Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Sudweeks, Jaye
Master of Science - MSc
University of British Columbia
The prevalence of cooperative behavior is a fundamental problem in evolutionary biology. Because cooperative behaviors impose a cost, cooperative individuals are vulnerable to exploitation by Defectors that do not contribute and instead benefit from the cooperation of others. The bacteriophage ϕ6 shows strategies of cooperation and defection during infection. While replicating inside a host cell, ϕ6 viruses produce essential proteins in the host cell cytoplasm. Because coinfection is possible, a virus does not have exclusive access to its own products. Cooperators viruses produce products while Defector viruses produce less and instead steal from Cooperators. Previous work found that ϕ6 is trapped in the prisoner’s dilemma, predicting that Defectors will replicate faster than Cooperators and out-compete them; cooperation is doomed. Still, cooperative ϕ6 exist, so there must be a mechanism that maintains cooperation. Defectors are advantaged when coinfection is common and there is ample opportunity to extort Cooperators. However, this advantage wanes when coinfection is less common, and Cooperators are advantaged when single infection is common. The rates of single infection and coinfection are modulated by the densities of hosts and viruses. We propose that environmental feedback, or the interplay between viral and host densities, might maintain cooperation in ϕ6 populations. We develop and analyze a mathematical model that incorporates environmental feedback and find that for many parameter combinations, environmental feedback leads to the co-existence of Cooperators and Defectors.
ENVIRONMENTAL FEEDBACK ALLOWS ESCAPE FROM THE PRISONER’S DILEMMAbyJAYE SUDWEEKSBS, North Carolina State University, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2022©Jaye Sudweeks, 2022The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Environmental feedback allows escape from the prisoner’s dilemmasubmitted by Jaye Sudweeks in partial fulfillment of the requirements for the degree of Masterof Science in Mathematics.Examining Committee:Christoph Hauert, Professor, Mathematics, University of British ColumbiaSupervisorAlun Lloyd, Professor, Mathematics, North Carolina State UniversitySupervisory Committee MemberLaura Parfrey, Professor, Zoology, University of British ColumbiaSupervisory Committee MemberiiAbstractThe prevalence of cooperative behavior is a fundamental problem in evolutionary biology. Be-cause cooperative behaviors impose a cost, cooperative individuals are vulnerable to exploitationby Defectors that do not contribute and instead benefit from the cooperation of others. The bacte-riophage ϕ6 shows strategies of cooperation and defection during infection. While replicating insidea host cell, ϕ6 viruses produce essential proteins in the host cell cytoplasm. Because coinfection ispossible, a virus does not have exclusive access to its own products. Cooperators viruses produceproducts while Defector viruses produce less and instead steal from Cooperators. Previous workfound that ϕ6 is trapped in the prisoner’s dilemma, predicting that Defectors will replicate fasterthan Cooperators and out-compete them; cooperation is doomed. Still, cooperative ϕ6 exist, sothere must be a mechanism that maintains cooperation. Defectors are advantaged when coinfectionis common and there is ample opportunity to extort Cooperators. However, this advantage waneswhen coinfection is less common, and Cooperators are advantaged when single infection is common.The rates of single infection and coinfection are modulated by the densities of hosts and viruses.We propose that environmental feedback, or the interplay between viral and host densities, mightmaintain cooperation in ϕ6 populations. We develop and analyze a mathematical model that in-corporates environmental feedback and find that for many parameter combinations, environmentalfeedback leads to the co-existence of Cooperators and Defectors.iiiLay SummaryCooperation abounds in nature- bees pollinate plants, meerkats rotate through guard duty, andbacteria work together to scavenge iron. However, many who study evolution find the prevalenceof cooperation surprising. Cooperative behaviors impose a cost, so the individual is incentivizedto not contribute while benefitting from the cooperation of others. The bacteriophage ϕ6 producesproteins in the host cell environment. As multiple viruses can infect the same cell, these products areshared, and cooperating viruses that produce products are vulnerable to exploitation by defectingviruses that steal from the common pool in a host cell. Previous work predicts that cooperation inϕ6 is doomed. However, cooperative ϕ6 exist; there must be a mechanism that allows cooperationto survive. We use mathematical modeling to show that fluctuating viral and host densities permitcooperation to survive in ϕ6 populations by vary the rates of coinfection and allowing Cooperatorsto avoid exploitation.ivPrefaceThis thesis was a collaborative effort between Christoph Hauert and me. Christoph posedthe original research question and we co-developed the model (Chapter 2) with some commentsfrom Eric Rozon. I performed the analysis (Chapter 3) and wrote the conclusions (Chapter 4) indiscussion with Christoph.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiChapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2. Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Chapter 3. Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1 Dynamics of Hosts and a Single Viral Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Dynamics with Single Infections Only, µ2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 Invasion Analysis: Cooperator Invasion of E∗D . . . . . . . . . . . . . . . . . . . . 143.2.2 Invasion Analysis: Defector Invasion of E∗C . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Numerical Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Dynamics with Double Infections only, µ1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Dynamics of the Full System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.1 Region I: uninfected I (µ < µSM ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.2 Region II: bi-stable (µSM < µ < µSC) . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.3 Region III: tri-stable (µSC < µ < µHB) . . . . . . . . . . . . . . . . . . . . . . . . 253.4.4 Region IV: bi-stable with limit cycle (µHB < µ < µSD) . . . . . . . . . . . . . 273.4.5 Region V & VI: uninfected II (µSD < µ < µT1) & uninfected III(µT1 < µ < µT2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28vi3.4.6 Regions VII & VIII: heteroclinic I (µ > µT2) & heteroclinic II (µ > µT3) 29Chapter 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Appendix A. Choosing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Appendix B. Stability Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Appendix C. Forms of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Appendix D. Creating Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Appendix E. No Host Density Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44viiList of Tables1 Table of parameter interpretation and values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6viiiList of Figures1 Single viral strategy: bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Single viral strategy: projections of phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Single viral strategy: limit cycle size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 µ2 = 0 : invasion analysis visual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 µ2 = 0: bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 µ1 = 0: numerical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Full system: bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Full system: dynamics of region I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Full system: dynamics of region II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410 Full system: dynamics of region III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611 Full system: dynamics of region IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2712 Full system: dynamics of regions V & VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813 Full system: dynamics of regions VII & VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2914 Supplemental figure:single viral strategy, no host density dependence . . . . . . . . . . . . . . 