Arts and Sciences, Irving K. Barber School of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Martignoni, Maria
Doctor of Philosophy - PhD
University of British Columbia
In the last decades, microbial inoculants have been used as organic fertilizers worldwide. One of the most widely used commercial products are arbuscular mycorrhizal (AM) fungi, as these fungi can associate with a vast variety of crops cultivated in agriculture. Despite the potential benefits for soil quality and crop yield associated with the use of AM fungi, experiments assessing the effective establishment of the fungi in the field have given inconsistent results. Additionally, it is not yet clear whether the introduction of commercial inoculants could lead to a biodiversity loss in the native fungal community, and ultimately have a negative impact on plant growth. Here I develop a series of ordinary and partial differential equation models to discuss the establishment and regional spread of the inoculants in the field from a theoretical point of view. I use linear and nonlinear stability analysis and extensive numerical simulations to study the spatio-temporal dynamics of a guild of mutualists (the fungal species) sharing a resource provided by the same partner (the host plant), constituting a shared resource for all fungi. My results shed light on the discordant experimental observations found in the literature, and allow to assess the risks and benefit of inoculation. I identify under which conditions inoculation can effectively boost productivity, when it has no significant effect on plant growth and on the native fungal community, and when it represents an invasion risk. More generally, the models provide important ecological insights into the mechanisms responsible for coexistence and competitive exclusion among mutualist guilds, and constitute a framework to predict the consequences of species manipulation in mutualist communities. Indeed, the models I have developed are simple enough to apply to a broad range of mutualisms found in nature, such as pollination or seed dispersal.
MATHEMATICAL INSIGHTS INTO MECHANISMSLEADING TO COEXISTENCE AND COMPETITIVEEXCLUSION AMONG MUTUALIST GUILDSbyMaria MartignoniA DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 2020c©Maria Martignoni, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Mathematical insights into mechanisms leading to coexistence and competitive exclusionamong mutualist guildssubmitted by Maria Martignoni in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics.Examining Committee:Prof. Rebecca Tyson, MathematicsSupervisorDr. Jimmy Garnier, MathematicsCo-SupervisorDr. Miranda Hart, BiologySupervisory Committee MemberDr. Dan Vollick, PhysicsSupervisory Committee MemberProf. Cigdem Eskicioglu, EngineeringUniversity ExaminerProf. Alan Hastings, MathematicsExternal ExamineriiAbstractIn the last decades, microbial inoculants have been used as organic fertilizers worldwide.One of the most widely used commercial products are arbuscular mycorrhizal (AM) fungi,as these fungi can associate with a vast variety of crops cultivated in agriculture. Despitethe potential benefits for soil quality and crop yield associated with the use of AM fungi,experiments assessing the effective establishment of the fungi in the field have given in-consistent results. Additionally, it is not yet clear whether the introduction of commercialinoculants could lead to a biodiversity loss in the native fungal community, and ultimatelyhave a negative impact on plant growth. Here I develop a series of ordinary and partialdifferential equation models to discuss the establishment and regional spread of the inoc-ulants in the field from a theoretical point of view. I use linear and nonlinear stabilityanalysis and extensive numerical simulations to study the spatio-temporal dynamics of aguild of mutualists (the fungal species) sharing a resource provided by the same partner(the host plant), constituting a shared resource for all fungi. My results shed light on thediscordant experimental observations found in the literature, and allow to assess the risksand benefit of inoculation. I identify under which conditions inoculation can effectivelyboost productivity, when it has no significant effect on plant growth and on the nativefungal community, and when it represents an invasion risk. More generally, the mod-els provide important ecological insights into the mechanisms responsible for coexistenceand competitive exclusion among mutualist guilds, and constitute a framework to predictthe consequences of species manipulation in mutualist communities. Indeed, the modelsI have developed are simple enough to apply to a broad range of mutualisms found innature, such as pollination or seed dispersal.iiiLay SummaryThe environmental and health risks associated with the extensive application of agrochem-icals lead to the imperative that we move toward new sustainable solutions for maintain-ing productivity in farmland. Beneficial plant-microbe interactions and, in particular,the widespread symbiosis between plants and arbuscular mycorrhizal (AM) fungi, offera promising alternative for sustainable agriculture worldwide. I developed a series ofmathematical models for plant and AM fungal growth to help to assess the establish-ment, spread and impact of the mycorrhizal inoculum on the native fungal communityand on plant productivity. My models provide a theoretical framework to determine thecircumstances under which the inoculated AM fungal species can coexist with the nativefungal community and effectively boost productivity, versus when inoculation constitutesa biodiversity risk and, ultimately, a detriment to crop yield. Although in my work Ifocus on plant-fungal interactions, my findings can be used to assess the impact of speciesintroduction in mutualistic communities in general (e.g. in pollinator communities), andsignificantly help to manage their conservation and restoration.ivPrefaceAll of the work presented herewith was conducted at the University of British Columbia,in Kelowna (Canada, BC), and at the University of Savoie-Mont Blanc, in Chambe´ry(France), where I had the opportunity to work from September to November 2018.I was the lead investigator, responsible for all major areas of model development,analysis and interpretation, as well as thesis composition. Prof. Rebecca Tyson wasinvolved throughout the project providing constant supervision and mentoring.Chapters 2, 3, 4 and 5: Prof. Tyson and Dr. Miranda Hart were involved in theearly stages of concept formation, provided advice and constant supervisory support ofthe theoretical, respectively ecological, aspects of the project, and contributed editorialsupport.Chapters 3 and 4: In addition to Prof. Tyson and Dr. Hart, also Dr. Jimmy Garnierprovided supervisory support by helping with concept formation, model analysis, andmanuscript composition. Dr. Garnier also significantly contributed to the mathematicalanalysis of the model presented in chapter 2.The article version of chapter 2 has been submitted on Nov 27, 2019, to ‘TheoreticalEcology’. The article version of chapter 3 has been accepted for publication on March2nd, 2020 to ‘Proceedings to the Royal Society, Series B’. Figs. A.1 and A.2 have beenreproduced with permission from Jansa et al. (2008).vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Mechanisms for coexistence within mutualist guilds . . . . . . . . . . 62.1 Background information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Model and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Formulating a model for a multi-species mutualism . . . . . . . . . 72.2.2 Quantification of resource exchange . . . . . . . . . . . . . . . . . . 92.2.3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Analysis of the one plant - one fungus system . . . . . . . . . . . . 122.3.2 Analysis of the one plant - multiple fungi system . . . . . . . . . . 232.3.3 The effect of differences in the resource exchange capacities of thefungi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.1 Modelling of mutualistic interactions . . . . . . . . . . . . . . . . . 382.4.2 Indirect parasitism among guild members . . . . . . . . . . . . . . . 402.4.3 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41viTABLE OF CONTENTS3 Predicting the consequences of the introduction of an alien mutualistinto a guild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Background information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Model and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.1 Formulating a model for competing mutualists . . . . . . . . . . . . 453.2.2 Investigating the effect of competition on mutualist guild dynamics 483.2.3 Investigating the impact of invasion on resource availability . . . . . 483.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.1 Analytical criteria for stability of the mutualist guild . . . . . . . . 493.3.2 Effect of competition on mutualist guild dynamics . . . . . . . . . . 663.3.3 The impact of invasion on resource availability (i.e. plant biomass) 693.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4.1 Competition and mutualist guild dynamics . . . . . . . . . . . . . . 693.4.2 The impact of invasion on resource availability . . . . . . . . . . . . 723.4.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 Investigating the spread of an alien mutualist in a field . . . . . . . . 784.1 Background information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Model and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.1 A model for investigating the persistence and spread of fungal in-oculants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.2 Investigating inoculum persistence and spread . . . . . . . . . . . . 834.2.3 Insights for agroecosystem management . . . . . . . . . . . . . . . . 834.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.1 Traveling wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.2 Investigating inoculum persistence and spread . . . . . . . . . . . . 954.3.3 Insights for agroecosystem management . . . . . . . . . . . . . . . . 964.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.4.1 Mathematical insights . . . . . . . . . . . . . . . . . . . . . . . . . 974.4.2 Investigating inoculum persistence and spread . . . . . . . . . . . . 984.4.3 Insights for agroecosystem management . . . . . . . . . . . . . . . . 1005 Fungal diversity and plant productivity . . . . . . . . . . . . . . . . . . 1055.1 Background information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Model and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.1 Q1 and Q2: The effect of varying αi and βi . . . . . . . . . . . . . . 1075.2.2 Q3: Qualitative reproduction of experimental results . . . . . . . . 1085.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3.1 Q1 and Q2: The effect of differences in phosphorus supply andcarbon sink strength . . . . . . . . . . . . . . . . . . . . . . . . . . 109TABLE OF CONTENTS5.3.2 Q3: Qualitative reproduction of experimental results . . . . . . . . 1125.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4.1 Q1 and Q2: The effect of differences in phosphorus supply andcarbon sink strength . