4 Model 4: Multilevel selection favors fragmentation modes that maintaincooperative interactions in multispecies communitiesReproduction is one of the requirements for evolution and a defining feature of life. Yet, across thetree of life, organisms reproduce in many different ways. Groups of cells (e.g., multicellular organisms,colonial microbes, or multispecies biofilms) divide by releasing propagules that can be single-celledor multicellular. What conditions determine the number and size of reproductive propagules? In mul-ticellular organisms, existing theory suggests that single-cell propagules prevent the accumulation ofdeleterious mutations (e.g., cheaters). However, groups of cells, such as biofilms, sometimes containmultiple metabolically interdependent species. This creates a reproductive dilemma: small daughtergroups, which prevent the accumulation of cheaters, are also unlikely to contain the species diversitythat is required for ecological success. In Chapter 5, I developed an individual-based, multilevel se-lection model to investigate how such multi-species groups can resolve this dilemma. By tracking thedynamics of groups of cells that reproduce by fragmenting into smaller groups, I identified fragmenta-tion modes that can maintain cooperative interactions. I systematically varied the fragmentation modeand calculated the maximum mutation rate that communities can withstand before being driven to ex-tinction by the accumulation of cheaters. I find that for groups consisting of a single species, the optimalfragmentation mode consists of releasing single-cell propagules. For multi-species groups we find var-ious optimal strategies. With migration between groups, single-cell propagules are favored. Withoutmigration, larger propagules sizes are optimal; in this case, group-size dependent fissioning rates can14prevent the accumulation of cheaters. This work shows that multi-species groups can evolve reproduc-tive strategies that allow them to maintain cooperative interactions.15Chapter 2Acculturation drives the evolution of intergroup con-flict2.1 IntroductionIntergroup hostility, ranging from intermittent skirmishes to large-scale conflicts, is a recurrent eventin human history (Keeley 1996; Otterbein 2004). It has been suggested that it contributed to shapingour species’ social behaviours (Bowles 2009) and that it played a role in our organization in complexsocieties (Turchin et al. 2013). Although intergroup conflict was not universal among nonstate soci-eties (Dennen 1995; Ember 1978), it was nonetheless frequent enough that 64 percent of hunter-gatherergroups engaged in violent conflict at least once every two years (Ember 1978). Intergroup conflict wasnot just frequent, but also costly, causing an estimated 14 percent of total mortality in hunter-gatherersocieties (Bowles 2009) (but see also Falk and Hildebolt 2017).Evolutionary anthropologists have used a variety of approaches and methods to understand theprevalence of intergroup conflict in the face of such costs (reviewed in Glowacki et al. 2017). Forexample, behavioral ecologists highlight the importance of reproductive benefits that can be acquiredby members of successful groups (Crofoot and Wrangham 2010). If the benefits of fighting are suffi-ciently large, the costs sufficiently low, or if the belligerent party has little to lose (i.e., the “value ofthe future” is low), then we should expect intergroup conflict (Enquist and Leimar 1990). The costs ofinitiating conflict could be low if group dynamics leads to vastly different group sizes, providing rela-tive safety to large groups (Manson and Wrangham 1991). Evolutionary psychologists, in turn, focuson the psychological mechanisms that underlie coalitional violence, based on the assumption that suchmechanisms were shaped by natural selection. For example, the “risk contract” theory suggests thatindividuals will be willing to engage in war if success is likely and if participants are rewarded (Toobyand Cosmides 1988), whereas the “male warrior hypothesis” argues that men have psychological traitsthat allow them to use intergroup conflict to protect or acquire reproductive resources (McDonald et al.2012). Anthropologists have noted that, apart from collective benefits such as territory, there are alsoprivate incentives to participating in group conflict (Chagnon 1988; Glowacki and Wrangham 2015;Glowacki and Wrangham 2013). In addition, groups can also sanction free-riders (Mathew and Boyd162011). Common to most of these approaches is the idea that the costs of conflicts are ultimately out-weighed by individual or collective benefits, often in the form of resources (including food, territory, ormates, Glowacki and Wrangham 2015; Lehmann 2011; Lehmann and Feldman 2008). Because of theiremphasis on reproduction-enhancing resources, many authors within these traditions (e.g., Crofoot andWrangham 2010; Lehmann 2011; Manson and Wrangham 1991; Wrangham and Glowacki 2012) pro-pose an evolutionary link between human intergroup conflict and the coalitional aggression behaviorsthat are prevalent among other animals, such as ants, termites, and chimpanzees.One prediction of this framework is that resource scarcity should be positively associated with thefrequency and/or magnitude of intergroup conflicts (Koubi et al. 2014). However, the empirical evi-dence for this association is mixed. For example, whereas land shortages (Ember 1982) and naturaldisasters (Ember and Ember 1992) seem to predict warfare, chronic resource scarcity and a lack of po-tential mates do not (Ember and Ember 1992). Overall, literature reviews (Koubi et al. 2014; Theisen2008) suggest that there is no robust quantitative link between resource scarcity and intergroup conflict.This raises the possibility that, in addition to resource-associated benefits, there may be alternativeexplanations for intergroup conflict. How can we explain the prevalence of this costly behavior in caseswhere it does not provide benefits to the winners? In genetic evolution, some alleles spread despite beingdetrimental to carriers by increasing their own rate of transmission, a process called meiotic drive (Lind-holm et al. 2016). Drivers underlie such costly phenomena as reduced sperm count in males (Price andWedell 2008) and spore killing in fungi (Raju 1994). In cultural evolution, a similar process of biasedtransmission between individuals has been invoked to explain the spread of costly behaviors such asfood taboos or religious rituals (Henrich et al. 2008). An analogous process of transmission of culturaltraits between groups (acculturation) could act as a driver for the evolution of costly group behaviorssuch as intergroup conflict in human societies.An example of acculturation is the adoption (either by coercion or imitation) of the victor’s culturaltraits following military defeats. For example, following the expansion of the Roman empire, nativeEuropean tribes adopted elements of Roman culture, such as language, eating habits (King 2001), re-ligion (Revell 2007), and laws (Sherman 1914). They also adopted Roman military techniques andweapons, occasionally using them against the Romans themselves (Dyson 1971). Modeling suggeststhat conflict-associated acculturation could be responsible for the spread of social traits and militarytechnology (Turchin et al. 2013), but could it also explain the tendency of groups to engage in costlyconflicts?In this paper, we will give a proof of principle that intergroup conflict can be explained even inthe absence of direct or indirect material benefits to either individuals or groups. In particular, weargue that acculturation can drive the evolution of costly antagonistic behaviors such as engagement(i.e., the tendency to initiate conflicts). Because we are dealing with human cultural traits and theireffects on population dynamics, we will use cultural evolutionary theory (Boyd and Richerson 1985;Cavalli-Sforza and Feldman 1981), in which individuals are characterized by cultural traits. Thesetraits can be transmitted between individuals and change over time by a process reminiscent of naturalselection. Furthermore, like Bowles et al. (2003) and Smaldino (2014), we consider that group dynamics17cannot be fully described solely by considering individual traits. Instead, we also consider group-leveltraits (sometimes called social institutions, Bowles et al. 2003, or ultrasocial norms, Henrich 2015;Turchin et al. 2013). Such group-level traits represent the phenotypic effects of generalized conventionsor organized structures which may differ among groups, and which are not reducible to individualbehaviors, such as professionalized government, monogamous or polygamous mating systems, resourceinheritance, etc. (Smaldino 2014). Much like individuals are bearers of genetic or cultural traits, groupsare bearers of institutions (Bowles et al. 2003). In our model, being a warrior is a property of individuals,but engagement and acculturation tendency are properties of groups.To simultaneously study the dynamics of individuals and groups, we use the framework of group(multilevel) selection (Simon et al. 2013). In this framework, the population and evolutionary dynamicsare determined by state-dependent event rates for both individuals and groups, resulting in a two-levelpopulation process (Simon et al. 2013). Individual-level events are births and deaths of individuals thatoccur within a given group and affect the composition of that group, i.e., the number of individualsand their cultural traits. Group-level events affect whole groups and consist of group fissions, groupextinctions, and interactions (conflicts). Following Simon et al. (2013), a trait evolves by (cultural)group selection if its evolution requires group-level events. The evolution of human culture by groupselection is called cultural group selection, particularly in the context of intergroup conflict (Henrich2004; Richarson and Henrich 2012; Wilson and Wilson 2007).2.2 Model descriptionConsider a population consisting of a number of distinct groups of individuals. For simplicity, theseindividuals are assumed to be haploid and asexually reproducing. The size and composition of groupsvary over time due to the birth and death of individuals (within-group dynamics). Similarly, the num-ber of groups changes due to group-level events, namely fissions, extinctions, and intergroup conflicts(between-group dynamics).2.2.1 Within-group dynamicsIndividuals belong to one of two classes—warriors or shepherds—whose within-group densities are de-noted by real variables x and y, respectively. Individuals are characterized by a cultural trait p ∈ [0,1],which is the probability that their offspring becomes a warrior (so that 1− p is the probability that anoffspring is a shepherd). This trait is inherited vertically from one’s parent, with some probability ofmutation. Later, we will allow for horizontal transmission of this trait through the process of accultura-tion. Although we refer to the two classes as “shepherds” and “warriors”, these names are just evocativelabels and do not need to refer to their literal meaning. More abstractly, we could refer to warriors as“intergroup conflict specialists”.To ensure density-regulation within groups, the birth rate of shepherds decreases exponentially withgroup density:b(x,y) = b1+b2e−(x+y)b3 , (2.1)18where the parameters b1, b2 and b3 can be interpreted respectively as minimum birth rate, differencebetween maximum and minimum birth rate, and sensitivity to density dependence.We assume that warriors have a lower birth rate than shepherds. This assumption implies thatproducing warriors represents a cost to shepherd individuals, with higher values of p correspondingto higher costs. In general, the birth rate of warriors is (1− c)b(x,y), with the parameter 0 ≤ c ≤ 1regulating the cost of warrior production. The worst-case scenario for the evolution of conflict occurswhen c = 1 (non-reproductive warriors); for this value, producing warriors is a “reproductive sink”for shepherds. For the majority of this paper, and unless otherwise stated, we will focus on this mostconservative case, but our results also generally apply to cases where c < 1.In general, death rates could also be density- (and class-) dependent, but for simplicity, we considerthe case of a constant per capita death rate d for all individuals. Due to the cost of producing non-reproductive warriors, within each group there is individual-level selection for low values of p. Thedifferential equations governing within-group dynamics are given in the Methods, Section 2.3.2.2.2 Between-group dynamicsGroup dynamics are governed by three types of events: fissions, group extinctions, and group interac-tions (conflicts). The rates at which these events occur in a given group depend in general on both thecomposition of the group and the number and composition of all other groups in the population. Theycan also depend on the group’s cultural traits. Just like individuals are characterized by a cultural traitp, groups are characterized by group-level traits, which may represent cultural institutions, fashions, orconventions. These institutions govern the collective behavior of the group and are not reducible to theproperties of the group’s constituent individuals. We consider two such cultural traits—engagement (q)and acculturation tendency (r)—which are relevant for conflict initiation and resolution, respectively.Group extinctionsWhen a group extinction occurs, the group is removed from the population. This event reflects thepossibility that whole groups of individuals collapse due to, for example, a bad harvest, intra-groupstrife, or natural catastrophes (but not due to a hostile interaction with another group, which is a separateevent). We assume that the group extinction rate is a decreasing function of the number of individualwithin the group, and an increasing function of the number of groups (see Appendix A for details).Thus, higher-density groups are less likely to go extinct, and extinctions are more frequent when thereare more groups, leading to regulation at the level of groups.FissionsWhen a group fissions, it divides into two autonomous daughter groups, which inherit the group-leveltraits (q and r) of their parent group, modified by normally distributed mutations with standard deviationσq, σr. The fission rate is linearly proportional to group density, so that bigger groups are more likely toundergo fission (see Appendix A for details). Because groups that produce many warriors have lowerdensities (see below), they also have lower fissioning rates. Thus, warrior production is counteracted19not just by individual selection but also by the faster proliferation of groups with fewer warriors.In the event of a fission, the division of individuals among the daughter groups is assortative withrespect to the value of the individual trait p, such that the most extreme individuals are more likely togroup together with individuals similar to themselves, whereas the most common individuals are morelikely to separate at random. The degree of assortment is regulated by a parameter s (see Appendix Afor details).Such assortment, called “associative” splitting by Hamilton (1975), is to be expected if splittingoccurs over internal political disagreements or if splitting occurs along family lines, both of which arecommon in human societies (Walker et al. 2014). Assortment increases variability between groups,enhancing the effect of group selection (Crow and Aoki 1982; Hamilton 1975).Group interactionsEarlier models of intergroup conflict assumed that individuals expressed a belligerence trait, and thatgroups initiated conflicts with a probability proportional to the average level of this trait within thegroup (Choi and Bowles 2007; Lehmann 2011; Lehmann and Feldman 2008). In contrast, in our model,the tendency to engage in conflicts is a group-level trait, q ∈ [0,1]. A given group initiates a conflict ata rate which is proportional to its engagement trait and to the total number of groups N:G(q,N) = γNq. (2.2)Once a focal group initiates a conflict, a rival group is chosen at random (i.e., there is no spatial struc-ture in the model). The focal group wins the conflict with a probability that depends on the differencebetween the numbers of warriors of the opposing factions (for more details, see Appendix A). The reso-lution of the conflict, in turn, depends on the victor’s acculturation tendency r ∈ [0,1]. With probability1− r, the defeated group is removed from the population (similar to an extinction event), otherwise itundergoes acculturation, i.e., it adopts the victor’s cultural traits (with mutations, as described above).Individuals within the group adopt variants of p in proportion to their frequency in the victorious group.We will also consider the robustness of our results to the possibility of “costly” acculturation, in whicha fraction of individuals die during the conflict prior to acculturation.2.3 MethodsIn order to simultaneously model evolution at two levels, we simulate within-group dynamics deter-ministically (as a system of differential equations), while modeling between-group dynamics stochasti-cally (as a Markovian process). Thus, our implementation follows a hybrid (partly deterministic, partlystochastic) method to simulate group selection dynamics (Puhalskii and Simon 2012; Simon and Nielsen2012).At the within-group level, the model is implemented by discretizing the trait space into k bins andtracking the density xi of shepherds of type pi,1 ≤ i ≤ k. Since warriors do not reproduce, there is noneed to keep track of their trait value (but the extension to the case when warriors also reproduce is20straightforward). This results in a system of coupled differential equations:x˙i =(k∑j=iµi jx j(1− p j)−k∑j=1µ jixi(1− pi))b(X ,y)− xid, (2.3)y˙ =k∑i=1xi pi b(X ,y)− yd,where µi j is the mutation rate from trait value j to i (with µii being the probability of no mutation), andX = x1+x2+ · · ·+xk. This system of equations is solved numerically in small time increments dt usingthe Euler method.While the densities within each group are updated deterministically, group events occur stochasti-cally based on their rates. The time increments dt are assumed to be small enough that the occurrenceof more than one group event per increment is unlikely. The total group rate (T ) is given by the sum ofall group event rates; the probability that a group event will occur in a given increment is then T × dt.The particular event type and the focal group are determined by weighted lottery, with probabilities pro-portional to the different group event rates. The relative velocity of the two time scales (group selectionand individual selection) can be regulated by adjusting the rate constants (ε , φ , γ). Multiplying themby a common factor regulates the overall frequency of group events and hence, the relative strength ofgroup selection vis-a`-vis individual selection.In the event of an acculturation, the relative densities xi/(X +y) of the defeated group were changedto match those of the victor, while keeping the total density (X +y) constant (where X = x1+x2+ · · ·+xk).2.3.1 Default parametersIn the numerical examples and figures throughout this article, unless otherwise stated, we used the fol-lowing parameter values.For the individual-level rates: b1 = b3 = 0.1, b2 = 3, d = 0.5; additional simulations show that the qual-itative results of the model are robust to the changes in group size that may result from small deviationsfrom these values. This is because changes in group size (which affect the intensity of individual-levelselection) also lead to changes in the equilibrium number of groups (which affects the intensity of groupselection).For the group-level rates: ε = 0.025, φ = 0.05, γ = 0.05. Variations about these values have no qualita-tive effect on our results, provided that large changes in the frequency of extinctions or interactions arecompensated by adjustments to the fission rate, and vice versa (to avoid extinction of all groups).Additional parameters: s = 10, gs = 10−4, ι = 10−5, σq = σr = 0.05, k = 101, dt = 0.01. The mutationrate was µi j = 0.01 if i = j±1, otherwise zero. The results we show are typical for the vicinity of theseparameter values.212.3.2 Language and dataThe simulations were performed in MATLAB R2017a (The MathWorks Inc. 2017) and the analyses andfigures with R 3.3.2. (R Core Team 2019). Simulation code is available online at http://www.zoology.ubc.ca/prog/henriques/2019 acculturation/.2.4 Results2.4.1 Population dynamics within groupsIn the absence of group events, the equilibrium within-group density of individuals, nˆ(p) = xˆ(p)+ yˆ(p),is lower for groups with higher mean p (see Appendix A for details). Thus, groups that produce higherfractions of warriors support fewer individuals, because warriors are non-reproductive (Fig. A.1). Notethat in the course of the two-level process, population dynamics within groups does not necessarilyhave time to reach the equilibrium (described by Eq. A.5), especially if the frequency of group events issufficiently high. Frequent fissioning events, on the one hand, and within-group evolution toward lowervalues of p, on the other, result in actual within-group densities that are lower than one would predictbased on the equilibrium condition above. This is illustrated in Fig. 2.1A, which compares realizedgroup densities for different values of p with the predicted equilibrium.Because increased p causes a decrease in the total group density, the maximal density of warriors isattained for an intermediate value of p, such that ∂pyˆ(p) = 0. With the reference parameters (see Section2.3), the number of warriors is maximized when p≈ 0.514 (Fig. A.2A).Since warriors do not reproduce, p evolves to a mutation-selection balance near zero in the absenceof group events, at which point groups consist almost exclusively of shepherds and have maximal den-sity. Thus, in our simulations, any significant production of warriors at equilibrium must result from theeffects of group events.2.4.2 Frequent intergroup conflict promotes warrior productionAs a starting point, we first consider engagement q to be a constant, rather than an evolving trait. Wealso set r = 0 (i.e., no acculturation), so that conflicts are always lethal for the loosing group. Theresults show that even small amounts of engagement are sufficient to sustain high levels of warriorproduction (Fig. 2.1B). This occurs despite the fact that high levels of warrior production entail costsnot only to the individual bearers of the trait, but also to their group (in the form of lower group sizes,which in turn reduce their fissioning rates). Thus (in contrast to previous models, Bowles 2009; Garcı´aand Bergh 2011) warrior production is not necessarily a form of within-group (or “parochial”) altruism.Rather, the relative group-level benefits and costs of warrior production will depend on the state of theglobal population at any given time.Contrary to what one might expect, it is not the case that higher amounts of engagement necessarilyentail higher warrior production. The population equilibrium depends not only on the level of engage-ment but also on the number and variance of groups, the group composition, the rates of fission and220510150.25 0.3 0.35 0.4 0.45 0.5Trait value (p)Within-group population sizeA0.00.20.40.01 0.02 0.04 0.08 0.16 0.32 0.64 1Engagement (q)Final trait value (p)B0 10−3 10−2 0.1 0.5 1 10 20 100Assortment parameter (s)CFigure 2.1: Evolution of warrior production with constant group-level traits. (A) Increased warrior productionleads to smaller groups. Five replicate simulations with constant q= 0.64,r= 0 were grouped together to calculatemean group density in groups with different mean p. Dark ribbon shows standard deviations and light ribbonindicates the full range of values. The dashed line shows estimated equilibrium values (Eq. A.5). (B–C) Effectof engagement and assortment on warrior production. Each point indicates a group’s mean p after equilibriumhas been reached, when q is constant and r = 0 (no acculturation). Panel B depicts the effect of varying q(engagement), with constant s = 10, while C depicts the effect of varying s (assortment parameter), with constantq = 0.5. For each treatment, the results of five replicate runs are shown. Whole-population extinction at highlevels of q has been prevented by enforcing fissions if the population drops to a single group.extinction, and the balance between individual- and group-level selection. For example, the number ofgroups and the between-group variance in mean p decrease with increasing conflict frequency, whichdecreases the effectiveness of group selection. Nevertheless, there is a minimum amount of engagementabove which warrior production is favored.Because the evolution of warrior production is a result of group selection, it depends on the main-tenance of variability between groups. Without associative splitting (i.e., when s = 0), daughter groupshave the same distribution in p as their parent groups. Therefore, although p can increase transiently(as long as standing variation exists), eventually between-group variation in p will be exhausted andindividual-level selection will dominate. Associative splitting (s> 0) prevents this outcome (Fig. 2.1C),with even modest amounts of assortment being sufficient to generate the evolution of at least somewarrior production.2.4.3 The evolution of warrior production can lead to whole-population extinctionDue to stochastic variations in the (finite) number of groups, the long-term demographic fate of theglobal population in our model is always extinction; nevertheless, the population will often persist fora long time in state in which the number of groups fluctuates stochastically around a predicted value(so-called “metastable” states, although here we refer to them, in slight abuse of terminology, simply asequilibria).Assuming that the population is monomorphic in q (engagement), r (acculturation), and p (warriorproduction), and that all groups have the same density n(p) = x(p) + y(p), we can approximate theexpected equilibrium number of groups Nˆ (Eq. A.7). This approximation is not a quantitatively accurateestimate for the realized number of groups at any given moment (see Fig. 2.2A, dashed line) due to the23010200.25 0.3 0.35 0.4 0.45 0.5Trait value (p)Number of groupsA02505007501000+0.1 0.3 0.5Trait value (p)Time to extinctionBFigure 2.2: Evolution of warrior production increases the risk of whole-population extinction. Increased warriorproduction leads to lower number of groups, which decreases time to extinction. (A) Five replicates simulationswith constant q = 0.64,r = 0 were grouped together to calculate mean group size for different values of meangroup p. Dark ribbon shows standard deviations and light ribbon indicates the full range of values. The dashedline shows estimated equilibrium values for group density (Eq. A.7, using mean group density in place of n(p)).(B) Time to whole-population extinction (simulations were interrupted at time t = 1000). For each value of p, weshow 50 replicate simulations. All trait values (p, q = 1, r = 0) were held constant.assumptions of monomorphic populations and identical groups. Furthermore, when p is evolving, thenumber of groups does not necessarily have time to reach the predicted equilibrium, especially whenthe number of groups is low. Qualitatively, however, our prediction agrees with the simulation data thatthe number of groups at equilibrium is a decreasing function of p (Fig. 2.2A). Thus, group selectionfavoring the production of warriors also results in a decrease, over time, in the number of groups atequilibrium. As a consequence, selection for increased p makes stochastic population-wide extinctionincreasingly likely. As shown in Fig. 2.2B, time to extinction decreases for higher warrior production.In deterministic evolutionary models, the adaptive evolution of a population toward its own ex-tinction is called evolutionary suicide (Ferrie`re 2000; Gyllenberg and Parvinen 2001), also known asDarwinian extinction (Webb 2003). Here, we have a similar (though not identical) process: the adaptiveevolution of a population toward smaller and smaller sizes, increasing the risk that stochastic variationswill cause whole-population extinction.The process of evolution toward extinction has often been studied in the context of self-interestedindividuals destroying common resources, a scenario which could be averted by group selection (Rankinet al. 2007). In contrast, our model provides an example of group selection causing (rather than prevent-ing) evolution of a population toward extinction.2.4.4 Without acculturation, intergroup conflict is not favoredIn the previous section, we considered engagement to be a constant group trait. While this does notimply that conflicts occur at a constant rate (because group interaction rates depend on the number ofgroups in the population), it means that groups have no control over how frequently they engage in them.In reality, not all human groups are likely to be equally prone to engage in conflict, which in our model24corresponds to variation in the group trait q. We modeled this variation by characterizing groups with atrait q (engagement) that influences the frequency with which they initiate conflicts.Groups with higher values of q engage in conflicts more often and, thus, are more likely to beeliminated from the population. Even high-engagement groups with higher numbers of warriors may bedefeated stochastically. Furthermore, defeating other groups does not increase a group’s birth rate (i.e.,fission rate), as would be the case if victories were associated with resource exchange. On the contrary,the groups that could conceivably benefit from higher engagement are those that have many warriorsand, therefore, lower densities and fission rates. Therefore, although groups may produce warriors ifconflicts are frequent, this is not an evolutionarily stable state, and groups that avoid initiating conflictsare more likely to persist and leave descendants. Thus, if winning group conflicts does not affect groupbirth rates, engagement (a group-level trait) and warrior production (an individual-level trait) coevolvetoward a “peaceful equilibrium” (Fig. 2.3A), at which q ≈ 0 and p ≈ 0. (For more details, see SectionA.4.)2.4.5 Acculturation drives the evolution of intergroup conflictWith acculturation, winning a conflict translates into a birth event for the winning group, because theloser of the conflict takes on the cultural traits of the winner. This turns out to tip the balance in favorof engagement: if conflicts are associated with the transmission of the victor’s culture, rather than theelimination of the defeated group (i.e., if r = 1), engagement and warrior production coevolve towardfrequent conflicts rather than peace (Fig. 2.3B). This occurs despite none of the individuals involved inthe conflicts earning any reproduction-enhancing benefit.Increased engagement does not provide either individuals or groups with any benefits, as it has noeffect on the probability of surviving a conflict. In fact, q is released from selection when r = 1, sinceno groups are eliminated as a result of conflicts, which explains some of the fluctuations and high levelsof variation in q (Fig. 2.3B; see also Section A.4.)On the other hand, cultures that are simultaneously engaging and warrior-rich maximize their ownreplication through acculturation. The reason why being engagement-prone and having warriors is asuccessful strategy when there is acculturation can be understood based on the following argument.Consider a population of N groups in which a number of groups ν have a more engaging and warrior-rich culture than the other groups. A more engaging group tends to win a single encounter with a lessengaging group. If the victory results in the elimination of the defeated group, the frequency of thehigh-engagement groups increases by νN−1− νN = νN(N−1) . However, if the victory results in the defeatedgroup taking on the culture of the victor, the frequency of the high-engagement groups increases byν+1N − νN = 1N . Thus, the increase in frequency tends to be much larger with acculturation (up to aboutN times larger if ν  N).Thus, the evolution of engagement occurs due to cultural drive, rather than due to any adaptiveeffect on its host groups. Although the level of engagement fluctuates widely, and sometimes drops tolow values, it is associated with an increase in warrior production, which rises to the value at whichwarrior density is maximized (p ≈ 0.514). This is a stable and reliable indicator that the majority of250.000.250.500.751.000 5 10 15 20 25Trait valueTraitpqrA0 5 10 15 20 25 B 0.000.250.500.751.000 10 20 30 40 50Time (thousands)Trait valueCFigure 2.3: Coevolution toward peace and coevolution toward conflicts. Time dynamics of a single replicate,showing the coevolution of the individual-level trait p (warrior production) and the group-level traits q (engage-ment) and r (acculturation). (A) Without acculturation (r = 0), q and p coevolve toward a peaceful equilibrium.(B) With maximal acculturation tendency (r = 1), q and p coevolve, leading to conflicts and maximizing warriorproduction (p ≈ 0.514). (C) Acculturation coevolves with the other traits, resulting in an outcome similar to B.Lines indicate population-wide rolling averages of group values (q, r) or of mean group values (p). Dark ribbonsshow rolling standard deviations and light ribbons encompass the full range of values. In order to make the tran-sient dynamics of p and q clear, mutational effect sizes in panel A (σq = 0.01) are smaller than in panels B and C(σq = 0.05).groups are prepared for conflict.2.4.6 Acculturation coevolves with engagement and warrior productionFinally, we consider the joint evolution of warrior production (p), engagement (q), and acculturation (r).Whereas the first trait characterizes individuals, the other two traits characterize groups. Starting from apeaceful population with small amounts of warriors or engagement, acculturation rapidly spreads, driv-ing engagement and warrior production along with it (Fig. 2.3C). This occurs despite the fact that thereare costs associated with high engagement: namely, during the initial increase of the acculturation trait,there is a risk (with probability 1− r) that a defeated group will be eliminated from the population. Fur-thermore, as discussed above, there are also costs (at both the individual and the group level) associatedwith high warrior production. Nonetheless, p evolves to maximize warrior production (p≈ 0.514).26To understand the rise of acculturation, consider the fate of a group with a mutation in this trait(r′), in a population that is monomorphic for the group trait values q, r. Furthermore, assume that thenumber of groups is large and has reached equilibrium (Eq. A.7). The only term in the group dynamicsthat depends on r′ is the rate at which resident groups are defeated and acculturated. If the mutantgroup defeats resident groups with a probability g (independent of r′, see Eq. A.4), this rate is given byG(q,N)r′g. Since the mutant is rare, the corresponding rate for the resident r is G(q,N)rg, and hencethe mutant’s rate of winning is higher than the resident’s if and only if r′ > r. Thus, invasion dynamicsfavors larger acculturation values.In summary, although the production of warriors and the evolution of conflicts are deleterious traitsfrom the perspective of individuals, they are advantageous from the perspective of the cultural variantsthemselves, since warrior-rich and engagement-prone groups with high acculturation tendencies aremore likely to succeed in spreading their cultural variants in the event of a group interaction.2.4.7 Overview of model behavior and robustnessSo far, we described how multilevel cultural selection can drive the coevolution of acculturation, en-gagement, and warrior production for a particular set of parameters. In this section we argue that thepatterns we discussed are robust and general. Overall, there are three main categories of possible out-comes in this model. The first outcome, illustrated in Fig. 2.3C and discussed above, is “coevolutiontoward conflict”, where r approaches 1, p approaches the value that maximizes warrior density, and qevolves to intermediate values and experiences large fluctuations. (Of these traits, q is the most sensi-tive to changes in parameters; most of the regimes discussed below decrease the equilibrium level of qcompared to Fig. 2.3C.) A second possibility is whole-population extinction, also discussed above. Thisoutcome is expected if the fission rate is too small compared to the group extinction rate, i.e. for lowgroup growth rates.The final possibility is that all traits evolve toward near zero values (“peaceful equilibrium”). Thiswill be the case if group selection is too weak relative to individual-level selection, which can occur, forexample, if fission-associated assortment is absent (see Fig. 2.1C). It can also occur if the frequency ofgroup events is very low, because one basic requirement for the increase in warrior production is thatgroup events should occur frequently enough relative to within-group population turnover (see SectionA.3). Keeping all other parameters identical, we obtained a peaceful equilibrium once the group eventrates were three orders of magnitude lower than the reference parameters (Fig. A.6). Group selectionwill also be weaker if the number of groups is small, but our results are fairly robust to this aspect.Small numbers of groups can be obtained either by changing the individual-level rate parameters or bychanging the group-level rate parameters. Within-group parameters that result in smaller groups willimply lower fissioning rates, and therefore fewer groups. In Fig. A.7A the population evolves towardconflict even though the individual-level parameters are such that the equilibrium number of groups isonly about 24, an order of magnitude lower than in Fig. 2.3C (214 groups). Small numbers of groups canalso be obtained by decreasing the fissioning rate. Fig. A.7B shows the dynamics of a population whosefissioning rate is one order of magnitude lower than the default parameter. Despite having only around2717 groups at equilibrium, it displays coevolution toward conflict. (Populations this small are at riskof population-wide extinction, a fate that was artificially averted in Fig. A.7B by preventing extinctionwhen the population was down to one remaining group.)Another potential challenge to coevolution toward conflict occurs when small increases in warriorproduction impose higher penalties on group size (and by extension, on group fissioning frequency).Although nˆ(p) is a strictly decreasing function of p (Eq. A.5), its shape depends on the choice ofparameters. In particular, the decrease in nˆ(p) for small values of p may be either accelerating (if 2b1 ≤d) or decelerating (otherwise), see Fig. A.2A–B. Our reference parameters result in an acceleratingfunction. Although decelerating functions pose higher challenges to any increase from low values ofp (because they correspond to steeper reductions in group density), additional simulations (Fig. A.4)confirm that coevolution toward conflict is robust to decelerating functions.Apart from changes to the main parameters of our model, we note also that the model is robustto generalizations. For example, coevolution toward conflict can still occur in the case of “costly”acculturation, in which a fraction of individuals die prior to acculturation (Section A.5, Fig. A.5). Thiscould correspond, for example, to scenarios in which one of the rival factions surrenders upon incurringsome level of casualties. Another generalization is the case in which warriors can reproduce (Fig. A.8).In this case, warriors inherit their parent’s trait value p. When they reproduce (at some fertility costc < 1), this trait value determines (as it does in shepherds) the proportion of offspring that will becomewarriors. Predictably, reducing the cost of warrior production makes the evolution of warriors lessstringent. Because warrior production imposes a smaller penalty on group size, the value of p at whichwarrior density is maximized is higher for lower values of c. Therefore, all else being equal, lowervalues of c lead to higher equilibrium levels of p.2.5 DiscussionThe most common explanation for the evolution and persistence of intergroup conflict in human societiesis that their costs are compensated by the acquisition of reproduction-enhancing resources, such as food,mates, or territory, held by other groups (Bowles 2006; Choi and Bowles 2007; Durham 1976; Gat2000; Lehmann 2011; Lehmann and Feldman 2008). Here, we have argued that the cultural evolution ofconflicts can be explained even in the absence of such benefits. As an alternative explanation, we showedthat conflicts can be maintained by cultural drive. In analogy with meiotic drive, cultural drive is theprocess by which a cultural trait spreads by promoting its own transmission at the expense of its host’sfitness. In our model, the particular mechanism underlying this process is acculturation: the impositionof the victor’s culture on defeated groups following