45ixAcknowledgementsI would like to extend my thanks and appreciation to many people. Christoph, thank you for yourguidance, patience, and general good humor over the last 3 years despite awful circumstances.Laura Parfrey, thank you for reading my thesis. I deeply appreciate your willingness to readmath! Alun Lloyd, thank you for reading my thesis, providing very helpful comments on anearlier draft, and your continued mentorship. Brandon Hollingsworth, thank you for years ofsupport, friendship, and guidance. Thanks especially for writing club, which helped keep memotivated when things were tough. Eric Rozon, thank you for fun, ideas, editing, and a shoulderto lean on. My family: Ivy, Jayce, and Jen– thank you for your constant love and support. Itmeans to world to me. And, of course, thanks to the Big House & Dunbar House for feeding me,gleefully participating in my schemes, and generally keeping me sane during 2020.xDedicationFor Clair & Marsha, Jay & Isabel, and the rest of my family.xiCHAPTER 1IntroductionThe evolution and maintenance of cooperation is a long-standing problem in biology. Thismay seem counterintuitive because cooperative behaviors abound in nature. To name only a fewexamples: vampire bats share blood, bacteria work together to scavenge iron, and meerkat sentinelsuse alarm calls to warn of predators [1–3]. However, upon further inspection, the prevalence ofcooperation is surprising. Regardless of the specifics of the interaction, cooperative behaviorsimpose a cost upon Cooperators, which makes Cooperators vulnerable to exploitation by Defectorsthat do not pay the cost to cooperate but still benefit from the cooperation of others.The fragility of cooperation is often captured through the mathematical metaphor of the pris-oner’s dilemma, which we describe through an example of a virus species with the potential forcooperative behavior. During replication, the viruses produce a shared pool of proteins in the hostcell cytoplasm [4]. This species has two strategies (genotypes): Cooperators that pay a cost to co-operate and produce proteins, and Defectors that produce fewer proteins and instead appropriatethe proteins created by Cooperators. Because strategies correspond to genotypes, an individualvirus cannot choose whether to cooperate or defect. The prisoner’s dilemma considers pairwiseinteractions within a population. The reproductive success of an individual is dependent on bothits own strategy and the strategy of its partner. We represent the reproductive potential of anindividual after all possible pairwise interactions (“payoffs”) in payoff matrix 1, which reads asfollows: Cooperators have payoff α when interacting with other Cooperators, but payoff β wheninteracting with Defectors. Similarly, Defectors have payoff γ when interacting with Cooperators,but payoff δ when interacting with other Defectors.(1)(Cooperator DefectorCooperator α βDefector γ δ)The ordering of the payoffs determines the characteristics of the interaction. The prisoner’sdilemma is a specific ordering such that γ > α > δ > β. Consider an individual in an interactingpair: if the partner is a Cooperator, the focal individual will have higher payoff from defectingrather than cooperating (γ > α), and if the partner defects, the focal individual will still havehigher payoffs from defecting rather than cooperating (δ > β). Defection is a dominant strategy1and always has higher payoff than cooperation. This creates a social dilemma, or a conflict ofinterest between the individual and the pair. If we extend this line of reasoning to a population ofCooperators and Defectors where individuals interact randomly, it follows that Defectors will alwayshave higher payoff than Cooperators and thus reproduce faster and out-compete Cooperators,driving Cooperators to extinction [5]. To see why this is true, consider the fates of Defectors andCooperators when either trait is at low frequency. When Defectors are at low frequency thereis ample opportunity to extort Cooperators, so Defectors will have high reproductive rates anddefection will increase in frequency. When Cooperators are at low frequency they are extorted andhave low reproductive potential; cooperation cannot spread from low frequency. The eliminationof cooperation is detrimental to the population as a whole. Pairs of Defectors have the lowestpayoffs of any combination, so as cooperation is eliminated, the reproductive rate of the populationdeclines.However, cooperative behaviors exist in nature, so there must be mechanisms that allow coop-eration to survive. Previous work has found potential avenues for cooperation to be maintained,including repeated interactions and direct reciprocity, spatial structure, punishment, voluntaryparticipation in interactions, and group selection [5–12]. In thesis we explore another potentialmechanism to maintain cooperation, inspired by experimental work with bacteriophage ϕ6.Previously, Turner and Chao found that ϕ6 is trapped in a prisoner’s dilemma [13]. The phage,which infects strains of Pseudomonas bacteria, replicates inside hosts cells and ultimately causeshost cells to lyse (rupture) and release ϕ6 viruses back into the environment [4, 14, 15]. Duringreplication ϕ6 viruses produce proteins in the host cell cytoplasm. Because multiple viruses cancoinfect the same cell, producing proteins in the host cell cytoplasm prevents a bacteriophage fromhaving exclusive access to its own materials, leading to strategies of cooperation and defection:Cooperator viruses donate proteins to the within-host common pool, while Defector viruses donateless while appropriating a greater share of common resources, and, as a consequence, replicate fasterthan Cooperators. Note that the shared proteins are not released into the environment when a celllyses and thus cannot be taken up by other cells. Turner and Chao measured the payoff matrixfor interactions between coinfecting Cooperators and Defectors and found that it conforms to theprisoner’s dilemma, indicating that cooperation in ϕ6 is doomed [16].We see an escape route for cooperation. We propose that environmental feedback, or the inter-play between host and viral densities, can maintain cooperation in ϕ6 populations. The first crucialidea here is that viruses have some chance of infecting a cell alone, which creates an environmentwhere Cooperators can shield themselves from extortion from Defectors. This idea is analogous toprevious work which has shown that group selection can maintain cooperation because Defectorsdo well in groups with Cooperators but suffer when with other Defectors, while in contrast Coop-erators suffer when in groups with Defectors but do well when in groups with other Cooperators.Here, though Defectors will have high rates of replication when coinfecting with Cooperators, the2rates will decrease when Defectors infect cells alone or with other Defectors, while in contrast Co-operators will thrive when singly infecting a cell. We can understand these differential replicativerates through the lens of context-dependence, or the idea that the costs or benefits associated withcooperation vary with the biotic or abiotic environment [17]. For ϕ6, cooperation is costly whenCooperators are in groups with other Defectors, but the costs decrease when Cooperators are aloneor in groups with other Cooperators.The second key idea is that the rates of single versus coinfection are modulated by the relativedensities of free-living virus and uninfected host cells [13, 18]. Turner and Chao’s work measuredthe reproductive output of coinfecting viruses, which required fixing the densities of viruses anduninfected host cells to control the probability of coinfection. We can think of these results asa snapshot in time, as in a natural system the densities of viruses and hosts would likely vary.Interestingly, previous work has shown that explicitly accounting for population dynamics in modelsfavors cooperation because varying population densities changes the group size, and Cooperatorsare more likely to be in groups of only Cooperators when group sizes are small [19]. It is possiblethe following cycle could emerge: viral densities are relatively high. Coinfection events are commonand Defectors are advantaged, leading Defectors to increase in frequency. However, Defectors havelower reproductive output than Cooperators, so as defection spreads viral reproduction decreases,leading to a decrease in viral density and subsequent reduction in coinfection events. This in turnadvantages Cooperators, Cooperators increase in frequency, and viral reproduction increases. Thisinterplay could settle into stable co-existence of Cooperator and Defectors, or could lead to cyclicbehavior where the densities of Cooperators and Defectors oscillate.To investigate the impact of environmental feedback on the maintenance of cooperation in thesystem, we develop a mathematical model which accounts for both single and double infectionevents and allows the densities of viruses and hosts to fluctuate.3CHAPTER 2Model DerivationWe track the densities of uninfected, Cooperator infected, Defector infected, and mixed infectionhost cells (HU , HC , HD, HM ), as well as free-living Cooperator and Defector viruses (VC , VD).Equation 2 describes the change in density for each of these variables.H˙U =logistic growth︷ ︸︸ ︷HU (r − ξHU )−single infection events︷ ︸︸ ︷µ1HU (VC + VD)−double infection events︷ ︸︸ ︷µ2HU(EV 2C + 2VCVD + V2D)(2a)H˙C =infection events︷ ︸︸ ︷HUVC (µ1 + µ2VC)−cell lysis︷︸︸︷dHC(2b)H˙D =infection events︷ ︸︸ ︷HUVD (µ1 + µ2VD)−cell lysis︷ ︸︸ ︷dHD(2c)H˙M =infection events︷ ︸︸ ︷2µ2HUVCVD −cell lysis︷ ︸︸ ︷dHM(2d)V˙C =viral replication︷ ︸︸ ︷λd (αHC + βHM )−degradation︷ ︸︸ ︷κdVC(2e)V˙D =viral replication︷ ︸︸ ︷λd (γHM + δHD)−degradation︷ ︸︸ ︷κdVD(2f)In the absence of viruses, uninfected hosts undergo logistic growth: the population of uninfectedhosts grows at a per-capita rate of increase r ( where r = bU − dU for per-capita birthrate bU andper-capita death rate dU ) which is decreased by competition at rate ξ.Uninfected host cells are susceptible to infection by viruses. Because empirical evidence showsthat the average number of ϕ6 viruses per infected host cell is low (between 2-3 phages per cell), weassume that host cells are infected by at most two viruses [20, 21]. This allows us to capture theimpact of double infected cells where Defectors can extort Cooperators while keeping our equationsrelatively simple and easy to work with. We do not consider spatial structure and assume infectionevents happen at a rate proportional the the densities of viruses and uninfected host cells. Unin-fected host cells are singly infected at the per-capita rate µ1(VC + VD), where µ1 is defined as thesingle infection rate constant.4We model double infection as a single event which happens at the per-capita rate µ2(VC+VD)2,where µ2 is defined as the double infection rate constant. Mechanistically, this means that doubleinfections occur when two viruses collide with and infect an uninfected host cell in a very,very short(negligible) time window. We assume that once this short time window