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4.2 Q3: Qualitative reproduction of experimental results . . . . . . . . 1155.4.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139List of Tables2.1 Brief description of the parameters of the model of Eq. (2.1). . . . . . . . . 92.2 Variables and parameters of the model of Eq. (2.1) . . . . . . . . . . . . . 122.3 Non-dimensional model parameters of the one plant - one fungus case . . . 132.4 Biomass-dependence of the gross benefit, cost and net effect functionalresponses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Non-dimensional model parameters of the one plant - multiple AM fungicase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Interactions schemes of different plant-fungi systems, and correspondingsteady states stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Description of variables and parameters of the model given in (4.7), withrespective measurement units. . . . . . . . . . . . . . . . . . . . . . . . . . 864.2 Possible outcomes following the introduction of fungal inoculants in a field 875.1 Standard values of the parameters of the model of Eq. (5.1) . . . . . . . . 117viiList of Figures1.1 Schematic representation of the mycorrhizal symbiosis . . . . . . . . . . . . 42.1 Phase-plane portraits of the model of Eq. (2.1) . . . . . . . . . . . . . . . . 152.2 Plant steady state as a function of model parameters . . . . . . . . . . . . 162.3 Fungal steady state as a function of model parameters . . . . . . . . . . . 172.4 Graphical representation of the functional responses of plant and AM fungi 212.5 Nullclines of the system of differential equations given in Eq. (2.21) for rp > 0 222.6 Nullclines of the system of differential equations given in Eq. (2.21) for dif-ferent biomass dependence of the maintenance costs of the plant (parametern) and of the fungus (parameter s) . . . . . . . . . . . . . . . . . . . . . . 232.7 Nullclines of the system of differential equations given in Eq. (2.21) forlinear maintenance cost of the fungus (parameter s = 1) . . . . . . . . . . . 232.8 Time series of plant and fungal growth when five fungal species coexist onthe same plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.9 Time series of plant and fungal growth, where fungal species differ in theirmutualistic quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.10 Interactions scheme of the direct and indirect interactions between a plantand its associated mutualist guild . . . . . . . . . . . . . . . . . . . . . . . 383.1 Phase-plane portrait of the model of Eq. (3.7) . . . . . . . . . . . . . . . . 643.2 Dependence of total fungal biomass and plant biomass at steady state, asa function of competition strength . . . . . . . . . . . . . . . . . . . . . . . 653.3 Ratio of plant to total fungal biomass at steady state . . . . . . . . . . . . 663.4 Plant and fungal growth over time in the presence of a complete cheater . 663.5 Direct interactions of a plant in association with two or three mutualists . 733.6 Possible outcomes following the introduction of an alien mutualist into aguild of coexisting mutualists . . . . . . . . . . . . . . . . . . . . . . . . . 743.7 Outcome of competition between a strong competitor and a community ofmutualists, for different initial biomass of the strong competitor and of thecommunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.8 Outcome of competition between a strong competitor and a community ofweak competing mutualists, when varying the resource exchange ability ofthe strong competitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76viiiLIST OF FIGURES3.9 Plant and fungal biomass over time, in response to the addition of a stronglycompeting alien mutualist . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1 Speed of invasion of the inoculated fungi (cases (1) and (3)) . . . . . . . . 924.2 Speed at which inoculants are displaced by the native community (case (2)) 934.3 Speed at which inoculants are displaced by the native community (case (4),N > 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4 Plant and fungal biomass density distribution resulting from the encounterof two strongly competing fungi . . . . . . . . . . . . . . . . . . . . . . . . 954.5 Spread of commercial inoculants through neighboring fields. . . . . . . . . 1024.6 Spatio-temporal dynamics when competition between the native fungalcommunity and the inoculants is strong . . . . . . . . . . . . . . . . . . . . 1034.7 Spatio-temporal dynamics when the inoculants are weakly competing fungi 1045.1 Schematic representation of the scenarios tested with the model . . . . . . 1085.2 Effect of differences in phosphorus supply and carbon sink strength on plantand fungal final sizes and average growth rates . . . . . . . . . . . . . . . . 1105.3 Effect of differences in phosphorus supply and carbon sink strength on plantand fungal growth over time . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.4 Plant biomass over time for different initial fungal biomassf . . . . . . . . . 1125.5 Experimental simulation showing the growth of medic and leek associatingwith different combinations of species of AM fungi . . . . . . . . . . . . . . 1135.6 Experimental simulation estimating plant and fungal biomass when theplant is associating with different combinations of fungal species . . . . . . 114A.1 Experimental results of fungal growth over time . . . . . . . . . . . . . . . 139A.2 Experimental results showing biomass at harvest time . . . . . . . . . . . . 140AcknowledgmentsI would first like to thank my supervisor Rebecca, that has been for me a constant sourceof inspiration, not only professionally but also at personal level. I will always be gratefulto her for her positivity and capacity to see and unleash my potential, what opened upso many new doors to my future. Having had Rebecca as supervisor has been a true giftin my life.A huge thank also to my co-supervisor Jimmy, for his passionate engagement and forthe extremely valuable guidance he has provided to me throughout the project.Thank you to my committee members: to Miranda, for believing in our collaborationsince the very beginning, and putting all her energy and contagious enthusiasm in it; andto Dan, for his availability and thought provoking questions.Finally thank you to my Cerg and Tylab friends, who alleviated the fatigue of myjourney toward graduation with their chatter and laughters, in the lab as well as on thehills and lakes around Kelowna.ixDedication‘Nothing truly valuable can be achieved exceptby the unselfish cooperation of many individuals’.A. EinsteinxChapter 1IntroductionThe extensive application of chemical fertilizers and enhancers has led to soil depletion,increased greenhouse gas emissions due to fertilizer production, and concerns about pos-sible toxic effects on organisms such as animal pollinators. These environmental prob-lems make it imperative that we move toward new environmentally friendly solutions toreduce the use of chemical fertilizers. Beneficial plant-microbe interactions, and partic-ularly the widespread symbiosis between plants and arbuscular mycorrhizal (AM) fungi,offer promising strategies for sustainable agriculture worldwide (Gianinazzi and Vosa´tka,2004). AM fungi are found inside the roots of the vast majority of terrestrial plants (seeFig. 1.1). The hyphae of these fungi grow out from the roots into the soil, where theyforage for nutrients that are limiting to plant growth (Smith and Read, 2010). In returnfor these nutrients, the plant provides carbohydrates to the fungi. AM fungi thus exert astrong influence on plant growth and fitness, and can be exploited as a biological tool forsustainable agriculture. Industry has capitalized on this discovery by raising AM fungicommercially, and selling AM fungal inoculants to farmers as a fertilizer.Despite the potential benefits of applying AM fungal inoculants in the field, relativelyfew studies have verified the impact of commercial fungi on the native fungal community,or its effectiveness at a large spatial scale (Bender et al., 2019, Lekberg et al., 2012, Niwaet al., 2018, Pellegrino et al., 2012). Mycorrhizal associations can range from mutualistic,to commensalistic, to parasitic, depending on plant and fungal genotypes and environ-mental conditions (Johnson et al., 1997, Jones and Smith, 2004, Klironomos, 2003).The assumption is that AM fungal inoculants will be mutualistic, but it is still unclearhow the functionality of commercial AM fungi, grown in-vitro and subjected to intensebreeding pressure, may differ from the way in which wild fungi operate and interact withhost plants. Commercial fungi have a higher sporulation rate (Jin et al., 2013), producespores that are often larger than those of wild AM fungi (Calvet et al., 2013), and showa propensity to invest more in their own reproduction at the expense of their mutualisticassociation with plants (Kokkoris et al., 2019). These adaptations may result in commer-cial fungi that are less mutualistic, and that have an important competitive advantage1over other native species of mycorrhizal fungi, possibly resulting in radical changes to thenative fungal community. Hence, by using AM fungal inoculants, growers are introducinga potentially invasive species to the environment, which could lead to a loss in below-ground diversity and a decrease in plant productivity (Hart et al., 2017, Ricciardi et al.,2017, Schwartz et al., 2006, Thomsen and Hart, 2018). It is therefore crucial to assess theconditions under which inoculation with commercial fungi is beneficial, and under whichit is harmful.In my work, I used a mathematical approach to study the plant-AM fungal communitydynamics and assess the risks and benefits connected with the application of mycorrhizalinoculants. Mathematical modeling and computer simulations can be illuminating tools inthe study of complex systems of interacting species. Mathematical approaches are efficientand relatively inexpensive, and can be used to verify hypotheses and generate testablepredictions when conventional field trials are logistically not feasible. Various theoreticaltechniques have been used to study how mutualistic interactions evolve in time, thatinclude game theory, population dynamics models, biological market models, and modelsof the evolution of virulence (Hoeksema and Bruna, 2000). Population dynamics is thetheoretical framework I used to investigate how plant and fungal growth is affected byinterspecific interactions.The first population dynamics models of mutualistic interactions dates back to the be-ginning of the nineteenth century (Gause and Witt, 1935). Here, interactions benefittedpopulations by increasing their densities independent of the actual population size. Even-tually, this never-ending positive feedback would lead to unbounded exponential growthof the interacting species (Gause and Witt, 1935, May, 1976, Vandermeer and Boucher,1978). Because of their unrealistic predictions, these mutualistic models were set asidefor years.Modelers started reconsidering mutualistic interactions only at the end of the 1970s.At this time, the problem of unbounded growth was addressed by incorporating somekind of