conflicts. Acculturation evolves as an alternativeto direct elimination of defeated groups and paves the way for the evolution of group engagement andwarrior production, ultimately resulting in the maintenance of intergroup conflict.Because intergroup conflicts are best described as group-level events (rather than the sum of indi-vidual behaviors), our model explicitly tracks population dynamics at two levels (Simon et al. 2013). Atthe lower level, individuals are characterized by their tendency to become warriors. At a higher level,groups are characterized by social norms or institutions such as engagement (the tendency to engage in28conflicts) and their tendency to acculturate defeated groups. The evolution of these traits is the resultof state-dependent events at both levels: births and deaths for individuals, and fissions, extinctions, andconflicts for groups. Our assertion that intergroup conflict is a group-level event (i.e. an event that canpotentially change the number of groups, Simon et al. 2013) does not imply that the actual process (andits outcome) is not driven by the properties of individuals (namely, by the number of warriors withineach competing faction, which is subject to cultural evolution). This is similar to the way group fissionsare generated by individuals assorting themselves among daughter groups, so that their outcome will bedetermined by the trait distribution in the parent group.Warrior production is necessary for winning conflicts, but it decreases the birth rate of individualsas well as the fission rate of groups, and engagement increases the risk of group elimination by conflict.In the absence of acculturation, i.e., when conflicts always result in the elimination of the defeatedgroup, the gain from victory in conflicts at the group level outweighs the cost of warrior production atthe individual level if conflicts are certain to occur frequently enough. As Darwin speculated in TheDescent of Man, when “tribes of primeval man, living in the same country, came into competition”, theones with “disciplined soldiers” would “spread and be victorious over the other tribes” (Darwin 1871).Note that within each particular group, the relative frequency of shepherds still increases, since thegains that come from the presence of warriors (in this case, survival) are equally enjoyed by all groupmembers. In this respect, participation in coalitional aggression is a collective action problem, wherethe lower fecundity of warriors is compensated by the (cultural) gains from competition with othergroups. (This process is similar to Gavrilets and Fortunato 2014, where individuals who contribute tocollective action at a cost have lower individual-level fitness when compared to their non-contributingcounterparts, but still increase in number because of the effects of intergroup competition.)However, if engagement is itself a cultural trait of groups that evolves under group selection, therate of engagement in risky conflicts evolves to zero, and as a consequence, so does warrior productionat the individual level. In the absence of acculturation, evolution will generate cautious, peaceful andlarge groups with few warriors.In contrast, with high acculturation tendencies, cultural variants that promote engagement get trans-mitted horizontally from group to group via group interactions, much like an epidemic spreads whenits host interacts with other individuals. This leads to coevolution of engagement at the group leveland warrior production at the individual level, the latter being selected by group selection despite stillbeing disadvantageous at the individual level. Moreover, when acculturation is itself an evolving grouptrait, cultural group selection leads to high acculturation levels, ultimately resulting in populations inwhich conflict is common and warrior production substantial. Thus, our model shows that conflicts canbecome frequent even when neither individuals nor groups benefit from it.Previous models had already shown that the prevalence of intergroup conflict could potentiallyunderlie the evolution of group-beneficial, but individually costly, human social behaviors (Bowles2009), as well as the evolution of group-level social institutions through acculturation of defeatedgroups (Turchin et al. 2013). Furthermore, past research had studied the co-evolution of group-levelinstitutions with individual social behaviors in the presence of intergroup competition (Bowles et al.292003). We extend this body of research by focusing on the origin of conflicts themselves, and showingthat, despite their costs, cultural group selection can favor their evolution when they are associated withthe horizontal transmission of social behaviors and institutions.In our model, acculturation tendency evolves to near one. At this point, very few cases of conflictare actually resolved with the death of defeated group members. This is (to some extent) reminiscentof modern wars in which casualties (relative to population size) are small (Gat 2013; Morris 2014).Nonetheless, the results of the model are not contingent on the assumption that acculturation is a blood-less business, as illustrated by the case of costly acculturation (Section A.5). Provided that group eventsare not too common, within-group dynamics eventually replenishes the defeated group’s warrior pool,rendering this type of demographic cost temporary.If the birth rate of warriors is equal or higher than that of shepherds, the conflict between individual-level and group level selection vanishes in our models, and the production of warriors will always befavored. Thus, in our context interesting questions only arise when warriors have a lower birth rate,and to gain conceptual clarity, we assumed the worst case scenario of no warrior reproduction for mostof the analysis. In real human societies, costs of warriors are unlikely to be that extreme, but may bepresent nevertheless, e.g. in the form a direct risk of injury or death (increases in death rate), as well astime and energy investments or social conventions that may be associated with decreases in birth rate.For example, in ancient Sparta, the members of the privileged Spartiate warrior caste tended to marrylate, leading to low birth rates (Hodkinson 2009). As an alternative biological interpretation, one caninstead consider the cultural trait p to be the fraction of lifetime that individuals are expected to dedicateto warrior activity (such as a military service during which individuals refrain from reproduction). Thereis also a possibility that warriors may reap direct benefits from conflicts and even achieve higher repro-duction than other individuals. For example, among the Yanomamo¨ (an indigenous population living inthe northern portion of the Amazon basin), men who kill an enemy during a raiding party may acquirea special status known as unokai. Across all age cohorts, unokai men have higher reproductive successthan their non-unokai counterparts, due to their increased success in finding mates (Chagnon 1988; al-though reproductive success only seems to be measured among successful unokai, i.e., excluding theindividuals who die in the attempt to achieve that status). Similarly, among the Nyangatom, a group ofnomadic pastoralists in East Africa, men who acquire livestock through raiding have higher access toreproductive opportunities, since Nyangatom marriage traditions require the exchange of bridewealth.Thus, Nyagatom elders who participated in raiding in their youth tend to have more wives and childrenthan other elders (Glowacki and Wrangham 2015). In cases like these, in which benefits outweigh thecosts, individual-level selection alone is capable of explaining the production of warriors.It would be extremely speculative to suggest that acculturation is anything more than a partial driverfor the prevalence of intergroup conflict in actually existing human societies. A complete picture of theorigin and maintenance of conflict must also incorporate individual-level benefits, such as the acquisitionof territory, access to females, or food (Bowles 2006; Choi and Bowles 2007; Gat 2000; Lehmann andFeldman 2008), as well as the ways in which individual benefits may feed back into group level eventsby increasing the chances of victory (Lehmann 2011). Nonetheless, the idea that cultural evolution30may promote the spread of conflict even in cases where it may be maladaptive has received relativelylittle attention. Conceptually, this type of explanation broadens the conditions under which we expectconflicts to occur, thus adding to, rather than replacing, current explanations. Perhaps in the future, acombination of both approaches will yield additional insights. For example, the dynamics of group sizeand number may in turn affect individual-level payoffs.Our model is, of course, not meant as a rigorous characterization of any particular human society ormoment in human history. Understanding the extent to which our results apply to, for example, hunter-gatherer societies, would require parameterizing our model with anthropological data and expanding itto include other relevant demographic processes such as migration and resource dynamics, as well assexual reproduction. This would be a challenging task that we did not attempt to undertake. Our modelis also unrealistic in considering primarily large-scale battles that result in the massacre of the defeatedgroup, an outcome that, in hunter-gatherer conflicts, was usually limited to cases in which one side wasgreatly outnumbered (Burch 2005). Nonetheless, our model illustrates the possibility that, even in theabsence of conflict-associated benefits, we can generate the same observed patterns as predicted by theresource-based, adaptive hypothesis that is paradigmatic in cultural evolution and biology. Importantly,the central ingredients of our explanation—social institutions and their horizontal transmission betweengroups—rely on uniquely human cognitive capacities, and therefore suggest a qualitative differencebetween the causes of human and non-human intergroup conflicts. This is in contrast with a long-standing tradition in biology of proposing an evolutionary continuity between antagonistic behavior inhuman and non-human animals (Crofoot and Wrangham 2010).Although we didn’t explore them in our model, it would be interesting to investigate the effectsof other types of events. For example, migration between groups, or even group fusions, could lowerinter-group diversity, and reduce the effect of group selection. Furthermore, although we have framedour model in terms of the evolution of human conflict, the same concepts and methods could be used toaddress other questions in cultural evolution. “Warrior production” in our model is an archetype for anyinstitution or cultural practice in which societies invest costly resources for the purpose of succeedingduring group interactions. In the same way, our modeling framework could accommodate other typesof group interactions, including mutually beneficial ones such as trading, which can provide a differentmechanism for the horizontal transmission of cultural variants between groups. Traditional (one-level)cultural evolutionary game theory has shown that costly behaviors (such as cooperation) can persistthanks to the frequency-dependent effects of interactions between individuals, provided that traits aretransmissible (social learning, Laland 2004). A multilevel framework for cultural evolution allows us toapply the same logic to the persistence of costly practices via their effect on interactions between groups,provided that these practices are transmissible (acculturation). We expect that this approach can yieldinsights much beyond the scope of human warfare, such as the cultural evolution of religion (Norenzayanet al. 2016).31Chapter 3Cooperation can promote rescue or lead to evolu-tionary suicide during environmental change3.1 IntroductionThroughout evolutionary history, populations have experienced constantly changing environmental con-ditions. Examples of large-scale environmental changes with profound evolutionary legacies include therise of atmospheric oxygen during the Proterozoic (Lyons and Reinhard 2014); continental movementslike the emergence of the Isthmus of Panama during the Pliocene (Stehli and Webb 1985); and climaticalterations such as the Pleistocene glaciations (Hewitt 2004). In recent times, natural populations are in-creasingly facing anthropogenic directional environmental shifts, ranging from habitat degradation andclimate change (reviewed in Gienapp et al. 2008; Hoffmann and Sgro` 2011; Sih et al. 2011) to exposureto increasing concentrations of antibiotics (Klein et al. 2018).Because populations are often locally adapted (Hereford 2009), environmental changes may leadto decreased fitness and to a decline in population size. Ultimately, this may lead to extinction, asthe population’s phenotype is unable to track the shifting environmental optimum. This fate may beprevented when genetic adaptation counters population decline and prevents extinction, a process calledevolutionary rescue (Bell 2017; Gomulkiewicz and Holt 1995). The effects of environmental change(and ultimately the possibility of evolutionary rescue) will, in general, depend on within– and between-species biotic interactions (Collins 2011; Harmon et al. 2009; Lavergne et al. 2010; Lawrence et al.2012; Tylianakis et al. 2008). Although most theoretical models of adaptation to changing environmentsignore ecological interactions—reviews of theory by Alexander et al. (2014) and Bell (2017) barelydiscuss the subject, only very briefly mentioning competition—there is a growing body of modelingwork on the effects on evolutionary rescue of ecological interactions, both within and between species,such as competition (e.g., Johansson 2007; Osmond and De Mazancourt 2012) and predation (e.g., Jones2008; Mellard et al. 2015; Osmond et al. 2017).Here we focus on how adaptation to a changing environment is affected by a different type ofinteraction, cooperation. Because high levels of cooperation can increase population mean fitness, andtherefore, population abundance (Chuang et al. 2009; Hauert et al. 2006a; Tekwa et al. 2017)—and32larger populations are both less prone to extinction from demographic stochasticity and can adapt fasterdue to a larger mutational input and less genetic drift—we predict that cooperation can reduce the chancethat a population will go extinct due to environmental change.Following a well-established game-theoretical tradition (reviewed in Archetti and Scheuring 2012),we will model cooperative interactions as public goods games: events in which individuals pay a cost tocontribute toward a public good. The benefit generated by this public good is then equally enjoyed by allinteracting partners. Public goods interactions can describe many instances of cooperative behavior innature (Levin 2014). For example, many microbes secrete extracellular molecules which can be utilizedby non-producers (Tarnita 2017; West et al. 2006); these include, for example, adhesive polymers inPseudomonas (Rainey and Rainey 2003), invertase in yeast (Gore et al. 2009), and indole (a signalingmolecule providing antibiotic resistance) in Escherichia coli (Lee et al. 2010). Similarly, in many costlyor risky behaviors such as cooperative breeding (Rabenold 1984), cooperative hunting (Bednarz 1972;Packer et al. 1990; Yip et al. 2008), alarm calls against predators (Clutton-Brock et al. 1999), or theformation of fruiting bodies in social amoebae (Strassmann et al. 2000), benefits are equally distributedto all participants. Even cancer cells share costly diffusible products, such as growth factors (Archettiet al. 2015; Axelrod et al. 2006). Thus, public goods interactions are a useful general framework tomodel cooperation, applicable to a wide range of natural scenarios.The public goods framework also permits a seamless articulation between evolutionary game dy-namics and ecological or demographic dynamics (such as changes in population size, as might be causedby environmental shifts). Models that make this connection explicit (called “ecological public goodsgames”, Gokhale and Hauert 2016; Hauert et al. 2006a, 2008; Parvinen 2010; Wakano et al. 2009) arebased on the feedback between population size and the fitness of cooperators. In these models, themarginal benefits of cooperation decrease with higher group sizes, meaning that cooperation is favoredonly when groups are small (Hauert et al. 2006a; this is sometimes called “weak altruism”, e.g., inFletcher and Doebeli 2009). This is also true (or true under some conditions) in other models of cooper-ation (Cornforth et al. 2012; Pen˜a and No¨ldeke 2018; Powers and Lehmann 2017; Shen et al. 2013), andit matches at least some empirical observations: for example, the amount of time that meerkats spendon guard decreases with group size (Clutton-Brock et al. 1999), and Pseudomonas “cheaters” that donot produce siderophores perform better at high cell densities (Ross-Gillespie et al. 2009). Therefore,if low population densities correspond to small interaction group sizes (e.g., Blank et al. 2012; Johnson1983; Pe´pin and Gerard 2008; Vander Wal et al. 2013), they allow cooperation to gain a foothold. Ascooperation increases, so does the population mean fitness, and, consequently, the population density.This feedback between ecology and evolution can lead to the maintenance of stable, intermediate fre-quencies of cooperators (Hauert et al. 2006a, 2008). Laboratory experiments in sucrose-growing yeast(Chen et al. 2014; Harrington and Sanchez 2014; Sanchez and Gore 2013) confirm the empirical rel-evance of these feedbacks between population density and the evolution of cooperation. Populationswhose growth is mediated by cooperative production of invertase approach intermediate densities andfrequencies of cooperators, consistent with ecological public goods dynamics (Sanchez and Gore 2013).Binary-action public goods games describe bimorphic populations where individuals are either co-33operators (who produce a fixed amount of public good) or defectors (non-producers). For example,whereas wild-type yeast cells produce invertase, mutant “cheaters” do not (Gore et al. 2009). Thisis the approach used in most models of ecological public goods games (Hauert et al. 2006a, 2008;Wakano et al. 2009). However, in some systems it may be more appropriate to model cooperation asa quantitative trait. For instance, meerkat females can provide more or less assistance with babysittingand pup feeding (Clutton-Brock et al. 2001). In continuous public goods games (Doebeli et al. 2004;Gylling and Bra¨nnstro¨m 2018; Killingback and Doebeli 2002; Molina and Earn 2017) individuals aredescribed by a continuous trait value that quantifies how much they invest into the public good. Parvi-nen (2010) showed that, if cooperation is modelled as a continuous trait, ecological public goods gamescan exhibit evolutionary branching—that is, the population can evolve into a highly cooperating and anon-cooperating strain. In other words, the continuous public goods framework can also (depending onthe parameters of the model) give rise to distinct cooperators and defectors, a process called “tragedy ofthe commune” by Doebeli et al. (2004), who first described it in the context of the continuous snowdriftgame. In our model, we will use the continuous public goods game. As we will see, depending on theshape of the function that describes the costs of cooperation, we may observe a tragedy of the commune.If this happens, the model will resemble a binary-action public goods game, where cooperation evolvesthrough changes in the relative frequencies of the two fixed strategies, rather than a shift in the value ofa quantitative trait (cf. Section B.3.1, particularly Eq. B.29). Thus, the same framework will allow usto investigate the interaction between cooperation and environmental change, both in those cases whereall individuals cooperate to a similar degree and in those cases where cooperators and defectors coexist.Under directional environmental change, species keep track of the moving environmental optimumat a certain phenotypic distance or “lag” (Bu¨rger and Lynch 1995; Lynch and Lande 1993). The sizeof the lag will depend on the speed of environmental change, as well as the population’s ability toadapt (Bu¨rger and Lynch 1995; Lynch and Lande 1993). The lag reaches an equilibrium when therate of environmental change equals the rate of adaptation, which, for mutation-limited evolution, isproportional to population size and mutation rate (Dieckmann and Law 1996). The larger the lag, thelower the size of the population. If the lag becomes too large, the population will get trapped in avicious cycle: it will grow so small that it cannot adapt fast enough to keep up with the environmentaloptimum. The result is an extinction vortex (Brook et al. 2008; Fagan and Holmes 2006; Gilpin andSoule´ 1986): an amplifying feedback that will ultimately lead the population to extinction (Johansson2007); in the quantitative genetics literature, a similar cycle exists with genetic variance taking the placeof population size (Bu¨rger and Lynch 1995; Osmond and Klausmeier 2017). However, a populationengaged in ecological public goods games could potentially break free from this vicious cycle, becausethe decline in population density would favor cooperation. This matches experimental evidence that, asenvironments deteriorate, public goods producers increase in frequency (Chen et al. 2014). Thus, ourhypothesis is that environmental change favors the evolution of cooperation, and that this process mayrescue a population that would otherwise have been unable to keep track with the optimum.343.2 Methods3.2.1 Model descriptionInheritance and life cycleWe model a population of individuals, each of which are characterized by a cooperation trait x > 0 anda functional quantitative trait y. The multidimensional phenotype~z = (x,y) is assumed to be completelygenetically determined. We assume clonal reproduction such that offspring inherit their parent’s phe-notype exactly with probability (1− µx)(1− µy); with probability µx a random normal deviate withstandard deviation σx is added to the cooperative trait and with probability µy a random normal deviatewith standard deviation σy is added to the functional trait. If such a mutation would result in a negativevalue of x, the offspring’s cooperative trait is set to zero. We ignore genetic correlations between x andy.Generations are non-overlapping, i.e., individuals are semelparous (they reproduce only once) andreproduce synchronously. Every generation, each individual survives to reproductive age with a prob-ability V (viability) that depends on the individual’s phenotype as well as the state of the environment(including both abiotic and biotic components). Should they survive, each gives birth to a Poisson num-ber of offspring with an expectation F (fecundity) that is also dependent on the individual’s phenotypeand the environment. Due to stochasticity in both survival and reproduction, the number of individualsin the population, n ∈ Z+0 , varies stochastically over time.Fitness componentsViability and fecundity are assumed to depend on four factors: environmental mismatch, density regu-lation, and the costs and benefits of cooperation. We discuss each factor in turn.Environmental mismatch The state of the environment, θ , is measured in the same scale as the func-tional trait, y. DefiningL (y)= |θ−y| as a phenotypic lag, we assume the fitness effect of environmentalmismatch is a Gaussian function of lag:M(y) = exp(−12 sL (y)2), (3.1)which is 1 whenL (y) = 0 and declines to 0 as lag increases, with larger values of s increasing the rateof this decline.Density regulation We implement density regulation similarly to Beverton and Holt (1957). The fitnesseffect of density is assumed to be a decreasing function of population size, n,D(n) =11+d n, (3.2)35which is 1 when n = 0 and declines to 0 as n→∞, with larger values of d increasing the rate of decline.Cost of cooperation Any investment into cooperation is assumed to come at a cost, with fitness effectC(x) = exp(−cxk), (3.3)which is 1 when x= 0 and declines to 0 as cooperation increases, with larger values of c and k increasingthe rate and concavity of this decline, respectively. Note that because this is a decreasing function, thecost of cooperation is higher when C(x) is smaller, and vice versa.Benefits of cooperation Each cooperative interaction begins with a focal individual assembling an in-teraction group. Every individual in the population has the same probability p of joining the interactiongroup (i.e., there is no spatial structure or heritable variation in interaction tendency). Sampling with-out replacement, the probability Pr(g|n) that an interaction group size will equal g is then given by ahypergeometric distribution. Here we approximate this with the simpler binomial distribution,Pr(g|n) =(n−1g−1)pg−1 (1− p)n−g, (3.4)which is accurate as long as group sizes are small relative to the total population size, g n. Thisincludes the possibility that the focal individual is the sole member of an interaction group (g = 1).Eq. 3.4 mechanistically connects population dynamics to public goods dynamics. Variations inpopulation size, n, will affect the group size, which in turn (see below) changes the incentives for co-operation. This feedback loop promotes the maintenance of cooperation at intermediate levels, and itmakes our model an example of an ecological public goods game (Gokhale and Hauert 2016; Hauertet al. 2006a, 2008; Parvinen 2010; Wakano et al. 2009). Whereas previous models implementedeco-evolutionary feedbacks by limiting reproductive opportunities, thus implicitly incorporating space(Gokhale and Hauert 2016; Hauert et al. 2006a, 2008; Parvinen 2010; Wakano et al. 2009), Eq. 3.4accomplishes the same qualitative effect via a different mechanism (by assigning each individual aprobability of joining interaction groups).In each interaction, every individual contributes to a common pool of public good. The total quantityis multiplied by a factor r, generating a benefit that is equally distributed among the members of thegroup. Therefore, for each interaction, the benefit to a focal individual with trait value x in a group withg−1 other individuals is (x+Σg−1i=1 xi)r/g, where the xi are the cooperative trait values of the non-focalindividuals. Here we assume there is little variation in trait values among the population, so that this canbe approximated byB(x|x,g) = rg(x+(g−1)x), (3.5)where x is the mean cooperative trait value in the population.Over the course of a lifetime we assume all individuals will have participated in many interactiongroups. Therefore, we can normalize the benefit (i.e., interpret r to be the interest rate per interaction36times the number of interactions) such that the fitness effect depends only on the average benefit receivedacross a lifetime,B(x|x,n) =n∑g=1Pr(g|n)B(x|x,g)= r[x+1− (1− p)nnp(x− x)],(3.6)which can vary from 0 to infinity. This assumption of many interactions is a simple caricature of aspecies which interacts at a much higher rate than individuals die. Relaxing this assumption, so that thegroup sizes experienced and therefore the benefits received each generation are stochastic, would be aninteresting extension.Our main results are robust to alternative functional forms for the cost and benefit (see Discussion).Partitioning fitness componentsWe partition these factors into viability and fecundity. We assume the benefits of cooperation, B(x|x,n),which in this framework are not bounded between 0 and 1, always affect fecundity while the other threecomponents multiplicatively decrease either this baseline fecundity or a baseline viability of 1. For in-stance, costs may affect fecundity while density and mismatch affect viability, giving F = B(x′|x,n)C(x)and V = D(n)M(y) (recall that higher values of C(x) correspond to smaller costs, Eq. 3.3). We refer tothe 23 = 8 different alternatives as “fitness partitions”.3.2.2 SimulationsWe have implemented the stochastic process described above as an individual-based (or agent-based)simulation in R, deposited in Dryad (Henriques and Osmond 2020). Note that we do not explicitlysimulate group size formation but instead directly use Eq. 3.6. To avoid numerical issues, cooperationtrait values below 10−6 are set to 10−6. We begin all simulations with monomorphic populations.3.2.3 Adaptive dynamicsTo gain deeper intuition we also analyzed an approximation of this individual-based process. As amodeling framework, we used adaptive dynamics (Dieckmann and Law 1996; Geritz et al. 1998; Metzet al. 1996) to follow the change, over time, of a population’s traits and density. This approach makes anumber of assumptions.First, adaptive dynamics assumes that evolution is much slower than ecological population dynam-ics (demography). This will be the case, for example, if evolution is limited by the arrival of raresmall-effect mutations, rather than occurring from standing genetic variation. This time-scale separa-tion allows the population to always reach ecological equilibrium (i.e., a monomorphic population atcarrying capacity) before the introduction of new advantageous phenotypes. Note that this implies thatwe are primarily concerned only with environmental changes that occur on the evolutionary time-scale,i.e., during the time it takes for a population to reach ecological equilibrium the environmental state37changes only by a negligible amount. If the environment were to change much faster than the evolu-tionary time-scale evolutionary rescue would be impossible and extinction certain; if the environmentwere to change much slower persistence would be trivial. As a consequence of the separation of time-scales, the population is nearly monomorphic, with nearly all individuals having a resident phenotype~z = (x,y). Mutations are so rare that, at any given time, no more than one mutant segregates in thepopulation; such mutants are denoted~z′ = (x′,y′). The distribution of group sizes formed by this mutantis therefore binomial (Eq. 3.4).Second, adaptive dynamics assumes that the population size is large and that there is no demographicstochasticity in the resident population. This is, of course, problematic in a model that purports todescribe the dynamics of populations that are approaching extinction. Quantitatively, this assumptionmeans that our approximation will get worse at small population sizes; however, as we will see, it stillcaptures the qualitative differences observed (on average) across the individual-based simulations.A final approximation that we make in order to translate the stochastic process to a deterministicdynamical system is that we redefine n as the expected population size, which is not in general aninteger and therefore violates the binomial sampling assumption in the model (Eq. 3.4). Despite this,we use this expected population size in the final line of Eq. 3.6 (and elsewhere) as an approximationof the underlying individual-based sampling process (section B.1.2), which qualitatively recapitulatesstochastic realizations of the individual-based process, as we will see below.3.3 Results3.3.1 Resident dynamicsUnder these adaptive dynamic assumptions, the average benefit of cooperation (Eq. 3.6) for a rare mu-tant with cooperative trait value x′ is B(x′|x,n), where the mean trait value, x, has been replaced by theresident trait value, x. Fitness, the expected number of offspring a focal individual leaves in the nextgeneration, is then W (~z′|~z) =V F = B(x′|x,n)C(x)D(n)M(y), which depends on both the focal and resi-dent phenotypes,~z′ and~z, as well as population size, n. (Again, recall that costs are highest, and hencefitness is lower, at small values of C(x).)For a given population size n, we can calculate the inter-generation change in the expected pop-ulation size: ∆n = n W (~z|~z)− n, where W (~z|~z) = B(x|x,n)C(x)D(n)M(y) is the fitness of a residentindividual. When the population size is constant, ∆n = 0, the population is at ecological equilibriumand will have size nˆ(~z) (Fig. 3.1A):nˆ(~z) =M(y)r x exp(−cxk)−1dif xmin ≤ x≤ xmax0 otherwise,(3.7)where xmin and xmax are, respectively, the minimum and maximum amounts of cooperation beyondwhich the population goes extinct.As expected, the population size is higher when public goods are cheaper to produce (low c) or38ℒ = 1ℒ = 0.5ℒ = 00 2 4 6 8 10 12 140100200300400xnxcritAp = 0.001p = 0.005p = 0.010.00.51.01.52.02.53.03.5x*xcritBp = 0.001p = 0.005p = 0.010.0 0.5 1.0 1.50100200300400ℒn (x* )ℒmaxFigure 3.1: The population size depends on distance to the optimum and on the amount of cooperative invest-ment. A: The population size at ecological equilibrium (nˆ) increases with cooperation (x) up to a maximum(xcrit, Eq. 3.9), and decreases for higher values. At very low (Eq. 3.8) or very high values of cooperative in-vestment, the population goes extinct. Population size also decreases with distance L (y) to the environmentaloptimum (Eq. 3.7), such that at very large lags (Eq. 3.10), the population goes extinct. B, Top: the equilibriumvalue of cooperation with a constant lag (x?, calculated from Eq. 3.11) increases for higher lags (L ) and smallerinteraction group sizes (smaller p). Bottom: corresponding population size at equilibrium (nˆ(x?)). Parameters:d = 0.01, s = 1, c = 0.3, r = 4, k = 1, p = 0.01 (except when otherwise stated in the figure).provide higher benefits (high r), as well as when density regulation is less intense (low d). Furthermore,even in a constant environment, a minimum amount of cooperation is necessary to sustain the populationand avoid extinction (Fig. 3.1A):xmin =k√−W (−ck(M(y)r)−k)ck, (3.8)whereW (·) denotes the Lambert W-function, also known as the product logarithm (Corless et al. 1996;Lehtonen 2016). As individuals invest more in cooperation, the population size increases (Fig. 3.1A),up to a critical value, xcrit:xcrit =k√1ck. (3.9)39Beyond this point, the costs of cooperation are so high that further investment leads to smaller popula-tions. Therefore, there is a maximum amount of cooperation, xmax beyond which the population cannotpersist (Fig. 3.1A).The equilibrium population size declines with the lag between the population’s trait and the envi-ronmental optimum (Fig. 3.1A). Substituting Eq. 3.1 into Eq. 3.7 and solving nˆ(~z) = 0 for L (y), wecan calculate the maximum value of lag above which the population goes extinct:Lmax =√2s(ln(rx)− cxk). (3.10)3.3.2 Evolution of cooperation for a constant lagWe now focus on the fate of a rare mutant with trait value x′ and consider the case of a constant lag(L (y) =L ) in a constant environment (i.e., θ is constant). Because we are concerned with adaptationin x and the lag is fixed, we will drop the dependence of various functions on y for the remainder of thissection. The mutant’s fitness is then given by W (x′|x) = B(x′|x, nˆ(x))C(x′)D(nˆ(x))M.The direction and strength of selection on cooperation can be measured by the selection gradient ofcooperative investment (the slope of the fitness landscape experienced by the mutant), which is given byS (x) =∂W (x′|x)∂x′∣∣∣∣x′=x=1− (1− p)nˆ(x)− ckpxknˆ(x)pxnˆ(x). (3.11)The evolutionary dynamics will be at equilibrium when there is no directional selection. At thispoint the selection gradient vanishes, S (x?) = 0, and x? is said to be an evolutionarily singular point(Geritz et al. 1998). Although an explicit analytical solution is impossible to calculate, we can solvefor x? numerically (Fig. 3.1B). Furthermore, we can analytically show that S (x) is positive when x issmall enough, declines with x when x < xcrit (at least for k ≥ 1, i.e., a sufficiently concave cost) andis negative when x = xcrit (section B.2.1). Therefore, in a monomorphic population, if x starts belowxcrit it will converge to the only singular point x? that exists below xcrit. Intuitively, selection doesnot drive cooperation above the value which causes decreases in population size. This convergence tothe singular point can be visualized from a pairwise invasibility plot (PIP), which indicates the mutantstrategies x′ that can invade any given resident strategy x (Geritz et al. 1998; Tienderen and Jong 1986).Fig. 3.2Ai and Fig. 3.2Bi show, for two sets of parameters, that the singular point is convergent stable.We are also able to show with implicit differentiation, again for k ≥ 1, that the equilibrium level ofcooperative investment, x?, increases with lag and decreases with group size, as driven by increases inp (see Fig. 3.1B for a numerical example and section B.2.2 for derivations).400 1 2012Resident (x)Mutant(x′ )A i0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.01.2Time ((μ σ2)-1)Cooperation(x)iiDCD0.00 0.05 0.10 0.15 0.20050100150200250Time ((μ σ2)-1)Populationsize(n)iiiDCD0 1 2 3 401234Resident (x)Mutant(x′ )B i1 10 100 1000 104 105051015202530Time ((μ σ2)-1)Cooperation(x)ii x1x21 10 100 1000 104 1050100200300400Time ((μ σ2)-1)Populationsize(n)iiin1n2Figure 3.2: Monomorphic evolutionary dynamics for a constant lag when the singular point x? is a stable point(A) and when it is a branching point (B). The leftmost panels are pairwise invasibility plots, where blue (red)areas correspond to mutants with higher (lower) fitness than the resident. The point at which the black linescross is the singular point (x?); this point cannot be invaded by any mutants in A, but it is invasible on bothsides in B. For visual reference, xcrit ≈ 2.74 in A and xcrit ≈ 44.44 in B. The other panels show the monomorphicevolutionary dynamics of cooperation (x, middle panels) and corresponding population size (n, rightmost panels).In A, a population starting away from the singular point evolves toward x? and remains at equilibrium. Purple andgreen curves correspond, respectively, to the fitness partitions D and CD (see section 3.3.2). In B, only one fitnesspartition (CD) is shown; a population starting at singular point branches into a defector (x1) and a cooperator (x2)strain. Parameters: d = 0.01, r = 4, c = 0.3, M = 1, p = 0.01. For A: k = 1.1; for B: k = 0.5.Evolutionary stabilityOnce the population reaches equilibrium there are two possible outcomes. If the singular strategy is afitness maximum then evolution will come to a halt and cooperation will remain constant at x? (evolu-tionarily stable strategy). Otherwise (if the singular strategy is a fitness minimum) the monomorphicpopulation can simultaneously be invaded by mutants on either side of x?. This leads the populationto split into two distinct and diverging phenotypes—a process called evolutionary branching (Geritzet al. 1998). Depending on the model parameters, x? can either be evolutionarily stable (Fig. 3.2A)or unstable (Fig. 3.2B). For individual-based simulations illustrating both possibilities, see Figs. B.1A(evolutionarily stable) and B.1B (evolutionary branching).The singular strategy will be a fitness minimum if∂ 2W (x′|x)∂ (x′)2∣∣∣∣x′=x=x?