passes, viruses institutedefenses that prevent further viruses from entering the host cell, meaning singly infected host cellscannot be converted into doubly infected host cells. Expanding the double infection term gives:µ2(V 2C + 2VCVD + V2D)The V 2C and V2D terms refer to double infections involving only Cooperators or Defectors. The2VCVD term refers to mixed infection events. The “2” corresponds to the two way a mixed infectionevent can happen: the host can be infected by a Cooperator then a Defector, or by a Defector andthen an Cooperator.We define an infection rate constant µ and set µ1 = µ and µ2 = µ2. The parameter µ captureshow competent viruses are at attaching to cells, given a collision. If single infection events, whichinvolve a meeting between a single virus and uninfected host, cell happen at a per-capita rateµ(VC + VD), then in our model setup-up it follows that double infection events, which involvetwo viruses infecting a uninfected host cell, should happen at per-capita rate µ2(VC + VD)2. Forexample, we can rewrite the term describing two Cooperators infecting an uninfected host cellas µ2HUV2C = µ2HUV2C = (µVC)(µVC)HU which captures two Cooperator viruses infecting anuninfected host cell in rapid succession.Once inside host cells, viruses replicate. The rate of replication of a given virus in dependent onits strategy as well as the strategy of a co-infecting virus, if any. We use the payoff values measuredby Turner and Chao (α = 1, β = 0.65, γ = 1.99, and δ = 0.83) as our strategy dependent replicationrates. These payoffs conform to the prisoner’s dilemma (γ > α > δ > β) but have been normalizedso that α = 1. We scale the rates of replication by a parameter λ, which allows us to considerrelatively faster or slower rates of viral replication while maintaining the proper ratio between thereplication rates of different strategies.The viruses produced inside cells are released as host cells lyse, which we assume happens ata per-capita rate d, regardless of infection type. We assume that a cell that is doubly infected bya given strategy will have the same overall contribution of free-living virus as a cell that is singlyinfected by that strategy, which is captured by the grouping of host cells that are singly and doublyinfected by the same viral strategy. We assume that virus is strictly lytic, meaning that infectedhost cells lyse without replicating and therefore do not contribute to the densities of infected hostcells.Free-living viruses degrade at a per-capita rate κd. Here, κ scales the life span of free-livingviruses relative to infected host cells. If κ < 1, free-living viruses in the environment have a longer5lifespan than infected host cells, whereas if κ > 1, infected host cells have a longer lifespan thanfree-living viruses.parameter value interpretationr 8.3 per-capita growth rate of uninfected host cellsd 8.3 per-capita death rate of infected host cellsκ 1scales the death rate of free-living virusesrelative to the per-capita death rate of infected host cellsλ 2 scales the rate of release of free-living viruses from lysing host cellsα 1 payoff to a Cooperator that co-infects with another Cooperatorβ 0.65 payoff to a Cooperator that co-infects with a Defectorγ 1.99 payoff to a Defector that co-infects with a Cooperatorδ 0.83 payoff to a Defector that co-infects with another Defectorξno competition: 0with competition: .1uninfected host cell competition rate constantµ varied infection rate constantµ1 µ1 = µ single infection rate constantµ2 µ2 = µ2 double infection rate constantTable 1. Table of parameter interpretation and values. Parameter values remainconstant throughout the paper, with the exception of µ (and accordingly µ1, µ2).6CHAPTER 3Analysis and ResultsOur basic question is whether environmental feedback can maintain cooperation, and if so, underwhat conditions. To answer this, we use µ as a bifurcation parameter and look at the dynamics ofthe model as µ varies. Specifically, we look at the stability of equilibria. We might find parameterranges for which all host cells and viruses die, ranges for which the system cannot support viruses,or ranges where only one viral strategy can co-exist with hosts; we might also find ranges whereCooperators and Defectors stably co-exist or co-exist in limit cycles.We first develop our understanding of environmental feedback by looking at a system that hasonly a single viral strategy. We then investigate the cases µ2 = 0 (only single infections) and µ1 = 0(only double infections), simplified cases where we have predictions for the fate of cooperation, totest and develop our intuitive understanding of the system. Finally, we analyze the full system.73.1 Dynamics of Hosts and a Single Viral StrategyWe first model hosts and a single viral strategy. There are no Cooperators or Defectors becausecooperation and defection have meaning only when there are multiple strategies. Instead there areuninfected hosts, infected hosts, and free-living viruses. There are two reasons to analyze this case:first, it is simpler than the full model, which enables us to build intuition for the dynamical behaviorof our model. Second, this model allows us to examine the impact of environmental feedback alonewithout multiple viral strategies.We might initially assume that higher infection rates should always be better for the viralpopulation. However, infection by lytic viruses such as ϕ6 causes host cells to lyse, and in additioninfected host cells do not replicate. This means that as viruses successfully spread through a hostcell population, they limit their progeny’s future potential for replication [22–24]. Even with asingle viral strategy, different infection rates could in principle lead to extinction of viruses, stableco-existence of hosts and viruses, or even cyclic behavior.Equations 3a-3c describe the dynamics of hosts and a single viral type. Here, HU is the densityof uninfected hosts, HI the density of infected hosts, and V the density of free-living viruses. Notethat equations 3a-3c are simply a subset of the full model equations (2a-2f) without multiple viralstrategies. To avoid introducing more parameters, we use the replication rate previously associatedwith Cooperators (λα).H˙U = rHU − ξH2U − µ1HUV − µ2HUV 2(3a)H˙I = HUV (µ1 + µ2V )− dHI(3b)V˙ = λdαHI − κdV(3c)Equation 3 is similar to standard models virus and host cell dynamics, but differs from previousmodels in that it does not account explicitly for the impact of infection events on viral densitiesand most interestingly through the inclusion of a double infection term [25, 26]. Equation 3 has4 biologically relevant (all components are non-negative, real numbers) equilibria: an extinctionequilibrium E∗0 = (HU , HI , V ) = (0, 0, 0), an uninfected equilibrium E∗U = (HU , HI , V ) = (rξ , 0, 0),and two endemic equilibria E∗E1 and E∗E2 where all hosts and viruses have non zero density. Here,endemic refers to a stable infected state where a viral infection is maintained in the host cellpopulation without further viruses being introduced. See Appendix C.1 for more details on theforms of E∗E1 and E∗E2.To explore the dynamics of Equation 3, we analyze the stability of the equilibria. We firstconsider E∗U to determine if viruses can successfully spread in an almost entirely susceptible (un-infected) host cell population. This is a standard analysis in infectious disease modeling that isaccomplished by considering R0, or the number of newly infected cells that arise from any one8infected cell when almost all other cells are uninfected. If R0 is less than one, each infected hostcell will produce on average less than one new infected host cell, and the virus will not spread. Incontrast, if R0 > 1, every infected cell will produce on average more than one new infected host cell,and the virus will spread. We adapt R0 to consider how many new viruses a given virus will giverise when most host cells are uninfected [25]. In the absence of viruses, host cells have a populationdensity rξ . The virus per-capita infection rate is µ1, and a virus lives for1κd , so each virus producesan average of rξκd infected cells. An infected cell produces free-living viruses at rate λdα, lives foran average of 1d time units, and so produces λα free-living viruses. Taken together, a given virusproduces rξκd×λα new viruses, giving R0 = µ1rλακdξ . If we solve for R0 < 1 (the virus cannot spread),we find that H∗U is stable when µ1 <dκξrλα .Through analysis of the Jacobian, we determine that E∗0 is never stable (Appendix B). Both E∗E1and E∗E2 produce stability conditions that are too complex to consider analytically. We proceedwith a numerical bifurcation using the parameters in Table 1. (See Appendix A for more details onparameter selection, and Appendix D for more details on the creation of the figure).Three bifurcations seen in Figure 1 lead to the four distinct dynamical regions illustrated inFigure 2. Note that trajectories in Figure 2 can cross because the figure shows two dimensionalprojection of three dimensional space. In Figure 1C, the shaded and unshaded regions indicatewhere single and double infections, respectively, are more common. In Equation 3, single infectionshappen at per-capita rate µ1V and double infections happen at a per-capita rate µ2V2. If we recallthat µ1 = µ and µ2 = µ2, then single and double infections occur at per-capita rates µV and µ2V 2,and single infections are more common than double infections when µ2V 2 < µV =⇒ V < 1µ .Interestingly, the line between the two regions cuts in between E∗E1 and E∗E2, meaning that E∗E1(dark blue line) has viral densities that favor single infections while the E∗E2 (light blue line) hasdensities that favor double infections. However, E∗E1 is never stable, and E∗E2 is only stable for ashort window (µS < µ < µH), after which the population exhibits cyclic