negative feedback in the population receiving the benefit, such as self-limitation,predation, or competition (Addicott, 1981, Dean, 1983, DeAngelis et al., 1979, 2012, Goh,1979, Heithaus et al., 1980, Johnson and Amarasekare, 2013, Pierce and Young, 1986,Ringel et al., 1996, Travis and Post III, 1979, Wolin, 1985). Negative feedback couldeffectively bound population growth, despite the fact that these models still presented alinear functional response for interspecific interactions (see floating frame), meaning thatthe benefits of mutualism keep on increasing linearly with the density of the populationgiving the benefit. Alternatively, modelers considered bounding population growth bysetting a limit in the benefit that a species can provide to the other, reached at highdensities of the receiving population (Holland and DeAngelis, 2006, Holland et al., 2002,Ingvarsson and Lundberg, 1995, Kot, 2001, May, 1976, Wright, 1989). The non-linearityof the benefit functional response (see floating frame) can lead to a stable equilibrium of2mutualistic coexistence, without needing to consider other sources of negative feedbacksuch as intraspecific self-limitation or interactions with a third species.Although the use of a non-linear functional response can lead to bounded growth,these type of models still fail to predict basic phenomenon empirically observed in pop-ulation dynamics of mutualists, such as fluctuations or damped oscillations (Addicott,1986, Bronstein, 2015, Morales, 2011). It is only in 2010 that Holland and DeAngelis at-tempted to fill this gap between theory and empiricism by developing a general theory forthe population dynamics of mutualism based on the consumer-resource mechanism (Hol-land and DeAngelis, 2010). Although the mechanism was already well known for studyingpredator-prey and competitive interactions (Abrams, 1987, MacArthur and Levins, 1967,MacArthur, 1984), its application to the modeling of mutualistic interactions had neverpreviously been explored. The consumer-resource mechanism makes the interplay of the-ory and experiments feasible, by relating the process of energy or nutrient transfer betweenthe consumer (e.g., the organism) and resource (e.g., any biotic or abiotic factor that af-fects the population growth of the organism). Hence, the rate of change in biomass ofthe interacting species due to their mutualistic interactions can be separated explicitlyinto the increase in biomass attributed to the resource received (the benefit functionalresponse of mutualism), minus the decrease in biomass caused by the resource given (thecost functional response of mutualism) (Agrawal and Fordyce, 2000, Chamberlain andHolland, 2008, Ferrie`re et al., 2007, Holland and DeAngelis, 2009, 2010, Holland et al.,2005, 2013, Kang et al., 2011, Liu et al., 2016, Wang and DeAngelis, 2011). Analyses andextensions of these models (Cropp and Norbury, 2018, 2019, Han and Dai, 2017), show adiverse range of population dynamics that were not recognized by any previous theory.The models presented in my thesis are also based on a consumer-resource mechanismof interaction, wherein both, the plant and the fungi, each act as consumer and as re-source. In my work, I explore how asymmetry in the benefit and cost functional responsesof mutualism can lead to coexistence of multiple mutualists. I use the consumer-resourceframework to include not only two interacting species, but a whole community of mutu-alists, sharing a resource supplied by the same plant and competing between each other.I show that the rates of resource exchange can not only be linked to changes in biomassof the interacting species, but can also be used as quantitative criteria to measure theresilience of a mutualistic community against invasion, or the spreading speed of a lessbeneficial mutualist (such as, perhaps, a competitive commercial fungus) in a landscape.In chapter 2, I present an ordinary differential equation model for the mutualistic in-teractions of a plant associated with multiple species of AM fungi which do not competedirectly. Using this model, I formulate ecological hypotheses explaining the coexistenceof a guild of mutualists on a single host. I show that parasitism, rather than compe-tition, could be the driving force behind community establishment. The dominance ofthe competition-driven perspective has long shaped ecological theory; my work presents3a novel and potentially transformative view of community dynamics.In chapter 3, I extend the model of chapter 2 to incorporate direct competitive inter-actions between fungi. The explicit consideration of competition gives important insightsinto the mechanisms leading to coexistence and competitive exclusion of mutualists withina guild. More specifically, I investigate the consequences for diversity in the native com-munity following the introduction of an alien mutualist, such as a commercially-developedfungus that is introduced as a fertilizer. This model allows for the formulation of concretecriteria to determine the benefits and biodiversity risks associated with the introductionof alien species in mutualist communities. Such information is critical for conservation orrestoration planning, and for growers interested in the long-term health of their soil.In chapter 4, I consider the model of chapter 3 with spatial movement. The spatialcomponent allows to understand conditions permitting the spread of the introduced inoc-ulants in the field, and characterize those factors affecting the spread velocity. Modeloutcomes are evaluated against experimental work, and used to explain some of theinconsistent observations on the establishment, persistence, and spread of mycorrhizalinoculants in the field.In chapter 5, I use the model presented in chapter 2 to discuss the effect of diversityamong fungal species on plant growth. I establish the correspondence between my theoryand real-world data by explaining observations reported in a recent experimental paper.My results demonstrate that the framework I have developed can significantly improveour understanding of the behaviour of mutualistic guilds and their associated host.Figure 1.1: Schematic representation of the mycorrhizal symbiosis. AM fungi (in blue) grow in theplant cortical cells (in black) and extend into the soil, where they help with the absorption of substancessuch as phosphorus (P), nitrogen (N), and water (H2O). In exchange, the plant provides fixed carbon tothe fungi. Nutrients exchange occurs inside the plant cells, where the fungi build branched structurescalled ‘arbuscules’. Hence the name ‘arbuscular mycorrhizal fungi’.4Functional responses (FR)- When studying the dynamics of interacting populations, we are interested inknowing how the rate of change in population density (or size) of a speciesvaries with the population density of the species it is interacting with. Func-tions providing this link are called functional responses. The rate of increasein the population density of a species due to its mutualistic interaction withanother species, can be referred to as the benefit functional response of mu-tualism (F (Mj)), with units of benefit per individuum per unit of time.- A linear functional response is observed when the benefit provided by a speciesto another increases linearly with the population density of the species pro-viding the benefit (see figure below, left plot). In this case, the rate of benefitcan be written as γijMjMi, where Mj is the population density of the speciesproviding the benefit and Mi is the population density of the species receivingthe benefit.- If the benefit provided by species j to i varies non-linearly with the popula-tion density of species j, then the functional response is said to have interspe-cific density dependence. A widely used functional response is the Michaelis-Menten function, i.e., the benefit can be written as γijMjMi/(cj + Mj). Inthis case, the rate at which the benefit is provided saturates at high den-sity of species j (see figure below, right plot). Indeed, for Mj large enough,the Michaelis-Menten function tends to the upper limit γij. Parameter cj isthe half-saturation constant, i.e. the population density Mj for which thefunctional response is half of its maximal value.- The rate at which the benefit is provided can also depend on the populationdensity of the species receiving the benefit. In this case, we say that thefunctional response presents intraspecific density dependence. In chapter 2 Iwill discuss how intra- and interspecific density dependence in the functionalresponses of mutualism can affect the growth dynamics of the interactingspecies.5Chapter 2Mechanisms for coexistence withinmutualist guilds2.1 Background informationWithin mutualist guilds, the principle of competitive exclusion seems not to apply. Indeed,within a guild, mutualists occupying closely related niches coexist on the same partner(Palmer et al., 2003). This fact presents a paradox: According to the principle of com-petitive exclusion one would expect less beneficial mutualists, i.e., species that provide alower amount of nutrient in return for the benefit received, to dominate and outcompetemore beneficial mutualists (Hardin, 1960, Palmer et al., 2003). Yet, mutualisms involvingonly two interacting species are relatively rare (Herre et al., 1999, Hoeksema and Bruna,2000). New evidence has revealed that even mutualisms previously thought to be species-specific, such as the fig tree-fig wasp mutualism, are now recognized to involve a guild ofmutualists with very similar habits and morphology (Herre et al., 1999, Hoeksema andBruna, 2000, Knowlton and Rohwer, 2003, Molbo et al., 2003, Pellmyr, 1999).The arbuscular mycorrhizal (AM) mutualism is particularly challenging to explain.AM fungi exist in the roots of most terrestrial plants (Smith and Read, 2010). Thehyphae of the fungi grow from the plant roots into the soil, where they are able toefficiently increase the absorption of nutrients that are limiting to plant growth (e.g.,phosphorus). In return for these nutrients, host plants provide carbon to the fungi. Innature, a single host plant often associates with dozens of AM fungal species, many ofthem functionally indistinguishable (Herre et al., 1999, Hoeksema and Bruna, 2000, O¨piket al., 2006). Guild members coexist tightly in space within the roots and in the soil(Aldrich-Wolfe, 2007, Bennett and Bever, 2009), rarely displaying the patchy distributiontypically associated with niche differentiation or aggregation (Powell and Bennett, 2016).There has been some investigation as to the mechanism for this coexistence. Nichedifferences (Batstone et al., 2018), colonization-competition trade-offs (Smith et al., 2018),6spatial structure (Wilson et al., 2003), or physio-evolutionary feedbacks (Bever, 2015), canfacilitate coexistence in trophic communities as well as in multi-species mutualisms. Noneof these mechanisms, however, can explain coexistence of species with a high degree ofniche overlap in the absence of spatial structure.One of the most popular explanations for coexistence of multiple fungal species hasbeen regulation of resource exchange (Palmer et al., 2003). This mechanism requires thatthe host favours guild members with more