> 0⇐⇒k(1+ cx?k)< 1, (3.12)(see Eqs. B.23–B.24 for more details). This means that the singular strategy will be a branching pointonly when the cost is sufficiently weak (small c) and convex (small k), and the equilibrium level ofcooperation is low.41If the singularity is a branching point the population branches into a defector strain and a cooperatorstrain (a process called tragedy of the commune, see Doebeli et al. 2004) with trait values x1 and x2,respectively (Fig. 3.2B). At evolutionary equilibrium x?1 and x?2 are values that do not depend on thelag (as shown in section B.3.1). In contrast to the case where branching does not occur (in which theequilibrium trait value changes with the lag in a quantitative fashion), after branching the trait valuesevolve to a stable equilibrium, so that cooperation evolves in response to environmental change onlythrough changes in their relative frequencies (cf. section B.3.1, particularly Eq. B.29). This situationresembles a binary-action public goods game.In this manuscript we will focus on the case where the singular strategy is evolutionarily stable(Fig. 3.2A). Individual-based simulations suggest that, although branching is possible (Fig. B.1B), theparameter range under which this process actually occurs, resulting in a stable dimorphic equilibrium,is relatively narrow when compared to the analytical predictions above. In particular the population sizeof the cooperator is often so small that stochastic extinction of that branch quickly occurs. Nonetheless,we present some remarks regarding the post-branching dynamics in section B.3.1.Monomorphic evolutionary dynamics for a constant lagThe selection gradient can also be used to determine the rate of evolution. Using a traditional adaptivedynamics approach (Dieckmann and Law 1996; Geritz et al. 1998; Metz et al. 1996), adapted for a lifecycle with non-overlapping generations (section B.1), we obtain a differential equation for the time-dynamics of cooperative investment (x˙):x˙ =1σ2W (x)nˆ(x)µxσ2x S (x), (3.13)where σ2W (x) refers to the variance in the resident’s reproductive success. Eq. 3.13 is analogous tothe canonical equation of adaptive dynamics (Dieckmann and Law 1996). In contrast to the canonicalequation, the rate of adaptation in Eq. 3.13 is inversely proportional to the variance in reproductive suc-cess, σ2W (x). Given the same expected number of offspring, variance in number of offspring increasesthe chance that an advantageous mutant is lost through genetic drift; therefore, variance in reproduc-tive success slows adaptation down (Appendix B, see also Durinx et al. 2008). Given that survivalis Bernoulli-distributed with expectation V and fecundity is Poisson-distributed with expectation F , thevariance in reproductive success is σ2W =W (1+F−W ). At ecological equilibrium (when W =V F = 1),this becomes σ2W = F = 1/V , which depends on how the fitness components are partitioned into viabil-ity and fecundity (Section 3.2.1). The intuition for this result is that, if population size is at equilibrium,any increase in fecundity must be balanced out by a decrease in viability, and vice versa. Additionally,the number of offspring is more variable when fecundity is high. Therefore, viability and fecundity haveopposite effects on reproductive variance.Substituting Eqs. 3.7 and 3.11 into Eq. 3.13 we are able to determine the time-dynamics of coop-eration. At this point, however, we need to be explicit about the fitness partition for the species weare studying. Because the rate of adaptation depends on the variance in reproductive success, σ2W (x),42Table 3.1: Possible fitness partitions in the model. Partitions are named after the initial letter of the modelelements (cost of cooperation, environmental mismatch, or density regulation) that decrease viability. Elementsthat do not decrease viability decrease fecundity instead. Abscissae indicate cooperation level (0 < x < xcrit,Eq. 3.9), and ordinates indicate nˆ(x)/σ2W (x). Parameters: d = 0.01, r = 4, c = 0.3, p = 0.01, M(y) = 1, k = 1.partitionFitness Viability includes...nˆ(x)/σ2W (x)Cost Mismatch Density regulationCMD X X X e−cxk M(y)− 1rxdCM X Xe−2cxk M(y)(−ecxk+M(y)rx)dCD X X e−cxk− 1M(y)rxdC Xe−2cxk(−ecxk+M(y)rx)dMD X X −ecxk+M(y)rxdrxM XM(y)(e−cxk M(y)rx−1)dD X 1−ecxkM(y)rxd∅ e−cxk M(y)rx−1dthe evolutionary trajectory will depend on how the different model elements—environmental mismatch(Eq. 3.1), cost of public good production (Eq. 3.3), and density regulation (Eq. 3.2)—are partitionedamong the two fitness components (viability and fecundity, see section 3.2.1). Each of the the threecomponents in our model can affect either viability or fecundity, for a total of eight possible partitions(Table 3.1). For example, the mismatch between the environmental state and the functional trait mayconceivably decrease either viability or fecundity, depending on the species. The choice of fitness par-tition will turn out to qualitatively affect the outcome of adaptation to changing environments, via itseffect on variance in reproductive success (σ2W ).Two qualitatively different behaviors will be observed, depending on whether the costs of publicgood production (which are high when C(x) is small, see Eq. 3.3) decrease viability (green curves inTable 3.1) or fecundity (purple curves). Throughout the text, we will illustrate the effect of fitnesspartition choice by contrasting two example fitness partitions (which we call CD and D, see Table 3.1).In partition CD, the costs of public good production decrease viability, whereas in partition D, theydecrease fecundity. As we will see, which fitness component is affected by density regulation andenvironmental mismatch has no qualitative effect. In both example fitness partitions, density regulationdecreases viability, whereas environmental mismatch decreases fecundity (Table 3.1).The viability of a resident individual with fitness partition D is VD =D(nˆ(x)) = exp(cxk)/(Mrx), andthat of a resident individual with fitness partition CD is VCD =C(x) ·D(nˆ(x)) = 1/(Mrx). As evolutiondrives the population toward higher levels of cooperative investment, the costs of cooperation increase43(i.e., lower values of C(x), which reduces 1/σ2W ), so that CD-species adapt slower and slower whencontrasted with D-species (Table 3.1). As we will see later on, this will lead to qualitative differences inthe capacity for evolutionary rescue, but in the case of adaptation toward a stable singular point with aconstant lag the long-term outcome of evolution is the same (Fig. 3.2A).3.3.3 Evolution of cooperation during environmental changeIn section 3.3.2, we considered the dynamics of cooperation when the functional trait is at some constantdistance to the environmental optimum. In the long term, populations that track moving environmentswill indeed always evolve to exhibit a constant lag (or, alternatively, they will fail to track the optimumand go extinct). However, the dynamics from section 3.3.2 do not take into account the feedback be-tween cooperation and lag distance. We will now track how populations evolve while the environmentaloptimum, θ , increases at some constant velocity v, i.e., θ = vt.We imagine that populations are initially well-matched to the environment (and at evolutionary equi-librium, see section 3.3.2). Recall that, if the evolutionary singular point x? is an evolutionary branchingpoint (Eq. 3.12), then the initially well-adapted population will be participating in a binary-action eco-logical public goods game (i.e., it will consist of two distinct strains: cooperators and defectors, whosefrequencies may fluctuate as the environment changes; for more details see section B.3.1). Otherwise, itwill be engaged in a continuous ecological public goods game (i.e., it will be a monomorphic populationcharacterized by a quantitative trait, describing the amount of cooperative investment). Here, we willconsider the latter scenario; for some remarks on the prior case see section B.3.1.The rate of evolutionWe again focus on the fate of a rare mutant, with phenotype ~z′ = (x′, y′) and fitness W (~z′|~z) =B(x′|~z)C(x′)D(~z)M(y′). The formula for the selection gradient of cooperation is given by Eq. 3.11(with the added dependence of M on y′). As for the functional trait, its selection gradient is given byS (y) =∂W (~z′|~z)∂y′∣∣∣∣~z′=~z= s(tv− y). (3.14)In contrast to section 3.3.2, there is now no endpoint to evolution because, if the population managesto change fast enough to avoid extinction, it will approach a state in which y is constantly evolving atthe same velocity as the environment (y˙ = v). Nonetheless, this implies there can be a “dynamical”equilibrium where the lag stabilizes at some constant value. To find the dynamical equilibrium, we areultimately interested in finding the values of x and L for which x˙ = 0 and L˙ ≡ v− y˙ = 0. As there isno explicit analytic solution for Eq. 3.11, we will solve for x andL numerically. Then, we will furtherclarify our results by analytically determining the maximum velocity of environmental change that thepopulation is able to track without going extinct (i.e., the critical rate of environmental change).The differential equation for the dynamics of the two traits~z= (x, y), with selection gradients~S(~z) =(S (x), S (y)), can be calculated similarly to Eq. 3.13 (cf. also section S1 and Eq. 6.1 in Dieckmann44and Law 1996):~˙z =1σ2W (~z)nˆ(~z)~µ~Σ~S(~z), (3.15)where~µ = (µx,µy)ᵀ is a vector of the mutation rates and~Σ=(σ2x 00 σ2y)is the variance-covariance matrixfor the mutational distribution (assuming no genetic correlation in the two traits).The evolution of cooperation rescues populations from extinctionIntuitively, one may expect that since increased phenotypic lag favors the evolution of cooperationand cooperation increases population size, the evolution of cooperation may counteract decreases inpopulation size caused by changing environments. If this is so, there may be rates of environmentalchange for which populations with a fixed level of cooperation tend to go extinct while those with anevolving level of cooperation do not, meaning that the evolution of cooperation facilitates evolutionaryrescue. This process can indeed be observed for some fitness partitions.For example, Fig. 3.3A shows the evolutionary dynamics of a population belonging to a species withfitness partition D (Table 3.1). The pink curves depict the dynamics of a population where cooperationis fixed at a constant value (µxσ2x = 0). At the beginning of the numerical simulation, the population isperfectly adapted to the environment, and the level of cooperation is at equilibrium. As the optimumstarts moving at a constant velocity, the population size decreases. For some time, adaptation in y keepsthe population afloat, by lowering the rate of increase in the lag and the rate of decrease in populationsize. But as the population becomes ever smaller, adaptation decelerates, limited by the supply ofbeneficial mutations. The positive feedback between decreased population size and slower adaptationleads the population to extinction. Compare this with the blue curves, which differ from the red curvesonly in allowing cooperation to evolve on a timescale more similar to that of the moving optimum (i.e.,larger µxσ2x , parameters in figure caption). The initial dynamics are very similar to the previous case: anincrease in the lag and a corresponding decrease in population size. But this very decrease in populationsize rewards higher cooperative investments. As cooperation increases it slows the rate of populationdecline and thus allows the population to adapt faster to the environmental change. Confirming ourintuition, the evolution of cooperation rescues the population and leads to a dynamical equilibrium(orange dot in Fig. 3.3A).If the environment changes too fast, no dynamical equilibrium will exist. Higher environmentalvelocities result in a narrower lag nullcline (black line in Fig. 3.3A’s stream plot); if the velocity is toohigh, the nullcline vanishes, meaning that the population will fail to track the optimum and will go ex-tinct regardless of the initial conditions (Fig. 3.4A). That said, species where cooperation can evolve willsurvive faster environmental changes compared to species where cooperation cannot evolve (Fig. 3.4A).Even when the velocities are low enough that both types of species can avoid extinction, species wherecooperation can evolve will equilibrate at higher population sizes and lower lags (Fig. 3.4A).A dynamical equilibrium may also not exist if the game parameters do not favor the evolutionof high cooperation. For example, larger interaction groups deter cooperation (because they decrease45Fitness partition CDFitness partition DFigure 3.3: Evolution of cooperation with a moving optimum, for a species with fitness partition D (panel A)or fitness partition CD (panel B). The stream plot to the left indicates the change in a population’s lag (L ) andcooperation (x) over time, for a population where cooperation evolves fast relative to the environmental change.The nullclines for lag (L˙ = 0) and cooperation (x˙ = 0) are in black and light grey, respectively. Within the darkgrey region, the population goes extinct. The thick blue curves are the result of numerical simulations, startingwith zero lag and with x = x?. For comparison, the thick pink curves are the result of numerical (deterministic)simulations, with identical starting conditions, where cooperation does not evolve (µx = 0). The panels on theright show the time-dynamics of cooperation, lag, and population size for the same simulations. Thin lines showthe mean cooperation value, mean lag, and population size of individual-based simulations (ten replicates for eachtreatment). Parameters: r= 10, c= 0.1, s= 1, k= 2, µx = µy = 10−4, σx =σy = 0.01, d = 0.0019, p= 5×10−4;for A: v= 5.8×10−6; for B: v= 4.5×10−6. In contrast to A, the velocity in B was chosen such that the nullclinesdo not intersect. Using the higher velocity in panel B would lead to extinction of both populations, whereas usingthe lower velocity in panel A would lead to persistence of both populations, see also Fig. 3.4.)46Fitness partition DFitness partition CDFigure 3.4: Lag (left) and population size (right) at the dynamical equilibrium, in populations with fitness partitionD (panel A) or fitness partition CD (panel B), for different velocities of environmental change (v). The curves areinterrupted at high velocities because the populations become extinct: at high v, the nullclines for lag (L˙ = 0)and cooperation (x˙ = 0) no longer intersect, so that there is no equilibrium lag. Note that in A, for high values ofv, the evolution of cooperation rescues populations that would otherwise undergo extinction; in contrast, in B, theopposite happens. The parameters are the same as in Fig. 3.3.the fraction of investment benefits that return to the cooperator). Increasing p makes groups larger, andtherefore it moves the cooperation nullcline (grey line in Fig. 3.3A’s stream plot) to the left, thus causingthe dynamical equilibrium to vanish. Finally, even if a dynamical equilibrium exists, the population willbe unable to reach it if it evolves too slowly in x. The pink curve in Fig. 3.3 represents the most extremesuch scenario (µx = 0 or σx = 0). Note that the position of the equilibrium itself does not depend on µxor σx, provided that µx > 0 and σx > 0.The evolution of cooperation causes extinctionSurprisingly, small differences in fitness partition can result in a diametrically opposite outcome. Con-sider now a species with fitness partition CD (Table 3.1). Recall that the only difference between the twofitness partitions is that in CD, the costs of cooperation decrease viability instead of fertility (as in D).In the one-dimensional dynamics, the differences between the two fitness partitions are relatively minor(Fig. 3.2A): although adaptation is slightly slower for CD, the evolutionary equilibrium is the same inboth fitness partitions. However, the differences between the fitness partitions turn out to have importantqualitative consequences for a population’s capacity to track a moving optimum.Because cooperation can slow the rate of adaptation in fitness partition CD (by increasing repro-ductive variance) it is possible that the evolution of cooperation could hinder persistence in a changingenvironment. To examine this possibility, Fig. 3.3B considers a scenario where the rate of environmental47change is small enough that a population with a fixed cooperative investment would persist (all other pa-rameters are the same as in panel A). Such a population is represented by the pink curve in Fig. 3.3B. Incontrast, a population with an evolving level of cooperation now goes extinct (blue curve in Fig. 3.3B).In effect, the changing environment causes the population to decline, which favors more cooperation toevolve. While increases in cooperation may temporarily make the population larger relative to a pop-ulation where cooperation does not evolve (Fig. 3.4B), speeding adaptation by increasing the supplyof mutations, increases in cooperation also increase the variance in reproductive success (and thus thestrength of genetic drift). Eventually the increase in the variance in reproductive success overwhelms thepositive effect of increased population size, making adaptation slower and thereby causing extinction.More generally, in stark contrast to fitness partition D, the evolution of cooperation in fitness partitionCD results in higher equilibrium lags (compare Fig. 3.4B for CD to Fig. 3.4A for D).The critical rate of environmental changeTo more formally understand the difference between D and CD, consider the rate of change in lag,L˙ = v− y˙. If a dynamical equilibrium exists, then at such an equilibrium, y˙ = v. From Eq. 3.15 we canwrite:y˙ =sL µyσ2yσ2W (~z)d(rxe−12 sL2−cxk −1), (3.16)which, for fitness partitions D and CD, equalsy˙ =sL µyσ2yrx− e−cxk+ 12 sL 2drxif DsL µyσ2yrx− ecxk+ 12 sL 2drxe−cxkif CD.(3.17)With fitness partition D, y˙ increases with x up to x = xcrit. Thus, the evolution of higher levels of coop-eration will always increase the population’s capacity to track the moving optimum—decreasing lag—because x will never evolve to values higher than xcrit (section B.2.1). In contrast, in fitness partition CDy˙ is maximized at a value of x below xcrit. Therefore, once a sufficiently high level of cooperation hasevolved, further increases will impair the ability of the population to track the moving environment—increasing lag.To understand how this leads to extinction we next turn to calculating the critical rate of environ-mental change. First, notice that Eq. 3.16 has a maximum in L , implying a saddle-node bifurcation inthe dynamical equilibrium lag. This bifurcation corresponds to the fact that an equilibrium lag ceases toexist at high velocities (Fig. 3.4); it is an “evolutionary tipping point” (Osmond and Klausmeier 2017).This bifurcation also explains why simulated populations rapidly go extinct (Fig. 3.3) as they can transi-tion quickly from large positive growth rates to large negative ones as they pass through the bifurcation.We can find the critical lag that maximizes the rate of evolution (Lcrit) by calculating the root of dy˙/dL :Lcrit =√2W( rx2 e1/2−cxk)−1s, (3.18)480.0 0.5 1.0 1.5 2.00.00.51.01.5xCriticalv(×104 )DCDFigure 3.5: Critical velocity (maximal velocity of environmental change for which the population is able totrack the moving optimum without undergoing extinction) as a function of the level of cooperative investment(x < xcrit), for fitness partitions D (purple) and CD (green). The evolution of increasing levels of cooperationmakes populations able to endure faster rates of environmental change in D, but it has the opposite effect in CD.The critical velocity is calculated by subbingLcrit (Eq. 3.18) into y˙ (Eq. 3.17). Parameters: r = 10, c = 0.1, s =1, k = 2, µx = µy = 10−4, σx = σy = 0.05, d = 0.0015, p = 5×10−4.where W again denotes the Lambert W-function.Having calculated the critical lag, we can plug it into Eq. 3.17 to obtain the maximum velocity ofenvironmental change that can be tolerated by the population before it goes extinct (for a given valueof x). Although the resulting expressions are not particularly insightful, a plot of this critical velocityconfirms that increases in x < xcrit have qualitatively different effects for the fitness partitions D and CD(Fig. 3.5). In particular, we see that the evolution of increased cooperation cannot drive extinction infitness partition D, since it never decreases the maximum rate of evolution in y (as seen from Eq. 3.17and argued above). In contrast, Fig. 3.5 shows that it is possible for increased cooperation to driveextinction in fitness partition CD, if cooperation evolves to such an extent that it lowers the critical rateof environmental change (e.g., beyond x ≈ 1.25 in Fig. 3.5). And this indeed can happen; increasesin lag lower the population size, driving increases in cooperation that stimulate yet further increases inlag and declines in population size, and thus more cooperation. Through this feedback cooperation caneventually reach levels that preclude a stable lag, as seen in Fig. 3.3. In other words, the evolution ofcooperation promotes a vicious cycle that drives the population to an extinction that would not haveoccurred if cooperation were fixed, an example of evolutionary suicide (Ferrie`re 2000; Gyllenberg andParvinen 2001). For this reason, in contrast to D, the lag nullcline in CD (black line of Fig. 3.3B)vanishes for high values of cooperation. Of course, equilibria may still exist if the velocity is small,which widens the area enclosed by the lag nullcline, or if the game parameters are such that equilibriumcooperation is low. For example, in Fig. 3.3, p is relatively small, so that the average group size isnp ≈ 3.5. In populations where a larger proportion of the population composes any one group (i.e.,larger p) a lower level of cooperation is favoured, so that the x nullcline in Fig. 3.3B moves to the leftand may intersect theL nullcline. In such a case, faster environmental change would be needed to seeevolutionary suicide.In general, for any fitness partition where the costs of cooperation decrease viability, nˆ(~z)/σ2W (~z)49will decline as x approaches xcrit, whereas for any fitness partition where the costs instead decreasefecundity, it will be maximized exactly at x = xcrit. This is because the slope at xcrit, ∂∂xnˆ(~z)σ2W (~z)∣∣∣x=xcritonlydepends on the derivative of V (~z) = 1/σ2W (~z) (since nˆ(~z) is maximized at xcrit). The viability, in turn, candepend on x only through C(x) and/or D(nˆ(~z)). If the viability is affected by neither (fitness partition∅),then its derivative will be zero. The same is true when the viability depends on x only through D(nˆ(~z)),again because nˆ(~z) is maximized at xcrit. Thus, all fitness partitions in which the costs of cooperationdo not affect viability have a maximum rate of evolution in y at x = xcrit. In contrast, when the viabilityis affected by the costs of cooperation, its derivative also depends on the derivative of C(x), which isalways negative. Therefore, if the costs of cooperation decrease viability, then nˆ(~z)/σ2W (~z) is decliningwith x at xcrit, and increasing cooperation can slow evolution.Throughout we assume the benefits of cooperation affect fecundity. If the benefits were insteadscaled in such a way to keep them bounded between 0 and 1 (e.g., B˜ = 1−exp(B)) we could also allowthem to affect viability. This would complicate our conclusions but the outcome can be intuited from theabove reasoning. On the one hand, if the benefits affected viability but the costs affected fecundity thenincreasing cooperation would speed up evolution (since the derivative of V would be positive), while onthe other hand, if both affected viability the effect of cooperation on evolution would be determined bythe slope of their product, B˜C, with respect to x.3.4 Extension to other types of social interactionsThe results above are consequence of a series of eco-evolutionary feedbacks (interactions between pop-ulation density and selection on trait values) that arise during the process of environmental change.Changes in one trait, x (in this case, cooperation), affect the evolutionary rate of a second trait, y, that isundergoing adaptation to a changing environment, through both its effect on population density and thestrength of genetic drift. The rate of evolution in both traits was found to be proportional to nˆ(~z)/σ2W (~z)(Eq. 3.15). As explored in the previous section, we can see the effect of x on the rate of evolution iny by taking the derivative of this quantity with respect to x, giving(∂ log nˆ(~z)∂x −∂ logσ2W (~z)∂x)nˆ(~z)/σ2W (~z).When this derivative is positive an increase in x will speed evolution in y, lowering the lag and favoringpersistence (potentially leading to evolutionary rescue). When it is negative an increase in x will slowevolution in y, increasing the lag and favouring extinction (potentially leading to evolutionary suicide).In this article, we were concerned with the effects of evolving cooperation. In our model of cooperation,the first term of the derivative is positive (since cooperation increases population size, ∂ log nˆ(~z)/∂x> 0)but the second term can be negative for some fitness partitions (since cooperation can also increase drift,∂ logσ2W (~z)/∂x > 0). The balance between these two terms underlies the diversity of results we havereported.While this work has been motivated by cooperation, the same framework can be used, more gener-ally, to explore other types of social interactions. As an example, we have implemented in our frameworka well-known model of competition (Matsuda and Abrams 1994). In this model, individuals are char-acterized by a costly trait x determining their resource preference and competitive ability. Individualswith more similar traits compete more strongly and individuals with larger traits out-compete those with50smaller (for details, see section B.5 and the accompanying Mathematica notebook, deposited in DryadHenriques and Osmond 2020). Matsuda and Abrams (1994) showed how this competition kernel leadsto the runaway evolution of extreme competitive abilities, despite the imposed cost, resulting in one ofthe earliest examples of evolutionary suicide.Here we add a second trait y to this model, and explore how the evolution of competitive ability x af-fects the ability of a population to track a moving environmental optimum through evolution in y. As inthe cooperation model, there are eco-evolutionary feedback loops between population density and traitvalues. In particular, population density decreases with higher levels of competition (Fig. B.11A). Thisimplies that the first term in our derivative of evolutionary rate above is now negative (∂ nˆ(~z)/∂x < 0),meaning increased competition will tend to slow the evolution of y through this pathway. However, asthe lag increases population size declines, lowering the equilibrium level of competition (Fig. B.11C,Supplemental Information). As a result, evolution leads to reduced competition and larger populationsizes than would occur if competition was fixed. In some cases, this means that the evolution of compe-tition (which, in static environments, can result in population extinction through evolutionary suicide)can actually rescue populations from extinction during environmental change (Fig. 3.6). Interestingly,however, despite the fact that in some fitness partitions the evolution of reduced competition results inmore genetic drift (∂ logσ2W (~z)/∂x < 0) and as a consequence increased lags, we did not find numeri-cal examples where the evolution of competition lowered the critical rate of environmental change andthus caused extinction in a changing environment (in contrast to our model of cooperation). This isparticularly interesting given this model was originally introduced as a model of evolutionary suicidein a constant environment. We expect that alternative functional forms of the fitness components wouldallow evolutionary suicide in changing environments as well, but leave this as a conjecture.3.5 DiscussionIn this manuscript, we have studied the role that the evolution of cooperation may play during adaptationto environmental change. By modeling cooperation as an ecological public goods game (Gokhale andHauert 2016; Hauert et al. 2006a, 2008; Parvinen 2010; Wakano et al. 2009), we were able to connect thedots between changes in environmental conditions and the evolution of social behavior. Our predictionwas that changes in the environment would promote the evolution of cooperation, which would com-pensate for decreases in population size and permit populations to keep up with moving environments.When the population is polymorphic (cooperators and defectors), we indeed observed these dynamics(see section B.3.1 in the Supplemental Information). However, when the population is monomorphicfor an intermediate level of cooperation, we found out that widely different outcomes are possible, de-pending on how the different factors that affect fitness (costs of cooperation, environmental mismatch,and density regulation) are partitioned between viability and fecundity.We contrasted a fitness partition where the costs of cooperation affect fecundity (D: Figs. 3.3A and3.4A), to one where they affect viability (CD: Figs. 3.3B and 3.4B). In the former case the evolution ofcooperation promoted rescue, while in the latter it led to evolutionary suicide. This seemingly paradox-ical result arose because, in the latter case, while cooperation increased census population sizes, it also510.0 0.5 1.0 1.5 2.0 2.5 3.001234Competition (x )Lag(ℒ)0.0.51.1.52.Competition(x)Evolving xStatic x012Lag(ℒ)0 2 4 6 8 10 120102030Time (millions)Populationsize(n)Figure 3.6: Evolution of competition with a moving optimum, for a species with viability equal to one anda fecundity that depends on competition (x) and lag (L ). The stream plot to the left indicates the change ina population’s lag and competition over time, for a population where competition evolves fast relative to theenvironmental change. The nullclines for lag (L˙ = 0) and competition (x˙ = 0) are in black and light grey,respectively. Within the dark grey region, the population goes extinct. The thick blue curves are the result ofnumerical simulations, starting with zero lag and with x at equilibrium. For comparison, the thick pink curves arethe result of numerical simulations, with identical starting conditions, where competition does not evolve (µx = 0).The panels on the right show the time-dynamics of competition, lag, and population size for the same simulations.Parameters: λ = 1000, a= 0.1, c= 1, s= 1, k = 2, µx = µy = 10−4, σx = σy = 0.01, v= 0.5×10−6. For moredetails on the model, and for parameter interpretation, see section B.5.increased variance in reproductive success, thus decreasing the effective population size (Hedrick 2005;Kimura and Crow 1963) and slowing down adaptation by increasing the effect of genetic drift. The evo-lution of a population toward extinction—evolutionary suicide (Ferrie`re 2000; Gyllenberg and Parvinen2001)—has often been discussed in the context of evolution favoring selfish individuals (Gyllenberg andParvinen 2001; Gyllenberg et al. 2002; Rankin et al. 2007). However, here we have shown the oppositeeffect, where the evolution of increased levels of cooperation underlies evolutionary suicide.Other models have previously shown that the fitness component being affected by cooperation (vi-ability or fecundity) has important consequences for the evolution of altruism. In spatially structuredpopulations, the “scaled relatedness coefficient” (i.e., a cost-to-benefit ratio of cooperation that incor-porates both local competition effects and effects of population structure, as in Lehmann and Rousset(2010) and Van Cleve and Lehmann (2013)) depends on life cycle properties, so that in certain models,the evolution of cooperation is contingent on whether the benefits and costs of sociality affect fecundity(“death–birth updating”) or survival (“birth–death” updating), cf. De´barre et al. (2014b), Grafen (2007),Ohtsuki and Nowak (2006), and Taylor et al. (2007).52Although we focused on models with non-overlapping generations, there are other possibilities. Atthe opposite end of the life-history spectrum, we may consider another extreme whereby generationsoverlap and in which individuals reproduce asynchronously by producing a single offspring at a time(as may be the case with microbes that reproduce by binary fission or by budding). Such a life-historyresembles a continuous-time birth–death process, and is commonly modelled using the standard canon-ical equation of adaptive dynamics (Dieckmann and Law 1996). A species with this life-history hasa rate of evolution that is proportional to n/2 (Dieckmann and Law 1996), rather than n/σ2W , whichis always increased by cooperation (up to xcrit, which is the maximum value of cooperation that willevolve). Thus, such a species will qualitatively behave like a species with fitness partition D, meaningthat the evolution of cooperation would favor persistence.There is, of course, a wide range of possible life-histories between synchronous semelparity andasynchronous budding. Species may be iteroparous, having multiple broods throughout their lives.Furthermore, in iteroparous species reproduction may be either synchronous (as with many perennialplants) or asynchronous (as with most mammals). The canonical equation of adaptive dynamics foriteroparous diploids and haplodiploids (Metz and Kovel 2013) is similar to Eq. 3.13 in that it includesthe term 1/σ2W , meaning that qualitatively, CD-type effects may be at play in such species (i.e., theevolution of cooperation may lead to evolutionary suicide in changing environments). The same is truefor physiologically (e.g., age or stage) structured populations (Durinx et al. 2008).Many of the analytical results discussed so far (e.g., Eqs. 3.7–3.11, 3.16–3.18) depend on the specificfunctional forms we chose in section 3.2.1, and in particular on the cost and benefit functions. However,our qualitative result—namely, that the evolution of cooperation as a response to environmental changecan either promote or hurt persistence, depending on the specific fitness partition—can be observed for avariety functional forms. The benefit function adopted in Eq. 3.5 assumes that the benefits of cooperationincrease linearly with the level of cooperative investment, x′+(g−1)x. This assumption of linearity isinherited from the continuous-strategy form of the traditional multiplayer prisoner’s dilemma. However,nonlinearity is mathematically more general, and empirically, nonlinear public goods are more commonthat linear public goods in biological systems (Archetti and Scheuring 2012). Two types of nonlinearbenefits that are particularly relevant from a biological point of view are discounted and synergisticbenefits (examples of which have been studied by Cornforth et al. 2012; Hauert et al. 2006b; Motro1991, and, in the context of ecological public goods games, by Gokhale and Hauert 2016). Discountedbenefits grow sub-linearly with investment (i.e., they are a concave function of cooperation), whereassynergistic benefits grow super-linearly (convex function). In section B.4.1, we provide examples ofnumerical simulations showing that our main results are robust to at least small deviations from linearbenefits.The same concern exists for the cost function. In this manuscript, we chose a cost function thatchanges non-linearly with cooperative investment x (Eq. 3.3), but linear costs (C(x) = 1− cx) are alsopossible. (We use the word “linear” in slight abuse of terminology; more precisely, we considereda piecewise linear function, C(x) = max(1− cx,0), to avoid nonsensical negative values.) Numericalsimulations show that our main results can also be obtained for linear costs (see section B.4.2). Note,53however, that this alternative functional form renders the evolutionary branching scenario impossible (atleast when benefits are also linear).Traditional models of public goods games assume that the cost of cooperation is additive (i.e., sub-tracted from fitness) rather than multiplicative. The choice of a multiplicative cost function in our modelhas a number of advantages: it constrains fitness to biologically meaningful (positive) values and it fa-cilitates comparison with continuous-time models. Ignoring, for simplicity, the fitness components thatdo not relate to cooperation, Wrightian fitness equals W (x′|x) = B(x′|x,n)C(x′). In continuous time wecan define m(·) = log(W (·)) as the Malthusian fitness (Crow and Kimura 1970; Wu et al. 2013); we thenhave m(x′|x) = log(B(x′|x,n))+ log(C(x′)). In particular, for the functional form in Eq. 3.3, we havelog(C(x′)) = −cx′k, which is comparable to previous implementations of the ecological public goodsgame (Hauert et al. 2006a), where costs are subtracted from benefits. Regardless, our main results canstill be obtained using additive costs (section B.4.2), provided that one is careful to avoid biologicallyimpossible fitness partitions, at which fecundity or viability turn negative.The evolution of cooperation is critically dependent on the size of the interaction groups. In ourmodel, cooperation is more beneficial when groups are small. The average group size is given by theproduct of the population size and the interaction probability p. Note that in many of our numericalsimulations (including some in Figs. 3.1–3.3), group sizes are often small, which allows for the main-tenance of cooperation. In some of these cases, the most frequent “group” consists solely of the focalindividual (which can hardly be described as cooperation). In this sense, our results do not depend onthe presence of cooperation, only on the evolution of trait x (see section 3.4). However, even in thosesimulations, cooperation in the sense of public good sharing still takes place, because there is variationin group size. Further, section 3.3.3 explains how our results are expected to hold regardless of thespecific parameter values chosen.Previous models (Bra¨nnstro¨m et al. 2011; Pen˜a 2012; Pen˜a and No¨ldeke 2016) have shown thatvariation in group size influences the evolution of cooperation. In our model the benefits of cooperation(Eq 3.6) are calculated using the entire distribution of group sizes and are therefore influenced by itsvariance (np(1− p)). We follow the approach of previous models of ecological public goods games(Gokhale and Hauert 2016; Hauert et al. 2006a, 2008; Parvinen 2010) in assuming that individualsengage in enough interactions that fitness depends only on the mean benefit accrued across all groupsizes. In the future it would be interesting to relax this assumption and study how the outcomes ofecological public goods games change when there are fewer interactions per generation. In such casesthe benefits of cooperation would be stochastic and perhaps then depend more strongly on the variancein group size.Overall, our results highlight that species engaging in cooperative interactions may exhibit disparateand counter-intuitive responses to environmental change. For example, mammals and birds engagingin public goods provisioning (although they are most often iteroparous) have a relatively low numberof offspring, and low reproductive variance. In contrast, many social insects, which engage extensivelyin cooperative interactions, have non-overlapping generations and very high numbers of offspring, im-plying the possibility of high reproductive variance. Whether reproductive variance indeed increases54with fecundity depends on the distribution of offspring number. In this model, we assumed that it does(by modeling fecundity as a Poisson distribution), which had a major influence in our results, but otherchoices could lead to different outcomes. Overall, whether the evolutionary dynamics of social behaviorwill ultimately be beneficial or detrimental during adaptation to environmental change is contingent onthe species’ specific life history.In the future, it would be interesting to extend the model to include other biologically realistic pos-sibilities, such as environmental noise and plasticity, as well as alternative features such as diploidy andsexual reproduction. Furthermore, it would be interesting to consider whether these results apply alsoto other mechanisms of cooperation. For example, if cooperation is modelled through kin selection,declines in population size can increase or decrease the degree of relationship between individuals (de-pending on life cycle and spatial structure, cf. Table 1 in Van Cleve and Lehmann 2013). This providesthe opportunity for dynamics very similar to the ones we have described. In contrast, other mechanismsof cooperation may affect adaptation in quite different ways. Particularly, in the case of between-speciescooperation (mutualism), environmental change would set up coevolutionary feedback loops (Northfieldand Ives 2013) that are not well described by our framework.55Chapter 4On the importance of evolving phenotype distribu-tions on evolutionary diversification4.1 IntroductionNatural selection can act on multiple traits at the same time. When this is the case, the population’sresponse to selection depends not only on the strength and direction of selection but also on the shape ofits phenotypic distribution (Lande 1979). But selection also changes the shape of this distribution, whichin turn may affect future dynamics. This creates the potential for a distribution–selection feedback thatcould render the outcome of evolutionary processes contingent on initial conditions and more generallyon past evolutionary dynamics. However, mathematical models often make simplifying assumptions,such as monomorphic populations, that erase the influence of the population distribution. In this article,we explore one such situation that arises in the context of evolutionary branching, the process by whichadaptive evolution splits a population into multiple strains (Doebeli 2011; Geritz et al. 1998; Metz et al.1992). Most analytical models of evolutionary branching disregard the phenotypic distribution; how-ever, we show that, in multidimensional phenotype spaces, the direction of branching can be influencedby the population’s phenotypic distribution, which in turn is affected by the direction of approach to thebranching point.Branching occurs when frequency-dependent selection, resulting from interactions between indi-viduals of the same species (e.g., competition or cooperation) or of different species (e.g., predation orparasitism), leads to phenotypic divergence (Doebeli and Dieckmann 2000). This is possible becausefrequency-dependent selection causes the fitness landscape experienced by a population to change as thepopulation’s trait value evolves. Hence, it is possible for natural selection to drive a population towarda fitness minimum (called an evolutionary branching point). If the population reproduces asexually, itthen splits into distinct phenotypic strains which diverge from one another.When the phenotype space is unidimensional, the outcome of evolutionary branching is often en-tirely determined by the shape of the fitness landscape at the branching point, regardless of the state ofthe population when it arrives at this point (but see De´barre and Otto 2016; Wakano and Iwasa 2013,who have shown that population size can also affect the outcome of branching: small populations are56unable to undergo branching