behavior with non-constantrates of single and double infection.For µ < µS (Figure 2A/E), viruses cannot spread through the host cell population, and virusesgo extinct. However, for µS < µ < µH (Figure 2B/F) the system is bistable; both E∗U and E∗E2 havebasins of attraction, and any trajectory that starts in an equilibrium’s basin of attraction is drawninto that equilibrium. This bistability is reminiscent of a backward bifurcation in epidemiologicalmodels. Backwards bifurcations are significant because they indicate a parameter range wherea disease can spread even though R0 < 1. Indeed, in our model, in this dynamical region highdensities of virus are able to invade and establish at the endemic equilibrium, even though R0 < 1until µ = µT . Beyond this, note that that the two endemic equilbria do not bifurcate directly fromE∗U at µS , which means that if viruses are able to invade there would be a sudden and non-trivialdecrease in the density of uninfected host cells. For µS < µ < µT , the system is still bistable. Thisparameter range exhibits cyclic behavior and even though R0 < 1, relatively low densities of virus9Figure 1. Bifurcation diagram for hosts and a single viral type (equations 3a-3c). Each panel shows a different component of the equilibria. Line colors indicateequilibria: light green, E∗0 ; dark green, E∗U ; dark blue, E∗E1; light blue, E∗E2. Dashedlines indicate that an equilibrium is unstable, while solid lines indicate stability.Dash-dotted vertical lines show bifurcations. The shaded/unshaded regions in Cindicate where single/double infections are more common. As µ increases, threebifurcations occur. First, at µS ≈ 0.033 there is a saddle node bifurcation and E∗E1and E∗E2 appear. Second, at µH ≈ 0.042 a Hopf bifurcation occurs as E∗E2 (lightblue) loses stability. Finally, at µT =κdξαλr = 0.05 a transcritical bifurcation occursas E∗E2 (dark blue) and the E∗U collide and exchange stability; Note that µT is ourpreviously determined R0 = 1 threshold. These three bifurcations lead to 4 distinctdynamical regions labeled in A. Region I: Uninfected. Viruses to extinct. RegionII: Bi-stability. All trajectories go to E∗U or E∗E2. Region III: Bi-stable limit cycles.All trajectories are drawn either into the stable limit cycle or into E∗U . Region IV:Stable limit cycles. All trajectories are drawn into the stable limit cycle.can invade and establish cycles of infection. For µ > µT , finally R0 > 1nand any level density ofvirus is able to invade an establish in limit cycles.Figure 3 shows the shape and size of the limit cycles surrounding E∗E2 for µ > µT . As µincreases, the limit cycles approach the axes, meaning that at different points in the course of acycle, viral and host densities approach zero. However, biological systems cannot fall arbitrarilyclose to zero. In reality, for high values of µ stochastic events would drive either viruses or bothhosts and viruses extinct.10Figure 2. Projections of phase space for the 4 stability regions indicated in Figure1. Note that trajectories may cross in this projection of a higher dimensional space.Dots show the position of equilibria. Open dots indicate unstable equilibria whilefilled dots indicate stable equilibria. Dot colors: light green , E∗0 ; dark green,E∗U ; dark blue, E∗E1; light blue, E∗E2. A & E Dynamics for Region I: Uninfected.µ = 0.020. All trajectories go to E∗U and viruses quickly die out. B & F Dynamicsfor region II: Bi-stability. µ = 0.035. Some trajectories go to E∗I2, while othertrajectories go to E∗U . Small densities of viruses cannot establish. C &GDynamicsfor region III: Bi-stable limit cycles. µ = 0.045 Some trajectories are drawn intoa stable limit cycle, outlined in orange, and move through different compositionsof viruses, infected, & uninfected host cells. Other trajectories are drawn towardsE∗U . Small densities of viruses cannot establish. D & H Dynamics for region IV:stable limit cycles. µ = 0.06. The population continues to exhibit cyclic behavior,but now all trajectories are drawn into the limit cycle outlined in orange.11Figure 3. The size of limit cycles as µ approaches the Hopf bifurcation at µH .Each closed loop indicates a limit cycle while the dot indicates the location of E∗E2.Color indicates the value of µ, which increases from red to blue. As µ increases,the limit cycles approach the axes, while as µ decreases to µH , the limit cyclescontract on E∗E2.123.2 Dynamics with Single Infections Only, µ2 = 0We reintroduce cooperation and defection and investigate the case µ2 = 0. In this case, there areno mixed infection cells (HM ), meaning that Defectors cannot extort Cooperators. Our intuition isthat cooperation should be able to survive in the absence of co-infection. Analyzing this simplifiedsystem will test our intuition and develop our understanding of the system as a whole.This system is described by the equations 4a-4e, a subset of the full model equations (2a - 2f)where µ2 has been set to 0. Note that when µ2 = 0, H˙M = −dHM , meaning that the density ofmixed infection cells will only decrease to 0. Because mixed infection cells will have only transientimpacts on the dynamics, we exclude H˙M .H˙U = rHU − ξH2U − µ1HU(VC + VD)(4a)H˙C = µ1HUVC − dHC(4b)H˙D = µ1HUVD − dHD(4c)V˙C = λdαHC − κdVC(4d)V˙D = λdδHD − κdVD(4e)There are 4 equilibria: an extinction equilibrium E∗0 = (HU , HC , HD, EC , ED) = (0, 0, 0, 0, 0), anuninfected equilibrium E∗U = (HU , HC , HD, EC , ED) = (rξ , 0, 0, 0, 0), a Cooperator only equilibriumE∗C , and a Defector only equilibrium E∗D.E∗C has the form:E∗C = (HU , HC , HD, VC , VD) =( d καλµ1,κ (αλµ1 r − d κ ξ)α2 λ2 µ12, 0,α λµ1 r − d κ ξαλµ12, 0)while E∗D has the form:E∗D = (HU , HC , HD, VC , VD) =( d κδ λµ1, 0,κ (δ λµ1 r − d κ ξ)δ2 λ2 µ12, 0,δ λ µ1 r − d κ ξδ λµ12)E∗0 is never stable and E∗U is stable for µ1 <κdξαλr (Appendix B). The stability conditionsassociated with E∗C and E∗D are too complicated to understand analytically. However, we can gainsome traction on stability of E∗C and E∗D by examining each viral strategy’s ability to invade theother. For example, if very small densities of Cooperators can invade a population of Defectors, weknow that the E∗D cannot be stable. Here, invasion means being able to grow from arbitrarily smalldensity.133.2.1 Invasion Analysis: Cooperator Invasion of E∗DWe first consider very small densities of VC and HC introduced to a population at E∗D. If HC andVC can grow at arbitrarily small densities, we know that E∗D is not stable.As we want to discuss the growth of HC and VC near E∗D, in equations 5a & 5b we substituteE∗D into H˙C and V˙C , the equations governing the growth of HC and VC .˙HD∗C =d κδ λVC − dHC(5a)˙V D∗C = αdλHC − dκVC(5b)Cooperators can invade if ˙HD∗C > 0 and˙V D∗C > 0 in the simultaneous limit HC , VC → 0. We’llfirst look for conditions where ˙HD∗C > 0, or where HC can increase in density.˙HD∗C > 0 =⇒ VC >λδκHC(6)There are many ways that HC , VC could go simultaneously to zero. HC could decrease fasterthan VC or vice versa, or both could decrease at the same rate. Figure 4 shows a graphicalrepresentation. Because we know that both HC and VC must be positive, we can visualize takingthe simultaneous limits as trajectories approaching the origin (E∗D) in the first quadrant.Figure 4. Taking the simultaneous limits HC , VC → 0 correspond to trajectoriesinto the origin (E∗D). The dotted line represents trajectories where HC decreasesfaster than VC , the solid line trajectories where both decrease at the same rate,and the dash dotted line trajectories where VC decreases faster than HC .14To account for all possible cases, we make the substitution HC = ρVC with ρ > 0; we thinkabout approaching E∗D from all possible directions where VC and HC maintain a constant ratioρ. In reference to Figure 4, this is equivalent to considering all lines with positive slope. As weconsider only arbitrarily small distances from E∗D, considering all possible values of ρ covers allpossible directions into E∗D.As VC → 0 (and accordingly, HC → 0 so that the ratio ρ is maintained), we find that condition6 met, and thus HC can increase, for any ρ such thatρ <κλδ(7)We’ll next consider conditions such that VC can increase from small densities.˙V D∗C > 0 =⇒αλκHC > VC(8)As before, to take the simultaneous limits we make the substitution HC = ρVC . As VC → 0simultaneously, condition 8 is met, and thus VC can increase, for any ρ such thatρ >κλα(9)Taking conditions 7 and 9 together, as VC → 0, both H˙C > 0 and V˙C > 0 for any ρ such that:κλα< ρ <κλδ(10)In the case of the prisoner’s dilemma, we know that δ < α =⇒ 1α < 1δ , so there are values of ρthat exist in this range. We have found a set of directions that are unstable, so we conclude thatE∗D is unstable.3.2.2 Invasion Analysis: Defector Invasion of E∗CWe’ll next consider the case where small densities of HD and VD are introduced into a populationat E∗C . If Defectors can invade, we know E∗C is not stable.In order to discuss the growth of HD and VD near E∗C , in equations 11a & 11b we substituteE∗C into H˙D and E˙D.˙HC∗D =dκαλED − dHD(11a)˙V C∗D = d δλHD − dκED(11b)15Analogously to the previous section, we want to find conditions where Defectors can increase indensity from arbitrarily small densities, or ˙HC∗D > 0 and˙V C∗D > 0 as HD, VD → 0 simultaneously.We first find conditions such that HD can increase:˙HC∗D > 0 =⇒ VD >αλκHD(12)As in the previous section, in order to take the the limits HD, VD → 0 simultaneously, we makethe substitution HD = ρVD; we think about approaching E∗C from directions where ED and VDmaintain a constant ratio ρ. As we take the simultaneous limits, we find that HD can grow for ρsuch that:ρ <κλα(13)We’ll follow the same line of reasoning with equation 11b to find conditions where VD can grow.