beneficial traits, thereby explaining the persis-tence of good mutualists. Such preferential allocation of resources has been documented inthe AM symbiosis (Bever et al., 2009, Hammer et al., 2011, Ji and Bever, 2016, Kiers et al.,2011) and has some theoretical support (Bachelot and Lee, 2018, Bever, 2015, Christianand Bever, 2018, Hoeksema and Kummel, 2003, Krˇivan and Revilla, 2019, Moeller andNeubert, 2016). However, experimental studies have yielded inconsistent results (Fitter,2006, Kiers and Van Der Heijden, 2006, Walder and van der Heijden, 2015) and no phys-iological basis for host discrimination has been identified. In addition, those experimentsshowing preferential allocation by the plant (allocation of plant carbon to fungi in pro-portion to the benefit received from each fungus) relied on an artificial spatial structurewhere the fungi were spatially segregated (Bever et al., 2009, Kiers et al., 2011, Walderet al., 2012). Thus, the problem of coexistence in the AM symbiosis has languished.I propose an alternative mechanism that stabilises coexistence of multiple AM fungiwithin a single host. I argue that indirect parasitism, not competition, is the main formof interaction observed among guild members, and that this interaction can explain theabsence of competitive exclusion. I show that stable coexistence of multiple AM fungalspecies on a single root system can be obtained without appealing to niche-based mecha-nisms, spatial segregation, or preferential allocation. My results provide a new perspectiveon how diversity within a guild of mutualists can be maintained.2.2 Model and methods2.2.1 Formulating a model for a multi-species mutualismTo investigate the dynamics of multi-species mutualisms I develop a model for the interac-tions between a host plant and its associated fungal mutualists. Resources exchanged arephosphorus (fungi to plant) and carbon (plant to fungi). Here, the changes in plant andfungal biomass are related to the amount of resource received, to the amount of resourcegiven, and to maintenance costs. All of the mutualists coexist on a single plant, i.e., thecarbon provided by the single plant supports the growth of all of the mutualists. Theamount of resource exchanged depends on the size of each fungal individual, and on thesize of the plant. I use ‘biomass’ as a measure of size. In the case of the plant, I as-sume that increasing biomass implies increasing the size of both below- and above-ground7parts. In my model therefore, this scenario can be represented by a series of differentialequations describing the change over time of plant biomass (p) and of the biomass (mi)of each of the N species of AM fungi (m1,m2, ..,mN):change inplant biomass︷︸︸︷dpdt= qhpphosphorusfrom AM fungi︷ ︸︸ ︷∑i(αimi)pd+ p− qcpcarbonto AM fungi︷ ︸︸ ︷∑i(βimi) p +intrinsicgrowth︷︸︸︷rp p −maintenancecost︷ ︸︸ ︷µp pn , (2.1a)dmidt︸︷︷︸change infungal biomass= qcmi βi pmi︸ ︷︷ ︸carbonfrom plant− qhmi αipd+ pmi︸ ︷︷ ︸phosphorusto plant− µmimsi︸ ︷︷ ︸maintenancecost, i = 1 .. N . (2.1b)A brief description of the parameters used in the model with the respective measurementunits is given in Tables 2.1 and 2.2. Here, I present an overview of the main features ofthe model.The first terms on the right sides of Eqs. (2.1a) and (2.1b) describe the gain in plant andfungal biomass per unit time due to the resource received. The second terms represent theloss in biomass per unit time due to resource given. The relationship between the specificfunctional forms chosen for these terms is a key component of the model, and so theseterms are discussed in detail in the following section. The last terms for both equationsdescribes the resource required for maintaining the existing biomass. The additionalterm in Eq. (2.1a), the intrinsic growth term, represents plant growth per unit time inthe absence of the fungi (Smith et al., 2003). AM fungi are obligate mutualists with nosaprobic ability and cannot survive in the absence of the host plant, so no intrinsic growthterm appears in Eq. (2.1b).Fungal species can be differentiated according to four specific traits: the efficiencywith which carbon and phosphorus are converted into biomass (qcmi and qhmi) (van Aarleand Olsson, 2003), the ability to provide phosphorus to the plant (αi) (Drew et al., 2003,Ravnskov and Jakobsen, 1995), the access to host carbon (βi) (Pearson and Jakobsen,1993, Zhu and Miller, 2003) or the maintenance cost (µmi) (Sylvia and Williams, 1992).Consistent with the literature, fungal species in the model can be distinguished accordinglyto these four traits.The literature provides little evidence about how the maintenance costs of plant andfungi depend on their respective biomasses (last term in Eqs. (2.1a) and (2.1b)). As bothdensity dependent and non-density dependent population growth are found in nature(Hassell, 1975), I investigate the effect of both linear (i.e. n = 1 and s = 1) and nonlinearforms of the maintenance cost term (i.e. n > 1 and s > 1).8Table 2.1: Brief description of the parameters of the model of Eq. (2.1).Parameter Descriptionαi Phosphorus exchange ability of fungal species iβi Access to host carbon of fungal species iµp Maintenance rate of the plantµmi Maintenance rate of fungal species iqhp Conversion factor of phosphorus into plant biomassqcp Conversion factor of carbon into plant biomassqhmi Conversion factor of phosphorus into fungal biomass of species iqcmi Conversion factor of carbon into fungal biomass of species irp Intrinsic growth rate of the plantd Half-saturation constantn Biomass dependence of the plant maintenance rates Biomass dependence of the fungal maintenance rate2.2.2 Quantification of resource exchangeNutrient exchange between the host plant and the fungal mutualists occurs inside theplant root cells, where the AM fungi build nutrient exchange zones called ‘arbuscules’(see Fig. 1.1) (Peterson and Guinel, 2000, Wang et al., 2017). It is reasonable to assumethat the amount of phosphorus transferred increases with increasing root colonization andwith increasing hyphal length (Douds et al., 2000, Sawers et al., 2017, Treseder, 2013);More arbuscules means more contact locations where nutrients can be exchanged, andlonger hyphae in the soil mean greater fungal access to soil phosphorus. In the model, Ithus assume a linear relationship between fungal biomass and phosphorus transfer.The percentage of root colonization by a fungal species is typically between 20% and50% of plant roots, reaching 80% at most (Hart and Reader, 2002, Klironomos and Hart,2002). That is, space availability in the roots seems not to be a factor limiting thepresence of AM fungi. Thus, I assume that the amount of root colonized by the fungidoes not depend on plant biomass when plant biomass is large. Consequently, at largeplant biomass the fungus will transfer a quantity of phosphorus that depends on its ownbiomass only, and not on plant biomass. That is, when plant biomass is large, fungi givephosphorus to the plant in proportion to the fungal biomass only. When plant biomass9is small, a linear relationship between phosphorus transfer and plant biomass is assumed,as less plant availability implies less habitat for the fungus and therefore less phosphorustransfer. Mathematically, these assumptions lead to the phosphorus transfer termPhosphorus transfer ∝ mi pd+ p. (2.2)where d is the half-saturation constant. When p d, phosphorus transfer dependslinearly on fungal and plant biomass, while when p d, phosphorus transfer dependson fungal biomass only. Eq. (2.2) corresponds to the second term of Eq. (2.1a) and tothe first term of Eq. (2.1b). Eq. (2.2) implies that in the presence of multiple mutualists,each of the mutualists supplies to the plant a quantity of phosphorus per unit time thatis proportional to its own biomass. Note that, in my model, the non-linearity in thephosphorus transfer term (Eq. (2.2)) is due to a limitation that the fungus experiencesat low plant biomass, and not a limitation in the benefit of mutualism set by the plant(i.e., the benefit is not saturating at large biomass of the species giving the benefit, asseen in previous models of mutualistic interactions (Holland and DeAngelis, 2010)). Thus,phosphorus transfer is limited by low plant biomass, but is independent of plant biomass(i.e., depends only on fungal biomass) when plant biomass is large.The literature shows that the plant supplies its associated mutualists with a proportionof the carbon it synthesizes (Douds et al., 2000, Graham, 2000, Treseder and Cross, 2006).It has been shown that carbon transfer from the plant to the AM fungi depends onphotosynthetic capacity and on the extent of root colonization by AM fungi (Thomsonet al., 1990, Vierheilig et al., 2002). In the model, I assume a linear relationship betweenplant biomass and carbon fixation. I further assume a linear relationship between carbonfixation and carbon transfer. Finally, I assume linear dependence of carbon transfer andfungal biomass. In sumCarbon transfer ∝ mip , (2.3)as indicated by the first and second terms of Eqs. (2.1a) and (2.1b). The choice of theterm for carbon transfer in Eq. (2.3) implies that when multiple mutualists are associatedwith the same host, each species is supplied with an amount of host carbon per unit timethat its proportional to its individual fungal biomass and to the existing plant biomass.The fungi do not directly compete for carbon, but each accesses a different proportion ofhost carbon depending on each fungus’s biomass.One can directly observe that the phosphorus and carbon transfer terms (Eqs. (2.2)and (2.3)) show a different dependence on the biomass of the supplying and receivingspecies. Resource exchange is therefore asymmetric, where carbon transfer depends onboth plant and fungal biomass, while phosphorus transfer depends on fungal biomass only,once the plant has reached a certain size. In other words, the fungus gives depending onits own biomass and takes depending on its own biomass and on plant biomass.10Although the dependence of the nutrient transfer terms (Eqs. (2.2) and (2.3)) on plantand fungal biomass is the same for all fungi, the exact amount of phosphorus and carbontransferred varies with the parameters αi and βi respectively, and can vary among species.Fungi characterised by a larger αi and lower βi parameters have a higher ability to transferphosphorus to the plant and take less carbon in return, and therefore can be said to bemore mutualistic than species having a smaller αi and larger βi.2.2.3 Model analysisI will analyse the model of Eq. (2.1) using both linear stability analysis and numericalsimulations (Case, 1999, Edelstein-Keshet, 1988). Numerical simulations were performedusing the built-in ode45 solver in MatlabR2017a.First, I will determine the conditions under which a stable mutualism can be estab-lished. These results are presented in sections 2.3.1 (one plant- one fungus case) and 2.3.2(one plant - multiple fungi case). Second, in section 2.3.3, I will investigate how differ-ences in the resource exchange ability of the fungi affects plant growth and the growth ofother guild members. For this purpose, I will study the system dynamics when all modelparameters are held at a constant value (given in the caption of Fig. 2.8) except thosethat quantify the exchange of nutrients between plant and fungi, i.e., αi and βi. Theseparameters represent, respectively, the ability of fungal species i to provide phosphorus tothe plant and the access of fungal species i to plant carbon. I will determine analyticallyhow the plant and fungal steady states depend on the overall phosphorus exchange abilityof the whole guild.Additionally I will solve Eq. (2.1) numerically and run the model to steady-state fordifferent combinations of the αi and βi parameters. Because host discrimination hasbeen one of the most popular explanations for coexistence in multi-species mutualisms(Fitter, 2006, Kiers and Van Der Heijden, 2006, Walder and van der Heijden, 2015), I aminterested in seeing how it affects the system dynamics. Numerically, I will consider thefollowing two cases:(a) No host discrimination: Access to host carbon is the same for all fungi, regardlessof the amount of phosphorus they provide (i.e., parameter βi is held constant forall species while αi varies). Since all of the species are characterised by the samecarbon transfer rate parameter, cheaters are those that provide little phosphorus(low αi).