due to demographic noise). When the phenotype space is multidimen-sional, however, fitness-based criteria are often not sufficient to predict long-term dynamics. Geneticcorrelations between traits, imposed by genes acting pleiotropically or by linkage disequilibrium, affectthe multidimensional stability conditions (Leimar 2009; Matessi and Pasquale 1996). Epistatic interac-tions between traits tend to destabilize equilibria, so that branching becomes more prevalent at higherdimensions (De´barre et al. 2014a; Doebeli and Ispolatov 2017). Finally, branching may occur before thepopulation reaches the equilibrium, in directions that are orthogonal to the selection gradient, when dis-ruptive selection overwhelms directional selection (Doebeli and Ispolatov 2017; Ispolatov et al. 2016;Ito and Dieckmann 2014, 2012).Here we show that the shape of the phenotypic distribution also plays a role in determining the direc-tion of branching. The shape of this distribution can be characterized by the genetic variance-covariancematrix (the G-matrix), which plays a central role in quantitative genetics theory, since it influences a pop-ulation’s ability to respond to selection (Lande 1979). However, the G-matrix is not necessarily stablein time, since the phenotypic distribution itself changes in response to selection (Arnold 1992) in waysthat are not yet fully understood (Arnold et al. 2008; Steppan et al. 2002). A series of simulation studies(e.g., Guillaume and Whitlock (2007), Jones et al. (2012), and Jones et al. (2003, 2004, 2007)) haverevealed that the G-matrix may be stable under some regimes (including correlational and stabilizingselection, moving optima, low migration, and correlated mutational effects), but this is not always thecase. Because the direction in phenotype space from which a population approaches the branching pointcan affect its phenotypic distribution, it can also affect the outcome of evolutionary branching. In par-ticular, we show that, in asexually reproducing populations, branching tends to occur, ceteris paribus,along a direction that is perpendicular to the direction of approach to the branching point.We investigate this phenomenon using, as a case study, the process of cooperative-based branch-ing. Cooperative interactions are those in which individuals incur some cost to provide benefits thataccrue equally to both interaction partners. They describe, for example, the production of common re-sources (public goods), such as invertase in yeast (Gore et al. 2009) or antibiotic resistance (indole) inEscherichia coli (Lee et al. 2010). Pairwise cooperative interactions have been modeled as an evolv-ing quantitative phenotypic trait by Doebeli et al. (2004), in a model called continuous snowdrift game(CSG). In this model, individuals pay a cost that increases with the amount of cooperative investmentand receive a benefit that depends on the total amount of public good produced in their interaction pair.Depending on the shape of the cost and benefit functions, Doebeli et al. (2004) showed that evolutionarybranching may occur: the initially monomorphic population evolves to some intermediary investmentlevel and subsequently splits into a high-investment “cooperator” strain and a low-investment “defector”strain.While the CSG provides one possible explanation for the evolutionary origin of cooperators anddefectors, it focuses on the production of a single public good. However, real organisms are engagedsimultaneously in several different interactions and tasks (Rueffler et al. 2012) each of which could bemodeled as a public goods game (Hashimoto 2006; Venkateswaran and Gokhale 2019). Because weare interested in branching in a multidimensional phenotype space, here we consider a simple extension57of the CSG for multiple dimensions: i.e., individuals play multiple continuous snowdrift games simul-taneously, and their fitness is the sum of the payoffs from each game. In such a case, the branchingpoint is a minimum of the fitness landscape in all directions, and hence the direction of branching isnot predictable. Despite its unpredictability, the branching direction could have important biologicalconsequences. For example, consider that the population is engaged in two simultaneous games andevolutionary branching occurs in both traits. One possible outcome is that the population may split intoa strain that performs both tasks, and another that performs none. Alternatively, diversification may re-sult in complementary strains, each of which produces only one of the public goods. This would resultin division of labor, with individuals of different strains relying on one another to share the products inwhich they specialize. Other outcomes, such as more than two co-existing strains, are also possible.In this manuscript, we show that the initial phenotype of the population (before it approaches thebranching point) influences the direction of evolutionary branching. We argue that during its approachto the branching point, the population accumulates phenotypic variance in the direction that is orthog-onal to movement (similar to the phenomenon described by Gomez et al. 2019). When the populationreaches the branching point, its phenotypic distribution, which is determined by past selection, in turndetermines the direction of evolutionary branching. We also show that the direction of evolutionarybranching has important long-term consequences, because it influences the stable state the populationwill eventually reach. In the CSG case, this can determine whether the equilibrium population con-sists of two specialist strains or of a cooperator and a defector strain. Analytical methods that ignorethe population’s phenotypic distribution are hence ill-prepared to predict the direction and outcome ofevolutionary branching.4.2 Methods4.2.1 Model descriptionOur model is a simple extension of the one-dimensional continuous snowdrift game (CSG) to multipledimensions. Asexual individuals live in a well-mixed population and their fitness is determined by thepayoffs of pairwise interactions.One-dimensional CSG In the original model (Doebeli et al. 2004), each individual is characterizedby a phenotype 0 < x < xmax, which determines its investment into a costly good that the individualproduces during an interaction. This good is shared by both interacting partners, who receive the sameamount of benefit, regardless of their contribution. Upon interacting with a partner with phenotypex, a focal individual with phenotype x′ receives a payoff P(x′,x) = B(x′+ x)−C(x′), where C(x) andB(x) determine, respectively, the cost of public good production and the benefit associated with itsconsumption. Both functions are assumed to be smooth and strictly increasing, satisfying B(0)=C(0)=0.58Multi-dimensional CSG We expand the original game so that individuals are engaged in the productionof more than one public good. Thus, instead of a single scalar phenotype, individuals are characterizedby a k-dimensional vector z = 〈z1, · · · ,zk〉, whose elements quantify the investment into each of the kcostly goods. Each good i ∈ [1,k] has its own cost, benefit, and payoff functions, Ci(zi), Bi(zi), andPi(z′i,zi). The payoffs associated with each public good are entirely independent, so that a focal individ-ual with phenotype z′ interacting with a partner with phenotype z earns a payoff P(z′,z) =∑ki=1 P(z′i,zi).As such, producing multiple goods incurs no trade-offs whatsoever; similarly, there is no synergism ordiscounting between each good’s benefits.Simplifying assumptions In most of the manuscript we will focus on the simplest possible version ofthis model, i.e., we consider only two public goods (with z = 〈z1,z2〉). Because we are interested instudying the effect of the initial phenotype on the branching process, we chose functions that ensurethat the fitness landscape is always perfectly symmetric, thus removing any selective effects that mayaffect the branching direction. Therefore, we assume that the public goods have the same exact effecton fitness (B1(x) = B2(x) ≡ B(x) and C1(x) =C2(x) ≡C(x)), so that they are entirely interchangeable.We relax both assumptions (k = 2 and equal fitness functions) later on. Following Doebeli et al. (2004),we use quadratic cost and benefit functions, Ci(x) = ci(x−dix2) and Bi(x) = ai(x−bix2) (and we dropthe index i ∈ {1,2} whenever it causes no confusion). Cost and benefit must be strictly increasing, sowe restrict these quadratic functions to values below their maxima. In particular, zi must be smaller thanthe maximum of B(2zi) and C(zi), i.e., (zi)max = min(1/(4b),1/(2d)).4.2.2 Adaptive dynamicsWe use adaptive dynamics (Dieckmann and Law 1996; Doebeli 2011; Geritz et al. 1998) to analyzethe evolution of the investment strategies before branching and to determine conditions for branching.Given a monomorphic population, with resident phenotype z, the growth rate of a rare mutant strategyz′ is given by the invasion fitness (Metz et al. 1992), w(z′,z) = P(z′,z)−P(z,z). Then, the direction ofevolution in phenotype space is given by the selection gradient, D(z), a vector whose ith entry is givenby Di(z) = ∂w(z′,z)/∂ z′i|z′=z. The selection gradient points from the resident toward the steepest uphilldirection of the fitness landscape, so that the time-dynamics of z is described by (Dieckmann and Law1996)dzdt= m(z)A(z)D(z), (4.1)where m(z) is the rate of occurrence of new mutations and A(z) =(σ2 00 σ2)is the mutational variance-covariance matrix (assuming that mutational effect size is drawn from an uncorrelated bivariate normaldistribution with very small standard deviation σ ). At evolutionary equilibrium z?, the selection gradientvanishes, D(z?) = 0 and directional evolution comes to a halt.Because the payoff of the joint games is simply the sum of the individual games, each element ofthe selection gradient vector is equal to the selection gradient of the corresponding one-dimensionalCSG, Di(z) = D(zi) = B′(2zi)−C′(zi). Accordingly, when the cost and benefit functions are equal for59all games, the approach to the equilibrium proceeds along a straight line. The equilibrium for each traitequals the equilibrium of the one-dimensional CSG, derived in Doebeli et al. (2004). With our choiceof quadratic cost and benefit function (section 4.2.1), there is a single evolutionary equilibrium for eachpublic good, at z?i =a−c4ab−2cd .Generally speaking, the full stability analysis for a multi-dimensional model such as ours requirescalculating and analysing the Jacobian matrix (convergence stability requires that the real parts of alleigenvalues are negative) and the Hessian matrix (evolutionary stability requires that the real parts ofall eigenvalues are negative), see Geritz et al. (2016). However, because in our model the total payoffis simply given by the sum of the payoffs associated with each public good, both the Jacobian andthe Hessian are diagonal matrices, with the nonzero elements being the quantities required to calculatestability along each individual game. Hence, the equilibrium is an attractor of the evolutionary dynamics(i.e., convergent stable) along dimension i provided that dD/dzi|zi=z?i = 2B′′(2z?i )−C′′(z?i )< 0. If this isthe case, then the population converges to the equilibrium, but its subsequent evolutionary fate dependson whether z? is a maximum or a minimum of the fitness landscape along each dimension i. If itis a maximum, ∂ 2/(∂ z′i)2w(z′i,z?i )∣∣z′i=z?i= B′′(2z?i )−C′′(z?i ) < 0, then the equilibrium is evolutionarilystable (meaning that it is an endpoint of evolution). If it is a minimum, then branching occurs. Theseconditions are identical to those given in Doebeli et al. (2004). Since we are interested in studyingmultidimensional evolutionary branching, we chose parameters for which the equilibrium is convergentstable but evolutionarily unstable, which is the case whenever cd/2b < a < cd/b; in all figures, we usethe values a = 1, b = 1.05, c = 0.9, and d = 1.65. Then, evolutionary branching causes the populationto divide into multiple clusters, which evolve toward the borders of the phenotype space. Becausethe phenotype space is two-dimensional, no more than three such clusters may result from the initialbranching event (Durinx et al. 2008), but subsequent branching events may occur.Once branching has occurred and the resident population has been replaced by multiple residentbranches, it is possible to use adaptive dynamics to study the fate of those branches; however, usingadaptive dynamics alone, it is not possible to predict the direction of branching (although we can make aprediction using the concept of the evolutionary branching line, Ito and Dieckmann 2012; see Discussionand Appendix C.4). In fact, since the fitness landscape is entirely symmetric about the evolutionarybranching point, it follows that at the evolutionary branching point all equidistant mutants have thesame fitness (Fig. 4.1). In other words, adaptive dynamics does not suggest any particular direction ofbranching.To understand the most likely direction of branching, we turned to computer simulations.4.2.3 SimulationsIn order to study the branching process, we perform three different kinds of simulations (explainedbelow), of which two are stochastic (individual-based simulation and oligomorphic stochastic simula-tion) and one is deterministic (partial differential equation numerical simulation), and we calculate thedirection of branching in these simulations. The reason for choosing these three methods is that theindividual-based and the partial differential simulations explicitly track the population’s phenotypic dis-60Figure 4.1: Fitness landscape during approach to the branching point (A) and at the branching point (B). A: Whenthe resident (orange dot) is away from the branching point (tip of arrowhead), it moves up the fitness landscape.B: When the resident (orange dot) is at the branching point, the fitness landscape in our model is symmetric aboutthe resident: all mutants equidistant to the resident have equal invasion fitness. Hence, invasion fitness alone doesnot predict any particular direction of branching.tribution, whereas the oligomorphic simulation ignores this information. For the stochastic simulations,we focus on the most frequent outcome where the population divides into two branches and discard therare replicates from our analyses that contained more than two branches at the end of the initial branch-ing event (this is the reason why some of the figures do not have a round number of replicates). Two-waybranching is the most common mode of adaptive diversification (Vukics et al. 2003), and simultaneousbranching into more than three clusters is not possible in a two-dimensional phenotype space (Dur-inx et al. 2008). However, subsequent branching events sometimes occur in our model after the initialseparation into two strains, leading to transient cases with three or even four coexisting branches, thatresolve back into dimorphisms after local extinction. The number of branches was determined using adensity-based spatial clustering algorithm (Ester et al. 1996).Individual-based simulations (IBS)Individual-based simulations, or IBS, are the most direct implementation of the birth–death process thatunderlies the evolutionary process. The differential equations from adaptive dynamics, as well as theother types of simulations we discuss in this manuscript, can therefore be seen as approximations oridealizations of this individual-based process. We simulated a population with a fixed size n, followingan algorithm adapted from Doebeli et al. (2004), in which interactions are modelled explicitly. Whenindividuals reproduce, their offspring’s trait values are drawn from a normal distribution centered at theparent’s value, with a small standard deviation σ . For more details about the algorithm and parameters,see Appendix C.1. The interested reader can also run individual-based simulations in a web applicationavailable at: https://tinyurl.com/TwoGamesSimulation (Hauert 2020).61Oligomorphic stochastic simulations (OSS)In contrast to the IBS, which exactly implements the birth–death process, the oligomorphic stochas-tic simulation (OSS) approximates this process under the assumption that the timescales of populationdynamics and evolutionary dynamics can be separated (Ito and Dieckmann 2007). The underlyingstochastic process (of which adaptive dynamics is a deterministic approximation) is a directed randomwalk called a “trait-substitution sequence” (Metz et al. 1992, 1996). Mutations are rare, so at any givenmoment of this random walk, the population consists of a single (or a few) monomorphic phenotypes atpopulation dynamical equilibrium (with constant total population size, n). Thus (in contrast to the IBS,where each strain may consist of a cloud of points in phenotype space) each strain consists of a singlepoint. A step of the random walk consists of the introduction (with some probability distribution) ofsuccessful mutant phenotypes that replace (or, in the case of evolutionary branching, coexist with) theirancestors. The probability of mutation at birth is µ and the mutational standard deviation is σ . Thus, theOSS is an extension of the monomorphic stochastic model (Dieckmann and Law 1996; Dieckmann et al.1995) that allows for evolutionary branching (Ito and Dieckmann 2007). For parameters and detailsabout the algorithm, adapted from Ito and Dieckmann (2007), see Appendix C.2.Numerical simulation of partial differential equations (PDE)While adaptive dynamics provides a deterministic description of the evolving populations, it approxi-mates each strain as a single monomorphic phenotype. Alternatively, we can describe the dynamics ofthe entire population distribution in phenotype space (in which case branching is the formation of mul-tiple modes in this distribution). This results in a deterministic system of partial differential equationsthat corresponds to a large-population limit of the individual-based model (Champagnat et al. 2008),often referred to as a PDE model. In order to numerically solve this system of equations, we discretizethe phenotype distribution, resulting in a system of coupled ordinary differential equations. Each com-bination of phenotype values, 〈z1,z2〉, is referred to as a phenotype class. The population density of eachclass changes over time deterministically, with the rate of change depending both on natural selectionand on the flow of mutations in and out of the class. Thus, we can track the entire population distributionover time. For details about the algorithm and parameters, see Appendix C.3.4.2.4 G-matrix orientation and direction of branchingIn the following sections, we summarize our results by focusing on three angles: the direction of ap-proach, the G-matrix orientation (in the IBS), and the direction of branching.Direction of approach. The direction of approach is the angle between the z1 axis and the line definedby the initial position and the branching point.G-matrix orientation. In the IBS, the population’s phenotypic distribution can be summarized by theG-matrix, i.e., the variance-covariance matrix of the population’s phenotype (or more rigorously, but620.060.080.100.06 0.08 0.10z1z 2A llllllllllllllllllllllllll lllllllllllllllllllllllllll0.060.080.100.06 0.08 0.10z1z 2BFigure 4.2: Examples of stochastic replicates of the individual-based (A) and oligomorphic (B) simulations. Ateach time point in the IBS, the population consists of many individuals forming a cloud in phenotype space,whereas in the OSS, the population consists of a single or a few monomorphic strains. The population starts atsome initial position and approaches the branching point before branching into two separate strains. The directionof approach (i.e., the angle between the z1 axis and the line defined by the initial position and the branching point)is −3pi/4 in A, and pi/4 in B. In A, dark red points indicate the onset of branching. In A and B, yellow pointsindicate the point at which we determine branching to be complete (and the angle between the yellow line and thez1 axis is the branching direction). Parameters: n= 10,000; σ = 2.5×10−5 (IBS) or 1.5×10−3 (OSS); µ = 10−2;for more parameters and details see Appendices A and B.in our case identically, its additive genetic component). The phenotypic distribution will be longestalong the direction given by the leading eigenvector of the G-matrix. This direction has been called the“genetic line of least resistance” (Schluter 1996) or the “G-matrix orientation” (Jones et al. 2003), and(for the two-dimensional case) we report it as the angle, in radians, between the eigenvector and thepositive direction of the z1 axis. For example, when the longest extent of trait variance occurs along thez2 direction, the population will have an orientation of pi/2; a population with strong positive covariancewill have an orientation of pi/4; and strong negative covariance corresponds to 3pi/4. Finally, when thelongest extent of variance occurs along the z1 direction, the population’s orientation is 0 (note that theG-matrix orientation is a periodic quantity, i.e., values close to pi are also close to 0).Direction of branching in stochastic simulations. In all stochastic simulations, we use the same cal-culation that we used to calculate the G-matrix orientation to determine the direction of branching. Wedifferentiate the onset of branching from the time point at which we consider branching to be complete.At the onset of branching, the population is often still monomorphic; the process is complete when thepopulation consists of multiple, well-differentiated strains moving in different directions (for examples,see Fig. 4.2). In our simulations, the onset of branching can be measured as the time point at which thevariance in fitness is at its lowest (dark red points in Fig. 4.2A). (When measuring the onset of branching,we exclude the first generation, in which the population has not yet accumulated variation by mutation.Note that this is not meant as a definition but simply as an operational criterion). In the IBS, we considerbranching to be complete when the population’s bimodality (measured by Hartigan’s dip statistic, HDS,63Hartigan and Hartigan 1985) is sufficiently high (HDS > 0.15); in the OSS, we consider branching to becomplete when the population remains polymorphic for ten consecutive successful mutant invasions. Itis at this time point that we measure the direction of branching (yellow points and line in Fig. 4.2).Direction of branching in deterministic simulations. In the deterministic simulations, branching wasconsidered to be complete whenever any of the population’s marginal distributions (i.e., the phenotypicdistribution along each of the coordinates) turned bimodal. This was considered to have occurred when-ever the local maxima of the marginal distribution were at least ten times larger than the minimum inbetween them. The direction of branching was calculated as the angle between the z1 axis and the linepassing through the local maxima of each branch.4.3 Results4.3.1 Perpendicular branchingSince the two public goods are interchangeable, it follows that when the resident population is at thebranching point the fitness landscape is perfectly symmetric (Fig. 4.1). As discussed in section 4.2.2,this implies that, following the framework of adaptive dynamics, all directions of branching should beequally likely. However, results from individual-based simulations (section 4.2.3) show that this is notthe case. Instead, branching tends to occur along the direction that is orthogonal to the direction ofapproach (Fig. 4.3A).Why is the distribution of branching directions not uniform? One possibility is that, by reducing thepopulation to a single, monomorphic point in phenotype space, adaptive dynamics discards informationthat is necessary for predicting the most likely direction of branching. Real populations consist of manyindividuals with slightly different phenotypes, and it is possible that the population’s distribution inphenotype space depends on the direction of approach to the branching point. The strength of naturalselection increases with variance in fitness (Fisher 1930). Hence, at the branching point, disruptiveselection will tend to be strongest along the direction of highest variance (i.e., the G-matrix orientation),and, all else being equal, we would expect branching to occur along that direction.To investigate this hypothesis, we implemented two alternative simulation methods: (i) an oligo-morphic stochastic simulation (section 4.2.3), in which each strain is monomorphic; and (ii) a partialdifferential equation model (section 4.2.3), in which the entire phenotypic distribution is deterministi-cally simulated.Fig. 4.3B shows that in the oligomorphic stochastic simulation (which simulates a trait-substitutionsequence, section 4.2.3), all branching directions are equally probable. This conclusion correspondsto the prediction of adaptive dynamics. In contrast to the OSS results, when we implement a partialdifferential equation (PDE) numerical simulation (which deterministically tracks the entire phenotypedistribution, section 4.2.3), it shows the population branching in a direction that is perpendicular to thedirection of approach (e.g., Fig. 4.4), similar to the IBS.The results from the PDE and OSS simulations are in agreement in our hypothesis that the popula-64−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi0.000.010.020.030.04FrequencyBranching direction with IBSA−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi0.000.010.020.030.04FrequencyBranching direction with OSSBFigure 4.3: In IBS, branching tends to occur perpendicularly to the direction of approach (A), but in OSS, alldirections of branching are equally likely (B). Each dark histogram shows the direction of branching for 600 (inA) or 8,500 (in B) replicates. The light histogram is a mirror copy of the same data and is included for ease ofvisual interpretation. The arrows show the direction of approach to the branching point (from left to right: pi/4,pi/2, 3pi/4, pi , and −3pi/4). Parameters: n = 10,000; σ = 2.5× 10−5 (IBS) or 1.5× 10−3 (OSS); µ = 10−2; formore parameters and details see Appendices A and B.tion’s phenotypic distribution plays a role in determining the direction of branching. To further investi-gate this effect, we now investigate how this distribution changes during the approach to the branchingpoint and its relation to the branching direction.4.3.2 Changes in the G-matrix orientation underlie the direction of branchingOur hypothesis is that during the approach to the evolutionary branching point, the population’s pheno-typic distribution changes. The low-fitness individuals at the tail end of the traveling wave are constantlyeliminated by natural selection. Simultaneously, because individuals on any given isocline of the fitnesslandscape are not under selection relative to each other, drift spreads the population outwards along theisoclines’ direction. Hence, we hypothesize that as the population approaches the branching point, itsG-matrix becomes oriented perpendicularly to the direction of approach, which is consistent with thePDE observations (Fig. 4.4) and with previous theoretical work on directional selection (Gomez et al.2019).This hypothesis is in line with the long-standing quantitative genetics result according to which, ifwe approximate the population’s distribution as bivariate normal, the G-matrix after selection is givenby G∗ = G(γ −DDT)G (Arnold 1992; Lande and Arnold 1983; Phillips and Arnold 1989). Here, γis a matrix describing the curvature and orientation of the adaptive landscape, γi, j = ∂ 2w/(∂ zi∂ z j), andD is again the vector of selection gradients (Eq. 4.1; this is often called β in the quantitative geneticsliterature). Imagine what happens to a population whose phenotypic distribution is initially perfectlysymmetric, with some additive genetic variance V along both axes and no covariance (which matches650.050.080.100.120.15z 20.050.10z 20.000.030.060.090.12z 20.000.030.060.090.12z 20.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.120.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.120.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.12 0.00 0.03 0.06 0.09 0.120.05 0.10z10.05 0.10z10.05 0.10z10.05 0.10z1ABCDFigure 4.4: In the PDE model, the population branches perpendicularly to the direction of approach. Each rowshows four illustrative time points (with time moving left to right) of a PDE numerical simulation. The population(whose phenotypic distribution is shown by the contour lines) approaches the branching point (intersection of theblack lines) from a given direction (A: 3pi/4, B: pi , C: −3pi/4, D: −pi/2), and then branches. Colors indicateinvasion fitness (red: negative values; blue: positive values; white: zero). For more details and parameter valuessee Appendix C.3.the initial condition in our system, where the population begins as an uncorrelated bivariate normal).For simplicity, assume that the population lies along the identity line (z1 = z2), such that D1 =D2 = D(for the general case we could use a rotated coordinate system, similar to the one used in Appendix C.4.)Furthermore, γ11 = γ22 = Γ, and γ21 = γ12 = 0. Hence, one round of selection results in the followingvariance-covariance matrix:G∗ =[V +(Γ−2D2)V 2 −2D2V 2−2D2V 2 V +(Γ−2D2)V 2], (4.2)6602004006000 pi4pi23pi4piG matrix orientation (radians)GenerationsG−matrix orientation duringapproach to equilibriumA0pi4pi23pi4pi0 pi4pi23pi4piG matrix orientation (radians)100 generations prior to branching onsetBranching direction (radians)G−matrix orientation predictsbranching directionB−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4piρ = 0.5ρ = −0.50.0000.0250.0500.0750.1000.0000.0250.0500.0750.100FrequencyBranching direction depends ondistribution at branching pointCFigure 4.5: IBS results show that changes in the G-matrix orientation (A) underlie the direction of branching(B, C). Panels A and B correspond to 500 IBS replicates approaching the branching point from direction −3pi/4(i.e., from the bottom-left of the phenotype space). A: as the population approaches the branching point, thepopulation’s phenotypic distribution becomes elongated along the direction that is perpendicular to movement.(In this case, the G-matrix orientation distribution becomes peaked at 3pi/4, consistent with the second andthird panels in Fig. 4.4C.) B: Although there is high stochasticity, the G-matrix orientation prior to the onset ofbranching predicts the eventual direction of branching. The orange line and yellow confidence bands result froma linear regression modified to take into account the circular variable space. C: All else being equal, differentG-matrix orientations cause different directions of branching. Each dark histogram shows the branching directionof 600 IBS replicates initialized at the branching point, with individuals drawn from a bivariate normal distributionwith a positive (top) or negative (bottom) Pearson correlation coefficient (ρ). The light histograms are a mirrorcopy of the same data and are included for ease of visual interpretation. Parameters: n= 10,000; σ = 2.5×10−5;for more parameters and details see Appendix C.1.which has a negative covariance. Because the selection gradient has the same direction as the identityline, this buildup of negative covariance implies that the distribution becomes elongated with the axis ofminor variation matching the direction of the selection gradient (as in Fig. 4.4).To test this hypothesis, we measure the G-matrix orientation in the IBS during approach to thebranching point. We find that, consistent with our expectation, the population’s distribution (whichstarts out symmetrical) rapidly becomes perpendicular to the direction of approach (Fig. 4.5A).If this change in phenotypic distribution indeed explains the pattern we observe, then the G-matrix orientation prior to branching should be a predictor of the branching direction. We measurethe G-matrix orientation 100 generations prior to the onset of branching, and find that it predicts thedirection of branching (Fig. 4.5B).While these measurements show that the G-matrix orientation correlates with the direction ofbranching, they do not demonstrate a causal connection between the two quantities. To show that thereis a causal relationship between the G-matrix orientation and the direction of branching, we initiatedIBS at the branching point, with individuals drawn from bivariate normal distributions with a Pearsoncorrelation coefficient of either ρ = 0.5 (G-matrix orientation: pi/4) or ρ =−0.5 (G-matrix orientation:673pi/4). Consistent with our predictions, the most common direction of branching matches the directionof highest variance (Fig. 4.5C). Naturally, this follows from the fact that the population’s mean trait isat the branching point (which is a fitness minimum), so that the phenotypes further away have a higherfitness.4.3.3 Effect of the branching direction in more than two dimensionsThe result that branching occurs perpendicularly to the direction of approach holds also for IBS in higherdimensions, i.e., when agents are engaged in three simultaneous games. To quantify the direction ofbranching in the three-dimensional system, we consider two perpendicular planes, P and Q (Fig. 4.6A),whose intersection we call I. Plane Q includes the initial position 〈z01,z02,z03〉, the equilibrium, andan arbitrary point 〈1,2,3〉. In other words, plane Q includes the direction of approach, i.e., the linedefined by the initial point and the equilibrium point (black arrowhead in Fig. 4.6A). Plane P, in turn, isperpendicular to the direction of approach and it also includes the equilibrium. Then, let the projectionof L (the line defined by the mean phenotypes of both strains, represented in black in Fig. 4.6A) on P becalled LP and the projection of L on Q be called LQ. We characterize the direction of branching usingtwo angles. The first, ϕ , is the angle between I and LQ. The second, ψ , is the angle between I andLP. We find that ϕ tends to be close to 0 or pi , meaning that branching occurs perpendicularly to thedirection of approach. In contrast, all values of ψ are equally likely (Fig. 4.6B), which is in agreementwith our hypothesis, since all lines in P are perpendicular to the direction of approach.4.3.4 Effect of the direction of approach for unequal gamesWe have shown that, when the two games are interchangeable, the direction of approach determines themost probable direction of evolutionary branching. While interchangeable games are not inconceivable(e.g., as the result of gene duplications, Ohno 1970), an arguably more likely scenario is that the twopublic goods have different fitness effects. We can quantify this difference by an asymmetry parameterα , such that P(z′,z) = (1−α)P1(z′1,z1)+αP2(z′2,z2), i.e., the weight of the z2 payoff on fitness is α .When α = 0.5, the two games are identical; when α > 0.5, good z2 contributes more to fitness thangood z1; and when α < 0.5, the opposite is true.The adaptive dynamics expectation is that for any α > 0.5 (α < 0.5), evolutionary branching is mostlikely to happen along the z2 (z1) direction. OSS simulations confirm this expectation and, furthermore,show that the larger the inequality between the games, the less stochastic the direction of branchingbecomes (Fig. 4.7A). IBS simulations show that the preferred direction of evolutionary branching isstill influenced by the direction of approach when α 6= 0.5 (Fig. 4.7B). Similar to what we found forα = 0.5 (Fig. 4.5), the direction of branching is affected by the G-matrix orientation. Initially, thepopulation’s phenotypic distribution becomes elongated in a direction that is perpendicular to the popu-lation’s movement (Fig. 4.7C). Over time, as it approaches the branching point, natural selection causesthe phenotypic distribution to elongate in one of the two directions, an effect that is strongest for highervalues of α (Fig. 4.7C).The results of IBS simulations are in qualitative agreement with PDE results, which show that the68AB−3π4 −π2−π40π4π23π4π−3π4 −π2−π40π4π23π4πϕ ψ0.000.020.040.06FrequencyFigure 4.6: Branching in three dimensions also tends to be perpendicular to the direction of approach. We quan-tify direction of branching in three dimensions with two angles, ϕ andψ . A: cartoons illustrating the interpretationof these angles (see text for details). The black arrowheads indicate the direction of approach, such that ϕ belongsto the same plane as the direction of approach (plane Q) and ψ belongs to a plane perpendicular to the directionof approach (plane P). The points show the population’s evolution after branching (bright blue points are abovethe plane and dark blue points are below it). B: the dark histograms show ϕ and ψ for 597 replicates (the lighthistograms are a mirror copy of the same data and are included for ease of visual interpretation). In the left-sidehistogram, the arrow illustrates the direction of approach; in the right-side histogram, the direction of approach isperpendicular to the circular coordinate plane. Whereas ψ has a uniform distribution, the frequency of ϕ peaks at0 and pi , meaning that branching tends to occur along a plane that is perpendicular to the direction of approach.Parameters: n = 10,000; σ = 2.5×10−5; for more parameters and details see Appendix C.1.direction of approach plays a role in determining the direction of evolutionary branching even for highvalues of α (Fig. 4.7D). Example numerical simulations for various values of α clarify that the directionof approach, which resembles a straight line when α ≈ 0.5, becomes curved as α increases (Fig. 4.7E).The stochastic IBS results are close to the PDE predictions (dashed line in Fig. 4.7B).The results presented in Fig. 4.7 show that the direction of approach still has an effect on the branch-ing direction when the two games are the same but have different fitness effects (provided that thisdifference is not too large). In biologically realistic scenarios, such as the simultaneous production ofdifferent enzymes, the cost and benefit functions will not be equal for the two games. Fig. 4.8A showstwo example cost and benefit functions, which define the payoffs P1(z′,z) and P2(z′,z). These resultin different selection gradients and evolutionarily stable strategies (z?1 and z?2, indicated by the verticallines). Despite the different payoff functions, IBS results show that the direction of approach still hasan effect on the branching direction (Fig. 4.8B). This result is most pronounced when the selection gra-dients and the curvatures of the fitness landscapes at equilibrium are roughly similar between the twogames, which in our case means that a1b1 ≈ a2b2 and c1d1 ≈ c2d2. To illustrate this, we held game 1constant and drew random parameter values for game 2: a2 and c2 were uniformly distributed between0 and 5; b2 and d2 were normally distributed with a mean of a1b1/a2 and c1d1/c2 (respectively) anda standard deviation of 0.1 times the mean. We then discarded any non-valid parameter combinations69−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4piα = 0.51 α = 0.58 α = 0.650.000.020.040.06FrequencyOSS branching direction with asymmetric gamesA−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4piα = 0.51 α = 0.58 α = 0.650.000.050.10FrequencyIBS branching direction with asymmetric gamesBInitial condition3000 generations before onset2000 generations before onset1000 generations before onset0 pi4pi23pi4piG matrix orientation (radians)α 0.51 0.58 0.65       G−matrix orientation during approach       to equilibrium with asymmetric gamesCl l l l l l l l l l l l l lllllllllllll l l l l l l l l l l l l lpi22pi35pi6pi0.00 0.25 0.50 0.75 1.00Asymmetry between traits (α)Angle of branchingPDE branching direction with asymmetric gamesDα = 0.51 α = 0.58 α = 0.650.050.060.070.080.09 0.050.060.070.080.09 0.050.060.070.080.090.050.060.070.080.090.10z1z 2       Example PDE dynamics with asymmetric gamesEFigure 4.7: Even when games have different fitness effects, the direction of approach plays a role in determiningthe direction of branching. In all panels, the direction of approach to the branching point is −3pi/4 (i.e., fromthe bottom-left of the phenotype space; indicated with arrows in panels A and B) and α is the weight of gamez2 when calculating fitness. A: Each dark histogram shows the branching direction in OSS for 7,950 replicates(light histograms are a mirror copy of the same data and are included for ease of visual interpretation). Wheneverα > 0.5, branching tends to occur along the z2 direction (pi/2); the distribution of branching directions becomesmore peaked as α increases. B: Each dark histogram shows the branching direction for 600 IBS replicates (lighthistograms are a mirror copy of the same data and are included for ease of visual interpretation). As α increases,the most common direction of branching gradually changes from 3pi/4 (perpendicular to direction of approach)to pi/2. The dashed lines are the PDE predictions. C: Frequency distribution of G-matrix orientation throughtime, as the population approaches the branching point, for different values of α . D: Branching direction forPDE simulations, for increasing values of α , showing that the direction of approach has an effect on the branchingdirection even for relatively high values of asymmetry. E: Example PDE numerical simulations with three differentvalues of α . In each panel, contour lines show the population’s phenotypic distribution at different points in time.Parameters: n = 10,000; σ = 2.5× 10−5 (IBS) or 1.5× 10−3 (OSS); µ = 10−2; for more parameters and detailssee Appendices A, B and C.