˙V C∗D > 0 =⇒δλκHD > VD(14)Similarly, as HD, VD → 0 simultaneously, VD can grow for ρ such that:ρ >κλδ(15)Comparing conditions 13 and 15, as VD → 0 (and thus HD → 0 to maintain the ratio ρ), VDand HD can grow simultaneously for ρ such that:κλδ< ρ <κλα(16)In a prisoner’s dilemma, α > δ, and accordingly 1α <1δ ; condition 16 can never be met. Thus,and VD and HD cannot increase at the same time. In addition, neither HD or VD can increase onits own. If HD alone increased, eventually condition 12 would be violated. Accordingly if VD aloneincreased, eventually condition 14 would be violated.So, VD and HD cannot increase alone or together. These are strong indicators that E∗C is stable.However, we cannot say this definitively and will confirm numerically.3.2.3 Numerical InvestigationFigure 5 shows a numerical bifurcation analysis of equations 4a-4e. See appendix D for more detailson how this figure was created. Figure 5 confirms that E∗C is stable for all values of µ for whichit is biologically relevant. In contrast, E∗D is always unstable. This matches our intuition: in thissystem where coinfection is not possible, Cooperators survive, and in fact prevail over Defectors.16Though there is no coinfection and accordingly no direct competition, viruses compete indirectlyfor a limited population of uninfected hosts. Cooperators, which have higher rates of replicationoutput when infecting a cell alone, are able to drive Defectors to extinction.Figure 5. Bifurcation diagram when µ2 = 0 (equations 4a-4e ). Each panelshows a different component of the equilibria. Line colors indicate equilibria: lightgreen, E∗0 ; dark green, E∗U ; blue, E∗C ; red, E∗D. Dashed lines indicate that anequilibrium is unstable, while solid lines indicate stability. As µ1 increases, thereare two transcritical bifurcations. First at µT1 =κdξαλr = 0.05, E∗U and E∗C collideand exchange stability. Note that µT1 is the condition R0 = 1 for a population ofonly Cooperators. Second at µT2 =κdξδλr = 0.0602, E∗U and E∗D collide, though bothequilibria remain unstable. Note that µT2 is the condition R0 = 1 for a populationof only Defectors.173.3 Dynamics with Double Infections only, µ1 = 0We next consider the case where there are only double infections (µ1 = 0). In this scenario,Defectors have ample opportunity to extort Cooperators. Our intuition is that cooperation will notsurvive in this system.The dynamics of this system are captured in equations 17a-17f.H˙U = rHU − ξH2U − µ2HU(V 2C + 2VCVD + V2D)(17a)H˙C = µ2HUV2C − dHC(17b)H˙D = µ2HUV2D − dHD(17c)H˙M = 2µ2HUVCVD − dHM(17d)V˙C = λd(αHC + βHM )− κdVC(17e)V˙D = λd(γHM + δHD)− κVD(17f)This system has up to 8 equilibria: an extinction equilibrium E∗0 = (HU , HC , HD, HM , VC , VD) =(0, 0, 0, 0, 0), an uninfected equilibrium E∗U = (HU , HC , HD, HM , VC , VD) = (rξ , 0, 0, 0, 0), two Co-operator only equilibria E∗C1 and E∗C2, two Defector only equilibria E∗D1 and E∗D2, and two mixedequilibria E∗M1 and E∗M2, where all hosts and viruses have nonzero density. See Appendix C.2 formore details on the forms of the Cooperator only and Defector only equilibria. We know that E∗0is always unstable (Appendix B.1) and that E∗U is always stable (Appendix B.2: when µ1 = 0, thestability condition for E∗U is always fulfilled).The purpose of the current analysis is to test our intuition about the fate of cooperation inthis simplified system, not a full understanding of the dynamics. As such, we only analyze thestability of the two Cooperator only and the two Defector only equilibria. This provides sufficientinformation on whether cooperation can survive in this system. The two Cooperator only andDefector only equilibria produce stability conditions that are too complicated for analytical stabilityanalysis. In addition, the equilibria are sufficiently complex that an invasion analysis is infeasible.We numerically determine the stability of the four equilibria.In figure 6, we find that none of these four equilibria are ever stable. This means that thepopulation will not exist at a stable population of only Cooperators or only Defectors. This doesnot preclude the possibility of a population of only Cooperators or only Defectors existing in alimit cycle, though a scan of the dynamics for 0.01 < µ2 < 0.1 did not reveal Cooperator only orDefector only limit cycles. If these cycles do exist, it’s likely that they are highly dependent oninitial condition.18Figure 6. The stability of the two Cooperator only and two Defector only equi-libria as µ2 varies. Each panel shows a different projection of the equilibria. Linecolors indicate equilibria: dark blue line, E∗C1; light blue line, E∗C2; dark red line,E∗D1; light red line, E∗D2. Dashed lines indicate that an equilibrium is unstable,while solid lines indicate stability. All four equilibria are unstable for all values ofµ2.193.4 Dynamics of the Full SystemWe now analyze the full system. We anticipate that for some µ values, environmental feed-back will maintain cooperation. The full system has up to 8 relevant equilibria: an extinctionequilibrium E∗0 = (HU , HC , HD, HM , VC , VD) = (0, 0, 0, 0, 0), an uninfected equilibrium E∗U =(HU , HC , HD, HM , VC , VD) = (rξ , 0, 0, 0, 0), two Cooperator only equilibria E∗C1 and E∗C2 , two De-fector only equilibria E∗D1 and E∗D1, and two mixed equilibria E∗M1 and E∗M2. See Appendix C.3for more details on the forms of the Cooperator only and Defector only equilibria. We examine thestability of these equilibria as µ varies. Only two equilibria, E∗0 and E∗U , are analytically accessible(Appendix B); E∗0 is never stable, and E∗U is stable for µ < µT2 =dκξrλα . Note that this is the condi-tion for R0 = 1 in a population of only Cooperators. The other other equilibria produce stabilityconditions that are too complex to treat analytically, and similarly invasion analysis is infeasible.We proceed with a numerical bifurcation analysis, shown in Figure 7. See appendix D for moredetails on how this figure was created. We can interpret equilibria E∗C1 and E∗C2 as a reductionback to the case of single viral type. Accordingly, the curves for the E∗C1 and E∗C2 in Figure 7have identical shapes to the curves in Figure 1, though the stability of the equilibria are different.Similarly, E∗D1 and E∗D1 correspond to populations of a single viral type with a different reproductiverate, and thus the red lines and blue lines in Figure 7 have very similar shapes, transformedslightly by different reproductive rates (λα for Cooperators vs λδ for Defectors). In Figure 7ABD,the red curves are shifted slightly to the right, indicating that Defector only populations requireslightly higher µ to be viable, but once established follow the same qualitative dynamical pattern asCooperator only populations. Notably, E∗M1 and E∗M2 (purple lines) appear before either E∗C1 andE∗C2 or E∗D1 and E∗D2; mixed populations can establish before either Cooperator only or Defectoronly populations.The shaded/unshaded areas in 7D denote where single/double infections are more common.In Equation 2, single infections happen at per-capita rate µ1(VC + VD) and double infectionshappen at a per-capita rate µ2(VC + VD)2. If we recall that µ1 = µ and µ2 = µ2, then single anddouble infections occur at per-capita rates µ(VC + VD) and µ2(VC + DV )2, and single infectionsare more common than double infections when µ2(VC + VD)2 < µ(VC + VD) =⇒ VC + VD < 1µ .The line separating the two regions (VC + VD <1µ ) lies close to all 3 saddle node bifurcations(µSM , µSC , µSD), though numerical investigation suggests that it does not cut exactly through thepoints.Notably, H∗M1 (light purple) is stable for some µ values, and the two Defector only equilibria(light and dark red) are never stable. Cooperation is not eliminated, and viruses have escapedthe dilemma! In the following sections, we examine the dynamics of the system for each of the 8dynamical regions labeled in 7A to determine the fate of cooperation in each region. Note thattrajectories that start with only HC and VC (or likewise only HD or VD) are simply a reduction20Figure 7. Bifurcation diagram for two viral strategies competing for hosts (equations 2a-2f).Each panel shows a different projection, while line colors indicate equilibria: light green, E∗0 ; darkgreen, E∗U ; dark/light purple, E∗M1/E∗M2; dark/light blue, E∗C1/E∗C2; dark/light red, E∗D1/E∗D2.Dotted lines indicate that an equilibrium is unstable, while solid lines indicate stability. In D, theshaded/unshaded regions indicate where single/double infections are more common. As µ increases,there are 7 bifurcations: first, at µSM ≈ 0.0293 a saddle node bifurcation as the two mixed equilibriaappear. Next, a saddle node at µSC ≈ 0.0329 as the two Cooperator only equilibria appear. Next,at µHB ≈ 0.03459 a Hopf bifurcation as H∗M2 loses stability. Next, at µSD ≈ 0.0397, a saddlenode bifurcation as the two Defector only equilibria appear. Next, at µT1 ≈ 0.0477 a transcriticalbifurcation as dark purple H∗M1 and dark blue H∗C1 collide, follow shortly by another transcriticalbifurcation at µT2 =dκξαλr = 0.0500 asH∗U and the dark blueH∗C1 collide and exchange stability. Notethat this is the condition R0 = 1 for a population of Cooperators. Finally, at µT3 =dκξδλr = 0.0602there is a transcritical bifurcation as H∗U and dark red H∗D1 collide. Note that this is the conditionR0 = 1 for a population of Defectors. These bifurcations lead to the 8 dynamical regions labeledin A. Region I: uninfected I. All trajectories go to E∗U . Region II: bi-stable. All trajectories go toE∗U or E∗M2. Region III: tri-stable. Trajectories go to E∗U , E∗M2, or a stable limit cycle. RegionIV: bi-stable with limit cycle. Trajectories go to either E∗U or a stable limit cycle. Region V:uninfected II. All trajectories go to E∗U . Region VI: uninfected III. All trajectories go to E∗U .Region VII: heteroclinic I. All trajectories cycle between E∗0 and E∗U . Region VIII: heteroclinic