(b) Host discrimination: More beneficial fungi (high phosphorus transfer) have accessto more host carbon, while fungi that are less beneficial (low phosphorus transfer),have a reduced access to host carbon (i.e., species with large/low parameter αi alsohave large/low βi).112.3 Results2.3.1 Analysis of the one plant - one fungus systemConditions for the existence of coexistence steady states: I am interested inderiving conditions for the existence of steady states for which plant and AM fungi coex-istence is observed. To facilitate computations, I reduce the number of model parametersin Eq. (2.1) and non-dimensionalise the equations by carrying out the substitutions givenin Table 2.3. I consider a version of the model for which rp = 0 and maintenance costsare non-linear (i.e., n = 2 and s = 2). Insights on the cases rp > 0 and n, s 6= 2 will begiven below (see Figs. 2.5, 2.6 and 2.7). We obtain the non-dimensional modeldPdτ= QMD + PP −M P − ηP P 2, (2.4a)dMdτ= P M − PD + PM − ηM M2. (2.4b)Nullclines of Eq. (2.4) can be found by setting dP/dτ and dM/dτ to zero and solving forP and M . We obtaindPdτ= 0 ⇐⇒ P = 0 or M = ηP P (D + P )Q−D − P , (2.5a)dMdτ= 0 ⇐⇒ M = 0 or M = P (P +D − 1)ηM (D + P ). (2.5b)The first nullclines in Eq. (2.5a) and in Eq. (2.5b) correspond to the absence of AM fungiand plant biomass respectively, and to the horizontal and vertical axes in the phase plane.The origin is therefore always a steady state.The second nullcline of Eq. (2.5a) is an hyperbola with vertical asymptote at P = Q−D(Fig. 2.1). In order for Eq. (2.5a) to allow positive values for M and P the asymptotemust be in the positive quadrant, i.e. I require Q −D > 0. This requirement yields thefirst condition for the existence of a steady state, which can be expressed in terms of theTable 2.2: Variables and parameters of the model given in Eq. (2.1), with respective measurement units.p plant biomass α phosphorus(unit time)×(unit fungal biomass) qhpplant biomassphosphorusm fungal biomass β carbon(unit time)×(unit fungal biomass)×(unit plant biomass) qcmfungal biomasscarbond plant biomass µp1(unit time)×(unit plant biomass) qcpplant biomasscarbonrp intrinsicgrowth rateµm1(unit time)×(unit fungal biomass) qhmfungal biomassphosphorus12Table 2.3: Description of the non-dimensional parameters used to translate Eq. (2.1) into Eq. (2.4) .P = qcmβqhmαp M = qcpβqhmαm τ = qhmα tD = qcmβqhmαd Q =qcmqhpqcpqhmηP =µpqcmβηM =µmqcpβoriginal parameters asα >qcpqhpβ d. (2.6)Expression (2.6) essentially states that for a coexistence steady state to occur, the abilityof the fungi to supply phosphorus to the plant (α) must be sufficiently large. At the sametime, the ratio between the conversion factors of nutrients given to received into plantbiomass (qhp/qcp), and the carbon sink strength (β) must be small enough. Moreover dmust be small, i.e., the threshold of plant biomass that determines when the benefit ofthe mutualism for the plant is limited by the plant or by the fungal biomass must berelatively low.The second nullcline of Eq. (2.5b) is also an hyperbola with asymptote at −D . In-tersections of the hyperbola with the horizontal axis occur at P = 0 and P = 1 −D. IfD > 1, the hyperbola assumes values in the positive quadrant for P > 0, while if D < 1the hyperbola assumes values in the first quadrant for P > 1−D (Fig. 2.1). In order forthe nullclines of Eqs. (2.5a) and (2.5b) to intersect, the asymptote at P = Q−D in (2.5a)must occur at P > 1−D. This condition gives a second requirement for the existence ofa steady state, that is Q−D > 1−D ⇔ Q > 1. In term of the original parameters, thiscondition translates toqcmqhpqcpqhm> 1. (2.7)Eq. (2.7) states that the product of the conversion factors of the nutrients received, i.e.,carbon for the fungi and phosphorus for the plant, must be larger than the conversionfactors of the nutrients given.An expression for the values of P ∗ at steady state can be derived by setting Eqs. (2.5a)and (2.5b) equal to each other and finding the roots of the polynomialηPηM(P∗ +D)2 − (Q−D − P ∗)(P ∗ +D − 1) = 0 . (2.8)Steady states occur atP ∗1,2 =Q+ 1±√(Q− 1)2 − 4QηPηM2(1 + ηPηM)−D. (2.9)The steady states P ∗ assume real values only if the expression under the square root isgreater than zero. This requirement translates into a third condition for coexistence, i.e.,13ηPηM < (Q− 1)2/4Q. In term of the original parameters this condition can be written asµpµm < β2 qcmqhp − qhmqcp4 qhmqhp. (2.10)Expression (2.10) essentially states that to allow stable coexistence of plant and AM fungithe maintenance rates for plant and AMF (µp and µm) can not be too large.Eq. (2.9) can have zero, one or two positive solutions depending on the choice of theparameters. The stability analysis conducted below and the graphical stability analysis ofFig. 2.1 show that when only one coexistence steady state exists (i.e., P ∗1 < 0 and P∗2 > 0),it is always an attractor and the origin is a saddle. When two coexistence steady statesexist the origin and P ∗2 are attractors and P∗1 is a saddle, where P∗1 < P∗2 . In this casethere are two distinct basins of attraction, one for the origin and one for the P ∗2 steadystate . The biological interpretation of this case is that below certain thresholds of plantand fungal biomass, mutual growth can not be observed.The steady state value of the plant (the positive root in Eq. (2.9)) corresponds tothe non-dimensionalised version of the upper bound to plant growth, and determines themaximal plant size at which plant growth saturates. This value increases with Q, anddecreases with D, ηP or ηM . Eq. (2.9) can be rewritten in terms of the original parametersasp∗1,2 =qhpαqcp+ αqhmqcm± α qhmqcm√(qhpqcmqcpqhm− 1)2− 4 qhpµpµmqhmq2cpβ22(β + µpµmqcmqcpβ) − d . (2.11)I am interested in determining how variations in all of the model parameters affect thevalue of the stable steady states p∗2 and m∗2, representing the final biomass of the plant andof the AM fungi respectively (the stability analysis will be given in the following section).Plots in Fig. 2.2 shows variations of p∗2 as a function of each parameter. From Eq. (2.5)one can derive an expression for the non-dimensional steady state M∗2 , that isM∗2 =ηP P∗2 (D + P∗2 )Q−D − P ∗2. (2.12)Or, in terms of the original parametersm∗2 =µpαqhmp∗2(d+ p∗2)( qcpαqhpβ− d− p∗2). (2.13)The plots in Fig. 2.3 show how m∗2 varies as a function of the model parameters.14Figure 2.1: Nullclines P˙ = 0 (dashed line) and M˙ = 0 (dotted line) of Eq. (2.1), where Q = 2,ηP = ηM = 0.3, N = 1 and D = 1.1 (left panel) or D = 0.8 (right panel). Blue dots represent theintersections of the nullcline M˙ = 0 with the horizontal axis. When D < 1 the non-zero interceptionoccurs for P > 0. When D < 1 the non-zero interception occurs for P < 0. Arrows represent qualitativelythe direction of the flow on the nullclines and nearby the steady states. The steady states at (P ∗,M∗)(left panel) and (P ∗2 ,M∗2 ) (right panel) are always stable. The left panel represents the situation forwhich only one positive steady state exists, and the steady state at the origin is a saddle. The right panelcorresponds to the case where two steady states occur. The steady state at (P ∗1 ,M∗1 ) is a saddle and theorigin an attractor. The asymptote of the nullcline P˙ = 0 occurring at P = Q−D is shown in red.15Figure 2.2: Steady state value p∗2 (or plant final biomass) as a function of the model parameters. Defaultparameter values are qhp = qcm = 2, qhm = qcp = 1, α = 0.6, β = 0.5, µm = µp = 0.3, d = 1.2.16Figure 2.3: Steady state value m∗2 (or fungal final biomass) as a function of the model parameters.Default parameter values correspond to those in Fig. 2.2.Stability of the coexistence steady states: I consider Eq. (2.4), describing variationin plant and fungal biomass over time, when the plant is associating with a single speciesof AM fungi. As shown in the previous section, as long as conditions (2.6), (2.7) and (2.10)are satisfied this system presents a minimum of one and a maximum of two coexistencesteady states (P ∗1,2,M∗1,2). I am interested in determining the stability of these coexistencesteady states. For practicality, I will initially refer to both steady states as (P ∗,M∗) anddistinguish the two different cases toward the end of the analysis. At steady state, it holdsthatP ∗(QP ∗ +D− 1)=ηP (P∗)2M∗, (2.14a)M∗(1− 1P ∗ +D)=ηM(M∗)2P ∗. (2.14b)17Hence the Jacobian of Eq. (2.4) at steady state can be written asJ(P ∗,M∗) =−(QM∗(P ∗ +D)2+ ηP)P ∗ P ∗(QP ∗ +D− 1)M∗(1− 1P ∗ +D)+P ∗M∗(P ∗ +D)2−ηMM∗=−(QM∗(P ∗ +D)2+ ηP)P ∗ηP (P∗)2M∗ηm(M∗)2P ∗+P ∗M∗(P ∗ +D)2−ηMM∗ .(2.15)One can first observe that the trace of J is always negative. The determinant of J canbe computed by using the relationships found in Eq. (2.14)det(J) =(QM∗(P ∗ +D)2+ ηP)P ∗ηMM∗ − ηP (P∗)2M∗(ηM(M∗)2P ∗+P ∗M∗(P ∗ +D)2)=P ∗(P ∗ +D)2(ηm(M∗)2Q− ηP (P ∗)2)=P ∗(P ∗ +D)2(P ∗M∗(1− 1P ∗ +D)Q− P ∗M∗(QP ∗ +D− 1))=(P ∗)2M∗(P ∗ +D)3((1 +Q)(P ∗ +D)− 2Q) .