(i.e., combinations resulting in equilibria beyond the edges of parameter space, non–convergent stableequilibria, or evolutionarily stable equilibria). This procedure ensures that the two games are different,but have comparable selection gradients and fitness landscape curvatures around their equilibria (al-though the two equilibria may be very distant in phenotype space). The histograms in Fig. 4.8C showthe resulting branching directions (each replicate corresponds to a unique random parameter draw forthe cost and benefit functions of game 2).700.000.050.100.150.200.00 0.05 0.10 0.15 0.20Investment strategy (z)Benefit or costGame12FunctionBenefitCostCost and benefit for games 1 and 2A−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi0.00.10.2FrequencyBranching direction with games 1 and 2B−3pi4−pi2−pi40pi4pi23pi4pi−3pi4−pi2−pi40pi4pi23pi4pi0.000.050.10FrequencyBranching direction with game 1 and random gamesCFigure 4.8: Even with different payoff (cost and benefit) functions, the direction of approach plays a role indetermining the direction of branching. A: Example cost (dashed) and benefit (solid) functions for game 1 (black)and game 2 (grey); the respective equilibria are indicated by the vertical lines. B: Each dark histogram shows theIBS branching direction for 560 replicates using the payoff functions from panel A (light histograms are a mirrorcopy of the same data and are included for ease of visual interpretation). The arrows indicate the direction ofapproach. C: Each dark histogram shows the IBS branching direction for 879 replicates (light histograms are amirror copy of the same data and are included for ease of visual interpretation). For each replicate, game 1 isthe same as in panel A while game 2 is randomly parameterized (see text for details). The arrows indicate thedirection of approach. Parameters: n= 10,000; σ = 2.5×10−5; µ = 10−2; a1 = 1; b1 = 1.05; c1 = 0.9; d1 = 1.65.In panels A and B, a2 = 1.19; b2 = 1.1; c2 = 0.9; d2 = 2.14. For more parameters and details see Appendix C.1.4.3.5 Eventual fate of the populationAfter the initial branching event, the two strains diverge toward the edges of phenotype space.Individual-based simulations reveal that, during this period of divergence, subsequent branching events(resulting in more than two strains) are common but transient. Eventually, the population will reach astable state. If we divide the phenotype space into quadrants separated by the horizontal and verticallines that cross the equilibrium, in principle, we would expect the population of strains to occupy anycombination of these quadrants at the end of the simulation. Our simulation results reveal that onlytwo of these combinations were observed (out of 4,000 replicates). The first outcome, which we term“division of labor” (DOL), involves two complementary strains (each strain is a specialist that invests7102505007501000−3π4π23π4 πDirection of approachNumber of replicatesOutcome Cooperator & defector Division of laborFigure 4.9: The initial direction of approach biases evolution towards one of the two possible stable states:“cooperator and defector” (one strain that cooperates on both traits and one that defects on both) or “division oflabor” (a pair of complementary strains that cooperate on one trait and defect on the other). No other outcomeswere observed. Parameters: n = 10,000; σ = 2.5× 10−5; µ = 10−2. For more parameters and details seeAppendix C.1.only into one of the goods, and not into the other good). The second outcome, which we term “coop-erator and defector” (CD), involves a generalist cooperator strain that invests into both goods, and a fulldefector that does not invest into either good.As a consequence of its effect on the branching direction, the angle of approach to the branchingpoint has a very strong influence on which of the two outcomes is eventually reached. Fig. 4.9 showsthat, whenever the approach occurs along the phenotypic axes, both outcomes are roughly equally likely,but when the approach is along a diagonal direction, the outcome is determined by the most likelybranching direction. Hence, the initial position of the population biases evolution towards one of thetwo possible stable states. This result suggests that division of labor between two traits can readilyevolve without any trade-offs, whenever both of the traits evolve from a small initial value.The two outcomes we discussed so far are only stable when mutations are small, as we assumethroughout this article. In Appendix C.6, we investigate what would happen if we allowed a large-effectmutation—specifically, a mutant occupying any one of the empty corners of the phenotype space—tooccur in a resident population with a CD or DOL configuration. We find that while the CD configuration isstable against invasion, the DOL configuration is susceptible to invasion by a third strain. The successfulthird strain will be a cooperator if zmax/2< z?, or a defector if zmax/2> z? (Appendix C.6). The resultingthree-strain equilibrium is stable against invasion by a fourth strain.724.4 DiscussionEvolutionary game theory studies simplified interactions (games) between individuals. These gamesrepresent common tasks that organisms perform in nature, such as the production of an enzyme (Goreet al. 2009), competition for a mating partner (Alonzo and Sinervo 2001), or participation in a huntinggroup (Packer and Ruttan 1988). Traditionally, these games are studied in isolation, and it is assumedthat the resulting evolutionary dynamics are independent between games. However, in nature, the sameindividual will engage in many different kinds of interactions, either simultaneously or throughout theirlifetime, which increases the trait space dimension and may lead to novel evolutionary outcomes thatare not predicted from each game’s individual dynamics (e.g., Brown and Taylor 2010; Ito and Doebeli2019; Mullon et al. 2016). Here, we have studied the case of evolutionary branching when a populationis engaged in multiple continuous snowdrift games.We have shown that there is a feedback between evolutionary dynamics and the population’s phe-notypic distribution: the branching direction can be influenced by the population’s phenotypic distri-bution, which in turn is affected by the direction of approach to the evolutionary branching point. Asa consequence, knowing that each trait undergoes branching is not sufficient to predict the direction ofbranching in multidimensional space.The dynamics of multiple simultaneous games cannot be predicted from the individual games, evenwhen their payoffs are additive. Analysis of discrete-strategy games reveals that when games have atleast three pure strategies, the resulting multi-game dynamics cannot be characterized from the sepa-rate analysis of each game (Chamberland and Cressman 2000; Hashimoto 2006; Venkateswaran andGokhale 2019). Together with these previous results, our study poses an important challenge becausemost of our understanding of frequency-dependent evolutionary dynamics is built on single-game stud-ies, despite the fact that multiple simultaneous games are arguably the most common scenario.Although we motivated our study by focusing on simultaneous continuous games, our results are,more broadly, relevant for our understanding of evolutionary branching in multidimensional spaces. Aswith multi-game dynamics, a growing number of studies has revealed that some properties of multidi-mensional evolutionary branching cannot be predicted from studying each dimension in isolation. Thesestudies have shown that correlations between traits (caused by pleiotropy, epistasis, or linkage disequi-librium) can influence the stability of equilibria and the direction of branching (De´barre et al. 2014a;Leimar 2009).Similarly to our result, previous studies (Doebeli and Ispolatov 2017; Ispolatov et al. 2016; Itoand Dieckmann 2014, 2012) have also found that branching can occur orthogonally to the selectiongradient. This phenomenon has been formally studied using the concept of evolutionary branching lines(Ito and Dieckmann 2012), the idea of which is that the strength of directional selection decreases as thepopulation approaches the branching point, and it may be overwhelmed by strong disruptive selectionorthogonal to the selection gradient. Although the evolutionary branching line approach predicts thesame branching direction as our model (Appendix C.4), we argue that our results are instead explainedby the effect of the population distribution because of the following two reasons. First, the predictionsof the evolutionary branching line can be tested by using the method of maximum likelihood invasion73paths (MLIP, Ito and Dieckmann 2014) or oligomorphic stochastic simulations. Both of these cannotpredict the observed branching direction: in our system, MLIP predicts branching along the approachingdirection (Appendix C.5), and the oligomorphic stochastic simulations do not exhibit any preferreddirection of branching (Fig. 4.3B). These results suggest that at least in our system those effects may bevery weak or easily overwhelmed by stochasticity. Second, our PDE model shows that the shape of thepopulation’s phenotypic distribution can underlie the direction of branching even under circumstanceswhere the evolutionary branching lines would make the opposite prediction (Appendix C.4, Fig. C.1).Branching is more likely to occur along the axis of greater variation, which acts as a “genetic line of leastresistance” for adaptive diversification (Schluter 1996). Not only does the phenotypic distribution affectevolutionary dynamics, but in turn it is also affected by the direction of movement (Fig. 4.5), creatinga distribution–selection feedback that is not captured in analytical models of evolutionary branching,which assume monomorphic populations.In order to more clearly demonstrate the importance of the phenotypic distribution, we have pur-posefully chosen a set of assumptions that removes any other causes for a non-uniform distribution ofbranching directions. Among those assumptions are: equal payoff functions, independent traits, andequal, uncorrelated mutations for each phenotype. Violating any of these would impose a preferredbranching direction, even for a monomorphic population. This is illustrated in our work by relaxingthe assumption that both traits are equally important for fitness. As the fitness effects become increas-ingly asymmetric, a particular branching direction becomes more likely even when the population ismonomorphic (Fig. 4.7A). However, our results also show that the population distribution can play arole even when this assumption is relaxed, provided that asymmetry is not too high. Correlated mu-tational effects would likely bias the direction of branching in a similar way, which would be worthexploring in further work.For future extensions it would be interesting to expand this model to a broader universe of game-theoretic scenarios, including N-player groups (Souza et al. 2009), competitive interactions (Ito andDoebeli 2019; Killingback and Doebeli 2002), synergy and discounting (Hauert et al. 2006b), or theeffect of spatial or social structure (Zhong et al. 2008).The branching direction has important long-term consequences because it leads the population to-ward different stable states. Hence, different directions of approach to equilibrium result in differentoutcomes. A similar phenomenon, albeit with a different mechanism, occurs in a one-locus two-allelemodel by Kisdi and Geritz (1999). In that model, the occurrence of branching slightly before the branch-ing point places the two strains in different basins of attraction (depending on which side of the equilib-rium branching takes place).In our model, there are two alternative evolutionary outcomes: one consists of a cooperator and adefector strain, and the other consists of two specialists strains, each of which produces one public goodand consumes the other (division of labor). This second outcome, which is most likely to occur wheneverboth evolving traits approach the branching point from the same direction, resembles the productionand sharing of metabolites that occurs, for instance, between microbes in multi-species biofilms (Eliasand Banin 2012). These interactions can certainly be interpreted as a (very) high-dimensional set of74simultaneous games where specialization and division of labor are common. As microbes in multi-species communities specialize in particular metabolites and functions, communities may give rise tonetworks of mutual dependency (Black Queen evolution, Morris 2015). Our model shows how this typeof mutual dependency may evolve even in the absence of the costs and trade-offs that are often invokedin other models of cooperation-based specialization (e.g., Wood and Komarova 2018).Our results exemplify how signatures of past selection can impact the response to current selectionpressures. Models that disregard these signatures (e.g., by assuming monomorphic populations) discardinformation that may be relevant for predicting evolutionary trajectories.75Chapter 5Multilevel selection favors fragmentation modesthat maintain cooperative interactions in multi-species communities5.1 IntroductionReproduction is a fundamental feature of life and the sine qua non of Darwinian evolution (Godfrey-Smith 2009; Lewontin 1970; Stearns 1992). Despite its centrality in natural selection, there appears to beno unique optimum approach to reproduction in multicellular organisms (Kondrashov 1994; Pichuginet al. 2017; Roze and Michod 2001; Van Gestel and Tarnita 2017). Whenever individual cells aban-doned solitary life to form groups—ranging from loose collectives (Claessen et al. 2014) to colonial andmulticellular organisms—they came up with a surprisingly diverse menagerie of of strategies for theproduction of reproductive propagues (Godfrey-Smith 2009).Many multicellular eukaryotes reproduce by undergoing single-cell bottlenecks. For example, sex-ually reproducing organisms produce unicellular gametes. Single-cell bottlenecks are also commonin plants and animals that reproduce asexually, such as the Amazon molly Poecilia formosa (Turneret al. 1980), several weevils of the Curculionidae family (Suomalainen 1969), and many angiospermspredominantly in the Asteraceae, Rosaceae, and Poaceae families (Bicknell and Koltunow 2004). Analternative to single-celled propagules is vegetative reproduction in which the offspring develops from amulticellular propagule. This type of fragmentation may involve specialized structures, such as conidia(in fungi) or gemmae (in algae, mosses and ferns) (Hughes 1971), or it may happen simply by budding(e.g., in hydra, Galliot 2012) or by fission (as in some flatworms, A˚kesson et al. 2002).This wide variation in modes of fragmentation is not limited to eukaryotes. Some bacterial ag-gregations, such as the clusters formed by Staphilococcus aureus, reproduce by releasing single-celledpropagules (Koyama et al. 1977). Others, such as filamentous cyanobacteria, reproduce vegetatively:dividing cells remain physically connected, and fragmentation of these aggregates creates new chains(Herrero et al. 2016). A single parent individual may also divide into two equally-sized multicellularoffspring. This occurs, for instance, in the multicellular collectives formed by magnetotactic bacteria76(Keim et al. 2004). Alternatively, one parent may simultaneously give rise to many equally-sized off-spring. The 16-celled colonial alga Gonium pectorale takes this strategy to the extreme, by dispersinginto 16 individual cells (Stein 1958).Why is there such wide variation in the number and size of reproductive propagules? A researchprogram initiated by Kondrashov (1994) attempted to answer this question by considering the evolu-tionary advantages of unicellular propagules relative to vegetative propagules. When offspring developfrom a single-cell propagule, they are genetically homogeneous. If the propagule carried any deleteri-ous mutation, its phenotypic effects will not be masked or compensated by wild-type cells. Therefore,reproductive bottlenecks ensure that natural selection is more efficient at eliminating deleterious mu-tations (Bergstrom and Pritchard 1998; Grosberg and Strathmann 1998; Kondrashov 1994; Roze andMichod 2001). Normally, in the absence of genetic recombination, deleterious mutations accumulatein the population in a process known as Muller’s ratchet (Muller 1964), becoming abundant and de-creasing the population’s mean fitness. This decrease in fitness is called mutation load (Agrawal andWhitlock 2012; Haldane 1937; Muller 1950). In extreme cases, the resulting maladaptation can leadto severe declines in population size, accelerating the accumulation of deleterious mutations by geneticdrift. This positive feedback, which may lead to extinction, is termed mutational meltdown (Lynch andGabriel 1990). By facilitating the purging of deleterious mutations, reproductive bottlenecks slow downthe detrimental effects of Muller’s ratchet (Bergstrom and Pritchard 1998) and reduce mutation load(Kondrashov 1994; Roze and Michod 2001).This family of models also informs our understanding of the evolutionary transition (Szathma´ryand Maynard Smith 1995) from unicellular life to multicellularity. That is because mutations that aredeleterious for the group may be beneficial for the mutant cell itself. Mutations of this type (whichRoze and Michod (2001) call “selfish mutations”) result in “cheater” cells whose fast growth comesat a cost to their group; they include, for example, cancer cells (Evan and Littlewood 1998). Selfishmutations (and cheaters more generally) instantiate a conflict between the direction of selection at thelevel of the cell and at the level of the group. Reproductive bottlenecks resolve this conflict by reducinggenetic variation among cells within offspring, and distributing the variation among progeny groups.This reorganization of genetic variation makes selection at the level of the group more effective thanselection at the level of the individual cell, thus furthering the evolutionary transition to multicellularity(Roze and Michod 2001).While this research program has been fruitful and insightful, it considers only a limited set of propag-ule production strategies (viz., the production of a propagule of varying size). Recent research byPichugin, Traulsen, and collaborators (Pichugin et al. 2019; Pichugin and Traulsen 2018; Pichugin et al.2017) has started to address the wide variety of strategies that exist in nature. Their models exhaus-tively analyse the fitness consequences of every mathematically possible partition of a homogeneousgroup. This approach is promising, but it becomes unwieldy to consider more complex scenarios, suchas big groups (which have many possible partitions), within-group density dependence, or the effect ofmutations, which, as we have seen, can have important consequences.All of the models described above focus on single-species groups, such as multicellular or colonial77organisms. However, multispecies communities also undergo fissioning and fragmentation. Such com-munities are particularly common in the microbial world: the majority of microorganisms belong forat least part of their life to multispecies groups, such as mixed biofilms that comprise up to thousandsof different species (Elias and Banin 2012). Just like single-species collectives, multispecies microbialcommunities display a wide variety of modes of fragmentation. Host-associated communities can bevertically transmitted between host generations or they can be recruited from the environment (both ofwhich are exemplified in insect microbiomes, Engel and Moran 2013). They can also be formed bya combination of both, which is the case, for example, for the human microbiome (Funkhouser andBordenstein 2013). Environmental communities can also form by recruitment (e.g., colonization ofmarine-snow particles, Datta et al. 2016) or from multicellular fragments which detach from maturecommunities (e.g., some bacterial biofilms, Claessen et al. 2014).Even though natural biofilms are heterogeneous communities of different microbial species, organ-isms within them are well adapted to group life. In mixed biofilms, individuals of different speciescommunicate with each other via quorum sensing (e.g., Moons et al. 2006; Riedel et al. 2001) andengage in cooperative interactions with each other (Elias and Banin 2012). Such interactions includecross-feeding, in which byproducts from nutrients that are metabolized by one species then serve as foodsource for a second species (reviewed in Seth and Taga 2014). Cross-feeding is so prevalent that manymicrobes have lost the ability to synthesize essential metabolites and became metabolically interdepen-dent (Tyson and Banfield 2005). This type of metabolic specialization allows communities to performotherwise incompatible tasks, such as photosynthesis and nitrogen fixation (Johnson et al. 2012).Much like selfish mutations in multicellular organisms, cheaters in mixed biofilms are manifesta-tions of a conflict between levels of selection. Some researchers have taken the view that group selectionpredominates in mixed biofilms, going so far as to call them “evolutionary individuals” (Day et al. 2011;Doolittle 2013; Ereshefsky and Pedroso 2015). Others warn that this comparison holds only as an anal-ogy, and that the conflict between levels of selection has not been resolved in mixed biofilms (e.g.,Clarke 2016; Coyte et al. 2015; Foster and Bell 2012; Nadell et al. 2009). Although the question ofwhether group- or individual-level selection predominates in mixed biofilms remains controversial, re-cent modelling work has started to address which conditions need to be met for multilevel selection tooperate in these communities (Roughgarden 2020; Van Vliet and Doebeli 2019).Here, we build on these previous models and further investigate how the conflict between levelsof selection can be resolved in both single- and multi-species groups. In particular, we focus on thequestion of how the mode of fragmentation of the group affects its ability to maintain cooperative in-teractions in the presence of recurrent mutations to cheating cell types. We developed an individualbased, multilevel selection model (Simon 2010; Simon et al. 2013) of groups consisting of cells thatinteract mutualisticly. Cells in the group stochastically give birth or die, while groups stochastically goextinct or reproduce. Whenever a group reproduces, it follows a predefined “fragmentation mode” thatdescribes how its constitutive cells are arranged over the offspring groups (Fig. 1). Both the size andthe number of offspring groups can vary continuously allowing us to fully investigate how the mode offragmentation affects the ability of the group to resist mutations.78snbig offspringmany offspringfew offspringsmall offspringComplete fragmentation(Filamentous cyanobacteria)Binary fission(Magnetotactic bacteria)Vegetative reproduction(S. aureus)Single-cell reproduction(G. pectorale)0 0.501Figure 5.1: All possible fragmentation strategies in our model can be described using a two-dimensional phasespace. Two parameters (s and n) determine the expected size and number of offspring, respectively (as a fractionof the number of cells in the parent group). Cartoons exemplify some modes of fragmentation. We refer to thethree corners of the triangle (complete fragmentation, single-cell reproduction, and binary fission) as archetypalmodes of fragmentation.Drawing from the lessons of models on single-species collectives, we expect that in multi-speciesgroups different fragmentation modes could have strong implications on how well the system can copewith mutants. For example, by decreasing variation within groups, we expect that reproductive bottle-necks intensify group selection and help purge cheater mutants. On the other hand, we expect that thebenefits of strong bottlenecks diminish in complex multi-species communities, because small propag-ules are less likely to contain all of the species that are metabolically interdependent.Our model confirms these expectations: we find that high-diversity communities face a reproduc-tive trade-off: tight bottlenecks result in the elimination of deleterious mutations, but they also decreasespecies diversity; the opposite is true when offspring groups are large. Using our model we also inves-tigated strategies that can relieve this trade-off: adding migration between groups reconciles single-cellbottlenecks with the need for diversity, while making group fragmentation rate dependent on groupsize increases the strength of group level selection thus reducing mutant load. Finally, we extend ourmodel to investigate how group fragmentation rates evolve and we show that groups always evolve tothe fragmentation mode that maximizes their resistance to mutants.5.2 MethodsWe developed an individual-based multilevel selection model (Simon 2010; Simon et al. 2013) to studya community consisting of a stochastically varying number G of multicellular groups. Within eachgroup i, there are Ni individual cells, which may belong to any of m different species. Regardless ofspecies, each cell is either wild-type (cooperator) or mutant (cheater). We denote the number of cellsof species j ∈ (1,m) within group i as N j,i, so that Ni = ∑mj=1 N j,i. (Note that any particular group may79contain fewer than m species; furthermore, species may go extinct, in which case the realized numberof species in the community will be less than m.)Community dynamics unfold simultaneously at both levels of biological organization: both thenumber of groups and the number of cells within each group vary over time. The number of cellswithin each group changes due to cell birth and cell death events, which occur at rates b j,i and di,respectively (section 5.2.1). Simulaneously, the groups themselves undergo fission (group birth) andextinction (group death) events, at rates Bi and Di, respectively (section 5.2.2). Finally, migration ofcells between groups occurs at a rate ν .The individual-based simulation was implemented using a Gillespie algorithm (Gillespie 1976).5.2.1 Cell dynamicsCell birth rate A cell’s birth rate depends on cooperative interactions with other cells in the samegroup. At a cost γ to themselves, wild-type cells (cooperators) contribute to these interactions, increas-ing the birth rate of others in the group. However, mutations turn cooperators into cheaters, who reapthe benefits of interactions without contributing (we ignore back-mutation). As a result, mutants alwaysmultiply faster than conspecific wild-type cells in the same group, but their presence slows the overallgrowth of the group.We consider two possibilities: single-species systems (m = 1), where groups resemble multicellularorganisms, and multispecies systems (m > 1), where groups are akin to mixed biofilms. When m = 1,cooperation occurs between individuals of the same species, whereas for m > 1, cooperation occursacross species, thus resembling obligate mutualistic interactions.Let nwtj,i denote the number of wild-type cells of species j in group i (so that the number of mutantsis nmutj,i = N j,i−nwtj,i). Then, the per capita realized birth rate of a mutant cell is equal tobmutj,i =nwtj,iNi, if m = 1,m(m−1)∏mk 6= jnwtk,iNi, otherwise,(5.1)where the term m(m−1) ensures that the total number of cells at equilibrium is the same across differentvalues of m. Eq. 5.1 means that, when m = 1, a cell can only grow in the presence of conspecificcooperating partners, and when m > 1 a cell of species j can only grow in the presence of cooperatingpartners of all m− 1 other species (nwtk,i > 0 for all k 6= j). Note that, when m > 1, growth does notdepend on the presence of conspecifics, because it is meant to represent mutualistic interactions such ascross-feeding. The realized birth rate of a wild-type cell is then bwtj,i = (1− γ) ·bmutj,i , where γ is the costof cooperation. Whenever a new cell is born, it stays in the same group as the parent. If a wild-typecell gives birth, with probability µ it generates a mutant cell of the same species and with probability(1−µ) the offspring remains wild-type. We only consider mutations from wild-type to mutant in samespecies: there is no back-mutation or mutations across species.80Cell death rate The per capita death rate di for all cells in group i is given by:di =1Kcells· (Ni)δind , (5.2)where Kcells is the within-group carrying capacity, and δind is an indicator variable that can take thevalues 0 or 1. If δind = 1, death rate increases linearly with group size, thus ensuring density regulationwithin the group; if δind = 0, death rate is constant (no density dependence).5.2.2 Group dynamicsGroup fission rate We use the term “fission rate” to refer to the rate at which groups reproduce (toavoid confusion with the birth of individual cells). All else being equal, groups with higher fissionrates are favored by group selection. We consider the possibility that bigger groups are more likely toreproduce: the rate increases (with constant of proportionality S) with Ni, up from a minimum value ofB0. In summary,Bi = B0+S ·Ni. (5.3)Modes of fragmentation When a group of size Ni reproduces, it fissions into a parental group and oneor more offspring groups, according to its mode of fragmentation. We assume that the offspring size ishomogeneous, i.e., all offspring groups have the same expected size (this reduces the space of possiblemodes of fragmentation when compared to the more exhaustive approach used by Pichugin et al. 2017).Our model accommodates a wide variety of modes, without attempting to be exhaustive. To do this, weconsider a triangular space of fragmentation strategies that is defined by two parameters (Fig. 5.1).The first parameter (0 < n ≤ 1) determines the expected number of cells that are transmitted tooffspring, as a fraction of the parent’s cell number. Whenever a group reproduces, we first draw thisnumber of cells, Noffspring, from a Poisson distribution with expectation n ·Ni, truncated at Ni. The secondparameter (0 < s≤ 0.5) determines the expected size of each offspring group, again as a fraction of theparent’s cell number. This size, σoffspring, is drawn from a Poisson distribution with expectation s ·Ni,bounded between 1 (the size of a single cell) and Noffspring. From these two quantities we then calculatethe total number of offspring groups, Goffspring = ceil(Noffspring/σoffspring). The first Goffspring− 1 off-spring groups are all assigned σoffspring cells, which are randomly sampled (without replacement) fromthe parent group. The final offspring group is assigned the remaining cells, again by random sampling(without replacement) from the parent group. Finally, we remove all cells that are transmitted to theoffspring groups from the parent group. The parameter s is analogous to the propagule size parameter inKondrashov (1994) and Roze and Michod (2001). The only valid combinations of parameters obey thecondition s < n < 1− s. The lower triangle is excluded because it is logically impossible, whereas theupper triangle is excluded because one of the groups that result from the fissioning process is arbitrarilylabelled as the parent group.81We refer to the three corners of this parameter space as archetypal fragmentation modes (Fig. 5.1):in binary fission, the parent group divides into two equally-sized groups; in single-cell reproduction, theparent group produces a single offspring group by releasing a unicellular propagule; finally, in completefragmentation, the entire parent group disperses into its constituent cells. Every mode of fragmentationdescribed in section 5.1 can be accommodated in this parameter space.Group extinction rate We use the term “extinction rate” to refer to the rate at which groups die (toavoid confusion with the death of individual cells). The extinction rate of group i can in general dependon the number of cells in the entire community Ntot = ∑i Ni (which we also refer to as “communityproductivity”) or the number of groups G, or a combination of these. The two different types of density-dependence are turned on or off by the indicator parameters δtot and δgroups, respectively. This results inthe per capita extinction rateDi =(GKgroups)δgroups·(NtotKtot)δtot, (5.4)where the parameters Kgroups and Ktot roughly scale the total number of groups and/or cells in thecommunity. In the main manuscript, we will focus on the case in which δgroups = 0 and δtot = 1, whichmeans that the extinction rate increases with the total number of cells in the community (Ntot). Weexplore a different choice of values for the indicator parameters in section D.2. Groups also go extinctwhenever the number of constituent cells drops to zero, e.g., due to stochastic sampling.Migration rate Each cell has a per capita migration rate ν . Whenever a cell migration event occurs,the migrant cell leaves its group and joins a second, randomly chosen group.5.2.3 Evolution of the fragmentation modeTo study the multilevel evolutionary dynamics of fragmentation mode, we characterize every cell in thepopulation with a quantitative phenotype vector (s,n). These traits could represent, for example, theproperties of the materials that are excreted into the extracellular matrix. The average trait value ofthe group determines the group’s position in state space (Fig. 5.1) and hence its mode of fragmenta-tion. Thus, the properties of the individual cells give rise to an emergent group property. When cellsreproduce, their offspring inherit their parent’s trait values; however, with a small probability µs or µn,small-effect mutations may occur. For computational efficiency, we discretize the phenotype space andassume that mutations always occur between adjacent phenotype bins. We do not allow mutants outsidethe bounds of the phenotype space: any such mutants are moved along the n-direction and placed on theboundary of the allowed space.820.000.250.500.751.00Fractional offspring number (n)Number of groups (G ) Productivity (N tot)1000 2000 3000 20000 400000.0 0.1 0.2 0.3 0.4 0.5Fractional offspring size (s)0.0 0.1 0.2 0.3 0.4 0.5Fractional offspring size (s)Figure 5.2: In the absence of mutations, with a single species, complete fragmentation into single cells maxi-mizes equilibrium population size, measured either as total number of groups (G) or as total productivity (Ntot).Parameters: µ = 0; all other parameters set to the default values (table D.1).5.3 Results5.3.1 Multilevel selection can maintain cooperators and avert mutational meltdownWe will first consider groups consisting of a single species. When there are no mutations, the strategyof complete fragmentation maximizes equilibrium population size, measured either as total number ofgroups (Fig. 5.2A) or as total number of cells (i.e., productivity, Fig. 5.2B). But what happens in thepresence of mutations?Because mutations constantly create cheaters, we expect slower growing wild-type cells to go extinctover evolutionary time in the absence of group-level events. Furthermore, because all cells depend onthe presence of cooperators to reproduce, this process is expected to lead to mutational meltdown: theextinction of the entire community due to the influx and spread of mutations. Indeed, when we setgroup-level rates to zero, we observe the accumulation of mutations, leading to the decrease in groupsize and eventual extinction of the community (Fig. 5.3A and black lines in Fig. 5.3B, C).However, group level selection can maintain cooperators and prevent meltdown. Cells in groupswith many cooperators multiply faster than cells in groups with few cooperators. Furthermore, whenthese groups fission, they give birth to offspring groups that also have higher frequency of cooperators.Bigger groups are also less likely to collapse due to the stochastic death of all its member cells (drivenby the accumulation of mutations and by genetic drift), and will therefore remain in the populationfor longer and have more opportunities to fission. Accordingly, when group-level rates are nonzero,selection at the level of the group can avert mutational meltdown (Fig. 5.3B, C).5.3.2 Complete fragmentation minimizes mutation loadPrevious theoretical work (Bergstrom and Pritchard 1998; Grosberg and Strathmann 1998; Kondrashov1994; Roze and Michod 2001) predicts that, in single-species systems, small offspring sizes should be8302,5005,0007,5001000 2000 3000TimeNumber of cellsWild−type cellsMutant cellsA1101001,0000 2000 4000 6000TimeNumber of groups (G) No group eventsSingle−cell reproductionComplete fragmentationBinary fissionB1001,00010,0000 2000 4000 6000TimeProductivity (Ntot)C010,00020,00030,00040,0000.01 0.10 1.00Mutation rate (µ)Productivity (Ntot)Single−cell repr.Complete frag.Binary fissionD0.000.250.500.751.000.0 0.1 0.2 0.3 0.4 0.5Frac. offsp. size (s)Frac. offsp. number (n)0.1 0.2 0.3 0.4Max. mutation rateE0.000.250.500.751.000.01 0.10 1.00Mutation rate (µ)Relative fraction of WT cellsSingle−cell repr.Complete frag.Binary fissionFFigure 5.3: When there are no group events, mutations cause community extinction, but group-level eventsprevent this fate, as exemplified by the three archetypal modes of fragmentation. A: Example replicate with nogroup events, showing fast initial rise in the number of mutants and the consequential decrease in the total cellnumber, resulting in extinction. B, C: Temporal dynamics of the number of groups (B) and total productivity (C)for different reproduction modes. Thin lines are moving averages of individual replicates; solid lines are averagesacross replicates. D: By simulating population dynamics for various values of the mutation rate (µ), we canidentify the value at which the population undergoes mutational meltdown–driven extinction (i.e., that strategies’maximum mutation rate). E: For every point in the strategy space, we calculated the maximum mutation rate(shown in panel D for the three archetypes). F: The percentage of wild-type cells at equilibrium decreases withmutation rate (µ); the plot depicts the fraction of wild-type cells relative to the equilibrium fraction when µ isvery small (µ = 0.01). Parameters: for the lines with group events, B0 = 0.01; all other parameters set to thedefault values (table D.1).most resilient against mutational meltdown, since single-cell bottlenecks expose harmful mutations tonatural selection. In agreement with these predictions, we found that the mode of group fragmentationhas important consequences for the capacity of the population to persist in the presence of mutations(Fig. 5.3D). In single-species communities (m= 1), the complete fragmentation archetype and strategiesclose to it are able to avoid mutational meltdown-driven extinction even for relatively high rates of mu-tation when compared to other strategies (Fig. 5.3D, E). This is because, although the average frequencyof wild-type cells per group always decreases with mutation rate, the relative decrease is smaller for thisarchetype (Fig. 5.3F), as would be expected given the role of tight bottlenecks in eliminating deleteriousmutations.840.000.250.500.751.00nm = 1 m = 2 m = 3 m = 4Uppertransect20000 40000Ntot0 10000 20000Ntot0 10000Ntot0 5000 10000Ntot0.00 0.25 0.50s0.00 0.25 0.50s0.00 0.25 0.50s0.00 0.25 0.50sA010,00020,00030,00040,00050,000s= 0.01n= 0.99s= 0.26n= 0.74s= 0.5n= 0.5Reproduction strategy(upper transect)Productivity (Ntot)No. species1234B0.000.250.500.751.000.01 0.10 1.00Mutation rate (µ)Rel. productivityNo. species1234C05,00010,00015,000s= 0.01n= 0.99s= 0.26n= 0.74s= 0.5n= 0.5Reproduction strategy(upper transect)Productivity (Ntot)Mutationrate0.0010.010.025DFigure 5.4: When community complexity is high, the productivity peak shifts away from unicellular bottlenecks.A: The color indicates equilibrium community productivity (Ntot). As the number of species (m) increases, thestrategy that maximizes Ntot moves rightward along the upper transect (pink line) of the strategy space. B: Equi-librium productivity as a function of strategies along the upper transect of the strategy space (corresponding tothe pink line from panel A; ranges from complete fragmentation, on the left of the x axis, to binary fission, onthe right), for different numbers of species. C: Equilibrium productivity decreases with mutation rate (µ); thisdecrease is faster for higher number of species (exemplified here for s = 0.1, n = 0.9.) D: Some strategies thatdo well with small µ (large offspring) are not viable when µ is large (shown here with m = 3). Parameters: allparameters are set to the default values, unless otherwise indicated (table D.1).5.3.3 Multispecies communities are more vulnerable to mutational meltdownWe have seen that, when single-species groups reproduce by complete fragmentation, they are mostresistant to mutational meltdown. Multispecies communities, in contrast, cannot resort to completefragmentation, since in small fragments there is a high chance that one or