II.All trajectories cycle between E∗0 and E∗U .21back to the case of a single viral strategy (Section 3.1). In the current discussion, we are notinterested in whether a single viral strategy can survive but rather the fate of cooperation canin the presence of defection. Going forth, we will consider only initial conditions with both viralstrategies. In addition, note that in subsequent figures trajectories may cross because the figuresshow two dimensional projections of higher dimensional spaces.223.4.1 Region I: uninfected I (µ < µSM)In this dynamical region only E∗0 and E∗U are biologically relevant. Figure 8 shows dynamicstypical for all µ values in this range. All trajectories are drawn towards E∗U , which is stable. Allviruses die, and only uninfected host cells remain.Figure 8. Projection of phase space for region I. µ = 0.02. Dots indicate equi-libria: light green, E∗0 ; dark green, E∗U . Open dots are unstable while closed dotsstable. In this dynamical region, all trajectories go to H∗U ; Virus goes extinct.233.4.2 Region II: bi-stable (µSM < µ < µSC)In the transition to region II, the mixed equilibria E∗M1 and E∗M2 (light and dark purple dots)appear at µSM . For all µSM < µ < µSC , the system is bi-stable. Figure 9 shows dynamicsrepresentative of this region. Depending on initial conditions, trajectories are drawn towards eitherE∗U (dark green dot) or E∗M2 (light purple dot). Both E∗U and E∗M2 have a basin of attraction, andall trajectories that start in an equilibrium’s basin of attraction will move towards that equilibrium.This means that perturbations in the initial composition of the population can lead to very differentpopulation outcomes. Some initial conditions will lead to the death of all viruses, while some willlead to the stable co-existence of Cooperators and Defectors, though a fairly large density of bothviral strategies would be required for viruses to establish.However, note that no Cooperator only or Defector only equilibria are biologically relevant forthis range of µ values. This means that influxes of only Cooperators or only Defectors would goextinct; mixed populations of Cooperators and Defectors can establish before either strategy is ableto establish on its own.Figure 9. Projection of phase space for region II. µ = 0.03. Dots indicate equi-libria: light green, E∗0 ; dark green, E∗U ; dark/light purple, E∗M1/E∗M2. Open dotsare unstable while closed dots stable. This dynamical region is bi-stable: sometrajectories go to stable E∗U while some go to E∗M2.243.4.3 Region III: tri-stable (µSC < µ < µHB)In the transition to region III, E∗C1 and E∗C2 (light and dark blue dots) appear at µSC . However,trajectories reveal unexpected stable and unstable limit cycles which indicate that a more compli-cated bifurcation that gives rise to limit cycles must occur between µSM and µSC . In future workwe will determine the bifurcation that gives rise to these limit cycles, but that analysis is outsidethe scope of this thesis.For µS < µ < µHB , the system is tri-stable. Figure 10 shows dynamics typical of this region.In Figures 10A/B, depending on initial configurations viral extinction, stable mixed co-existence ofCooperators and Defectors, or periodically fluctuating co-existence of Cooperators and Defectorsare possible. Figures 10C/D show a closer look at the stable and unstable limit cycles.25Figure 10. Phase diagram for dynamical region III. µ = 0.035. Dots indicate equilibria: lightgreen, E∗0 ; dark green, E∗U ; dark/light purple, E∗M1/E∗M2; dark/light blue, E∗C1/E∗C2. Open dotsare unstable while closed dots are stable. The system is tristable: trajectories are drawn into E∗U ,a stable limit cycle (outlined in orange), or E∗M2. A&B show a broad view of the dynamics. Thered rectangle in B indicates the borders of C, which shows a closer look at the stable limit cycleoutlined in orange. The red rectangle in C indicates the borders of D. InD, some trajectories spiralout towards the stable limit cycle while some trajectories spiral inwards towards E∗M2, indicatingthe presence of an unstable limit cycle.263.4.4 Region IV: bi-stable with limit cycle (µHB < µ < µSD)In the transition to region IV, a Hopf bifurcation occurs at µHB as the unstable limit cyclescollides with E∗M2 (light purple). The unstable limit cycle disappears and E∗M2 loses stability.For µHB < µ < µSD, the system is bi-stable. Figure 11 shows dynamics typical of this region.All trajectories are drawn either towards E∗U or towards the stable limit cycle surrounding H∗M2.Depending on initial condition, viruses will either go extinct or Cooperators and Defectors willco-exist with fluctuating densities.Figure 11. Projection of phase space for dynamical region IV. µ = 0.0355. Dotsindicate equilibria: light green, E∗0 ; dark green, E∗U ; dark/light purple, E∗M1/E∗M2;dark/light blue, E∗C2/E∗C2. Open dots are unstable while closed dots are stable.Some trajectories are drawn towards E∗U , while some are drawn into the stablelimit cycle outlined in orange. There is no unstable limit cycle.273.4.5 Region V & VI: uninfected II (µSD < µ < µT1) & uninfected III (µT1 < µ < µT2)In the transition to region V, E∗D1 an E∗D2 (dark and light red) appear at µSD while the stable limitcycle surrounding E∗M2 disappears. In the transition to region VI, E∗M1 and E∗C1 (dark purple andblue) collide at µT1, so E∗M1 is not present in region VI. However, this collision does not result inany stability changes, and all trajectories are still drawn into E∗U .Figure 12 shows dynamics typical for these regions. Though there is some transient cyclicbehavior, all trajectories are drawn into E∗U ; Viruses go extinct.Figure 12. Projection of phase space for dynamical regions V & VI. µ = 0.04.Dots indicate equilibria: light green, E∗0 ; dark green, E∗U ; dark/light purple,E∗M1/E∗M2; dark/light blue, E∗C1/E∗C2; dark/light red, E∗D1/E∗D2. Open dots areunstable, while closed dots are stable. After transient cyclic behavior, trajecto-ries are drawn into E∗U ; viruses cannot survive in this system. This figure showsdynamics from region V. In the transition to region VI, E∗M2 collides with E∗C2at µT1. However, there is no stability change and the dynamics in region VI areessentially the same.283.4.6 Regions VII & VIII: heteroclinic I (µ > µT2) & heteroclinic II (µ > µT3)In the transition to region VII, E∗U and E∗C1 collide at µT2, causing E∗U to lose stability. In thetransition to region VIII, E∗U and H∗D1 collide at µT3, but this collision does not result in a stabilitychange, and the dynamics remain the same overall. Figure 13 shows dynamics typical of thesetwo regions. The system goes through boom and bust cycles where all hosts and viruses approachzero density before uninfected hosts are able to recover, leading to a rise in infected host and viraldensities and a subsequent crash.Figure 13. Time series representative of the dynamics of regions VI & VII. Solidlines indicate the host cell densities, while dotted lines represent the density of free-living virus. Line colors: dark green, HU ; dark blue, HC ; dark red, HD; purple,HM ; light blue, VC ; light red, VD. The system crashes to close to E∗0 beforeuninfected host cells rebound to E∗U . This is quickly followed by an increase ininfected host cells and viruses, leading to another crash, and another rebound. Notethat the “period”, or time that it takes the system to crash before recovering toH∗U , lengthens. The dynamics of this region suggest a hetero-clinic cycle connectingE∗0 E∗U .29However, we cannot place too much weight on the transient dynamics in Figure 12. The behaviorof the trajectory in Figure 12 suggests a heteroclinic cycle, a dynamical behavior where a cycleconnects two equilibria [27, 28]. Here it appears that the cycle connects E∗U and E∗0 . However,both these equilibria have zero components (E∗0 has entirely zero components, while E∗U has onenon-zero component). As densities get very small, there is significant potential for numerical errorin the software that solves our equations.If what we are seeing is indeed a heteroclinic cycle, the trajectory will continue to cycle betweenthe two equilibria, though the length of the cycle will lengthen, meaning that as time goes on, thetrajectory will stay near each equilibrium for longer time spans. However, real populations cannotfall arbitrarily close to zero [29]. In reality, after a few cycles either virus would go extinct, leavingonly uninfected hosts, or all hosts and virus would die, leading to total extinction.30CHAPTER 4ConclusionsOur key result is that environmental feedback maintains cooperation for a range of µ values.Most mechanisms that promote Cooperation (e.g. repeated iteration, spatial structure) do soby allowing Cooperators to interact with other Cooperators at frequencies that are higher thanwould be expected in random interactions, permitting Cooperators to avoid extortion and sharethe benefits of cooperation only with other Cooperators [30–32]. Similarly, environmental feedbackpermits single infection events which allow Cooperators to hide from the extortion of Defectorsby infecting a cell alone. Previous work offers ϕ6 an escape from prisoner’s dilemma through theevolution of more efficient Cooperators. In [16], efficient Cooperators that evolve separately inpopulations with low rates of coinfection are able survive when introduced to Defectors. In [33]“ultra-Cooperators” emerge in response to “ultra-Defectors” for high rates of coinfection. However,both assume constant rates of coinfection. Our results provide an easier escape: if both host hostand viral densities, and thus the rate of coinfection, fluctuate, even relatively inefficient cooperationcan be maintained in the presence defection. It should be noted that though we explore a specificstudy system in this thesis, environmental feedback would be a relevant