(2.16)The sign of det(J) depends therefore on the sign of (1 + Q)(P ∗ + D) − 2Q, where thecoexistence steady state (P ∗,M∗) is stable if P ∗ + D ≥ 2Q/(1 + Q), and it is a saddlepoint otherwise.From the analysis conducted above, we know that X = (P ∗ + D) are roots of thepolynomial T (X) = ηPηMX2 − (Q − X)(X − 1) (see Eq. (2.8)). I use X+ and X− todenote the positive and negative root of T (X), whereX± =1 +Q±√(1−Q)2 − 4QηPηM2(1 + ηMηP ). (2.17)To determine the stability of the steady states it is sufficient to show thatX− ≤ 2Q/(1 +Q) ≤ X+ . (2.18)To prove Eq. (2.18), I want to understand the location of 2Q/(1 + Q) with respect tothe roots of the polynomial T (X). For this purpose, I need to determine the sign of18T (2Q/1 +Q). I obtainT (2Q/(1 +Q)) = ηPηM4Q2(1 +Q)2−(Q− 2Q(1 +Q))(2Q(1 +Q)− 1)= ηPηM4Q2(1 +Q)2− Q(Q− 1)(1 +Q)(Q− 1)(1 +Q)= (4ηPηMQ− (Q− 1)2) Q(1 +Q)2= −∆ Q(1 +Q)2≤ 0 ,(2.19)where ∆ corresponds to the expression under the square root in Eq. (2.17). Because T (X)is an upward parabola, Eq. (2.19) implies that Eq. (2.18) holds true. One can concludethat the coexistence steady state (P ∗2 ,M∗2 ) associated to X+ is stable (as Tr(J)< 0 anddet(J) > 0), and the steady state (P ∗1 ,M∗1 ) associated to X− is a saddle point (as Tr(J)< 0and det(J) < 0).Graphical insights into the establishment of mutualistic interactions: The es-tablishment of a mutualistic relationship between a plant and one fungal species canoccur as long as (i) the plant receives enough nutrients to support its growth (i.e.,α > (qcpβd)/qhp, see Eq. (2.6)), (ii) plant and AM fungi are able to make effective useof the resource received (i.e., (qcmqhp)/(qcpqhm) > 1, see Eq. (2.7)), and (iii) maintenancecosts are kept low (i.e., µpµm < β2(qcmqhp − qhmqcp)/(4qhmqhp), see Eq. (2.10)). As longas these three conditions are satisfied, the two nullclines in the phase plane of Fig. 2.1intersect and coexistence is observed.From the phase plane in Fig. 2.1 we can also notice that when fungal biomass is largecompared to plant biomass, fungal biomass will decrease and plant biomass will increase.The plant will therefore be exploiting the fungi, till its biomass is large enough to allowfungal growth. Similarly, when fungal biomass is low compared to plant biomass, the fungiwill not be able to support plant growth. In this case, before showing mutual growth,plant biomass will decrease (suggesting parasitic fungi) and fungal biomass will increase.Plant growth saturates at the point where there is a balance between the benefit andcost of the mutualism. This balance is reached as a consequence of the asymmetry inphosphorus and carbon transfer (Eqs. (2.2) and (2.3)). Fig. 2.4 demonstrates how thebenefits and costs of the mutualism are distributed across the plant and fungus, as afunction of plant or fungal biomass. The figure shows plots of the three key functionalresponses of the mutualism: the gross benefit (GB), cost (C), and net effect (NE), whereNE = GB - C. These functions are given in Table 2.4. In the right panel of Fig. 2.4,19one can see that when plant biomass is small, the gross benefit functional response of theplant is larger than the cost functional response. This situation results in a positive neteffect on plant growth, and therefore in an increase in plant biomass. When plant size islarge, however, the gross benefit functional response decreases, while the cost functionalresponse remains constant, eventually leading to a negative effect on plant growth. Plantbiomass is therefore expected to stabilise around a certain value for which the plant grossbenefit and cost functional responses are equal to each other. That is, the plant stopsgrowing when the net effect (see Table 2.4) is zero. Consequently, plant growth is limitedby its interaction with the fungal mutualists.Fungal growth is not limited by its interaction with the host plant. As long as plantbiomass is large enough, the net effect of mutualism for the fungi remains positive (seeleft panel of Fig. 2.4). Without any kind of self-limitation (s = 1) fungal growth isunlimited (see last term in Eq. (2.1b)). Fungal growth is limited for s > 1. In eithercase, the model predicts plant-fungi coexistence (see Fig. 2.7). Plant-fungus coexistenceis observed whether the maintenance cost per unit time of the plant (see last term inEq. (2.1a)) is linearly proportional to plant biomass (n = 1) or increases non-linearly(n > 1) with plant biomass (see Fig. 2.6), and for a positive intrinsic growth rate rp > 0(see Fig. 2.5).Table 2.4: Biomass-dependence of the gross benefit (GB), cost (C) and net effect (NE) functionalresponses of plant (p) and a species of AM fungi (mi), characterizing the model described in Eq. (2.1).The GB and C functional responses of plant and fungi are obtained by dividing Eqs. (2.2) and (2.3),representing the biomass-dependence of phosphorus/carbon transfer per unit time, by the biomass of thespecies receiving (GB) or providing (C) the nutrient. The net effect functional response is calculated asthe difference between the gross benefit and cost functional responses (NE = GB − C).GB C NEPlant (p) mid+p mi mi (1d+p− 1)AM fungi (mi) ppd+p p (1− 1d+p)20Figure 2.4: Qualitative graphical representation of the functional responses of plant and AM fungi givenin Table 2.4. Functional response curves of gross benefit (GB), cost (C) and net effect (NE = GB − C) ofa species of AM fungi (left panel) and plant (right panel) are plotted as a function of plant, respectivelyfungal, biomass. Note that the GB functional response of the plant depends on both, fungal and plantbiomass, making the NE functional response of the plant negative at large plant biomass and positivewhen plant biomass is small. Plant growth will therefore stabilise about a certain value of p for whichplant GB and C functional responses are equal to each other.Facultative mutualism for the plant (rp > 0) and variation in themaintenance costs (n, s 6= 2)The analysis conducted above considered the case where the intrinsic growth rate of theplant is zero (rp = 0), and maintenance costs increase non-linearly with plant and fungalbiomass (s, n = 2). Here I investigate the behaviour of systems where rp > 0 and n, s 6= 2.We consider the modeldpdt= qhpαmpd+ p− qcpβmp+ rpp− µppn, (2.20a)dmdt= qcmβpm− qhmiαpd+ pm− µmms. (2.20b)Nullclines of Eq. (2.20) for s > 1 are given by:p˙ = 0 ↔ m = µppn−1 − rp(qhpαd+p− qcpβ) , (2.21a)m˙ = 0 ↔ m = s−1√1µm(qcmβp− qhmα pd+ p). (2.21b)A graphical representation of the nullclines of Eq. (2.21) is given in Fig. 2.5, for the casewhere rp > 0, and in Figs. 2.6, for the cases n = 1 and s = 2 (left figure), n = 3 and21s = 2 (middle figure) and n = 2 and s = 3 (right figure). We can see that for all thesecases a stable steady state exists, where the phase planes qualitatively resemble Fig. 2.1(considered in the analysis).When s = 1, the nullcline of Eq. (2.21b) corresponds to a vertical axis atp = −qcmβd+ qhmα + µm +√(qcmβd− qhmα− µm)2 + 4qcmβµmd (2.22)For d large enough, the nullclines do not intersect (see Fig. 2.7, left panel), what causesfungal biomass growth unbounded, while plant growth is limited. For d small enough,intersection of the nullclines results into an unstable steady state (see Fig. 2.7). Thissituation corresponds to the right panel of Fig. 2.1.Therefore, graphically we can observe that plant-fungus coexistence is observedwhether the plant maintenance cost increases linearly (n > 1) or non-linearly with plantbiomass (n = 1), and for rp > 0. Without any kind of self-limitation (s = 1) fungal growthis unlimited, however also in this case plant-fungus coexistence is observed. Further an-alytical work should provide the exact parameter range for which nullclines intersection(and thus mutualism establishment) is observed when rp > 0 and n, s 6= 2. Conditionsfor mutualism establishment for the case rp = 0 and n, s = 2 are derived in the analysisabove (see Eq. (2.6), (2.7), and (2.10)).Figure 2.5: Nullclines p˙ = 0 (dashed line) and p˙ = 0 (dotted line) of the system of differential equationsgiven in Eq. (2.21) for rp > 0. Arrows represent qualitatively the direction of the flow nearby the stablesteady state (p∗,m∗). Parameter values are rp = 0.1, qhp = 3, qcm = 2, qcp = qhm = 1, α = β = 0.4,µm = µp = 0.3, d = 1.2, s = n = 2.22Figure 2.6: Nullclines p˙ = 0 (dashed line) and p˙ = 0 (dotted line) of the system of differential equationsgiven in Eq. (2.21) for different biomass dependence of the maintenance costs of the plant (parametern) and of the fungus (parameter s). Arrows represent qualitatively the direction of the flow nearby thestable steady state (p∗,m∗). Parameter values are n = 1 and s = 2 (left figure), n = 3 and s = 2 (middlefigure), and n = 2 and s = 3 (right figure). Other parameters corresponds to those of Fig. 2.5, except forrp = 0.Figure 2.7: Nullclines p˙ = 0 (dashed line) and p˙ = 0 (dotted line) of the system of differential equationsgiven in Eq. (2.21) for linear maintenance cost of the fungus (parameter s = 1). When d is large enough,no coexistence steady state is observed (left figure) and fungal biomass increases unbounded. When d issmall enough, nullclines intersection results into an unstable steady state. Arrows represent qualitativelythe direction of the flow nearby the stable steady state (p∗,m∗). Parameter values are n = 2 and rp = 0.Other parameters corresponds to those of Fig. 2.7.2.3.2 Analysis of the one plant - multiple fungi systemPlant and fungal biomass as a function of the number of fungal species N Letus consider a system consisting of a plant (p) and N fungi (mi). I assume that all AMfungi have identical parameters. Then the non-dimensionalised model of Eq. (2.4) can be23rewritten asdPdτ=[( QD + P− 1) N∑i=1Mi − ηPP]P , (2.23a)dMidτ=[(1− 1D + P)P − ηMiMi]Mi . (2.23b)The coexistence steady state are solutions toM∗i =1ηMiP ∗(D + P ∗ − 1)D + P ∗, (2.24a)N∑i=1M∗i = ηPP∗ (D + P∗)Q−D + P ∗ . (2.24b)Since all Mi are identical, Eq. (2.24) becomesM∗i =1ηMiP ∗(D + P ∗ − 1)D + P ∗, (2.25a)M∗i =ηPNP ∗(D + P ∗)Q−D + P ∗ . (2.25b)Combining those two equations one obtains an expression for the plant steady statesP ∗ =Q+ 1±√(Q− 1)2 − 4QηP ηMiN2(1 +ηP ηMiN)−D . (2.26)One can observe from Eq. (2.26) that the positive root of P ∗ is an increasing function ofN , where the value of P ∗+ is bounded and approaches the asymptote Q−D (see Fig. 2.5)as N tends to infinity, i.e.,P ∗(N →∞)→ Q−D . (2.27)The value of the steady state M∗i is also increasing in N , whereM∗i =1ηMiP ∗(N)(1− 1D + P ∗(N))(2.28)andM∗i (N →∞)→Q−DηMiQ− 1Q. (2.29)Hence, the total fungal biomass tends to infinity for large N , i.e.,N∑i=1M∗i (N) ∼ NQ−DηMiQ− 1Q→∞ as N →∞. (2.30)24The results of Eq. (2.30) can be explained by the fact that the model presented in thischapter does not take competition between fungi into account. Indeed competition shouldset a limit to fungal biomass at large fungal density. A version of the model includingcompetition between fungi is presented in chapter 3.Conditions for the existence of coexistence steady state: In this section, I willderive conditions for the existence of coexistence steady states in a system consisting ofone plant (p) and multiple AM fungi (mi, where i = 1 .. N). Similarly to the previoussection, for simplicity, I consider the case where the intrinsic growth rate of the plant iszero (i.e. rp = 0) and plant and fungi have non-linear maintenance costs (n, s = 2). Iassume that different AM fungi have different resource exchange capacities αi and βi, anddifferent maintenance rate µmi . The system of ordinary differential equations representingthis situation is given bydpdt= qhp∑i(αimi)pd+ p− qcp∑i(βimi)p− µp p2, (2.31a)dmidt= qcmiβi pmi − qhmiαipd+ pmi − µmim2i , i = 1 .. N . (2.31b)I am interested at looking the steady state where there is coexistence of all species(p∗,m∗1, ...,m∗N), , where the mi are solutions tom∗i =p∗µmi(qcmβi − qhmαid+ p∗). (2.32)I non-dimensionalise Eq. (2.32) by using the same substitutions given in Table 2.4 formi and µmi . However, the non-dimensionalisation of p and d are expected to include theparameters of all fungi present. Hence I defineα¯ =1NN∑i=1αi , (2.33a)β¯ =1NN∑i=1βi . (2.33b)The quantities α¯ and β¯ correspond to the arithmetic mean of the resource exchangecapacities αi and βi of all AM fungi present. Eq. (2.33) can be used to non-dimensionaliseEq. (2.31a) by