more heterospecific typesare missing. In the extreme case of unicellular fragments, the offspring cell can never grow becauseof the absence of mutualistic interactions. In the absence of mutations, multispecies communities arethus most productive when groups have larger offspring, which allows offspring to maintain a varietyof species. In other words, for increasing species number, the productivity maximum moves rightwardalong the upper diagonal of the phenotype space (Fig. 5.4A, B).Because multispecies communities need larger offspring, they are also more vulnerable to mutants(Fig. 5.4C). Therefore, there is a trade-off between resistance to mutational meltdown (low offspringsize) and the maintenance of mutualistic interactions (high offspring size, Fig. 5.4D).855.3.4 Size-dependent fragmentation rate and migration prevent mutational meltdownin multispecies communitiesWe have seen that (consistent with previous studies) multicellular organisms can reduce mutation loadand prevent mutational meltdown by reducing propagule size (section 5.3.2), but that this strategy isnot available in multispecies communities, since it deprives cells in daughter groups of their mutual-istic partners (section 5.3.3). What reproductive strategies, then, allow for more complex multispeciescommunities to resolve this trade-off?One alternative is size-dependent fragmentation rate. Larger groups may be more likely than smallerones to undergo fission and thus produce offspring groups (in our model, this is achieved by increasingthe slope parameter S in Eq. 5.3). Because all groups have equal extinction rate, groups with higher fis-sion rate are favored by group selection. Hence, increased size-dependence in fissioning rate intensifiesthe effectiveness of group selection in purging mutants, since groups with many mutants grow slowerand thus take longer to reproduce. By increasing the importance of group selection relative to individual-level selection, size-dependent fissioning should allow complex communities to withstand high rates ofmutation. To test this hypothesis we calculated, for different values of S, the highest mutation rate thatcommunities can withstand, over the entire strategy space. In other words: for each fragmentation modewe found the highest mutation rate that the population can handle before if collapses; we then selectedthe fragmentation mode for which this mutation rate is highest. We will later see that this strategy is alsoan evolutionary attractor, so we expect that this is the mode of fragmentation of groups at evolutionaryequilibrium. The results confirm our prediction that at higher values of S, communities can survive inthe presence of higher mutation rates (Fig. 5.5; see also Fig. D.2 for a broader range of parameters).Another alternative is migration of cells between groups, which is common in bacterial biofilms. Inmultilevel selection theory, migration is often considered to be detrimental to group selection, becauseit decreases variance between groups and increases the role of individual-level selection. However, inmulti-species systems, low to intermediate levels of migration can be beneficial, because they allowsmall offspring to recruit individuals from the environment and, thus, achieve the diversity necessary formutualistic interactions (Fig. 5.5; see also Fig. D.3 for a broader range of parameters).Size-dependent fragmentation and migration are thus two possible solutions to the challenge ofresisting the accumulation of mutations while also maintaining species diversity within groups. Size-dependent fragmentation increases the strength of group selection, which purges deleterious mutations,thus making small offspring sizes unnecessary. In contrast, migration allows for the recruitment ofheterospecific cells, which allows diversity to be maintained even when offspring sizes are small.5.3.5 Strategies that maximize community productivity are evolutionary attractorsReal communities vary in their mode of fragmentation. In fact, experiments in a variety of species—including Chlamydomonas reinhardtii (Herron et al. 2019; Ratcliff et al. 2013b), Saccharomyces cere-visiae (Ratcliff et al. 2013a), and Pseudomonas fluorescens (Hammerschmidt et al. 2014)—show thatthe mode of fragmentation can rapidly evolve under selection pressure.Fragmentation mode has implications for quantities such as community productivity (total number86m = 1 m = 2 m = 3 m = 4S=0, ν=0S=0, ν=1S=2, ν=00 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.500.5100.5100.51Frac. offsp. size (s)Frac. offsp. number (n)−10.0−7.5−5.0−2.5Maximummutationrate(log)Figure 5.5: Each panel shows, for each position in the strategy space, the maximum mutation rate a populationcan experience before going extinct (similar to Fig. 5.3E). Columns (from left to right) depict increasing number ofspecies. Rows depict: no size-dependent fragmentation and no migration (top); migration and no size-dependentfragmentation (center); size-dependent fragmentation and no migration (bottom). Grey squares correspond tocommunities for which the maximum mutation rate is outside the range of our simulations or numerical errors.For each value of mutation rate, we assessed ten replicates per fragmentation mode; we then calculated the mean(across replicates) of the logarithm of maximum mutation rate. Each point in the figure corresponds to a nearest-neighbour average of this quantity. Parameters: all parameters are set to the default values, unless otherwiseindicated (table D.1). Figures D.2 and D.3 comprise a wider range of values of S and ν and depict raw values ofmaximum mutation rate (rather than nearest-neighbour averaged values).of cells), number of groups, and total frequency of mutants in the community. However, different strate-gies may maximize each of these quantities. For example, when fragmentation rate is size-dependent,the number of groups is maximized when the slope of the fission function (S) is small (close to thecomplete fragmentation archetype), whereas the total community productivity is maximized when S islarge (close to the binary fission archetype). Given that both the number of groups and the number ofcells affect selection in different ways at both levels of organization, it is not obvious which mode offragmentation is favored by natural selection under different conditions.To answer this question, we simulated the multilevel evolutionary dynamics (section 5.2.3). Acrossa diverse set of parameters, we found that strategies that maximize community productivity (i.e., totalnumber of cells) are evolutionary attractors and endpoints of evolution (for some examples, see Fig.8700.51m = 1S  = 0.1m = 1S  = 4m = 2S  = 0.1m = 2S  = 401K2KN tot00.51Fractional offspring number (n)0500G00.510 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5Fractional offspring size (s)02040N iFigure 5.6: Evolution of fragmentation mode maximizes total number of cells (Ntot, top row) rather than otherquantities such as number of groups (G, middle row) or average group size (Ni, bottom row). Red lines representevolutionary trajectories of the average mode of fragmentation over time (darker shades of red correspond toearlier time points). For each value of S, we assessed five initial phenotypes; trajectories that rapidly went extinctare not depicted. Parameters: all parameters are set to the default values, unless otherwise indicated (table D.1),except:except: Kcells = 100,B0 = 0.01,Ktot = 30,000 (when S= 0) or Ktot = 10,000 (otherwise), µs = µn = 10−2.5.6). Therefore, when complexity is low, we expect evolution to lead to tight bottlenecks, but formore complex communities with more than one species, we expect group selection to promote theevolution of binary fission. When there are multiple species, the evolutionary outcome will depend onwhat mechanisms are in place to resolve the trade-off between maintaining complexity and eliminatingdeleterious mutations; for example, size-dependent fragmentation favours the evolution of binary fission(Fig. 5.6).5.4 DiscussionReproduction is the defining characteristic of life, yet organisms across the tree of life reproduce inmany different ways. In organisms that have single-celled gametes, there is less within-organism varia-tion than in organisms with many-celled propagules, which increases the strength of natural selection inremoving deleterious mutations (Bergstrom and Pritchard 1998; Grosberg and Strathmann 1998; Kon-drashov 1994; Roze and Michod 2001). Hence, single-celled bottlenecks allow populations to reduce88mutation load. In this paper, we studied how this process affects the mode of fragmentation of multi-species collectives of organisms, such as microbial biofilms. We found that complex communities facea reproductive dilemma. On the one hand, their persistence relies on aligning the Darwinian interests ofthe group and its individual cells, which can be achieved by small reproductive bottlenecks (not unlikesingle-species organisms). On the other hand, due to stochastic sampling, small daughter groups lackthe species diversity that multispecies communities rely on for ecological success. There is a tug-of-warbetween two competing selection pressures affecting daughter group size: maintaining species diversitywhile reducing mutation load. We explored two alternative solutions to this dilemma: migration (whichmakes small groups viable) and size-dependent fragmentation (which reduces load even within largegroups).Migration of cells between groups allows small groups to acquire species diversity. Due to stochasticsampling, some newly born groups are free of mutants, but may lack mutualistic partners. Thanks tomigration, these groups will be able to recruit individuals of other species, whose presence is necessaryfor cell growth. Mutants also migrate, which may be detrimental for some groups. As long as thecombined stochastic processes of birth and migration create some groups that have all species but nomutants, those groups will be favored by group-level selection. In nature, many mixed biofilms grow bya mechanism of “co-colonization” that resembles this process: one species often plays the role of theinitial colonizer and other, mutualistic species later join (Elias and Banin 2012).Strategies that intensify the effectiveness of group-level selection relative to individual-level selec-tion can provide alternative ways to eliminate mutant cells. If larger groups are more likely than smallerones to fission and produce offspring, group size becomes favored by multilevel selection. Since groupswith fewer mutants grow faster, size-dependent fragmentation rates allow large, species-diverse groupsto persist in the face of high mutation rates.Migration and size-dependence are only two plausible solutions to the dilemma of fragmentation inmultispecies communities. They are instructive in that each resolves the dilemma by relaxing one of thetwo conflicting selection pressures, but alternative solutions are, of course, entirely possible. One exam-ple is selective group extinction, which could have identical effects to selective group fission. Anotherexample is the segregation of cooperators and defectors during fission events, also known as associa-tive splitting (Bowles 2006; Eshelt and Cavalli-Sforza 1982; Haldane 1932; Hamilton 1975; Simon andPilosov 2016; Wright 1943). In fact, when Kondrashov (1994) first proposed that small propagule sizereduces mutation load, he pointed out that the effect will be most effective when mechanisms are in placeto ensure that the propagules are as homogeneous as possible. Hence, mutation load is minimized whenonly very related cells are recruited to form a propagule (i.e., associative splitting). Multicellular collec-tives are able to increase relatedness between propagule cells by creating segregated germ lines. Suchcomplex mechanisms, however, are not required to achieve associative splitting. Even simple aggrega-tions of cells can ensure that their propagules are maximally related by maintaining spatial structure. Ifdaughter cells remain close to their parents, then group fragments that break off from the mother groupwill be highly homogeneous. This process could potentially allow large daughter groups to eliminatemutation load, although such groups would presumably also be homogeneous in terms of their species89composition. Another simple mechanism is differential adhesion, where each species remains tightlyattached to one (or very few) of the other species, which could ensure multispecies propagules.By incorporating migration and size-dependent fission rates, we observed that cooperative interac-tions can be maintained even for multi-species communities and for very high cooperation costs andmutation rates. Both migration and size-dependent fission are simple and realistic processes that arepresent in many natural microbial communities. Hence, our model suggests that cross-species coopera-tive interactions could potentially be stable in natural systems. This result stands in contrast to the viewthat cooperation is likely to be unimportant in microbial communities (e.g., Foster and Bell 2012).The results we discussed are facilitated by the stochastic nature of our model. When newly born,small groups are created, sampling variation allows for the birth of groups that consist entirely of wild-type cells. In an infinite-size continuous limit, there would always be some fraction of mutants in everygroup. Because mutants grow faster than wild-type cells, even small fractions of mutants could posechallenges to the persistence of the community.Our model also encompasses single-species groups, and we can contrast it to previous studies offragmentation mode. We recover the classic result of Kondrashov (1994) (which has since been ex-panded by many other authors, e.g. Bergstrom and Pritchard 1998; Grosberg and Strathmann 1998;Roze and Michod 2001), viz. that by producing small propagules, single-species groups are able to per-sist in the presence of high mutation rates. We can also recover the result from Pichugin et al. (2019),Pichugin and Traulsen (2018), and Pichugin et al. (2017) who used an analytical model to calculate theoptimal fragmentation mode for simple single-species groups in the absence of mutations. Specifically,we recover their main finding that the optimal fragmentation mode depends on the rate functions usedfor cell birth and death (Fig. D.4). The main strength of our multilevel selection framework is that itis highly versatile: not only can we can combine these different strands of the previous literature ina single mathematical framework, but in addition we can use it to study multi-species groups, a topicwhich has received little attention so far. The versatility of our model also makes it easy to extend it infuture work to further investigate how more complicated rate functions (e.g., nonlinear size-dependencein fission rate or different types of fission-associated costs) can explain the variety of fragmentationmodes in nature.90Chapter 6ConclusionAbout thirty years ago there was much talk that geologists ought only to observe and nottheorise; and I well remember some one saying that at this rate a man might as well go into agravel-pit and count the pebbles and describe the colours. — Charles DarwinI conclude the thesis by reiterating the major results from the previous four chapters. For each ofthem, I also reflect upon what they meant to me, what lessons I have drawn from my research over thepast six years, and future direcitons.6.1 Chapter 2: Acculturation drives the evolution of intergroup conflictIn Chapter 2, I considered the question of intergroup conflict. Superficially, this may appear out ofplace in a thesis about cooperation. But intergroup conflict is, in some ways, a collective action problem(Gavrilets and Fortunato 2014). After all, warriors risk their lives to protect their group-mates, whichmeans they are, in a manner of speaking, “parochial” altruists (Choi and Bowles 2007; Garcı´a and Bergh2011). Furthermore, individuals in society allocate resources to intergroup conflict, which certainlyconstitutes a sacrifice when we consider potential alternative uses for those resources.In this chapter, I have argued that the cultural evolution of conflicts can be explained even in theabsence of benefits derived from resource acquisition. I attacked the problem by developing a modelof multi-level cultural selection: At the lower level, individuals are characterized by their tendency tobecome “warriors”, who participate in conflicts. At a higher level, groups are characterized by socialnorms or institutions such as the tendency to engage in conflicts and the tendency to spread their cultureupon defeating a rival group. The evolution of these traits results from the interplay of state-dependentevents at both levels: births and deaths for individuals, and fissions, extinctions, and conflicts for groups.As an alternative to the resource-acquisition hypothesis, I showed that conflicts can be maintainedby acculturation, a mechanism analogous to meiotic drive (in which a cultural trait spreads by promot-ing its own transmission at the expense of its host’s fitness). Acculturation is the imposition of thevictor’s culture on defeated groups following conflicts. The model shows that acculturation evolves asan alternative to directly eliminating defeated groups, and in doing so, makes way for the evolutionand maintenance of intergroup conflict. While I do not argue that this mechanism is necessarily the91historical cause for intergroup conflicts in human societies (I leave exercises of this sort to experts in thearea, such as anthropologists and historians), I believe that this model provides proof-of-concept thatthe cultural evolution of intergroup conflict does not imply that conflicts are somehow net-beneficial forthe individuals who engage in them.This chapter was an important learning experience for me for many reasons. It was the first completeproject—and the first publication—of my doctorate, as well as the first publication in which I was a leadauthor. It gave me a degree of confidence and self-reliance that I did not have before and, emotionally,marked a transition point in my PhD. It was in the course of this project that I started to think as amodeller. I began to appreciate the importance of anchoring eco-evolutionary models on an underlyingbirth-death process, a lesson that I learned from my three co-authors (Doebeli et al. 2017).In the course of doing this project, I learned about cultural evolution (Cavalli-Sforza and Feldman1981) and related it to the concept of genetic conflict (Burt and Trivers 2006), a topic which I hadexplored, during my Master’s degree, in the context of social insect societies (Quin˜ones et al. 2019).Finally, I also learned most of what I know about multi-level selection, under the mentorship of mysupervisor, Dr. Michael Doebeli, and from Dr. Burton Simon. Both of them authored some of the mostclear writing and illuminating perspectives on what is often a murky and controversial topic (including,among others, Simon 2010; Simon et al. 2013; Simon and Pilosov 2016; Van Veelen et al. 2014).The combination of cultural evolution and multilevel selection theory opens up an enormous num-ber of avenues for future research. In principle at least, it seems possible to model the combination ofinteractions between groups and within groups in some sort of multilevel game theory framework. Sucha framework could incorporate not only conflict between groups but also mutually beneficial interac-tions, such as trading. Trading can also provide a different mechanism for the horizontal transmission ofcultural variants between groups. It would also be interesting to investigate the effects of other types ofevents, such as group fusions, or alternative implementations of some of the events we already consider(we explore the consequences of alternative modes of fissioning in Chapter 5). Finally, even though ourmodel is one of cultural evolution, we considered only vertical transmission within groups. Culturalevolution theory provides many alternative modalities of cultural transmission, which could interact ininteresting ways with horizontal transmission of cultural traits between groups.6.2 Chapter 3: Cooperation can promote rescue or lead to evolutionarysuicide during environmental changeIn Chapter 3, I investigated how the evolution of cooperation affects adaptation to environmental change.As a starting point, I used the ecological public goods game (Gokhale and Hauert 2016; Hauert et al.2006a, 2008; Parvinen 2010; Wakano et al. 2009), which relates cooperation to population density. Inthis model family, high cooperation increases fitness, and this causes the population to grow; but withbig populations, the marginal benefits of cooperation decrease, and as a consequence, the level of co-operation decreases as well. This link between cooperation and population size maintains intermediatelevels of both. I investigated how a moving environmental optimum (Bell 2017; Lynch and Lande 1993)interacts with these ecological game dynamics. My hypothesis was that changes in the environment can92promote the evolution of cooperation, which compensates for decreases in population size and permitspopulations to keep up with moving environmental optima. Surprisingly, I discovered that, depending onhow the different factors that affect fitness are partitioned between viability and fecundity, cooperationcan also have the opposite effect and hinder rescue.I have sadly forgotten who the person was that first described the “peanut butter and jelly approach tomathematical biology”. The idea is simple: take two well-established theories, say, evolutionary rescueand public goods games. Then, mash them together, like peanut butter and jelly. I don’t know if I shouldbe embarrassed to admit this, but this chapter falls squarely within this time-honored tradition. I hadbeen reading a lot about moving optimum models—something about the dynamic interplay betweenecology and evolution sparked my interest—and having discussions about evolutionary rescue withDr. Matt Osmond (then a senior PhD student) and Dr. Hildegard Uecker. But it was when I watchedMatt’s exit seminar and learned about his research showing that predators can help prey adapt andpersist in changing environments (Osmond et al. 2017) that the metaphorical peanut butter and jellycame together in my mind. The effect of ecological interactions (such as predation) on the capacityof populations to adapt to environmental change is mediated by changes in population density. And Iknew, from discussions during our joint lab meetings with Dr. Christoph Hauert’s lab, that there is atwo-way interaction between cooperation and population density. I reached out to Matt with the ideafor this project and he was immediately receptive.On a technical level, this project taught me much about two different families of models: movingoptimum models and ecological public goods games. Both of these extended my appreciation for theinterplay between population density and evolution and helped me find common themes and bridgesbetween models that I previously considered to be unrelated. For example, the very same mechanismthat maintains cooperation in ecological public goods games also underlies the results of games withloners (Hauert et al. 2002) and with group-size variation (Pen˜a 2012; Pen˜a and No¨ldeke 2016).This article fits into a flourishing family of models that investigate the effect of incorporating eco-logical realism into moving optimum models. In particular, an increasing number of papers look intothe adaptive effect of intra- and interspecific interactions. The most commonly studied interactions arepredation and competition (Klausmeier et al. 2020); my model investigates intra-specific cooperation.This leaves a gap in our understanding that could be readily filled by future research: inter-specificcooperation (mutualism).6.3 Chapter 4: On the effect of evolving phenotype distributions onevolutionary diversificationIn this chapter, I considered the consequences of evolutionary branching when the population plays mul-tiple games simultaneously. Traditionally, games are studied in isolation; however, in nature, the sameindividual will engage in many different kinds of interactions, which increases the trait space dimensionand may lead to novel evolutionary outcomes. I found that there is a feedback between evolutionarydynamics and the population’s phenotypic distribution. As the population approaches the evolutionarybranching point, natural selection changes the population’s phenotypic distribution. Depending on the93direction of approach, the distribution at the equilibrium will be different. Furthermore, the shape ofthis distribution in turn influences the direction of evolutionary branching. Finally, depending on thedirection of branching, the final outcome of evolution may be different as well.What originally drew me into this project was an interest in the evolution of division of labor: amutual dependence between different strains, where each produces one public good from which theother benefits. For example, microbes in multi-species communities specialize in particular metabolitesand functions, so that the community as a whole depends on these networks of interdependence (Morris2015). Usually, the evolution of division of labor is explained by invoking costs and trade-offs betweenthe different products being exchanged (e.g., Wood and Komarova 2018). The idea is that, while it maybe cheap to produce a large amount of each public good, it could be disproportionately expensive ormetabolically impossible to produce large amounts of both public goods simultaneously. But in ourmodel, division of labor straightforwardly results from branching along two phenotypic dimensions,each representing a continuous snowdrift game (Doebeli et al. 2004). All that is required is that, initially(during the monomorphic stage of directional selection), production of both goods is either below orabove the equilibrium.Although some reviewers have found the results of this project to be fairly “obvious”, it was byfar the most technically challenging of my PhD. I had to develop multiple alternative implementationsof the model, from individual-based simulations to oligomorphic stochastic simulations and even asurprisingly complicated partial-differential simulation model. The project dragged on and off for years:while it superficially looked like a simple model, it ended up being unexpectedly complex. I don’t thinkI ever felt as stuck or demotivated in my academic life and that discouragement started affecting myself-confidence and self-image. Key to my eventual success was the contribution of my collaborator Dr.Koichi Ito, a post-doc in the lab who, with enormous generosity and patience, helped me struggle withthe cumbersome mathematics that had kept me from making progress. I learned more from this projectthan from any other, not only technically but also personally. It taught me how to persevere even whenI felt immovably stuck, and it taught me how important collaboration can be.For potential future directions, I would be interested in further exploring the effect of dimensionality.As microbes in multi-species communities specialize in particular metabolites and functions, communi-ties may give rise to networks of mutual dependency (Black Queen evolution, Morris 2015). It would beinteresting to understand under what conditions the simultaneous coevolution of multiple public goodstraits can lead to these mutually interdependent communities.6.4 Chapter 5: Multilevel selection favors fragmentation modes thatmaintain cooperative interactions in multispecies communitiesThe most elementary property of life is the capacity to reproduce. Every living being reproduces, yetacross the tree of life, there are many diverse reproduction strategies, from single-cell reproduction (e.g.,the production of gametes, as in humans) to the production of large multicellular fragments. Becauseorganisms that have single-celled gametes have less within-organism variation than organisms withmany-celled propagules, natural selection would be more effective at removing deleterious mutations94in the former (Bergstrom and Pritchard 1998; Grosberg and Strathmann 1998; Kondrashov 1994; Rozeand Michod 2001). Hence, single-celled bottlenecks allow populations to reduce mutation load.But what about the mode of reproduction of multispecies collectives of organisms, such as microbialbiofilms? To answer this question, we put together a multilevel selection individual-based model, whereindividual cells reproduce and die within groups which can themselves undergo fission or extinction.We found that communities with many species face a reproductive dilemma. On the one hand, to persistin the face of mutations, they must rely on small reproductive bottlenecks (not unlike single-speciesorganisms). On the other hand, due to stochastic sampling, small daughter groups lack the speciesdiversity that multispecies communities rely on for ecological success. This dilemma—maintainingspecies diversity while reducing mutation load—can be solved in different ways. Our model illustratesthis with two alternative solutions: migration (which makes small groups viable) and size-dependentreproduction (which reduces load even within large groups).The idea for this project came up after a series of brainstorming sessions with my collaborator Dr.Simon van Vliet, then a post-doc at the Doebeli lab. Simon had just finished working on a fascinatingmultilevel selection model investigating whether or not cooperation between microbes in the gut mi-crobiome could affect the evolution of the host (Van Vliet and Doebeli 2019). The implementation ofSimon’s model had, in turn, been inspired by my earlier project (Henriques et al. 2019) (Chapter 2 inthis thesis), so we both thought it would be fun to try and join forces. Diversity in microbial commu-nities is Simon’s specialty, and I was interested in investigating the consequences of different fissioningmodes in a multilevel selection model. Working with Simon was incredibly fun and I learned a lot aboutmicrobial communities and the surprisingly controversial ongoing debate about the role of cooperativeinteractions in biofilms.A model like this can be extended in many different directions. 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For each value of p, f (p) is the fraction of focal groupindividuals with trait value p that are allocated to the first daughter group (all other individuals with traitvalue p are allocated to the second daughter group). We used the following fissioning function:f (p) =(p+∆pmax− p˜)s(p+∆pmax− p˜)s+(−p+∆pmax+ p˜)s . (A.3)Here, p˜ is the modal value of p in the parent group. The value of ∆pmax is related to the range of traitvalues in the parent group. Ignoring values of p with negligible densities (i.e., densities below a smallfraction ι 1 of the total group density), this value is given by ∆pmax =max(p˜−min(p),max(p)− p˜).111s=0.01s=0.1s=1s=10s=1000121p~ − ∆pmax p~ p~ + ∆pmaxpf(p)Ags=0.001gs=0.5gs=1gs=20.000.250.500.751.00−10 −5 0 5 10yj−yig(yi, y j)BFigure A.1: Fissioning function (A, Eq. A.3) and probability of winning a conflict (B, Eq. A.4) for differentvalues of the parameters s and gs, respectively.The parameter s determines the steepness of the fissioning function at its inflexion point. It regulatesthe amount of assortment during fissioning, from s = 0 (no difference between the daughter groups) tos→ ∞ (individuals assort themselves into daughter groups based on their distance—in trait space—tothe most common trait value in their group). Fig. A.1A shows the fissioning function f (p) for differentvalues of s.A.1.3 Probability of winning conflictOnce a focal group i initiates a conflict, a rival group j is chosen at random. The focal group wins theconflict with probability:g(yi,y j) =[1+ e(y j−yi)/gs]−1. (A.4)Otherwise, the rival group wins. The parameter gs determines the steepness of g(yi,y j), with shallowerfunctions allowing for groups with fewer warriors to have a better chance of winning conflicts. Fig. A.1Bshows the probability of winning g(yi,y j) for different values of gs.A.2 Basic model outcomesThis section provides more detailed arguments for basic dynamic properties of the 2-level model usedin the main text.A.2.1 Within-group equilibriumFor simplicity, consider a group that is monomorphic in p. The full deterministic system of equationsgoverning within-group population dynamics is given in the Methods section of the main text (Eq. 2.3),but in the monomorphic case it simplifies to a system of two equations describing the time derivatives1120.2 0.4 0.6 0.805101520xyn = x+y^^^ ^ ^0.1 0.2 0.3 0.4 0.5 0.6 0.705101520p (warrior production) p (warrior production)DensityA BFigure A.2: Eq.uilibrium densities of the within-group dynamics for a monomorphic group. The green line(nˆ) shows the solution of Eq. A.5; the blue and orange lines show the within-group equilibrium densities ofshepherds (xˆ) and warriors (yˆ), respectively. A: Default parameters (see Methods). B: Alternative parameters:b1 = 0.45,b2 = 2,b3 = 0.2,d = 0.5. Because 2b1 > d, the group density (green curve) is decelerating for lowvalues of p. This implies that the initial evolution of p causes sharper decreases in group density.of shepherd and warrior density (x˙(p) and y˙(p), respectively). By setting the right-hand side of theseequations to zero, we can calculate the equilibrium density nˆ(p) = xˆ(p)+ yˆ(p):nˆ(p) =1b3log[b2(1− p)d−b1(1− p)], (A.5)with xˆ(p) = (1− p)nˆ(p) and yˆ(p) = pnˆ(p) (Fig. A.2A). Thus, groups that produce higher fractionsof warriors support fewer individuals, because warriors are non-reproductive. The choice of birth anddeath rate functions we used also entails a maximum value of p above which the group is not sustainableand goes extinct. This critical value is given by setting nˆ(p) (Eq. A.5) to zero:pcrit =b2+b1−db2+b1. (A.6)For the reference parameters (see Methods), pcrit ≈ 0.839 (Fig. A.4A).A.2.2 Between-group equilibriumBy equating the rate at which new groups are created (by fission, F(x,y), Eq. A.2) to the rate at whichgroups are removed from the population (either by extinction, E(x,y,N), Eq. A.1), or by conflict,G(q,N)(1− r), Eq. 2.2, main text), we obtain:Nˆ =n(p)2φn(p)γq(1− r)+ ε . (A.7)113A.3 Time-scale of loss of within-group variationOne basic requirement for the evolution of warrior production in our model is that group events needto happen at a fast enough rate compared to individual-level events. Intuitively, if individual-level ratesare too fast, the within-group dynamics leads to the loss of warrior production in all groups, and, as aconsequence, there is no between-group variation in warrior production on which group selection couldact. To get a sense of the required magnitude of the relative time scales of individual and group levelevents, we make the following argument.Consider a group with a bimorphic population consisting of some individuals with cultural traitp, n(p) = x(p) + y(p), and some other individuals with a larger value of cultural trait p′ = p+∆p,n(p′) = x(p′)+ y(p′). The growth rates of the two variants, w(p) and w(p′), are given by:w(p) =(b1+b2e−(n(p)+n(p′))b3)(1− p)−d,w(p′) =(b1+b2e−(n(p)+n(p′))b3)(1− p′)−d.(A.8)The decrease in frequency of p′ follows from the dynamics of the in-group birth-death process. Let ρ =n(p′)n(p) . Assuming that the dynamics of ρ have only a negligible effect on the birth rate b= b(n(p)+n(p′)),we can track the rate of change in ρ:d ln(ρ)dt=dn(p′)n(p′)dt− dn(p)n(p)dt= b∆p (A.9)Thus, the time-scale at which variation within a bimorphic group is lost is roughly proportional to 1/∆p.Group selection can only operate when group events happen faster than this time-scale. In particular,sufficiently frequent fission events can maintain or even increase the in-group fraction of higher-p indi-viduals. Since the p′ population decreases in frequency due to the individual-level dynamics, we willhave n(p′) < n(p); then, according to the fission function (Eq. A.3), most p′ variants will end up in asingle daughter group. In addition, this p′-enriched group is also filled by some p individuals. Suffi-ciently frequent fissioning events can bring the fraction of p′ close to 1/2, thus allowing the larger-pvariant to take a hold in a progressively smaller fraction of groups.A similar argument holds for polymorphic groups. In this case, the time-scale at which variationis lost is again roughly proportional to 1/∆p, where ∆p is the smallest phenotypic distance betweenvariants of p (Fig. A.3).A.4 Selection gradients at the between-group levelIn this section, we study the group-level evolutionary dynamics, in order to understand the evolutionarytrends of the traits (p,q,r). Consider the rate of change in the number of rare groups N(p′,q′,r′) in abackground of N(p,q,r) groups.∂N(p′,q′,r′, t)∂ t= N(p′,q′,r′, t)(−ε Nn(p′)+φn(p′)+G)(A.10)1142505007501000125050 100 150 2001 ∆ pτ 0.95Figure A.3: In the absence of group events, the time-scale at which variation is lost within a polymorphic groupis, roughly, linearly proportional to 1/∆p. The figure shows the time τ0.95 required for 95 percent of the group’sdensity to be concentrated at p = 0. The initial frequencies follow a normal distribution N (0.5,0.1) in p (dis-cretized into bins of width ∆p), truncated at zero and pcrit (Eq. A.6), with initial group density Σini(pi) = 1.Mutations were neglected; otherwise the dynamics of xi(t) and yi(t) follow Eq. 3 (Main text), with the defaultparameters described in Methods. The points span three orders of magnitude (from ∆p = 0.005 to ∆p = 0.1)The first two terms describe the spontaneous extinction and fission events and their rates depend on thegroup size n(p), which in turn depends on p. These two terms favor evolution towards faster growinggroups and thus, smaller p, because smaller p generate larger n(p). The third term, G, describes gain orreduction in the number of primed groups resulting from conflicts. The rate of conflicts per group withtraits (p′,q′,r′) isγ(q′+q)N, (A.11)which represents the sum of conflicts initiated by the primed group, q′N and of conflicts initiated by anyof the N resident groups with the primed group, qN.From the definition Eq. A.4, the probability to win a conflict for a group with a small excess in thenumber of warriors ∆y is12+∆ygs. (A.12)ThusG = γ(q′+q)Nr′(12+∆ygs)− γ(q′+q)N(12− ∆ygs)= (A.13)Nγ(∆q+2q)12gs(2∆y(1+ r′)−gs(1− r′)), (A.14)where the first term describes the gain in the number of primed groups as a result of acculturation afterwinning the conflict, while the second term describes the reduction in the number of primed groups dueto a loss in a conflict. Analyzing this term allows us to make qualitative conclusions about evolutionarytrends of (p,q,r):115• Without acculturation, r = 0, the level of engagement q will evolve towards zero, making the Gterm irrelevant. This is clear for small ∆y, and for finite ∆y the corresponding term (g− 1/2 inEq. A.4) never exceeds 1/2. After this, the first two terms in Eq. A.10 drive the increase in thein-group population growth rate, which requires the reduction of p to zero.• When r is allowed to evolve, it grows to its maximum, r = 1. At the same time, y also growsalmost to its maximum which is achieved at popt ≈ 0.514 (see the main text). However, y maynot quite reach its maximum (and p may stay a bit below popt) due to the effect of the death andfission terms in Eq. A.10, which favor faster growing groups and thus smaller p.• The selective pressure on engagement may vary through the course of evolution depending onparticular evolutionary trajectories of y and r. When the term in the brackets of Eq. A.14 isnegative, the engagement level should decrease, and vice versa.• However, when both y and r reach their maxima, the effect of the term G disappears, selection onengagement ceases, and engagement becomes an essentially randomly evolving trait. However,when fluctuations in p (and subsequently y) and r make the brackets in Eq. A.14 non-negligible,engagement is under selection again and may exhibit directional changes.A.5 Costly acculturationIn the main version of the model, group conflicts can be resolved in one of two ways: first, with aprobability equal to the victor’s trait value r, the defeated group undergoes acculturation; second, withprobability 1− r, the defeated group is eliminated from the population. At equilibrium, in the scenarioof ‘coevolution toward conflict’ (Fig. 2.3C), groups still expend costly resources in the production ofconflict specialists (warriors), but conflicts are resolved without mortality. Here, we analyze the effectsof ‘costly’ acculturation, i.e. the possibility that acculturation is accompanied by the death of part of thegroup’s individuals.We simulated three different scenarios, corresponding to different types of cost (Fig. A.5A).• Type i: a fraction pi of the defeated group’s warriors are eliminated prior to the process of ac-culturation. This could correspond to a situation in which the group surrenders after incurring anumber of casualties.• Type ii: similar to type i, but the cost pi is a fraction of all individuals in the group, warriors andshepherds alike. Both civilians and combatants are indiscriminately targeted during the conflict.