mechanism to supportcooperation in any horizontally acquired symbiont.Another interesting result from our work is that increased infection rates do not necessarily leadto higher densities of free-living virus. Referencing Figure 3, for a single viral type we see thatas µ increases, the limit cycles approach the axes. This means that increasing the infection rateconstant increases the chance of stochastic events driving either viruses or both hosts and viruses toextinction. In addition, when we consider two viral types, high µ values lead to extinction viruses(regions V & VI) and heteroclinic cycles which predict the eventual extinction of either viruses orboth hosts and viruses (regions VII & VIII). Overall, viral populations face lower extinction risks forlower µ values, indicating that selective pressures might lead to lower µ values. These results echoprevious work on the evolution of virulence (defined as increased host mortality due to infection),which finds that if transmission is linked to virulence then in some circumstances selection willlead to intermediate virulence [23]. Though discussions of the evolution of virulence often centeraround non-lethal phages, previous work has made a link to lytic phages [34, 35]. Selective pressurestowards lower infection rates have important implications for the fate of cooperation because lowerinfection rates allow more single infections, advantaging Cooperators.31A final point to highlight is the surprisingly rich dynamical behavior of our model. Already, oursimplified model with only single viral type exhibits limit cycles. With the inclusion of two viraltypes, the model showed stable and unstable limit cycles and heteroclinic cycles. In future work,we will seek a fuller understanding of the more complex bifurcations and behaviors of the model.We see a few other interesting extensions. Our work with a single viral type extends a previousmodel by incorporating double infections [26]. However, that work analyzed the impact of burst size(in our model, λα), not infection rate, on the dynamics, so we cannot directly compare our workto theirs. In future work, we would like to analyze our simplified model with only single infectionsto investigate the impact of double infection on the ecological dynamics of viruses and hosts, evenwithout multiple viral types.Another potential extension would be to consider the impact of environmental feedback on thesnowdrift game, which is another mathematical metaphor for cooperation. Tthe snow drift game isan ordering of the payoff matrix so that γ > α > δ > β. In the snow drift game payoff is maximizedwhen partners have opposite strategies. In contrast to the prisoner’s dilemma, both strategies caninvade when rare, leading to the stable co-existence of Cooperators and Defectors. However, becausea population with both Cooperators and Defectors has lower average fitness than a population ofonly Cooperators, the snow drift game is still a social dilemma, and it is relevant to examinewhich mechanisms promote cooperation and lead to higher relative proportions of Cooperators[30]. Interestingly, mechanisms that promote cooperation in the prisoner’s dilemma do not alwayshelp cooperation in the snow drift game [30]. For example, spatial structure promotes cooperationin the prisoner’s dilemma, but can lead to lower prevalence of Cooperators in the snow drift game[10]. We are interested in examining the impact of environmental feedback on the prevalence ofcooperation in the snow drift game.32Bibliography1. Wilkinson, G. S. Reciprocal food sharing in the vampire bat. Nature 308, 181–184 (1984).2. Griffin, A. S., West, S. A. & Buckling, A. Cooperation and competition in pathogenic bacteria.Nature 430, 1024–1027 (2004).3. Clutton-Brock, T. H. et al. Selfish sentinels in cooperative mammals. science 284, 1640–1644(1999).4. Turner, P. E. & Chao, L. Prisoner’s dilemma in an RNA virus. Nature 398, 441–443 (1999).5. Nowak, M. A. & May, R. M. Evolutionary games and spatial chaos. Nature 359, 826–829(1992).6. Trivers, R. L. The evolution of reciprocal altruism. The Quarterly review of biology 46, 35–57(1971).7. Axelrod, R. & Hamilton, W. D. The evolution of cooperation. science 211, 1390–1396 (1981).8. Clutton-Brock, T. H. & Parker, G. A. Punishment in animal societies. Nature 373, 209–216(1995).9. Fehr, E. & Ga¨chter, S. Altruistic punishment in humans. Nature 415, 137–140 (2002).10. Hauert, C. & Doebeli, M. Spatial structure often inhibits the evolution of cooperation in thesnowdrift game. Nature 428, 643–646 (2004).11. Hauert, C., De Monte, S., Hofbauer, J. & Sigmund, K. Volunteering as red queen mechanismfor cooperation in public goods games. Science 296, 1129–1132 (2002).12. Traulsen, A. & Nowak, M. A. Evolution of cooperation by multilevel selection. Proceedings ofthe National Academy of Sciences 103, 10952–10955 (2006).13. Turner, P. E. & Chao, L. Sex and the evolution of intrahost competition in RNA virus φ6.Genetics 150, 523–532 (1998).14. Vidaver, A. K., Koski, R. & Van Etten, J. L. Bacteriophage φ6: a lipid-containing virus ofPseudomonas phaseolicola. Journal of virology 11, 799–805 (1973).15. Mindich, L. et al. Isolation of additional bacteriophages with genomes of segmented double-stranded RNA. Journal of bacteriology 181, 4505–4508 (1999).16. Turner, P. E. & Chao, L. Escape from prisoner’s dilemma in RNA phage Φ6. The AmericanNaturalist 161, 497–505 (2003).3317. Bronstein, J. L. Conditional outcomes in mutualistic interactions. Trends in ecology & evolu-tion 9, 214–217 (1994).18. Sokal, R. R. & Rohlf, F. J. Biometry (Freeman and Company, San Francisco, 1988).19. Hauert, C., Holmes, M. & Doebeli, M. Evolutionary games and population dynamics: main-tenance of cooperation in public goods games. Proceedings of the Royal Society B: BiologicalSciences 273, 2565–2571 (2006).20. Olkkonen, V. M. & Bamford, D. H. Quantitation of the adsorption and penetration stages ofbacteriophage φ6 infection. Virology 171, 229–238 (1989).21. Turner, P. E., Burch, C. L., Hanley, K. A. & Chao, L. Hybrid frequencies confirm limit tocoinfection in the RNA bacteriophage ϕ6. Journal of virology 73, 2420–2424 (1999).22. Turner, P. E., Cooper, V. S. & Lenski, R. E. Tradeoff between horizontal and vertical modesof transmission in bacterial plasmids. Evolution 52, 315–329 (1998).23. Lenski, R. E. & May, R. M. The evolution of virulence in parasites and pathogens: reconcili-ation between two competing hypotheses. Journal of theoretical biology 169, 253–265 (1994).24. Messenger, S. L., Molineux, I. J. & Bull, J. Virulence evolution in a virus obeys a trade off.Proceedings of the Royal Society of London. Series B: Biological Sciences 266, 397–404 (1999).25. Nowak, M. A. & Bangham, C. R. Population dynamics of immune responses to persistentviruses. Science 272, 74–79 (1996).26. Beretta, E. & Kuang, Y. Modeling and analysis of a marine bacteriophage infection. Mathe-matical biosciences 149, 57–76 (1998).27. Kazarinoff, N. D. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields(John Guckenheimer and Philip Holmes). SIAM Review 26, 600 (1984).28. Podvigina, O. Heteroclinic Cycles in Nature. Izvestiya, Physics of the Solid Earth 56, 117–124(2020).29. May, R. M. & Leonard, W. J. Nonlinear aspects of competition between three species. SIAMjournal on applied mathematics 29, 243–253 (1975).30. Doebeli, M. & Hauert, C. Models of cooperation based on the Prisoner’s Dilemma and theSnowdrift game. Ecology letters 8, 748–766 (2005).31. Queller, D. C. Kinship, reciprocity and synergism in the evolution of social behaviour. Nature318, 366–367 (1985).32. Fletcher, J. A. & Zwick, M. Strong altruism can evolve in randomly formed groups. Journalof theoretical biology 228, 303–313 (2004).33. Chao, L. & Elena, S. F. Nonlinear trade-offs allow the cooperation game to evolve from Pris-oner’s Dilemma to Snowdrift. Proceedings of the Royal Society B: Biological Sciences 284,20170228 (2017).34. Bull, J. Optimality models of phage life history and parallels in disease evolution. Journal oftheoretical biology 241, 928–938 (2006).3435. Abedon, S. T., Hyman, P. & Thomas, C. Experimental examination of bacteriophage latent-period evolution as a response to bacterial availability. Applied and environmental microbiology69, 7499–7506 (2003).36. Dennehy, J. J. & Turner, P. E. Reduced fecundity is the cost of cheating in RNA virus φ6. Proceedings of the Royal Society of London. Series B: Biological Sciences 271, 2275–2282(2004).37. LaBauve, A. E. & Wargo, M. J. Growth and laboratory maintenance of Pseudomonas aerug-inosa. Current protocols in microbiology 25, 6E–1 (2012).38. Strogatz, S. H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry,and engineering (CRC press, 2018).39. Lotka, A. J. Elements of physical biology (Williams & Wilkins, 1925).35APPENDIX AChoosing ParametersWe used the game parameters (α, β, γ, δ) experimentally determined by Turner and Chao [4].We planned to vary µ1, µ2 throughout analysis. When looking for values for the parameters relatedto population dynamics (r, d, κ, λ, ξ), though we found some sources measuring parameters similarto those in our model (for example, [36, 37]), these measurements were done on lab strains insteadof wild populations, were not necessarily the same in interpretation as the parameters in our model,and were often very small, creating the potential for numerical issues (e.g. equilibria values veryclose to 0). The model is intended as a theoretical exploration of a concept, not an exact descriptionof biology, so we ultimately prioritized choosing parameters that allowed for easier analysis of ourmodel (e.g. equilibria that are sufficiently far from 0).Initial parameter selection was done for the case ξ = 0. We started with this case in the hopesthat the system would be more analytically accessible (See Appendix E for further