carrying on the substitutions illustrated in Table 2.5.Eq. (2.32) can therefore be rewritten asM∗i =1ηMiP ∗(biai− 1P ∗ +D), (2.34)25Table 2.5: Description of the non-dimensional parameters used to translate Eq. (2.31a) into Eq. (2.36).These parameters correspond to the ones defined in Table 2.4, where the quantities α and β have beensubstituted by α¯ and β¯ (see Eq. (2.33)).P = qcmβ¯qhmα¯p Mi =qcpβiqhmαimi τ = qhmα¯ tD = qcmβ¯qhmα¯d Q =qcmqhpqcpqhmηP =µpqcmβ¯ηMi =µmiqcpβiwhere ai = αi/α¯ and bi = βi/β¯.From Eq. (2.34) one can see that in order for the steady state M∗i to be in the firstquadrant, bi/ai−1/(P ∗+D) needs to be positive. One can therefore derive a first conditionfor the existence of a coexistence steady state, that isP ∗ +D >aibi. (2.35)By substituting Eq. (2.34) and the parameters of Table 2.5 into Eq. (2.31a), one obtainthat the value of the steady state P ∗ is a solution toN∑i=1(Qaibi− (P ∗ +D))(biai(P ∗ +D)− 1)aiηMi− ηP (P ∗ +D)2 = 0 . (2.36)There must therefore exist i0 ∈ 1, . . . , N such that Qai0/bi0 − D > 0, otherwise thepolynomial of Eq. (2.36) is always negative for P ∗ > 0. Hence a second condition for theexistence of a coexistence steady state is that there must exist an i0 such thatαi0 >qcpqhpdβi0 . (2.37)This condition corresponds to Eq. (2.6) of the one plant - one fungus case. One can seethat when multiple AM fungi are present, it is sufficient that one of the fungi satisfiesEq. (2.6). Combining the conditions given in Eqs. (2.35) and (2.37) one obtainsQai0bi0− (P ∗ +D) > Qai0bi0− ai0bi0=ai0bi0(Q− 1) > 0 . (2.38)Hence one obtain the unique condition Q > 1, or, in terms of the original parametersqcmqhpqhmqcp> 1 , (2.39)which corresponds to condition (2.7), stated for the one plant - one fungus case.26By solving Eq. (2.36) one obtains an expression for the steady state P ∗, that isP ∗ =(1 +Q)∑Ni=1aiηMi±√∆2(∑Ni=1biηMi+ ηP) −D , (2.40)with∆ = (1 +Q)2(N∑i=1aiηMi)2− 4Q(N∑i=1a2ibiηMi)(N∑i=1biηMi+ ηP)(2.41)One therefore obtains a third condition for the existence of a coexistence steady state,i.e.,(1 +qcmqhpqcpqhm)2(N∑i=1qcpαiβiµmi)2− 4qcmqhpqcpqhm(N∑i=1qcpα2iµmi)(N∑i=1qcpβ2iµmi+µpqcm)> 0 . (2.42)The expression of Eq. (2.42) involves the product and sum of the resource exchangecapacities αi and βi. By setting a = α/√µm = (α1/√µ1, . . . , αn/√µn) and b = β/√µ =(β1/√µ1, . . . , βn/√µn), condition (2.42) can be rewritten as:(1 +Q)2(1 +Cov(a,b)ab)2− 4Q(1 +V ar(a)a2)(1 +µpNqcmqcpb2 +V ar(b)b2)> 0 .(2.43)We can observe that condition (2.43) is more likely to be satisfied when the covariancebetween a and b and the means of a and b are large, and when the variances of a and bare small. Additionally, the condition is less likely to be satisfied when the maintenancecost of the fungi is large.In sum, we can see that conditions for the existence of a coexistence steady state there-fore depend on the resource exchange ability of the whole guild. Altogether, conditions(2.37), (2.39), and (2.43) suggest that as long as a good mutualist is present (αi large)and the average mutualist quality of all species is large enough, (i.e., α is large enoughand β is small enough) a coexistence steady state exists.Let us assume now that all fungi have the same resource exchange capacities α and βand the same maintenance cost µm, condition (2.43) reduces toµpµmβ2qcmqcpN=ηPηMiN<(1−Q)24Q, (2.44)which corresponds to condition (2.10) for the case where multiple AM fungi are present.This condition tells us that in the presence of multiple species, guild survival can beguaranteed even when maintenance costs are higher than what could be tolerated whenonly one fungal species is present.27Conditions for stability of the coexistence steady state: Let us consider a systemconsisting of one plant (p) and two AM fungal species (m1 and m2), differing only in theirresource exchange capacities αi and βi. We writedpdt= qhp(α1m1 + α2m2)pd+ p− qcp(β1m1 + β2m2)p− µp p2, (2.45a)dm1dt= qcmβ1 pm1 − qhmα1 pd+ pm1 − µmim21, (2.45b)dm2dt= qcmβ2 pm2 − qhmα2 pd+ pm2 − µmim22. (2.45c)I am interested at looking at the steady state (p∗,m∗1,m∗2). The Jacobian of Eq. (2.45)at steady state is given byJ(p∗,m∗1,m∗2) =−qhp(α1m∗1 + α2m∗2)p∗(d+ p∗)2− µpp∗︸ ︷︷ ︸:=−Aqhpα1p∗d+ p∗− qcpβ1p∗︸ ︷︷ ︸:=C1qhpα2p∗d+ p∗− qcpβ2p∗︸ ︷︷ ︸:=C2m∗1(qcmβ1 − qhmα1d+ p∗+ qhmα1p∗(d+ p∗)2)︸ ︷︷ ︸:=B1−µmm∗1 0m∗2(qcmβ2 − qhmα2d+ p∗+ qhmα2p∗(d+ p∗)2)︸ ︷︷ ︸:=B20 −µmm∗2.(2.46)We know that A > 0, B1 > 0, and B2 > 0 (see Eq. (2.32)). The sign of C1 andC2 is unknown, however we can notice that these terms increase for increasing αi anddecrease for increasing βi. We can therefore say that Ci (where i = 1, 2) increase forincreasing resource exchange capacity of the fungal species. The characteristic polynomialof Eq. (2.46) is given byP (λ) = λ3 + λ2 µm(m∗1 +m∗2) + A︸ ︷︷ ︸:=a1+λ (µ2mm∗1m∗2 + µm(m∗1 +m∗2)−B1C1 −B2C2)︸ ︷︷ ︸:=a2+Aµ2mm∗1m∗2 − µmm∗1B2C2 − µmm∗2B1C1 = 0︸ ︷︷ ︸:=a3.(2.47)According to the Routh-Hurwitz conditions for stability (Murray, 2007), in order forRe(λ) > 0 the following three conditions need to be satisfied(i) a1 > 0, (ii) a3 > 0, and (iii) a1a2 − a3 > 0 . (2.48)One can directly see that a1 > 0, i.e., (i) is always satisfied. Hence we obtain the following28two conditions for stability of the coexistence steady state(ii) Aµmm∗1m∗2 −m∗1B2C2 −m∗2B1C1 > 0 , (2.49a)(iii) µm(m∗1 +m∗2)(µ2mm∗1m∗2 + µm(m∗1 +m∗2) +A)−B1C1(A+ µmm∗1)−B2C2(A+ µmm∗2) > 0 .(2.49b)As long as conditions (2.49), together with the conditions for existence of the coexistencesteady state derived previously in (2.37), (2.39) and (2.43), a stable steady state of multi-species coexistence exists. Numerically, we observe that the conditions for stability arealways satisfied for the range of parameters for which a coexistence steady state exists.Numerical simulations and multi-species coexistence: When considering a plantassociating with N fungal species, either coexistence or extinction of all mutualists isobserved. And as long as (i) at least one of the fungal species provides enough nutrientsto the plant (i.e., the inequality αi > (qcpβid)/qhp of Eq. (2.6) is satisfied for at leastone species), and as long as (ii) plant and fungi are able to make effective use of theresource received (i.e., (qcmqhp)/(qcpqhm) > 1, Eq. (2.7)), the mutualism can establish andcoexistence is observed. Additionally, for mutualism establishment, it must hold(iii) (1 +Q)2(1 +Cov(α/√µm,β/√µm)(α/√µm)(β/√µm))2− 4Q(1 +V ar(α/√µm)(α/√µm)2)(1 +µpNqcmqcp(β/√µm)2+V ar(β/√µm)(β/√µm)2)> 0 , (2.50)where α = (α1, . . . , αN) and β = (β1, . . . , βN) represent respectively the ability of thewhole guild to transfer phosphorus and get carbon from the plant, µm = (µ1, . . . , µN)represents the maintenance cost of the guild, Cov(.) is the covariance, V ar(.) is thevariance, and α/µm and β/µm represent the mean of the individual resource exchangecapacities divided by the maintenance cost of each of the fungi present. (a full derivationof Eq. (2.50) is provided above, Eq. (2.43)). Condition (iii) is more likely to be satisfiedwhen the covariance between α and β and the means α and β are large, and whenthe variances V ar(α) and V ar(β) are small. Additionally, the condition is less likely tobe satisfied when the maintenance cost µm of the fungi is large. Altogether, conditions(i)-(iii) tell us that it is the overall exchange capacity of all mutualists that determineswhether coexistence or extinction of the whole guild will be observed. Plant associationwith a sufficiently beneficial mutualist can compensate for the presence of a minimallybeneficial species, or even for a species that does not provide any nutrients to the plant (i.e.αi = 0), as long as the plant receives on average enough nutrients to support its growth,and as long as guild members are similar enough in their ability to transfer phosphorusand get host carbon.29Fig. 2.8 shows typical results of simulation experiments where the carbon supplied by asingle plant is shared among fungi characterized by different resource exchange capacities.In particular, I investigate the two cases described in the methods: No host discrimination(Fig. 2.8a) and host discrimination (Fig. 2.8b). As expected from the model analysis,coexistence is observed whether or not the plant rewards more mutualistic fungi with morecarbon. The growth dynamics, however, differ. When all fungi have access to the sameamount of carbon independent of the phosphorus supplied, less mutualistic fungi reacha higher abundance than more mutualistic ones (Fig. 2.8a). Mutualists supplying morephosphorus and accessing more carbon reach a higher biomass than less mutualistic fungi(Fig. 2.8b). The presence of more mutualistic fungi increases plant biomass, and thereforeresource availability. If the plant is supported by enough high quality mutualists, thenless mutualistic species can exploit the plant’s resource and, indirectly, the more beneficialspecies. More insights into how plant and fungal growth depends on the resource exchangeability of the mutualists are provided below.Figure 2.8: Plant and fungal growth over time where (a) Fungi differ only in their phosphorus supplyto the plant and access to host carbon is the same for all species and (b) Fungi supplying the plant witha large/small amount of phosphorus (αi) have high/low access to host carbon (βi). In both scenariospecies 1 provides the greatest net benefit to the plant, and species 5 provides the lowest. The plant pis represented as a dashed line, while different fungal species mi are dotted lines, with the numbers closeto each curve indicating the species number. Numerical solutions are obtained by solving Eq. (2.1) fori = 1 .. 5 where (a) αi decreases from 0.62 to 0.38 in steps of size 0.08, while βi = 0.4 for all species. (b)αi increases from 0.40 to 0.48 in steps of size 0.02, for i = 1 .. 5 and βi decreases from 0.38 to 0.30 in stepsof size 0.02. Other parameters are qhp = 3, qcmi = 2, qcp = qhmi = 1, rp = 0.04, µp = 0.3, µm = 0.8,n = s = 2 and d = 1.2, for all species. Initial abundances are p(0) = 0.2, mi(0) = 0.02.302.3.3 The effect of differences in the resource exchange capaci-ties of the fungiAnalytical insights: I want to investigate how differences in the phosphorus exchangeability of the mutualists impact plant growth and the growth of other guild members. I willconsider a guild of mutualists having different phosphorus exchange abilities (parameterαi), while all other parameters are held constant across all fungi. I will first study how thevalue of the plant at steady state (p∗) depends on the phosphorus exchange ability of allspecies present (i.e., how p∗ depends on α = (α1, α2, . . . , αN)). This will give insights intohow conditions for mutualism establishment depend on the resource exchange capacityof the whole guild. For this purpose, I consider a plant (p) associating with multiplefungal species (mi), differing in their ability to transfer phosphorus (αi) to the plant, andcompare the biomass of the plant at steady state (p∗) when the plant is associating withN identical fungi (i.e., αi = α), to the biomass of the plant at steady state when the plantis associating with N fungi differing in their resource exchange capacities. When multiplespecies are present, we know from Eq. (2.40) that the plant steady state corresponds top∗(α) =qhmα¯qcmβP ∗(α)− dwith P ∗(α) =1 +Q+√(1 +Q)2 − 4Q 1N∑Ni=1 a2i(1 +ηPηMN)2(1 +ηPηMN) .