• Type iii: the defeated group loses a fraction pi of warriors; additionally, the victorious group losesan amount of warriors equal to half the number of casualties of the defeated group.Although these three scenarios do not exhaust the possibilities, they show that, even if acculturationis associated with moderate levels of mortality, groups can still evolve to engage in conflicts. Theequilibrium level of engagement (qˆ) evolves to nonzero values (which decrease with the severity of the116cost, Fig. A.5A). For example, Fig. A.5B shows the coevolutionary dynamics of a typical populationwith type i cost, pi = 0.2. Despite the high mortality, groups evolve intermediate levels of engagement(qˆ≈ 0.133). Furthermore, conflicts still occur frequently enough that groups invest heavily in producingwarriors (as illustrated by the red line in Fig. A.5B).Not all scenarios are equally costly, and the equilibrium levels of engagement reflect these differ-ences. When only warriors from the defeated group are eliminated (type i), the equilibrium engagementis fairly high even for extremely high values of pi (Fig. A.5A, type i). Although this may seem surpris-ing, note that even substantial losses of warriors can be replenished quickly by within-group dynamics.Costs of types ii and iii are far more severe. Whereas low levels of pi produce appreciable levels ofengagement, the same is not true for values equal to or above ca. 10 percent mortality (Fig. A.5A, typesii, iii). In both cases, for the same value of pi , the number of casualties is significantly higher than inscenario i (and, in the case of type iii, both groups bear the costs). Additionally, in type ii costs, the deathof shepherds significantly reduces the group’s growth rate (the same is not true when only warriors die).In stark contrast to type i, the death of shepherds reduces the stock of potential parents from which thewarrior pool may be replenished. In all cases, the equilibrium level of engagement is higher than whatwould be reached without the evolution of acculturation (i.e. if r were held constant at zero; see greyline in Fig. A.5A).To underline the point that selection is operating to favor high engagement, we set up a simplecompetition experiment. We ran simulations with initial conditions equal to the end-state of Fig. A.5A),type i, with pi = 0.2. We randomly replaced each group’s engagement value with either q = 0.154 orq = 14 × 0.154 (0.154 being the average value of q at equilibrium for type i, pi = 0.2, cf. Fig. A.5A).We also removed the possibility of mutations in q. Thus, in this simple experiment, there are only twoalleles of q: high engagement and low engagement. In 38 out of 39 replicates, the high-engagementstrain out-competed the low-engagement strain, demonstrating selection in favor of high engagement(we note that because of the finite number of groups, the system is stochastic, which explains the onerun in which the low engagement strain went to fixation).To understand the dynamics of costly acculturation, note that at the group level, the introduction ofcasualties (whether among warriors or shepherds in the defeated group) has a similar qualitative effect asa reduction in r. An acculturated group is still produced, and, over time, due to within-group dynamics,it will grow to its equilibrium size. However, before it reaches equilibrium, it has a lower probabilityof splitting and higher probability of death as a result of conflicts or extinction. Thus, the introductionof pi > 0 is equivalent to a lower number of acculturated normal size groups, i.e. lower r. In sectionA.4, we estimated the selection gradients of the group-level dynamics for the basic version of the model.To accommodate costly acculturation, Eq. A.14 can be modified by multiplying r by a monotonouslydecreasing function of pi , r→ h(pi)r, reflecting a deficit in normal size groups. To underscore this point,we simulated our main model (without costly acculturation) with the restriction that acculturation islimited to the interval 0≤ r ≤ 0.85. The resulting dynamics (Fig. A.5D) resemble those observed in thecostly acculturation scenarios (Fig. A.5B).117A.6 Supplementary figures0.000.250.500.751.000 5 10 15 20 25Time (thousands)Trait valueTraitpqrFigure A.4: Coevolution toward conflict can occur for different growth rate functional forms, including when nˆ(p)is decelerating for low values of p (which occurs when 2b1 > d). The figure shows example population dynamicsfor b1 = 0.45,b2 = 2,b3 = 0.2,d = 0.5 (the same parameters that are used for Fig. A.2B). All other parametersare identical to the default parameters. Lines indicate population-wide rolling averages of group values (q, r) orof mean group values (p). Dark ribbons show rolling standard deviations and light ribbons encompass the fullrange of values.118lll lll lll lll lll lll llllllllllll lll lll lllllllll lll lll lll lll lll lllllllll lll lll lll lll llllll lll lll lll lll lll lllllllll llllll lll lll lllType iiiType iiType i0.010.050.100.200.300.400.500.00.20.40.00.20.40.00.20.4Cost (pi)Equilibrium trait valueTrait l lp qA0.000.250.500.751.000 5 10 15 20 25Time (thousands)Trait valueTraitpqrB0.000.250.500.751.000 1 2 3 4 5Time (thousands)Frequency of  high−q strainC0.000.250.500.751.000 5 10 15 20 25Time (thousands)Trait valueTraitpqrD Figure A.5: Coevolution toward conflict can occur for costly acculturation. A: equilibrium trait values for en-gagement (qˆ, in green) and warrior production ( pˆ, in red), calculated as the time-averaged median values of qand p for time 15–20 thousand, for different values of cost (pi). For each value of cost, we show three replicates.Bars indicate the 25% and 75% quantiles. For details about the three types of cost (i, ii, iii) see section A.5. Forcomparison, the grey horizontal line shows the median value of qˆ (averaged over three replicates) when r is heldconstant at zero. Light grey bars indicate the 25% and 75% quantiles. B: example dynamics for a type i populationwith pi = 0.2. C: competition experiment between high-engagement and low-engagement strains (see section A.5for details). C: example dynamics for a population without cost of acculturation, but with acculturation limited tothe interval 0≤ r ≤ 0.85. For figures B–D all other parameters are identical to the default parameters. In figuresB and D, lines indicate population-wide rolling averages of group values (q, r) or of mean group values (p); Darkribbons show rolling standard deviations and light ribbons encompass the full range of values.1190.000.250.500.751.00Trait valueTraitpqr0.000.250.500.751.00Traitvalue0.000.250.500.751.000 5 10 15 20 25Time (thousands)TraitvalueABCFigure A.6: Sufficiently rare group events lead to a peaceful equilibrium. The three panels show example popu-lation dynamics when the group rate parameters (ε , φ , γ) are one (A), two (B), or three (C) orders of magnitudesmaller than the default group rate parameters described in the Methods. (All other parameters are identical tothe default parameters.) Lines indicate population-wide rolling averages of group values (q, r) or of mean groupvalues (p). Dark ribbons show rolling standard deviations and light ribbons encompass the full range of values.1200.000.250.500.751.00Trait valueTraitpqr0.000.250.500.751.000 5 10 15 20 25Time (thousands)Trait valueABFigure A.7: Coevolution toward conflict occurs even for populations with low numbers of groups. A: Exampledynamics when b1 = 0.45,b2 = 3,b3 = 0.2 (all other parameters are identical to the default parameters described inthe Methods.) The number of groups at equilibrium (mean number of groups between time 20000 and time 25000)is 23.747. B: Example dynamics when φ = 0.005 (all other parameters are identical to the default parametersdescribed in the Methods.) The number of groups at equilibrium (mean number of groups between time 20000and time 25000) is 17.399. In B, population-wide extinction was averted by artificially preventing extinctionwhen the population was down to one remaining group. In both panels, lines indicate population-wide rollingaverages of group values (q, r) or of mean group values (p). Dark ribbons show rolling standard deviations andlight ribbons encompass the full range of values.1210.000.250.500.751.000 5 10 15 20 25Time (thousands)Trait valueTraitpqrFigure A.8: Example dynamics when warriors can also reproduce, at a cost c = 0.5. All other parameters areidentical to the default parameters described in the Methods. Lines indicate population-wide rolling averages ofgroup values (q, r) or of mean group values (p). Dark ribbons show rolling standard deviations and light ribbonsencompass the full range of values.122Appendix BSupplementary Information for Chapter 3B.1 Canonical equation modified for non-overlapping generationsThe canonical equation of adaptive dynamics, derived in Dieckmann and Law (1996), describes thedynamics of the mean evolutionary path that results from a continuous-time birth-death process. Herewe follow the same steps to obtain a recurrence equation which resembles the canonical equation, butis more appropriate to describe populations with discrete, non-overlapping generations.Let x represent the current mean trait value of the evolving population (which is assumed to be quasi-monomorphic). The random variable V (x′,x) determines whether a newborn individual with trait valuex′ will reach reproductive age. This will happen with probability V (x′,x), referred to as viability, so thatV (x′,x) ∼ Bernoulli(V (x′,x)). When she reproduces, her expected number of offspring (fertility) willbe F(x′,x); the realized number of offspring is random variableF (x′,x) (independent of V (x′,x). Fol-lowing conventional practice, we assume in the main manuscript thatF (x′,x)∼ Poisson(F(x′,x)), butthe appropriate distribution will depend on the species being studied (for example, in vertebrate species,the most appropriate distribution to describe variation in number of offspring may be the two-parameterPoisson-Consul distribution, also known as generalized Poisson, Kendall et al. 2010). Individuals aresemelparous (they die upon giving birth). Thus, the Wrightian fitness is W (x′,x) =V (x′,x) ·F(x′,x).B.1.1 Stochastic description of trait substitution sequencesWe assume that a successful mutation will rise to fixation before a new mutation occurs, such that nomore than one mutation segregates in the population. In reality, even the most successful mutations willtake many generations to achieve fixation, but we model the process as if successful mutations achievefixation instantly. This corresponds to a separation of time-scales between ecological and evolutionarydynamics (Geritz et al. 1998).Based on these assumptions, the evolutionary dynamics consists of a sequence of substitution eventsin which a mutant x′ replaces a resident x. Dieckmann and Law (1996) call this the trait substitutionsequence. Because mutation and selection depend only on the present state of the population, the traitsubstitution sequence is Markovian. To describe this sequence of substitutions, we first need to calculatethe probability per generation of the trait substitution x→ x′, which we call transition probability, Px′,x.123Transition probabilitiesBased on these assumptions, the transition probability per generation from x → x′ is given by twofactors: first, the probability per generationM (x′,x) that the mutant enters the population; second, theprobability of fixationS (x′,x):Px′,x =M (x′,x) ·S (x′,x). (B.1)In a given generation, mutations can occur in any newborn individual. The expected number of new-borns is W (x,x)nˆ(x), where nˆ(x) is the equilibrium population size (censused before viability selection).The fraction of births that give rise to mutations is µ . The probability distribution function of mutanttrait values around the resident is M(x′− x). Collecting these terms we obtainM (x′,x) =W (x,x)nˆ(x)µM(x′− x), (B.2)which, for small µ , approximates the probability per generation that one mutant of type x′ enters thepopulation.The probability of fixationS (x′,x) depends on demographic stochasticity. Mutants appear as singleindividuals, and thus fixation is dependent on avoiding stochastic extinction when rare. We make twoassumptions: first, that the resident population size nˆ(x) is large enough that there’s a negligible riskthat the population will go accidentally extinct; second, that as long as an advantageous mutant escapesstochastic extinction, it will achieve fixation (‘invasion implies fixation’).The probability of avoiding stochastic extinction can be calculated using discrete-time branchingprocess theory (Allen 2010). The key parameters are the expectation of the per capita number of mutantoffspring—i.e., the mutant’s Wrightian fitness, W (x′,x)—and the variance of the same quantity—i.e.,the mutant’s reproductive variance, σ2W (x′,x):W (x′,x) = E[V (x′,x)F (x′,x)] =V (x′,x)F(x′,x) (B.3)σ2W (x′,x) = Var[V (x′,x)F (x′,x)]. (B.4)We will also represent the resident’s reproductive variance as σ2W ≡ σ2W (x,x) = Var[V (x,x)F (x,x)].Following example 4.4 in page 172 of Allen (2010), the fixation probability for a mutant that startsas a single copy is approximately (to second order, assuming that the single mutant’s lineage has aprobability of extinction close to one) equal toS (x′,x) =1− exp(−2(W (x′,x)−1)σ2W (x′,x))if W (x′,x)> 10 otherwise.(B.5)Eq. B.5 is also used in other variants of the canonical equation of adaptive dynamics (e.g., Durinxet al. 2008; Metz and Kovel 2013).124Stochastic dynamicsUsing the transition probabilities, we can describe the dynamics of x in evolutionary time. We define thetransition rate px′,x as the limit of Px′,x/∆t for small ∆t. The unit of time is one generation (i.e., ∆t = 1),which is very small in relation to the evolutionary time scale, px′,x ≈ Px′,x. Hence, we can describe thedynamics of x in continuous time asddtPr(x, t) =∫Px,x′ Pr(x′, t)−Px′,x Pr(x, t) dx′, (B.6)where Pr(x, t) denotes the probability that the trait value is x at time t. The same conclusion can bereached, much more rigorously, by rescaling time in units of µ nˆ and taking the small µ and large nˆlimits (see Remark 1 in Champagnat 2006).B.1.2 Mean pathImagine a large number N of trait substitution sequences, x1(t),x2(t), . . . ,xN(t). The mean path is thendefined as〈x〉(t) =∫xPr(x, t)dt. (B.7)Taking the derivative, using equation B.6, and simplifying we can thus describe the dynamics of themean path asddt〈x〉=∫xddtPr(x, t)dt=∫x∫Px,x′ Pr(x′, t)−Px′,x Pr(x, t)dx′dx=∫∫(x′− x)Px′,x Pr(x, t)dx′dx. (B.8)Eq. B.8 is, by definition, the mean of the quantity called the first jump moment,a(x) =∫(x′− x)Px′,xdx′. (B.9)Thus,ddt〈x〉= 〈a(x)〉(t). (B.10)As long as the deviations of the stochastic path from the mean path are small or the nonlinearity inthe first jump moment is weak, such that we can treat the first jump moment as being approximatelylinear, we can approximate the mean path as (Dieckmann and Law 1996, p. 593 and references within)ddt〈x〉 ≈ a(〈x〉(t)). (B.11)We call this approximation the deterministic path.125B.1.3 Deterministic pathSubstituting Eq. B.7 into Eq. B.11, using Eqs. B.1, B.2, and B.5 (and dropping the angle brackets) weobtaindxdt=W (x,x)nˆ(x)µ∫+(x′− x)M(x′− x)[1− exp(−2(W (x′,x)−1)σ2W (x′,x))]dx′, (B.12)where the range of integration has been restricted to advantageous mutations (i.e., W (x′,x)> 1).Note that the rate of evolution depends on the fertility and viability of all possible advantageousmutant trait values, x′, as indicated by the range of integration. In order to transform this global couplinginto a local one we apply a Taylor expansion to the probability of fixation (the term within squarebrackets), when x′ ≈ x:1− exp(−2(W (x′,x)−1)σ2W (x′,x))≈ (x′− x) 2σ2WdW (x′,x)dx′∣∣∣∣x′=x. (B.13)This approximation implies that the mutational step size (x′− x) is very small compared to the otherterms. In particular, the approximation ceases to be valid if the reproductive variance σ2W approachesthe order of the mutation size.Substituting this result into Eq. B.12:dxdt= 21σ2Wnˆ(x)µdW (x′,x)dx′∣∣∣∣x′=x∫+(x′− x)2M(x′− x)dx′. (B.14)Finally, we focus on the mutation process. As long as dW/dx is not zero there is directional selectionto leading order; then, if M(x′− x) decays fast enough as x′ departs from x (i.e., small mutations), weonly need to consider the effect of selection up to leading order, which means that, since mutations aresymmetric about x, mutations are advantageous with a probability of 1/2. Since the density of beneficialand deleterious mutations are equal, we can remove the restriction to the domain of integration bymultiplying the right-hand side of Eq. B.14 by 1/2. By definition, the second moment of M(x′− x) isthe variance in mutation size, σ2µ =∫(x′− x)2M(x′− x)dx′. This concludes our calculation:dxdt=1σ2Wnˆ(x)µσ2µdW (x′,x)dx′∣∣∣∣x′=x. (B.15)The main qualitative difference between this and the traditional (overlapping generations) canonicalequation of adaptive dynamics (Dieckmann and Law 1996) is that the rate of evolution is inverselyproportional to the reproductive variance (i.e., the variance of the per capita number of offspring). Thisis similar to other variants of the canonical equation of adaptive dynamics (e.g., Durinx et al. 2008; Metzand Kovel 2013).If we assume that the number of offspring follows a Poisson distribution, F (x′,x) ∼Poisson(F(x′,x)), thenσ2W (x′,x) =W (x′,x)(1+F(x′,x)−W (x′,x)). (B.16)126It follows that, at ecological equilibrium, σ2W = F(x,x). Then the rate of evolution is proportional to1/F(x,x) (or equivalently, since W (x,x) =V (x,x) ·F(x,x) = 1, to the viability, V (x,x)).B.2 Equilibrium cooperation for a constant lagWhen the lag is constant,L (y) =L , the selection gradient of cooperation,S (x), is given by Eq. 3.11(main text). When this gradient is equal to zero, the evolutionary dynamics of cooperation comes toan equilibrium, x?. Although an explicit expression for x? is impossible to calculate, in this section weexplore some properties of this equilibrium. First, we will show that (at least for k ≥ 1) there is onlyone equilibrium value of cooperation. Then, we will show and that this equilibrium is smaller than xcrit.Finally, we will explore how the equilibrium value depends on parameters such as M and p.B.2.1 Proof: there is only one equilibrium value of cooperation below xcritTo prove there is only one equilibrium value of cooperation below xcrit, we show that the selectiongradient i) is positive at small enough x, ii) declines monotonically with x while x < xcrit and iii) is nogreater than zero at x = xcrit. Thus there is only one value of x below (or at) xcrit whereS (x) = 0. Thisimplies any x below xcrit will converge to this one equilibrium.The selection gradient is given by Eq. 3.11 in the main text,S (x) =1− (1− p)nˆ(x)− ckpxknˆ(x)pxnˆ(x). (B.17)The denominator is always positive. As x approaches xmin from above nˆ(x) goes to zero, meaningthat for small enough x the numerator approaches 1. Thus the selection gradient is positive at small x,implying that cooperation increases from small values.The derivative of the selection gradient isdS (x)dx=−n(1− p˜n+ c(k−1)knpxk)+ x(1− p˜n+ p˜n log p˜n)dndxn2 px2(B.18)where p˜≡ (1− p) and n≡ nˆ(x), for brevity. The denominator is always positive. The first term in thenumerator is always positive when k≥ 1. Finally, the second term in the numerator is positive wheneverdn/dx > 0, which is true up to xcrit. Thus, for k ≥ 1, the selection gradient declines monotonically withx below xcrit.When x = xcrit the selection gradient becomes −[nˆ(x)p− 1+(1− p)n](ck)1/k/(nˆ(x)p). This is nogreater than zero whenever nˆ(x)p ≥ 1− (1− p)nˆ(x). Using the properties of a binomial distribution wecan show that this is indeed always true:nˆ(x)p =nˆ(x)∑k=1k Pr(K = k)≥nˆ(x)∑k=1Pr(K = k) = Pr(K ≥ 1) = 1− (1− p)nˆ(x). (B.19)Thus, the selection gradient is never positive at xcrit, implying that either selection will cause cooperation127to decline from xcrit or that xcrit is a singular point.B.2.2 Effect of lag and p on the equilibrium value of cooperationAlthough it is impossible to give an explicit expression for x?, we can use implicit differentiation tostudy how x? changes with respect to different parameters (such as p and M).Effect of p on the equilibrium value of cooperationStarting with the selection gradient of cooperation (Eq. 3.11, main text), we treat x as a function of p:S (x(p)). At equilibrium, S(x?(p)) = 0; we differentiate both sides of this equation with respect to p,and solve for x?′(p):x?′(p) =−nx?[1− (1− p)n−1(1+(n−1)p)]p(n(1− p˜n+ c(k−1)knpx?k)+ x?(1− p˜n+ p˜n log(p˜n)) dndx?) , (B.20)where n≡ nˆ(x?) and p˜≡ (1− p), for brevity.The denominator is always positive as long as k≥ 1 and provided that dn/dx? > 0, which is true forx? < xcrit (see section B.2.1). The numerator, in turn, is negative if(n−1)p < (1− p)1−n−1, which isalways true provided that n > 1.Therefore, x?′(p) < 0, i.e., the equilibrium value of cooperation decreases with larger values of p.This result matches numerical simulations, such as Fig. 3.1B (in the main text).Effect of M on the equilibrium value of cooperationStarting with the selection gradient of cooperation (Eq. 3.11, main text), we treat x as a function of M:S (x(M)). At equilibrium, S(x?(M)) = 0; we differentiate both sides of this equation with respect to M,and solve for x?′(M):x?′(M) =−x?(1− p˜n+ p˜n log(p˜n)) ∂n∂Mn(1− p˜n+ c(k−1)knpx?k)+ x?(1− p˜n+ p˜n log(p˜n)) ∂n∂x?, (B.21)where n≡ nˆ(x?) and p˜≡ (1− p), for brevity.The sign of the numerator is −∂n/∂M, which is negative (from Eq 3.7), while the denominator ispositive provided that k ≥ 1 and ∂n/∂x? > 0 (which is true up to xcrit). Since M decreases with the lag,the equilibrium value of cooperation increases with the lag. This result matches numerical simulations,such as Fig. 3.1B (in the main text).B.3 Some remarks on post-branching (polymorphic) dynamicsAs described in section 3.3.2 of the main text, the singular point x? can be a branching point if∂ 2W (x′|x)∂ (x′)2∣∣∣∣x′=x=x?> 0. (B.22)128For notational simplicity, let us write B(x) as short-hand for B(x|x, nˆ(x)), and let us ignore M(y) sinceit does not depend on x. Then, we can write W (x′|x) = B(x′)C(x′)D(nˆ(x)), and the condition abovebecomes equal toD(nˆ(x?))(2B′(x?)C′(x?)+C(x?)B′′(x?)+B(x?)C′′(x?))> 0, (B.23)where primed functions represent derivatives with respect to x. The first factor (density) is alwayspositive. B′(x) is also always positive. When the benefits are linear, B′′(x) = 0. Thus, we obtain thebranching condition from Eq. 3.12 in the main text (for more details see the accompanying Mathematicanotebook, deposited in Dryad, Henriques and Osmond 2020):B(x?)C′′(x?)+2B′(x?)C′(x?)> 0⇐⇒k(1+ cx?k)< 1. (B.24)This means that the singular strategy will be a branching point only when the cost is sufficiently weak(small c) and convex (small k), and the equilibrium level of cooperation is low. If that is the case, thenafter converging to the singular point, the population will branch into two separate strains (with traitvalues x1 and x2), which will evolve in opposite directions (Fig. 3.2B, main text). Individual-basedsimulations illustrate these two behaviors (Fig. B.1).As mentioned in the main text, individual-based simulations indicate that the parameter range underwhich evolutionary branching occurs and results in a stable dimorphic equilibrium is narrower than theanalytical predictions suggest. In particular the population size of the cooperator is often so small thatstochastic extinction of that branch quickly occurs. Nonetheless, for the sake of completeness, here wepresent some numerical results regarding the deterministic dynamics after branching.B.3.1 Polymorphic dynamics for a constant lagFor simplicity, we will first focus on the case of a constant lag, L (y) =L , in a constant environment,v = 0. Because we are concerned with adaptation in x and the lag is fixed, we will drop the dependenceon y for the remainder of this section.The benefits of cooperation now depend on both phenotypes. In a group with g1 type 1 residentsand g2 type 2 residents the benefit to the focal individual (with phenotype x′) isB2(x′|x1,x2,g1,g2) = r1+g1+g2 (x′+g1x1+g2x2). (B.25)The probability Pr(g|n) that the interaction group size is g = g1+g2 is the same as Eq. 3.4 (main text),with n = n1+n2 the population size, and the probability of g1 of those being of type 1 isPr(g1|g,n1,n2) =(gg1)(n1n1+n2−1)g1( n2−1n1+n2−1)g−g1. (B.26)1290.00.51.01.50 250 500 750 1 000 Time (thousands)Cooperation (x)A01002003000 250 500 750 1 000 Time (thousands)Population size (n)0123450 250 500 750 1 000 Time (thousands)Cooperation (x)BDefectorsCooperators0250050007500100000 250 500 750 1 000 Time (thousands)Population size (n)Figure B.1: Individual-based simulations showing cooperation values (left) and population size (right) for bothmonomorphic and polymorphic dynamics, with static environments. A: Monomorphic approach to the equilib-rium (indicated with a dashed line). B: Evolutionary branching, starting from the evolutionary branching point. Inboth panels, colors represent different replicates (10 replicates per panel). In the left-side panels, each dot indicatesan individual (75 random individuals per replicate per time plot were plotted). Thick lines indicate strain averages;in B, individuals were assigned to different strains based on whether their trait value was higher or lower than thebranching point. Parameters in A: d = 0.01, r = 4, c = 0.3, k = 1.1, p = 0.01, µ = 0.001, σ = 0.01, M = 1; inB: d = 5×10−4, c = 0.05, r = 9, p = 0.01, k = 0.5, µ = 0.01, σ = 0.01, M = 1.Therefore, the lifetime average benefit isB2(x′|x1,x2,n1,n2) =n−1∑g=0g∑g1=0Pr(g|n) Pr(g1|g,n1,n2)B2(x′|x1,x2,g1,g2)=rn(n−1)p[(n1x1+(n2−1)x2)((1− p)n−1+np)− ((1− p)n−1)(n−1)x′].(B.27)The lifetime average benefits allow us to calculate the Wrightian fitnesses in the dimorphic popu-lation, W (x′|x1,x2) = B2(x′|x1,x2, nˆ1, nˆ2)C(x′)D(nˆ)M, as well as the selection gradients for each of thetwo strains,S (xi) = ∂W (x′|x1,x2)/∂x′|x′=xi , i ∈ {1,2}. We use these gradients to numerically evaluate1300.0 0.5 1.0 1.5 2.0 2.50100200300400500Lag (ℒ)Strainsize(n i )A Defectors (n1 )Cooperators (n2 )0.0 0.5 1.0 1.5 2.0 2.50100200300400500Lag (ℒ)Populationsize(n )B Static frequenciesChanging frequenciesFigure B.2: Equilibrium strain sizes in a bimorphic population with a constant lag. A: The equilibrium fre-quency of cooperators (solid line) is higher for large lags; however, since the frequency of defectors (dashedline) decreases, the total population size (blue line in B) is smaller at large lags. Nonetheless, evolution inthe relative frequencies of cooperators and defectors allows populations to persist at much higher lags com-pared to a population where the frequency of cooperation is not an evolving trait (pink line in B). Parameters:d = 0.01, r = 4, c = 0.3, k = 0.5, p = 0.01.the evolutionary dynamics (Fig. 3.2B) using a coupled system of two differential equations (each similarto to Eq. 3.13), describing the change in x1 and in x2.After branching, the strategies diverge until the edge of the phenotype space is reached at x?1 = 0. Intime, the other strain also reaches an equilibrium x?2 (Fig. 3.2B, main text). At this point, the populationwill consist of a cooperator strain (with x?2 > 0) and a defector strain (with x?1 = 0), which does notcontribute to public good production. Thus, starting from a continuous phenotype space, we observe theevolutionary origin of defectors and cooperators (a tragedy of the commune, Doebeli et al. 2004).The equilibrium value of cooperation x?2 can be found by setting x?1 = 0. The cooperator’s selectiongradient can then be shown to simplify to (see supplementary Mathematica file, deposited in Dryad,Henriques and Osmond 2020)S (x2) =C(x2)x2(W (x2|0,x2)C(x2)(1− ckxk2)−W (0|0,x2)). (B.28)At equilibrium,S (x?2) = 0 and the fitnesses of both strains are one, so that the solution is determinedby1− ck(x?2)k =C(x?2), (B.29)which depends only on c and k. In particular, the equilibrium investment of the cooperator strain doesnot depend on the magnitude of that lag. Thus, when populations consist of two discrete phenotypes(defectors and cooperators), they adapt to mismatches between the functional trait y and the environ-mental optimum only by changing the relative frequencies of each strain. Because of this property,once evolutionary branching has occurred and bimorphic evolutionary equilibrium has been reached,our model resembles an binary-action ecological public goods game. When this is the case, cooperatorscoexist with defectors in the same population, but their strain sizes change as they track the movingoptimum.131For any given lag, at ecological equilibrium, W (x1|x1,x2) =W (x2|x1,x2) = 1, we can obtain a (nu-merical) solution for the strain sizes. The overall population size decreases with lag (Fig. B.2B, bluecurve), but the number of cooperators is higher at large lags (Fig. B.2A). This change in the frequenciesof cooperators and defectors allows populations to withstand much larger lags, and at higher populationsizes, than they would if cooperation simply occurred at a given (constant and lag-independent) fre-quency. The pink curve in Fig. B.2B illustrates the population size for a reference population where thefrequency of cooperators is kept constant at the same value that is displayed by the evolving populationat equilibrium (blue curve) whenL = 0.At very high lags, extinction occurs for the defector strain (Fig. B.2A). When this happens, thepopulation reverts back to the single strain case. Monomorphic dynamics are then expected to drive thepopulation back to the evolutionary branching point, and branching will occur again, resulting in cyclesof diversification and extinction.B.3.2 Polymorphic evolution during environmental changeWe now turn to the evolution in the frequency of cooperators when the environment is changing at somevelocity v, and populations consist of two discrete strains (cooperators and defectors).We are interested in tracking the evolution in the functional trait y, whose selection gradient is givenby Eq. 3.14 (main text). A dynamical equilibrium will exist whenever the lag becomes constant, i.e.L˙ ≡ v− y˙= 0. We only need to keep track of the sizes nˆ1, nˆ2 of each strain, since the trait values x1 = 0and x2 = x?2 (Eq. B.29) do not evolve. However, we do need to keep track of each strain separately,because (since they have different sizes and payoffs) they evolve at different rates.The differential equations for the dynamics of the two strains’ lags are given byL˙i = v− 1σ2W (yi)nˆiµyσ2y sLi, i ∈ {1,2} (B.30)where strain sizes, nˆi, are the solution to W (~z′i|~z1,~z2) = B2(x′i|~z1,~z2)C(x′i)D(~z1,~z2)M(y′i) = 1, ∀i∈ {1,2}.Numerically evaluating Eq. B.30 shows that, as the environment changes, the distance to the opti-mum becomes much higher for cooperators than for defectors, which is not surprising since the coop-erator strain size always seems to be smaller than the defector strain. Nonetheless, increasing distancesfrom the optimum lower population size and hence favour cooperators, increasing their frequency. Asan example, Fig. B.3 shows both the population size and the lag becoming stabilized after a period of os-cillatory behavior, achieving dynamical equilibrium (Fig. B.3). The different effects of fitness partitionchoice seen in the continuous games case are no longer present, because the value of x is not evolving(nonetheless, fitness partition choice does affect the relative rate of evolution of the two strains).Although the equilibrium cooperator frequency is much higher at fast velocities than at slow ve-locities, there can be a very slight decrease in the cooperator frequency at intermediate velocities ofenvironmental change (Fig. B.4A). This reflects the tension between opposing effects of environmen-tal change. On the one hand, the decline of population size (Fig. B.4B) favours cooperation; on theother hand, the smaller population size of the cooperators means a slower mutational input and larger1320 3 6 9 12 150.00.10.20.30.4Time (thousands)CooperatorfrequencyA0 3 6 9 12 1551050100500Time (thousands)Populationsize(n)DefectorsCooperatorsB0 3 6 9 12 150.0050.0100.0500.1000.5001Time (thousands)Lag(ℒ)DefectorsCooperatorsC0 10 20 30 40 500.00.20.40.60.8Time (thousands)CooperatorfrequencyA0 10 20 30 40 50151050100500Time (thousands)Populationsize(n) DefectorsCooperatorsB0 10 20 30 40 500.0010.0100.1001Time (thousands)Lag(ℒ)DefectorsCooperatorsCFigure B.3: As the environment changes, the frequency of cooperators increases (A) which, after a period ofoscillations, stabilizes the population size (B) and the distance to the optimum (C). Top row (purple): fitnesspartition D; bottom row (green): fitness partition CD. Parameters: d = 0.02, r = 4, c= 0.3, s= 1, p= 0.005, k =0.5, µy = 0.05, σy = 0.1. For D: v = 5×10−3; for CD: v = 9×10−4.lag (Fig. B.4C). At either extreme (slow and fast velocities) cooperators will therefore equilibrate athigher relative frequencies when compared to intermediate velocities. At even higher velocities of en-vironmental change, one of the strains goes extinct before any equilibrium is reached. In the numericalsimulations we have attempted, this is always the defector strain, because it experiences big oscillationsin population size (as seen in Fig. B.3B, top row). After the extinction of the defector strain, the systemreverts back to the monomorphic case, explored in detail in the main text. In our numerical results (suchas the examples illustrated in Fig. B.4) this reversal to monomorphism occurs at such high velocities ofenvironmental change that no dynamical equilibrium can be reached, and the population goes extinct.B.4 Alternative cost and benefit functional formsB.4.1 Nonlinear benefitsWe can generalize Eq. 3.5 (main text) so that the benefits B(x|x,g) increase nonlinearly as a functionof the total cooperative investment X ≡ (x+(g− 1)x). For example, we can use the quadratic formB(x|x,g) = rg(X +bX2). When b > 0, the benefits increase super-linearly with cooperative investment,and when b < 0, they increase sub-linearly (for X ≤ −1/(2b)). For b = 0, the model reduces to thelinear case.Using this quadratic form, the average benefits areB(x|x,n) =n∑g=1Pr(g|n)B(x|x,g) (B.31)(see the accompanying Mathematica notebook (Henriques and Osmond 2020) for the full expression).133● ● ● ● ● ●● ●● ●0 1 2 3 40.000.050.100.150.200.250.30Velocity (v × 103 )EquilibriumcooperatorfrequencyA ○ ○ ○ ○○○ ○ ○ ○ ○● ● ● ●● ● ● ● ● ●0 1 2 3 451050100500Velocity (v × 103 )Equilibriumpopulationsize(n )○ Defectors● CooperatorsB○ ○ ○ ○ ○ ○○ ○ ○○● ●● ●● ● ●● ● ●0 1 2 3 40.00.51.01.52.0Velocity (v × 103 )Equilibriumlag(ℒ )○ Defectors● CooperatorsC● ● ● ● ● ● ●●●1 2 3 4 5 6 7 8 90.00.10.20.30.40.5Velocity (v × 104 )EquilibriumcooperatorfrequencyA ○ ○ ○ ○ ○ ○○○ ○● ● ● ● ●● ● ● ●1 2 3 4 5 6 7 8 951050100500Velocity (v × 104 )Equilibriumpopulationsize(n )○ Defectors● CooperatorsB○ ○ ○ ○ ○ ○ ○ ○○● ● ●● ●● ●● ●1 2 3 4 5 6 7 8 90.00.51.01.52.02.5Velocity (v × 104 )Equilibriumlag(ℒ )○ Defectors● CooperatorsCFigure B.4: Frequency of cooperators (A), strain sizes (B) and distance to the optimum (C) at dynamical equi-librium, for different velocities of environmental change (v). Top row (purple): fitness partition D; bottom row(green): fitness partition CD. Parameters: same as in Fig. B.3.We can now follow the same approach that was used in the main text to calculate the population sizeat ecological equilibrium (Eq. 3.7 in the main text):nˆ(~z) =M(y)rx(1+bx(1− p))− exp(cxk)d exp(cxk)−M(y)bprx2 if xmin ≤ x≤ xmax0 otherwise,(B.32)where xmin (xmax) is the minimum (maximum) value of cooperative investment below (above) whichthe population goes extinct. We cannot find an analytical solution for xmin or xmax when b 6= 0. Incomparison to the linear case, super-linear (sub-linear) benefits result in larger (smaller) populationsizes (Fig. B.5A).When the lag is constant, so that only x evolves, we can again follow the same procedures as inthe linear case (similarly to Eq. 3.11 in the main text) to calculate the invasion fitness and the selectiongradient. The full analytical expressions are long and cumbersome, and we do not reproduce them here,but they can be found in the accompanying Mathematica notebook, deposited in Dryad (Henriques andOsmond 2020). Using these quantities, we can then (numerically) find the evolutionary equilibrium ofcooperation and the corresponding population density (Fig. B.5B–C). Overall, non-linearity may resultin either increased or decreased values of cooperation at equilibrium (x?), depending on lag and p.Surprisingly, at high lags, super-linearity (sub-linearity) leads to higher (lower) equilibrium cooperation(Fig. B.5B). At low lags, the same may be true if p is low (Fig. B.5B1), but the opposite is true if pis high (Fig. B.5B2). In general, regardless of the value of p, equilibrium densities are higher withsuper-linear benefits, and lower with sub-linear benefits (Fig. B.5C).We can now consider the case when the environmental optimum is changing. Just like in the maintext, we can calculate selection gradientsS (x) andS (y), based on Eq. 3.15 (main text). The selection134b = -0.02b = 0b = 0.020100200300400nA1b = -0.02b = 0b = 0.022.62.83.03.23.43.6x*B1b = -0.02b = 0b = 0.020100200300400n (x* )C1b = -0.02b = 0b = 0.020 2 4 6 8 10 12 140100200300400xnA2b = -0.02b = 0b = 0.020.0 0.5 1.0 1.51.52.02.53.03.5ℒx*B2b = -0.02b = 0b = 0.020.0 0.5 1.0 1.50100200300400ℒn (x* )C2Figure B.5: Effects of small deviations from linearity with a constant lag, for low (top) and high (bottom) valuesof p. In comparison to the linear benefits case, super-linearity (positive b, black) results in increased populationdensity at ecological equilibrium nˆ, and sub-linearity (negative b, light gray) has the opposite effect (A). Bothsuper- and sub-linearity may result in either increased or decreased values of cooperation at equilibrium (x?),depending on lag and p. In general, at high lags, super-linearity (sub-linearity) leads to higher (lower) equilibriumcooperation (B). At low lags, the same may be true (if p is low, B1), or the opposite may be true (if p is high, B2).Regardless of the effect on x?, super-linearity (sub-linearity) always leads to a higher (lower) population densityat evolutionary equilibrium (C). Parameters: s = 1, r = 4, c = 0.3, k = 1; for A:L = 0.5; for 1: p = 0.001; for2: p = 0.005.gradient of y is again equal to s(tv− y). Using numerical simulations, we can find the dynamicalequilibrium where the lag stabilizes at some constant value. At this equilibrium, x˙ = 0 and L˙ ≡ v− y˙=0. Just like in the main text, the qualitative effect of cooperation depends on the fitness partition; again,we exemplify our results using fitness partitions D (Fig. B.6A) and CD (Fig. B.6B).These numerical results show that, for benefit functions with small deviations from linearity, ourmain result—namely, that the evolution of cooperation can either promote rescue or drive populationsto extinction during environmental change—remains valid. The main qualitative difference is that therange of environmental velocities at which cooperation promotes rescue is increased when benefitsincrease super-linearly with cooperation.B.4.2 Linear costsIn this section, we consider an alternative to the cost function introduced in Eq. 3.3 (main text). Weassume that the cost increases linearly with cooperative investment:C(x) ={1− cx if x≤ 1/c0 otherwise.