details). Wechose our parameters in reference to the Cooperator only equilibria E∗C and Defector only equilibriaE∗D in the case µ2 = 0. These equilibria have fairly approachable forms:E∗C = (HU , HC , HD, HM , VC , VD) =(dκαλ1µ1,καλ1µ1, 0, 0,rµ1, 0)E∗D = (HU , HC , HD, HM , VC , VD) =(dκδλ1µ1, 0,κrδλ1µ1, 0, 0,rµ1)We chose parameter values that simplified the equilibria and ensured they were far from zero.For simplicity, we set κ = 1. We then set d = 1δ = 8.3 and r =1δ = 8.3. As κ and α were both setto 1, the densities of infected hosts and viruses differ by only a scalar term (e.g. HC =1λEC). Weset λ = 2 so that the density of viruses was higher than the density of infected hosts.We chose ξ = .1 to ensure that the Uninfected equilibrium (HU =rξ ) was also sufficiently farfrom zero.36APPENDIX BStability CalculationsAn equilibrium is stable if the real parts of the eigenvalues of the Jacobian evaluated at thatequilibrium are all negative [38]. The Jacobian of our system has the following form, where ET =EC + ED:r − 2ξHU − ET (µ2ET + µ1) 0 0 0 −HU (µ1 + 2µ2ET ) −HU (µ1 + 2µ2ET )EC (µ1 + µ2EC) −d 0 0 HU (µ1 + 2µ2EC) 0ED (µ1 + µ2ED) 0 −d 0 0 HU (µ1 + 2µ2ED)2µ2ECED 0 0 −d 2µ2EDHU 2µ2ECHU0 dαλ 0 dβλ −dκ 00 0 dδλ dγλ 0 −dκB.1 Stability of E∗UThe six eigenvalues of Jacobian evaluated at the E∗0 are:r−d2(κ±√κ2 − 4αλ)−d2(κ±√κ2 − 4δλ)−d37As r > 0, we know that the E∗0 is unstable for all biologically relevant parameter values. Thereis always at least one unstable direction: in the absence of viruses, any non-zero uninfected hostpopulation will grow.B.2 Stability of E∗UIn the case of a single viral type, this stability condition can be found by considering R0 (3.1).In general, the six eigenvalues of the Jacobian evaluated at E∗U are:−d−r−d(1 + κ)2±√d2√ξ√dξ(κ− 1)2 + 4αλµ1r−d(1 + κ)2±√d2√ξ√dξ(κ− 1)2 + 4δλµ1rAs r, d > 0, first two eigenvalues are always negative.We next consider the eigenvalues−d(1 + κ)2±√d2√ξ√dξ(κ− 1)2 + 4αλµ1rAs d, ξ, αλ, µ1, r > 0, the value under the radical is always positive, and these two eigenvaluesare never imaginary. Furthermore,−d(1 + κ)2−√d2√ξ√dξ(κ− 1)2 + 4αλµ1ris clearly negative, so we need only consider−d(1 + κ)2+√d2√ξ√dξ(κ− 1)2 + 4αλµ1rwhich is negative when µ1 <κdξαλr .We next consider−d(1 + κ)2±√d2√ξ√dξ(κ− 1)2 + 4δλµ1rThese two eigenvalues are identical to the prior two, with the exception of a δ in the last term underthe radical. Following the same arguments as above, we know that these two eigenvalues are never38imaginary. Furthermore,−d(1 + κ)2−√d2√ξ√dξ(κ− 1)2 + 4δλµ1ris always negative, and−d(1 + κ)2+√d2√ξ√dξ(κ− 1)2 + 4δλµ1ris negative when κdξαλr <κdξδλr .In the prisoner’s dilemma, δ < α =⇒ 1α < 1δ . For the payoff ordering we are concerned with,H∗U is stable for µ1 <κdξαλr .39APPENDIX CForms of EquilibriaC.1 Single viral type: endemic equilibriaThere are three endemic equilibria which we denote as E∗E1, E∗E2, and E∗E3. We solve for theseequilibria using the software Maple. solving for the equilibria requires finding the roots of a cubicpolynomial, which in Maple yields terms that include I, defined as I =√−1. Terms includingI appear in E∗E2 and E∗E3, which have parallel forms. This means that E∗E2 and E∗3 have thepotential to be imaginary. We know from our numerical analysis that one of these equilibria isalways biologically irrelevant, but further work is required to determine which, meaning that E∗E1in the text does not necessarily correspond to E∗E1 in this Appendix. Note that in all three equilibria,V = αλκ HI .E∗E1 = (HU , HI , V )=(3(21/3TS1/3 + 2r9 S2/3 − 22/3αλU)22/3,κ(6(2)1/3α2λ2µ2r + 24/3α2λ2µ21 − 4µ1α1/3 + 22/3S2/3)6α2λ2µ2S1/3,αλκHI)E∗E2 = (HU , HI , V )=(3(−21/3TS1/3(I√3 + 1) + 4r9 S2/3 − 22/3αλU(I√3− 1))4ξS2/3,κ(22/3S2/3(I√3− 1)− 6αλ( 4µ13 S1/3 + 21/3αλ(µ2 + µ213 )(I√3 + 1)))12µ2α2λ2S1/3,αλκHI)E∗E3 = (HU , HI , V )=(3(22/3αλU(I√3 + 1) + 4r9 S2/3 + 21/3TS1/3(I√3− 1))4ξS2/3,κ(−22/3S2/3(I√3 + 1) + 6αλ(− 4µ13 S1/3 + 21/3αλ(µ2 + µ213 )(I√3− 1)))12µ2α2λ2S1/3,αλκHI)40Where:R =√(−12µ32r3−3µ2µ1r2)λ2α2−54µ1(µ2r+2µ219 )καdξλ+81d2κ2µ2ξ2µ29S = (9αλµ2µ1r + 2µ31αλ− 27dκµ2ξ + 3R)α2λ2T = −αλµ1r9+ κξd−RU = −µ1R+2rαλ(µ2r +µ216 )3+ dκµ1ξC.2 No Single Infections, µ1 = 0C.2.1 Cooperator only equilibriaThere are three Cooperator only equilibria, which we’ll refer to as E∗C1, E∗C2, and E∗C3. We solve forthese equilibria using the software Maple. Finding these equilibria requires finding the roots of acubic polynomial, which in Maple yields terms that include I, defined as I =√−1. Note that E∗C2and E∗C3, which have parallel forms, both have terms including I. This means that E∗C2 and E∗C3have the potential to have imaginary components. We know from our numerical analysis that oneof the three equilibria is always biologically irrelevant, but further work is required to determinewhich. (Note that this means that E∗C1 in the text does not necessarily correspond to E∗C1 in thisAppendix.) In all three equilibria, VC =αλκ HC .E∗C1 = (HU , HC , HD, HM , VC , VD)=(−(12)4/3α2λ2r2µ2 + 9(12)2/3dκξB1/3 − 122/3AB1/3 + 12rB2/336ξB2/3,κ121/3(rα2λ2121/3µ2 +B2/3)6α2λ2µ2B1/3, 0, 0,αλκHC , 0)E∗C2 = (HU , HC , HD, HM , VC , VD)=(8r3 B2/3 + 122/3B4/3(I√3 + 1) + 43121/3λ2α2r2µ2(1− I√3)8ξB2/3,121/3κ(B2/3(I√3− 1)− 121/3λ2α2µ2r(I√3 + 1))12B1/3α2λ2µ2, 0, 0,αλκHC , 0)41E∗C3 = (HU , HC , HD, HM , VC , VD)=(8r3 B2/3 + 43121/3λ2α2r2µ2(I√3 + 1) + 122/3B4/3(1− I√3)8ξB2/3,121/3κ(121/3λ2α2µ2r(I√3− 1)−B2/3(I√3 + 1))12α2λ2µ2B1/3, 0, 0,αλκHC , 0)Where:A =√−12r3α2λ2µ2 + 81d2κ2ξ2B = α2λ2µ2(−9dκξ +A)C.2.2 Defector only equilibriaThere are also three Defector only equilibria, denoted E∗D1, E∗D2, and E∗D3. The Defector onlyequilibria have the same form as the Cooperator only equilibria but with the substitution α = δ.C.3 Full SystemThe mixed equilibria of the full equation are too complicated to have any meaning upon visualinspection. There are three Cooperator equilibria, as well as three Defector only equilibria. Theseequilibria represent a reduction back to the case of a single viral type. As such, the three Cooperatoronly equilibria are exactly identical to the equilibria from Appendix C.1, and the three Defectoronly equilbria have the same forms as C.1 but with the substitution α = δ.42APPENDIX DCreating Bifurcation DiagramsWe create each bifurcation diagram through the same basic process. First, we substitute allparameter values except µ (i.e. r, d, κ, λα, β, γ, δ, ξ) into the equations. We then iterate over µvalues. For each µ value, we do the following:• Plug µ into the model equations where relevant• Numerically solve for the equilibria• Numerically calculate the stability of the equilibria using the JacobianWhen creating the bifurcation diagrams, we plot an equilibria only if all its components are realand non-negative.43APPENDIX ENo Host Density DependenceIn equations 18a-18f, we set ξ = 0 in an attempt to make our system more analytically accessible.We expect that viruses will keep the host population from exploding, eliminating the need for anexplicit density dependence term in the host equations. A similar assumption is made in the Lotka-Volterra predator-prey model, which also lacks a density dependence term in the host population[39].H˙U = rHU − ξH2U − µ1HU(EC + ED)− µ2HU(E2C + 2ECED + E2D)(18a)H˙C = HUEC(µ1 + µ2EC)− dHC(18b)H˙D = HUED(µ1 + µ2ED)− dHD(18c)H˙M = 2µ2HUECED − dHM(18d)E˙C = λd(αHC + βHM )− κdEC(18e)E˙D = λd(γHM + δHD)− κdED(18f)As in our analysis of the model with density dependence, we first consider a single viral strategy.This system is described by the equations in Section 3.1 with ξ = 0 and has two biologically relevantequilibria: E∗0 = (HU , HI , EV ) = (0, 0, 0) and an endemic equilibrium E∗E :E∗E = (HU , HI , EV ) =(d κ(√µ12+4µ2 r−µ1)2αλµ2 rκ(√µ12+4µ2 r−µ1)2αλµ2√µ12+4µ2 r−µ12µ2)We know from Appendix B that H∗E is always unstable (ξ = 0 does not change this result), butcannot gain traction on the stability of H∗I analytically. We proceed with a numerical investigationof the dynamics.We determine numerically that E∗E is unstable for the parameter values of Table 1 along withµ = .1 (and correspondingly µ1 = .1, µ2 = .01). In Figure 14, we use the same parameter values tonumerically solve the differential equations and examine trajectories of the system. All trajectorieslead back into H∗E ; neither hosts nor viruses are able to survive, and the system goes extinct.44Note that even though trajectories lead into E∗0 , the equilibrium is unstable. There is at least one(and perhaps only one) unstable direction: in the absence of viruses, the uninfected host populationwill explode and grow without bound.Figure 14. Dynamics of a single viral strategy in the absence of host densitydependence. µ = .1. Dots indicate equilibria: Light green, E∗0 ; blue, E∗E . Allequilibria are unstable. All trajectories lead back to E∗0 . Note that E∗E is still un-stable because there is at least one unstable direction: in the absence of viruses, thepopulation of uninfected hosts will grow without bound. Overall, the introductionof viruses crashes the uninfected host cell population, leading to the extinction ofboth hosts and viruses.We see similar trajectories for all other values of µ that we investigate. It seems that in theabsence of host density dependence, the system crashes when viruses are introduced. When weexamine the cases µ1 = 0 and µ2 = 0 without host density dependence, we find the same dynamicof the entire population crashing to extinction. The same is true for trajectories of the full system45without host density dependence. It seems that without density dependence, hosts are not able tosurvive the introduction of viruses.46
Thesis/Dissertation
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Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Hauert, Christoph
Environmental feedback allows escape from the prisoner's dilemma
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