(B.33)135Evolving xStatic xb = -0.02b = 0.020.00.51.01.5Equilibriumlag(ℒ)AEvolving xStatic xb = -0.02b = 0.020100020003000400050006000Equilibriumpopulationsize(n)Evolving xStatic xb = -0.02b = 0.020 1 2 3 4 5 60.00.51.01.5v×106Equilibriumlag(ℒ)BEvolving xStatic xb = -0.02b = 0.020 1 2 3 4 5 60100020003000400050006000v×106Equilibriumpopulationsize(n)Figure B.6: Lag (left) and population size (right) at the dynamical equilibrium, in populations with fitness par-tition D (panel A) or fitness partition CD (panel B), for different velocities of environmental change (v). Dashedlines represent populations where benefits are slightly sub-linear (b =−0.02), whereas solid lines represent pop-ulations where benefits are slightly super-linear (b = 0.02) The curves are interrupted at high velocities becausethe populations become extinct: at high v, the nullclines for lag (L˙ = 0) and cooperation (x˙ = 0) no longer in-tersect, so that there is no equilibrium lag. Note that in A, for high values of v, the evolution of cooperationrescues populations that would otherwise undergo extinction; in contrast, in B, the opposite happens. Parameters:r = 10, c = 0.1, s = 1, k = 2, µx = µy = 10−4, σx = σy = 0.05, d = 0.0019, p = 5×10−4.Multiplicative costsWith multiplicative costs, C(x) multiplies one of the fitness components (viability or fecundity, depend-ing on the choice of fitness partition), just like in the model in the main text. If this is the case, thenfitness W (~z|~z) = B(x|x,n)C(x)D(n)M(y) and the equilibrium population size (Fig. B.7A) equalsnˆ(~z) =M(y)rx(1− cx)−1dif xmin ≤ x≤min(xmax,1/c)0 otherwise,(B.34)where 1/c is the value of x at which the cost becomes equal to zero (Eq. B.33), and xmin and xmax(which are the minimum and maximum values of cooperation such that the population does not goextinct) are equal to 2/(M(y)r±√M(y)r(M(y)r−4c)). Just like in the main text, the population sizeis maximized at an intermediate value of cooperation, xcrit = 1/(2c) (Fig. B.7A). Also like in the mainmodel, the population will never evolve towards values of x higher than xcrit. Because xcrit < 1/c, thediscontinuity in Eq. B.33 is never actually reached by gradual evolution; therefore, in the rest of thissection, we simplify the presentation and the calculations by re-defining C(x) ≡ 1− cx and ignoringvalues of x above 1/c.136ℒ = 1ℒ = 0.5ℒ = 00.0 0.5 1.0 1.5 2.0 2.5 3.0050100150200250xnAp = 0.001p = 0.005p = 0.010.0 0.5 1.0 1.50.00.51.01.5ℒx*Bp = 0.001p = 0.005p = 0.010.0 0.5 1.0 1.5050100150200250ℒn (x* )CFigure B.7: Ecological and evolutionary dynamics are qualitatively similar between the nonlinear multiplica-tive cost scenario (main text) and the linear multiplicative cost scenario. A: The population size at ecologicalequilibrium (nˆ) is maximized at intermediate values of cooperation x, and decreases with distance L (y) to theenvironmental optimum. B: The equilibrium value of cooperation with a constant lag (x?) increases for higherlags (L ) and smaller interaction group sizes (smaller p). C: Corresponding population size at equilibrium (nˆ(x?)).Parameters: d = 0.01, s = 1, c = 0.3, r = 4, p = 0.01.Again, we start by building intuition, by considering a constant lag. Evolution occurs in x, but noty. Omitting the argument in M(y), we can write the selection gradient of cooperation:S (x) =Mrnˆ(x)(1+dnˆ(x))p((1− (1− p)nˆ(x))(1− cx)− nˆ(x)pcx). (B.35)At evolutionary equilibrium, S (x?) = 0. Although we are unable to analytically calculate x?, nu-merical simulations show that x? increases with lag, up to the point at which x? = xcrit (Fig. B.7B–C).For higher lags, the population goes extinct. Solving nˆ(xcrit) = 0 forL , we obtain the maximum valueof lag above which the population goes extinct:Lmax =√2sln( r4c). (B.36)Qualitatively, these results are similar to the ones presented in the main model, except that with linearcosts, evolutionary branching is impossible.Now, consider a moving optimum, θ = vt, and evolution of both traits x and y. Just like in themain model, we try to find a dynamical equilibrium where the lag stabilizes at some constant value.At this equilibrium, x˙ = 0 and L˙ ≡ v− y˙ = 0. The differential equation for the dynamics of the twotraits, with selection gradients S (x) and S (y), follows Eq. 3.15 (main text). In order to numericallyfind the dynamical equilibrium, we must specify a fitness partition; just like in the main model, fitnesspartitions where the costs of cooperation decrease viability behave qualitatively differently from otherfitness partitions. When the costs of cooperation decrease fecundity, the evolution of cooperation pro-motes persistence, and when they decrease viability, the evolution of cooperation has negative effectson persistence. (For examples of both model behaviors, see Fig. B.8.) The explanation is the same asfor the case with nonlinear costs.1370 1 2 3 4 50.00.51.01.52.02.5Cooperation (x )Lag(ℒ)A0123Cooperation(x)Evolving xStatic x012Lag(ℒ)0 1 2 3 4 5 60200040006000Time (millions)Populationsize(n)0 1 2 3 4 50.00.51.01.52.02.5Cooperation (x )Lag(ℒ)B01234Cooperation(x)Evolving xStatic x012Lag(ℒ)0 1 2 3 4 5 6 7 8 9 100200040006000Time (millions)Populationsize(n)Figure B.8: Evolution of cooperation with a moving optimum in a model with linear multiplicative costs, for aspecies with fitness partition D (panel A) or fitness partition CD (panel B). The stream plot to the left indicates thechange in a population’s lag (L ) and cooperation (x) over time, for a population where cooperation evolves fastrelative to the environmental change. The nullclines for lag (L˙ = 0) and cooperation (x˙= 0) are in black and lightgrey, respectively. Within the dark grey region, the population goes extinct. The thick blue curves are the resultof numerical simulations, starting with zero lag and with x = x?. For comparison, the thick pink curves are theresult of numerical simulations, with identical starting conditions, where cooperation does not evolve (µx = 0).The panels on the right show the time-dynamics of cooperation, lag, and population size for the same simulations.Parameters: r = 10, c = 0.1, s = 1, µx = µy = 10−4, σx = 0.025, σy = 0.01, d = 0.002, p = 5× 10−4; for A:v = 6.5×10−6; for B: v = 4.7×10−5. In contrast to A, the velocity in B was chosen such that the nullclines donot intersect.)138ℒ = 1ℒ = 0.5ℒ = 00 2 4 6 8 10050100150200xnA1p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0ℒx*B1p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0020406080100ℒn (x* )C1ℒ = 1ℒ = 0.5ℒ = 00 2 4 6 8 10050100150200xnA2p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.51.01.52.0ℒx*B2p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4020406080ℒn (x* )C2Figure B.9: Ecological and evolutionary dynamics are qualitatively similar between the nonlinear multiplicativecost scenario (main text) and the linear additive cost scenario. A: The population size at ecological equilibrium(nˆ) is maximized at intermediate values of cooperation x, and decreases with distanceL (y) to the environmentaloptimum. B: The equilibrium value of cooperation with a constant lag (x?) increases for higher lags (L ) andsmaller interaction group sizes (smaller p). C: Corresponding population size at equilibrium (nˆ(x?)). The toprow (A1–C1) is for fitness partition CMD; the bottom row (A2–C2) is for fitness partition CD. Parameters: d =0.01, s = 1, c = 0.3, r = 4, p = 0.01; for A: c = 0.1.Additive costsWith additive costs, C(x) is subtracted from one of the fitness components (viability or fecundity, de-pending on the choice of fitness partition). If this is the case, fitness (and therefore the populationdensity at ecological equilibrium) will depend on the fitness partition. Furthermore (because C(x) cangrow without limit as x increases), fitness partitions for which cooperation can evolve to extremely highvalues are biologically meaningless, since they would result in negative viabilities or fecundities.Since our aim in this section is simply to demonstrate that our main results do not rely on the choiceof a multiplicative cost, we will examine two example fitness partitions; following the same conventionas in the main text, we will call them CMD and CD. In the former, cooperation helps persistence, whereasin the latter, it hurts persistence.The population size at ecological equilibrium is again given by the solution to ∆n= nW (~z|~z)−n= 0.For fitness partition CMD, W (~z|~z) = B(x|x,n) · (D(n)M(y)−C(x)); for fitness partition CD, W (~z|~z =(B(x|x,n)M(y)) · (D(n)−C(x)). For CMD, the equilibrium size (Fig. B.9A1) is thennˆ(~z) =M(y)rx− crx2−1d(1+ crx2)if xmin ≤ x≤ xmax0 otherwise,(B.37)where xmin and xmax (which are the minimum and maximum values of cooperation such that the pop-ulation does not go extinct) are given by 2/(M(y)r±√r(M(y)2r−4c)). Just like in the main text,the population size is maximized at an intermediate value of cooperation, xcrit = 1/√cr (Fig. B.9A2).139Also like in the main model, the population will never evolve towards values of x higher than xcrit. Atecological equilibrium, viability equals 1/(rx), which never turns negative.For CD, the equilibrium size (Fig. B.9A2) isnˆ(~z) =M(y)rx(1− cx)−1d(1+M(y)crx2)if xmin ≤ x≤ xmax0 otherwise,(B.38)where xmin and xmax are given by 2/(M(y)r±√M(y)r(M(y)r−4c)). For this fitness partition, xcrit =1/√M(y)cr (Fig. B.9A2). At ecological equilibrium, viability equals 1/(M(y)rx), which never turnsnegative.To start building intuition, we consider a constant lag. Evolution occurs in x, but not y. Omitting theargument in M(y), we can write the selection gradient of cooperation:S (x) =(1− p˜nˆ(x))r(M− cx(1+dnˆ(x)))nˆ(x)(1+dnˆ(x))p− crx if CMDMr((1− p˜nˆ(x))(1− cx(1+dnˆ(x)))nˆ(x)(1−dnˆ(x))p − cx)if CD,(B.39)where p˜ = (1− p).At evolutionary equilibrium, S (x?) = 0. Although we are unable to analytically calculate x?, nu-merical simulations show that x? increases with lag, up to the point at which x? = xcrit (Fig. B.9B1, B2,C1, C2). For higher lags, the population goes extinct.Now, consider a moving optimum, θ = vt, and evolution of both traits x and y. Just like in the mainmodel, we try to find a a dynamical equilibrium where the lag stabilizes at some constant value. At thisequilibrium, x˙ = 0 and L˙ ≡ v− y˙ = 0. The differential equation for the dynamics of the two traits, withselection gradients S (x) and S (y), follows Eq. 3.15 (main text). Numerical simulations show that,depending on the fitness partition, the evolution of cooperation may either promote evolutionary rescue(Fig. B.10A, fitness partition CMD) or evolutionary suicide (Fig. B.10B, fitness partition CD). Thus ourmain qualitative results do not depend on multiplicative costs.B.5 The evolution of competition during environmental changeIn this section, we use the framework described in this article to study the effect of the evolution ofa different type of social interaction: competition. Individuals in a population of size n compete witheach other for resources. They are characterized by traits x (competition) and y (a functional trait thatis adapting to environmental change). The maximum possible viability is one, whereas the maximumpossible fecundity is λ . Just like in the cooperation model, other factors can decrease fecundity orviability from these maximum values. Here, we consider three such factors: the effect of competition(A(x′|x,n)), the cost of competitive ability (C(x′), Eq. 3.3 in the main text), and the mismatch betweenthe trait value y and the environmental optimum (M(y′), Eq. 3.1 in the main text). We will model the1400.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0Cooperation (x )Lag(ℒ)A0.0.10.20.30.40.5Cooperation(x)Evolving xStatic x0.0.250.50.751.1.251.5Lag(ℒ)0 1 2 3 4 5 602004006008001000Time (millions)Populationsize(n)0.0 0.5 1.0 1.5 2.00.00.51.01.52.02.5Cooperation (x )Lag(ℒ)B0.0.51.1.52.2.5Cooperation(x) Evolving xStatic x0.0.51.1.52.Lag(ℒ)0 1 2 3 4 5 6 7 8050010001500Time (millions)Populationsize(n)Figure B.10: Evolution of cooperation with a moving optimum in a model with linear additive costs, for a specieswith fitness partition CMD (panel A) or fitness partition CD (panel B). The stream plot to the left indicates thechange in a population’s lag (L ) and cooperation (x) over time, for a population where cooperation evolves fastrelative to the environmental change. The nullclines for lag (L˙ = 0) and cooperation (x˙= 0) are in black and lightgrey, respectively. Within the dark grey region, the population goes extinct. The thick blue curves are the resultof numerical simulations, starting with zero lag and with x = x?. For comparison, the thick pink curves are theresult of numerical simulations, with identical starting conditions, where cooperation does not evolve (µx = 0).The panels on the right show the time-dynamics of cooperation, lag, and population size for the same simulations.Parameters: r= 10, c= 0.1, s= 1, µx = µy = 10−4, σx =σy = 0.01, d = 0.002; for A: p= 0.008,v= 1.7×10−6;for B: p = 0.001, v = 3.1× 10−6. In contrast to A, the velocity in B was chosen such that the nullclines do notintersect.)141effect of competition asA(x′|x,n) = exp(−a ·α(x′,x)n), (B.40)where a determines how quickly the fitness component declines with competition and α(x′,x) is thecompetition kernel (Matsuda and Abrams 1994),α(x′,x) = exp(−(x′− x)β − (x′− x)24ς2). (B.41)In this equation, β is a constant that determines the degree of competitive asymmetry: individuals withlarger values of x are better competitors (note that we consider only x ≥ 0, since, when x < 0, the costfunction, Eq 3.3, increases with x for odd values of k, which is biologically nonsensical). The parameterς determines the range of resources individuals compete for, i.e., how rapidly competitive effects declineas the trait values become more dissimilar.With these quantities, we can define fitness as W (~z′|~z) = λA(x′|x,n)C(x)M(y), and (following thesame procedure as for Eq. 3.7 in the main text), determine the population size at ecological equilibrium:nˆ(~z) =1a(log(λM(y))− cxk)if x≤ xmax0 otherwise,(B.42)where xmax = (log(λM(y))/c)1/k. As expected, the population size decreases with competition as wellas with lag (Fig. B.11A).Following the same steps as in the main manuscript, we start building intuition by considering theeffects of evolving competition in the case of a constant lag (M(y) = M). We use the same approach aswe used for Eq. 3.11 (main text), and calculate the selection gradient of a mutant in x:S (x) = β log(λM)− cxk−1(k+ xβ ). (B.43)Although we are unable to analytically calculate the evolutionary equilibrium of competition (i.e.,the value x? such that S (x?) = 0) for any k, we can for special cases (e.g., when k is any non-zerointeger). Taking k = 2 as an example, we find x? =√1/β 2+ log(λM)/c−1/β , which decreases withlag (Fig. B.11B–D). Further, we can implicitly differentiate S (x) with respect to M, which shows thatequilibrium competition decreases with lag for any k > 1. This makes intuitive sense; at higher lagspopulation density is smaller, decreasing the need to compete for resources.Now, we consider the simultaneous evolution of x and y. Just like in the cooperation model, we tryto find a a dynamical equilibrium where the lag stabilizes at some constant value. At this equilibrium,x˙ = 0 and L˙ ≡ v− y˙ = 0. The differential equation for the dynamics of the two traits, with selectiongradients S (x) and S (y), follows Eq. 3.15 (main text). In order to numerically find the dynamicalequilibrium, we must specify a fitness partition, by assigning the different factors (effect of competition,cost of competition, and environmental mismatch) to each of the two fitness components (viability andfecundity).142ℒ = 2ℒ = 1ℒ = 00.0 0.5 1.0 1.5 2.0 2.5 3.00100200300400500600700xnA ℒ = 2ℒ = 1ℒ = 00.0 0.5 1.0 1.5 2.0-4-202468x(x)Bβ = 1β = 2β = 30.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.00.51.01.52.02.5ℒx*C β = 1β = 2β = 30.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50100200300400ℒn (x* )DFigure B.11: Competition x decreases the ecological equilibrium of population density nˆ (A), creating an eco-evolutionary feedback loop. For constant values of lag L (B–D) the selection gradient of competition S (x)is decreasing in x (B). Increased lags lead to smaller values of competition (C) and to decreased equilibriumpopulation densities (D). Parameters: s = 1, λ = 1000, a = 0.01, c = 1, k = 2, β = 1.As an example fitness partition, let V = 1 and F = λA(x′|x,n)C(x′)M(y). For such a fitness partition,the rate of evolution is proportional to nˆ(~z)/σ2W (~z) = nˆ(~z). This increases as competition declines, sothat the evolution of reduced competition leads to faster adaptation. Thus, while a population whose xdoes not evolve might go extinct due to increases in lag, a population where x is an evolving trait canpersist through evolutionary rescue (Fig. 3.6 in the main text).In contrast, let’s imagine a fitness partition where V = A(x′|x,n) and F = λC(x′)M(y). Forsuch a fitness partition, the rate of evolution is proportional to nˆ(~z)/σ2W (~z) = nˆ(~z)A(x?|x?, nˆ). In thiscase, while the evolution of decreased competition again increases population size, nˆ(~z), it also in-creases the strength of genetic drift, here seen by the decrease in the amount of resources acquired,1/σ2W (~z) = A(x?|x?, nˆ), with x?. Together, the evolutionary rate nˆ(~z)/σ2W (~z) for a given lag can declinewith decreases in x, which suggests the evolution of competition might drive extinction in this fitnesspartition. However, the maximum evolutionary rate across all lags, i.e., the critical rate of environmentalchange, appears to always increase as competition declines, just pushing this maximum rate of evolu-tion to larger lags. Therefore, we find that while the evolution of decreased competition in a changingenvironment can increase the steady-state lag, the evolution of decreased competition does not driveevolutionary suicide.143Appendix CSupplementary Information for Chapter 4C.1 IBS algorithmThe population consists of a fixed number n of individuals. Each individual is defined by their invest-ment strategies z (we focused mostly on the two-dimensional case z = 〈z1,z2〉, but we also consideredthe three-dimensional case z = 〈z1,z2,z3〉). The population is updated asynchronously. At every timestep, two randomly chosen focal individuals with phenotype z and z′ interact with randomly chosenpartners p and p′, earning payoffs P(z, p) and P(z′, p′), respectively. The first focal individual, z, thendies, and the empty position is taken by the offspring of whichever of the two focal individuals has thehighest payoff. Each element of the offspring’s phenotype vector is drawn from a normal distribution,centered at the parent’s phenotype value, with a small standard deviation σ . The simulation is endedwhenever an individual’s strategy reaches the edges of phenotype space.For all figures, we used the following parameters: n = 10,000, σ = 2.5×10−5, and initial distanceto the branching point = z?1/2.C.2 OSS algorithmThis algorithm is adapted from Appendix A in Ito and Dieckmann (2007). For most of the simulation,the population consists of a single resident monomorphic strain. As the simulation unfolds, mutantstrains are introduced. When this happens, there are three alternatives: (i) the mutant may invade andbecome the new resident; (ii) the mutant may fail to invade and be removed from the population; or(iii) the strains may be mutually invasible, in which case all strains will be resident (i.e., the popu-lation becomes polymorphic). Hence, although a small number of strains may coexist, each strain ismonomorphic (in contrast with the IBS, where each strain resembles a cloud in phenotype space).1. At the beginning of the simulation, the population consists of a single monomorphic resident strainwith a two-dimensional phenotype. More generally, throughout the simulation, the populationwill consist of a list of N resident strains with two-dimensional phenotypes Z = 〈Z1, · · · ,ZN〉.Each entry of Z is a strain, i.e. a vector of trait values, Zi = 〈z(i)1 ,z(i)2 〉. The equilibrium frequencydistribution nˆ = 〈nˆ1, · · · , nˆN〉 is proportional to P−1J (where J is a column vector of ones and Pi, j144is the payoff of strain i upon interacting with strain j).2. Then, an invasion attempt occurs. The mutant emerges from strain i with probability νi/ν , whereνi = µ nˆi and ν = ∑Ni=1 νi. The mutant phenotypes, Z′i , are normally distributed about the parentalstrain strategy. Time t is updated by ∆t =−(1/ν) lnρ , where ρ is uniformly distributed between0 and 1.3. The payoff of the mutant is given by P(Z′i ,Z) = ∑Nj=1 P(Z′i ,Z j)nˆ j. The payoff of the resi-dents is calculated similarly: P(Z′1,Z) = ∑Nj=1 P(Z′1,Z j)nˆ j (since, by definition, all residentshave the same payoff, we arbitrarily pick resident Z1 for the calculation). If the mutant’s pay-off is higher than the resident’s, the probability that the mutant replaces the resident is givenby S (Z′i ,Z) = [P(Z′i ,Z)−P(Z′1,Z)]/P(Z′i ,Z), otherwise the probability is zero. To acceleratethe simulation we work with a modified invasion probability, S˜ (Z′i ,Z) = 1− exp(−χS (Z′i ,Z)),where the coefficient χ adjusts the magnitude of the fitness effect.4. The invasion attempt is successful if ρ < S˜ (Z′i ,Z), where ρ again is a uniformly distributedrandom number between 0 and 1. To accelerate the simulation, if there have been over η unsuc-cessful invasions, set χ to χ×ξ . Similarly, if an invasion was successful in less than η/2 invasionattempts, set χ to χ/ξ . If mutant Z′i was not successful, return to step 2.5. If the mutant was successful, we test for mutual invasibility by calculating equilibrium frequencydistributions, nˆ′ = 〈nˆ1, · · · , nˆi, · · · , nˆN〉, for a population where the mutant replaces the resident,Z′ = 〈Z1, · · · ,Z′i , · · · ,ZN〉. If all elements of nˆ′ are larger than a threshold ε , and S (Zi,Z′)< 0 atnˆ′, there is no mutual invasibility. The mutant takes the resident’s place: replace Zi with Z′i and nˆwith nˆ′.6. Otherwise, the number of strains in the resident population, N, increases by one, and the mutantbecomes a new element of Z. Calculate the new resident equilibrium frequencies, nˆ. Shouldany frequency fall below the threshold ε , remove that strain from the population. Begin a newiteration by returning to step 2.For all figures, we used the following parameters: n=∑Ni=1 nˆi = 10,000, µ = 10−2, σ = 1.5×10−3,ε = 10−4, χ = 100, η = 10, and initial distance to the branching point = z?1/2.C.3 PDE algorithmWe used PDE simulations to study the two-dimensional model. First, we discretized phenotype spaceinto a square matrix with classes of width and height h. At any point in time t, the payoff of everyphenotype class 〈z1,z2〉 is given by P1,2 =∑p,q n1,2(t)P(〈z1,z2〉,〈zp,zq〉), where n1,2(t) is the populationdensity of the phenotype class. The density of a phenotype class can change either by natural selectionor due to the flow of mutations in and out of the class. Selection depends on the fitness of phenotype〈z1,z2〉, which equals w1,2 =max(sP1,2,0), where the parameter s characterizes the strength of selection.Mutations occur at a rate µ , and their effect size is assumed to be small, such that a mutant produced by145phenotype 〈z1,z2〉 must be a von Neumann neighbour of 〈z1,z2〉. The von Neumann neighbourhoodNcomprises the immediate non-diagonal (cardinal) neighbouring classes. Further, the set excludes classesbeyond the boundaries of phenotype space. Together, selection and mutation allow us to update everyclass density according to the coupled system of equationsdn1,2(t)dt=(1− µh2)n1,2(t)w1,2+ ∑〈p,q〉∈Nµ4h2np,q(t)wp,q−n1,2(t)w− ζh2 , (C.1)where w is the mean population fitness and ζ  1 is a change threshold. The change threshold partiallymimics the effect of demographic stochasticity with finite population sizes, by preventing the low-density edges of the population distribution from rapidly spreading outward (densities were truncatedat zero to avoid nonsensical negative values). This constrains the population distribution to a discretenumber of compact clusters (strains), as opposed to a wide distribution over the entirely of the phenotypespace. Consequently, ζ regulates the phenotypic variance of the strains.For all figures, we used the following parameters: h = z?1/100, µ = 0.01× (h/100)2, s = 100,ζ = 2× 10−7× (h/100)2. To numerically simulate the PDE system, we used the adaptive step-sizeRunge–Kutta–Fehlberg method (Fehlberg 1969; Press and Teukolsky 1992) with permissible absoluteerror = 10−5, permissible relative error = 10−4, error order = 4, and step order = 5.C.4 Branching condition along evolutionary branching lineIn this section, we derive the condition for the occurrence of evolutionary branching in a directionallyevolving population following Ito and Dieckmann (2012). The payoff of a mutant z′ = 〈z′1,z′2〉 in aresident population z = 〈z1,z2〉 is:w(z′,z) = a(1−b(z′1+ z1))(z′1+ z1)− c(1−dz′1)z′1+ a(1−b(z′2+ z2))(z′2+ z2)− c(1−dz′2)z′2 (C.2)The selection gradients for two traits, D1 and D2, areD1 =∂w∂ z1 z′=z= a− c− (4ab−2cd)z1 (C.3)D2 =∂w∂ z2 z′=z= a− c− (4ab−2cd)z2 (C.4)respectively. Consider a new rotated coordinate system 〈x,y〉:(xy)=(cosθ −sinθsinθ cosθ)(z1z2)(C.5)146The selection gradients on these new coordinate axes, Dx and Dy, areDx =∂w∂x∣∣∣∣x′=x,y′=y=∂x∂ z1D1+∂x∂ z2D2= (a− c)(cosθ − sinθ)− (4ab−2cd)x (C.6)Dy =∂w∂y∣∣∣∣x′=x,y′=y=∂Y∂ z1D1+∂y∂ z2D2= (a− c)(sinθ + cosθ)− (4ab−2cd)y (C.7)respectively. We can always find an adequate rotation angle θ ∗ under which Dx = 0 for any focalpopulation 〈x,y〉.According to Ito and Dieckmann (2012), the condition for the occurrence of evolutionary branchingunder directional selection (i.e., Dy > 0) isDx = 0 (C.8)Cxx < 0 (C.9)σγxx|Dy| >√2 (C.10)whereCxx =∂∂xDx= −4ab+2cd (C.11)γxx =∂ 2w∂x2∣∣∣∣x′=x,y′=y= −2ab+2cd (C.12)and σ is the mutation distribution. (C.8) is always satisfied when θ = θ ∗. Additionally, because now wefocus on the situation where evolutionary branching occurs at the singular point, (C.9) is also satisfiedwith our focal parameter values. By substituting (C.7) and (C.12), (C.10) is rewritten asσ(−2ab+2cd)|(−4ab+2cd)(y− y?)| >√2 (C.13)where y? is the singular point, i.e.,y? =(a− c)(sinθ + cosθ)4ab−2cd . (C.14)1470.000.050.10z 20.000.050.10z 20.00 0.05 0.10 0.00 0.05 0.10 0.00 0.05 0.100.00 0.05 0.10z10.00 0.05 0.10z10.00 0.05 0.10z1Figure C.1: In the PDE model, for some initial phenotypic distributions, branching may occur in directions thatare different from those predicted by the evolutionary branching line approach. Each panel is an illustrative timepoint of a PDE numerical simulation. The phenotypic distribution is shown by the contour lines. Colors indicateinvasion fitness (red: negative values; blue: positive values; white: zero). For more details and parameter valuessee Appendix C.3.Since (C.13) is always satisfied when the resident is sufficiently close to the singular point (x?,y?),evolutionary branching occurs along the x direction, i.e., orthogonal to the selection gradient (Ito andDieckmann 2012).Despite this prediction, partial-differential equation simulations show that, depending on the initialphenotypic distribution, branching can occur in the opposite direction (Figure C.1).C.5 Maximum-likelihood invasion pathMaximum likelihood invasion path (MLIP, Ito and Dieckmann 2014) is one of the methods for investi-gating the dynamics around the evolutionary branching line. Here, we show that the MLIP cannot predictthe branching direction observed in our individual-based simulations.Hereafter, we represent the traits using the new rotated coordinate axes from Appendix C.4. Byusing Eqs. (C.6), (C.7), and (C.12), the payoff of a mutant (x+δx,y+δy)in a resident population (x,y)148is represented asP(x+δx,y+δy,x,y) = P(x,y,x,y)+Dxδx+Dyδy+12γxxδ 2x +12γyyδ 2y (C.15)By using condition (C.8) and the fact that γxx = γyy, the invasion fitness of the mutant, w(x+δx,y+δy,x,y) isw(x+δx,y+δy) = Dyδy+12γxx(δ 2x +δ2y ). (C.16)Now we assume that the mutation probability follows a two-dimensional normal distribution, i.e.,M(δx,δy) =12piσ2exp(− 12σ2(δ 2x +δ2y )). (C.17)Then, the probability density that the focal mutant emerges and successfully invades at the nextinvasion event is represented asP(x+δx,x+δy) = Tµ nˆM(δx,δy)w(x+δx,y+δy) (C.18)where T is expected waiting time for the next invasion event, µ is the probability of the mutation and nˆis the population size (see Eqs. (8a) and (8b) in Ito and Dieckmann 2014).When we randomly choose which mutant invades following (C.18), this method is equal to theoligomorphic stochastic simulation (Appendix C.2). On the other hand, in the MLIP, it is assumed thatthe mutant which has the maximum invasion probability density will be chosen in the next invasionevent. Such a mutant can be derived by investigating the extremum of Eq. (C.18) for δx and δy. Themutant with maximum invasion probability density, (x+δ ∗x ,y+δ ∗y ), should satisfy∂P∂δx∣∣∣∣δx=δ ∗x ,δy=δ ∗y= 0 (C.19)∂P∂δy∣∣∣∣δx=δ ∗x ,δy=δ ∗y= 0 (C.20)By substituting Eq. (C.16) and Eq. (C.17), we can rewrite conditions (C.19) and (C.20) asδ ∗x (γxx−1σw(x+δ ∗x ,y+δ∗y ,x,y)) = 0 (C.21)andδ ∗y γxx+Dy−δ ∗y1σw(x+δ ∗x ,y+δ∗y ,x,y) = 0 (C.22)respectively.If δ ∗x 6= 0, condition (C.21) is satisfied when and only when w(x+ δ ∗x ,y+ δ ∗y ,x,y) = σγxx. By149substituting this into Eq. (C.22), we obtain Dy = 0. Because we are using a rotated coordiane systemsatisfying Dx = 0, this condition is satisfied when and only when the population reaches the singularpoint; in other words, the mutant with δ ∗x 6= 0 cannot invade into the resident population until the pop-ulation reaches the evolutionary singular point. Consequently, under directional selection (i.e., Dy > 0)the MLIP always predicts the invasion of mutants whose trait value for x-axis is same as that of theresident. Because the MLIP predicts the occurrence of evolutionary branching when the invaded mutantand resident strains can coexist, the direction of branching always occurs along the y direction. Thiscontradicts the analytical prediction from Appendix C.4 and the simulation results in the main text.C.6 Invasion of large-effect mutants into a two-strain residentLet the population consists of two resident strains Z = 〈Z1,Z2〉 with two-dimensional phenotypes,i.e., Zi = 〈z(i)1 ,z(i)2 〉. With quadratic cost and benefit functions, the population diversifies into a co-operator (with phenotype zmax) and a defector (with phenotype 0) for each individual game. Resultsfrom individual-based simulations suggest that gradual evolution with small mutations in two dimen-sions (Fig. 4.9) results in one of two possible resident populations: either a cooperator and a defector,ZCD = 〈〈0,0〉,〈zmax,zmax〉〉, or two complementary specialists, ZDOL = 〈〈0,zmax〉,〈zmax,0〉〉.At the equilibrium frequencies nˆ = 〈nˆ1, nˆ2〉, all strategies have the same expected payoff P, i.e.,PJ = Pnˆ, (C.23)where Pi, j is the payoff of strain i upon interacting with strain j and J is a column vector of ones.Because P is a scalar value, we can re-write this equation asnˆ = PP−1J. (C.24)Since nˆ is the vector of frequencies, the sum of all elements of nˆ is one, i.e.,JT nˆ = PJT P−1J = 1. (C.25)Therefore,P =1JT P−1J. (C.26)Hence, the equilibrium frequency distribution is equal tonˆ =P−1JJT P−1J, (C.27)which is proportional to P−1J.We now investigate the success of large-effect mutants occupying the empty corners of the parameterspace.150The CD state is stable against the invasion of either specialist mutant. For the resident configurationZCD, we have the payoff matrixPCD =[0 B(zmax)B(zmax)−C(zmax) B(2zmax)−C(zmax)], (C.28)which results in equilibrium frequenciesnˆ1 =B(zmax)−B(2zmax)+C(zmax)2B(zmax)−B(2zmax)nˆ2 =B(zmax)−C(zmax)2B(zmax)−B(2zmax) . (C.29)The mean payoff of a mutant Z′ = 〈z′1,z′2〉 is given byP(Z′,ZCD) = nˆ1[(1−α)(B(z′1)−C(z′1))+α(B(z′2)−C(z′2))]+ nˆ2[(1−α)(B(zmax+ z′1)−C(z′1))+α(B(zmax+ z′2)−C(z′2))]. (C.30)Plugging in nˆ1 and nˆ2 from Eq. C.29, α = 1/2, (z′1 = 0,z′2 = zmax) or (z′1 = 0,z′2 = zmax), and notingthat B(0) =C(0) = 0, we findP(Z′,ZCD) =B(zmax)(B(zmax)−C(zmax))2B(zmax)−B(2zmax) . (C.31)Using a similar approach, we can calculate the mean payoff of a resident, say 〈0,0〉. We take theright-hand side of Eq. C.30 and replace 0 for z′1 and for z′2. Plugging in nˆ1 and nˆ2 from Eq. C.29,α = 1/2, and noting that B(0) =C(0) = 0, we find that the mean payoff of the resident exactly equalsthe right-hand side of Eq. C.31. Since the resident and the mutant have identical mean payoffs, a raremutant is not favored by natural selection and can only invade the resident population by neutral drift.The DOL state is vulnerable to invasion by a cooperator or by a defector, depending on parameterchoice. For the resident configuration ZDOL, we have the payoff matrixPDOL =[αB(2zmax)−αC(zmax) B(zmax)−αC(zmax)B(zmax)− (1−α)C(zmax) (1−α)(B(2zmax)−C(zmax))], (C.32)which, for α = 1/2, results in equally frequent strains, nˆ1 = nˆ2 = 1/2 (which is a consequence of themodel’s symmetry).151Again assuming α = 1/2, the mean payoff of a specialist (resident) strain is given byP(〈0,zmax〉,ZDOL) = 12(B(zmax)−C(zmax)2+B(zmax)2)+12(B(2zmax)−C(zmax)2)=B(2zmax)+2B(zmax)−2C(zmax)4. (C.33)The mean payoff of a defector mutant is given byP(〈0,0〉,ZDOL) = 12(B(zmax)2)+12(B(zmax)2)=B(zmax)2, (C.34)and the mean payoff of a cooperator mutant is given byP(〈zmax,zmax〉,ZDOL) = 12(B(2zmax)−C(zmax)2+B(zmax)−C(zmax)2)+12(B(zmax)−C(zmax)2+B(2zmax)−C(zmax)2)=B(zmax)−B(2zmax)−2C(zmax)2, (C.35)The defector mutant can invade wheneverP(〈0,0〉,ZDOL) > P(〈0,zmax〉,ZDOL)2C(zmax) > B(2zmax)0 < zmax(c−a+ zmax(2ab− cd)). (C.36)Since zmax is always positive and the branching condition implies cd < 2ab < 2cd, it follows that thedefector can invade wheneverzmax >a− c2ab− cd = 2z?zmax2> z?. (C.37)The cooperator mutant can invade whenever P(〈zmax,zmax〉,ZDOL) > P(〈0,zmax〉,ZDOL), which by anidentical calculation is true whenever zmax < z?.Both three-strain configurations are stable against invasion by a fourth strain. We can use thesame procedure that we used above to show that the invasion fitness of a defector mutant in aresident population consisting of two specialists and a defector as well as the the invasion fit-ness of a cooperator mutant in a resident population consisting of two specialists and a co-operator are equal to zero. Hence, a fourth strain is not favored by natural selection and152could only invade by genetic drift. See the Mathematica notebook AppendixF.nb, availableat https://github.com/GilHenriques/2020 simultaneous branching, for the fullderivation (for readers without access to this proprietary software, we also included a .pdf version ofthe same file).153Appendix DSupplementary Information for Chapter 5D.1 List of parametersTable D.1 lists the parameters used in the model and their interpretation, and indicates the default valuesused for producing all figures (unless otherwise indicated in the text).Table D.1: List of parameters and default values that were using for producing figures.Parameter Default valueParameters affecting community complexitym Initial (maximum) number of species 1Parameters affecting mutation and migrationγ Cost of cooperation 0.01µ Rate of mutation 0.001ν Rate of migration 0Parameters affecting within-group density regulationKcells Within-group carrying capacity 100δind Within-group density-dependence (indicator variable) 1Parameters affecting group fission rate (GFR)B0 Minimum GFR 0.05S Slope of group size-dependence for GFR 0Parameters affecting group extinction rate (GER) and between-group density regulationKgroups Scales the total number of groups in the community 0Ktot Scales the total number of cells in the community 105δgroups Dependence of GER on number of groups (indicator variable) 0δtot Dependence of GER on the community’s number of cells (indicator variable) 1Parameters affecting mode of fragmentationn Fractional offspring number (0, 1]s Fractional offspring size (0, 0.5]1540.000.250.500.751.00nm = 1 m = 2 m = 35000 10000 0 5000 0 2000 40000.00 0.25 0.50s0.00 0.25 0.50s0.00 0.25 0.50sAllllllll llll ll lllll l l l lll lllll lll l l l l l l l llll ll ll lllllllllllllllll llllllll ll llll lllll lllll llllllllllllllll02,0004,0006,0008,000s= 0.01n= 0.92s= 0.248n= 0.71s= 0.486n= 0.5Reproduction strategy(upper transect)Productivity (Ntot)Mutationratelll0.0010.010.025BFigure D.1: When extinction rate grows with number of groups instead of number of cells, the main qualitativemodel result does not change: increased community complexity shifts the productivity peak away from smallbottlenecks and toward binary fission. A: The color indicates equilibrium community productivity (Ntot). As thenumber of species (m) increases, the strategy that maximizes Ntot moves rightward along the upper transect ofthe strategy space. B: Some strategies that do well with small µ (large offspring) are not viable when µ is large(shown here with m = 2). Parameters: δtot = 0, δgroups = 1, Kgroups = 103; all other parameters are set to thedefault values (table D.1).D.2 Density regulation based on number of groupsAs the community grows, the extinction rate (Eq. 5.4) increases, which maintains the number of cellsbounded at some upper limit. In the main text, we focused on the case in which δtot = 1 and δgroups = 0,meaning the the extinction rate increases with the number of cells in the community. Alternatively, it isconceivable that rate could instead increase with the number of groups in the community, in which caseδtot = 0 and δgroups = 1. This kind of density-dependence provides qualitatively similar results to the oneexplored in the main text. For example, it is still the case that increasing the number of species moves theproductivity optimum away from complete fragmentation and toward binary fission, reflecting a trade-off between resistance to mutational meltdown at low offspring sizes and the maintenance of mutualisticinteractions at high offspring sizes (Fig. D.1).D.3 Effect of migration and of size-dependent fragmentation onmaximum mutation rateIn section 5.3.3, we showed that size-dependent fragmentation rate and migration can allow multispeciescommunities to prevent mutational meltdown (Fig. 5.5). The same results are shown in Fig. D.2 for abroader range of values of S (slope of size-dependence), and in Fig. D.3 for a broader range of values ofν (migration rate).D.4 Recovering the results of Pichugin et al. (2017)In Pichugin et al. (2017), cells within group i grow with a birth rate b(Ni) = 1+Mg(Ni), where g(Ni) =[(Ni−1)/(Nmax−2)]κ . Here, Nmax is the maximum cell number in a group, M is the maximum benefitof group life, and κ is a “complementarity” parameter that measures how each additional cell increasesthe benefit. When the number of cells reaches Nmax, the group fissions according to a given fissioning155m = 1 m = 2 m = 3 m = 4S = 0.0S = 0.1S = 2S = 40 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.500.5100.5100.5100.51Frac. offsp. size (s)Frac. offsp. number (n)−10.0−7.5−5.0−2.5Maximummutationrate(log)Figure D.2: Size-dependent fragmentation rate allows multispecies communities to resist mutational meltdown.Each panel shows, for each position in the strategy space, the maximum mutation a population can experiencebefore going extinct (similar to Fig. 5.3E and Fig. 5.5). Columns (from left to right) depict increasing numberof species. Rows (from top to bottom) depict increasing slope of size-dependence. Grey squares correspond tocommunities for which the maximum mutation rate is outside the range of our simulations or numerical errors.The white polygons are convex hulls containing the five highest values of maximum mutation rate (to help guidethe eye). For each value of mutation rate, we assessed ten replicates per fragmentation mode; the figure showsthe mean (across replicates) of the logarithm of maximum mutation rate. Parameters: all parameters are set to thedefault values, unless otherwise indicated (table D.1).strategy, or “partition”. Pichugin et al. (2017) tested all mathematically possible partitions for a givenNmax, and measured their group-level fitness. They found that the complementarity parameter κ isthe main determinant of group-level fitness. Overall, if there are diminishing returns to the benefit ofadditional cells (κ 1), binary fragmentation is the most fit partition, but if there are increasing returns(κ  1), unicellular propagule production is the most fit. Other parameters, such as maximum benefitM, have a smaller influence on group-level fitness.Our framework is general enough to accommodate Pichugin et al. (2017) as a special case. We set156m = 1 m = 2 m = 3 m = 4ν = 0.0ν = 0.01ν = 0.1ν = 10 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.500.5100.5100.5100.51Frac. offsp. size (s)Frac. offsp. number (n)−10.0−7.5−5.0−2.50.0Maximummutationrate(log)Figure D.3: Low to intermediate migration rates allow multispecies communities to resist mutational meltdown.Each panel shows, for each position in the strategy space, the maximum mutation a population can experiencebefore going extinct (similar to Fig. 5.3E and Fig. 5.5). Columns (from left to right) depict increasing number ofspecies. Rows (from top to bottom) depict increasing migration rates. Grey squares correspond to communitiesfor which the maximum mutation rate is outside the range of our simulations or numerical errors. The whitepolygons are convex hulls containing the five highest values of maximum mutation rate (to help guide the eye).For each value of mutation rate, we assessed ten replicates per fragmentation mode; the figure shows the mean(across replicates) of the logarithm of maximum mutation rate. Parameters: all parameters are set to the defaultvalues, unless otherwise indicated (table D.1).the group extinction rate to zero, and the birth rate of a cell in group i to bi(Ni) = 1+g(Ni), with g(Ni) =[(Ni− 1)/(Kind− 2)]κ . Finally, we set the group fission rate to zero if Ni < Kind, else Bi = B0 = 106.When we measure group growth rate we observe the same result as Pichugin et al. (2017) (Fig. D.4A).The same pattern is observed when measuring cell growth rate (Fig. D.4B), which makes sense because,as soon as group size has equilibrated, both cells and groups have to grow at the same rate.1570.000.250.500.751.00nκ = 0.025 κ = 0.05 κ = 0.1 κ = 1 κ = 10 κ = 20 κ = 400.75 0.80 0.85 0.75 0.80 0.85 0.75 0.80 0.55 0.60 0.65 0.45 0.50 0.45 0.50 0.440.460.480.500.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5sA0.000.250.500.751.00nκ = 0.025 κ = 0.05 κ = 0.1 κ = 1 κ = 10 κ = 20 κ = 400.75 0.80 0.85 0.75 0.80 0.85 0.75 0.80 0.55 0.60 0.65 0.46 0.48 0.50 0.46 0.48 0.50 0.44 0.46 0.48 0.500.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5s0.0 0.1 0.2 0.3 0.4 0.5sBFigure D.4: In the absence of group-level density-dependence, if the cell birth rate increases with group size,the complementarity parameter κ determines which strategy maximizes fitness—measured either as group-levelgrowth rate (A) or as cell-level growth rate (B). For κ  1 (diminishing returns), binary fragmentation maxi-mizes fitness, whereas for κ 1 (increasing returns), single-cell reproduction maximizes fitness. The color scaleindicates growth rate.158

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