(2.51)Considering that∑i αi = Nα and ai = N , the term∑Ni a2i can be written as1NN∑i=1a2i =1NN∑i=1(ai − 1N∑iai)2+(1N∑iai)2=V ar(α)α2+ 1 , (2.52)where V ar(α) is the variance of α, defined byV ar(α) =1NN∑i=1(αi − α)2 with α = 1NN∑i=1αi . (2.53)The value of the plant at steady state can be rewritten asp∗(α) =qhmα¯qcmβP ∗(α)− dwith P ∗(α) =1 +Q+√(1 +Q)2 − 4Q(1 +V ar(α)α2)(1 +ηPηMN)2(1 +ηPηMN) .(2.54)31where we require p∗ > 0 and(1 +Q)2 − 4Q(1 +V ar(α)α2)(1 +ηPηMN)> 0 . (2.55)Hence mutualism establishment and the final size of the plant p∗(α) depend on the meanas well as the variance of the phosphorus exchange capacities of all mutualists present.One can directly observe that p∗(α) decreases with respect to the variance. Additionally,if the AM fungi are identical (i.e., V ar(α) = 0), p∗ increases linearly with the value of α¯,andp∗(α) ≤ p∗((α, . . . , α)) ≤ p∗((max(αi), . . . ,max(αi))) for any α ∈ (0,∞)N . (2.56)Let us assume that a guild consist of N fungal species, where one of these speciesdoes not provide any benefit to the plant and can be regarded as a complete cheater(i.e., αi = 0) while the other (N − 1)-fungal species have the same phosphorus exchangecapacity α. In this case, the mean and variance of α can be expressed asα¯ = αNN − 1 and V ar(α) = α2N − 1N2. (2.57)By substituting Eq. (2.57) in (2.55) we obtain that mutualism establishment in the pres-ence of a cheater is possible as long asp∗(α) =qhmαNqcmβ(N − 1)P∗(α)− d > 0with P ∗(α) =1 +Q+√(1 +Q)2 − 4Q(NN − 1)(1 +ηPηMN)2(1 +ηPηMN) .(2.58)Condition (2.58), in combination with Eq. (2.37), tells us that mutualism establishmentin the presence of a complete cheater can be observed as long as the fungal speciespresent provide enough phosphorus to support plant growth. Condition (2.58) is morelikely to be satisfied in the presence of a higher number of fungal species.Impact on plant biomass: To better understand how the resource exchange capac-ity of the guild affects plant growth and the growth of other guild members, I considerthe case N = 2, where a plant is associating with two AM fungi. I want to comparethe biomass of the plant and the total fungal biomass at steady state when the plantis associating with two individuals that are good mutualists (species 1), with twoindividuals that are less beneficial mutualists (species 2), or with a combination of species321 and species 2. I assume that species 1 is a better mutualist with respect to species 2,with α1 > α2. I define α1 = α+  and α2 = α, where  ∈ (0,∞). In this case the varianceand mean of the resource exchange capacities can be computed asα = α +2and V ar(α) =24, (2.59)and the plant biomass at steady state is given byp∗(α, α + ) =qhm(α + /2)qcmβP ∗()− dwith P ∗() =1 +Q+√√√√(1 +Q)2 − 4Q(1 + ( /2α + /2)2)(1 +ηPηM2)2(1 +ηPηM2) .(2.60)From Eq. (2.56) we already know thatp∗(α, α + ) < p∗(α + /2, α + /2) < p∗(α + , α + ) . (2.61)Eq. (2.61) tells us that plant biomass at steady state is larger when the plant is associatingwith two individuals of species 1, then when the plant is associating with a combinationof species 1 and species 2.I want now to determine whether p∗(α, α) is larger or smaller then p∗(α, α+ ), i.e., Iwant to understand if plant biomass at steady state is larger when the plant is associatingwith two individuals that are less beneficial mutualists (species 2), or with a combinationof species 1 and species 2.I first consider the case where  is large. For →∞ we have,P ∗() ∼ 1 +Q+√(1 +Q)2 − 4Q (1 + ηPηM)− 4Q2(1 + ηP ηM2) = P ∗∞ > 0 ,and p∗(α, α + ) ∼ qhm(α + /2)qcmβP ∗∞ − d .(2.62)Thus for  large enough P ∗ has a lower bound, however p∗ increases with increasing . Ican conclude that Eq. (2.62) holds true for  large enough.Let us now consider the case where  is small. By using a Taylor expansion, we havethe following estimates for plant biomass at steady statep∗(α, α + ) = p∗(α, α) +qhm2qcmβP ∗(0)+O() (2.63)33One can conclude that also for small  it holdsp∗(α, α) < p∗(α, α + ) . (2.64)Thereforep∗(α, α) < p∗(α, α + ) < p∗(α + , α + ) for any  > 0 . (2.65)Eq. (2.65) states that the biomass of the plant at steady state is at highest when theplant is associating with a good mutualist (species 1), is at lowest when the plant isassociating with a less beneficial mutualist (species 2) and it is between these two valueswhen the plant is associating with a combination of good and less beneficial mutualists(species 1 and 2).Impact on fungal biomass: I consider now how differences in the phosphorus ex-change ability of the fungi in a guild affect fungal biomass. The fungal biomass of eachspecies at steady state is a function of the phosphorus exchange ability α and of plantbiomass at steady state, and it is given bym∗(αi,p∗(α)) =p∗(α)µm(qcmβ − qhmαid+ p∗(α)). (2.66)I want to understand if fungal biomass at steady state is increased or decreased inthe presence of a species that has different phosphorus exchange ability. I assume thatfungal species 1 has a higher phosphorus exchange ability that fungal species 2, withα1 = α +  and α2 = α. Let us first consider the fungal biomass of the more beneficialspecies (species 1) when the plant is associating with two good mutualists. Using therelationship of Eq. (2.54), i.e. p∗ = qhmαqcmβP ∗ − d, we obtainm∗(α+,p∗(α+,α+)) =p∗(α + , α + )µm(qcmβ − qhm(α + )d+ p∗(α + , α + ))=p∗(α + , α + )µm(qcmβ(d+ p∗(α + , α + ))− qhm(α + )d+ p∗(α + , α + ))=p∗(α + , α + )µm(qhm(α + )(P∗ − 1)d+ p∗(α + , α + )).(2.67)where P ∗ has been defined in Eq. (2.54) and does not depend on α if the fungal species areidentical, as in this case V ar(α) = 0. Similarly, the biomass of species 1 when the plantis associating with a combination of a good (species 1) and a less beneficial mutualist(species 2) can be expressed asm∗(α+,p∗(α,α+)) =p∗(α, α + )µm(qhm(α + )(P∗()− 1)− qhm 2P ∗()d+ p∗(α, α + )). (2.68)34where P ∗() is given in Eq. (2.60). As P ∗() < P ∗, and p∗(α, α + ) < p∗(α + , α + )(see Eq. 2.65), one can directly say thatm∗(α+,p∗(α,α+)) < m∗(α+,p∗(α+,α+)), (2.69)i.e., the fungal biomass of a good mutualist is decreased by the presence of a less beneficialmutualist.Let us first consider the fungal biomass of the less beneficial species (species 2) whenthe plant is associating with two individuals of species 2:m∗(α,p∗(α,α)) =p∗(α, α)µm(qhmα(P∗ − 1)d+ p∗(α, α)). (2.70)The fungal biomass of species 2 when the plant is associating with a combination of agood (species 1) and a less beneficial mutualist (species 2) is given bym∗(α,p∗(α,α+)) =p∗(α, α + )µm(qhmα(P∗()− 1) + qhm 2P ∗()d+ p∗(α, α + )). (2.71)I want to show thatm∗(α,p∗(α,α)) < m∗(α,p∗(α,α+)), (2.72)i.e., the fungal biomass of the less beneficial mutualist is increased by the presence of agood mutualist. We know that P ∗() → P ∗∞ for  large enough (Eq. (2.62)). Hence thevalue of P ∗() has a lower bound, however p∗(α, α + ) increases with increasing . I canconclude that Eq. (2.72) holds true for  large enough. If  is small, P ∗() can be expressedasP ∗() = P ∗(0)− Q2α2√(1 +Q)2 − 4Q(1 + ηP ηM2)2 +O(), (2.73)where P ∗(0) = P ∗. Hence, for  small, Eq. (2.71) can be rewritten asm∗(α,p∗(α,α+)) 'p∗(α, α + )µm(qhmα(P∗ − 1)d+ p∗(α, α + ))+p∗(α, α + )µm(d+ p∗(α, α + ))(− Q2α2√(1+Q)2−4Q(1+ ηP ηM2)2 + qhm2P ∗()).(2.74)Because p∗(α, α) < p∗(α, α + ) (see Eq. (2.64)), I can conclude that also for  smallenough it holdsm∗(α,p∗(α,α)) < m∗(α,p∗(α,α+)), (2.75)i.e., fungal biomass of a less beneficial fungus increases in the presence of a more bene-ficial species. Hence, in sum, the biomass of a fungus decreases in the presence of a lessbeneficial mutualist (Eq. (2.75)), and increases in the presence of a more beneficial mu-35tualist (Eq. (2.69)). These results are supported by numerical simulations, as illustratedin Fig. 2.9.Indirect parasitism and numerical simulations: The results of the analysis aboveare supported by numerical simulations. Fig. 2.9 compares plant and fungal growth whenthe plant is associated with a more mutualistic fungus (species 1), a less mutualisticfungus (species 2), or a combination of the two species. When a less mutualistic fungus(species 2) is considered in pairwise association with the plant, the growth rates and finalsizes of the plant and the fungus are lowest. When the plant is considered in pairwiseassociation with the more mutualistic fungus (species 1), growth rates and final sizes arehighest. If species 1 and 2 are both present, the growth rate and final size of the plantand species 1 are lowered, while the growth rate of species 2 is increased. The presenceof species 2 negatively affects the growth of species 1, while the presence of species 1positively affects the growth of species 2. Indirect interactions between species 1 and 2are therefore parasitic, with the less mutualistic fungus (species 2) exploiting the moremutualistic one (species 1). The larger the difference in net benefit provided by guildmembers, the stronger the indirect parasitic interactions.These indirect relationships are shown schematically in Fig. 2.10. The diagram qual-itatively represents the interactions of a plant and three fungal species differing in theirresource exchange capacities. Species 1 is assumed to be a better mutualist than species2 and 3, and species 2 is a better mutualist than species 3. Pairwise direct interactionsof the plant with each of the three AM fungal species are mutualistic, where the plantprovides equal access to the resource to all three fungi, and receives resource from eachspecies according to its resource exchange ability. Consequently, pairwise indirect interac-tions among fungal species are parasitic, since the less mutualistic species take advantageof the plant growth resulting from the benefit provided by the more mutualistic species.In other words, species 1 is exploited by 2 and 3, and species 2 is exploited by 3.The model predicts none of the interactions to be competitive, i.e., disadvantageousfor both parties. Instead, in the pairwise indirect interactions between fungal mutualists,one of the two always benefits. In terms of Fig. 2.10, each parallel set of arrows betweenfungi represents interactions that are parasitic, i.e. positive for the less mutualistic speciesand negative for the more mutualistic one.36Figure 2.9: Growth over time of a single plant associating with two species of AM fungi. Species 1is more mutualistic and provides the plant with more phosphorus than species 2 (α1 > α2). (a) Accessto host carbon is the same for both species (β1 = β2) or (b) Species 1 have access to more host carbonthan species 2 (β1 > β2). In both cases, the upper figures represent plant biomass over time whenthe mycorrhizal association is between a plant and two individuals of the more mutualistic species (thindashed curves, number 1), of the less mutualistic species (thin dashed curves, number 2) or a combinationof species 1 and species 2 (thick dashed curves, 12). The bottom figures represent fungal biomass overtime when the fungal species are considered in separate association with the plant (thin dotted curves) orgrown simultaneously on the same plant (thick dotted curves). Parameters are: α1 = 0.45 and α2 = 0.35,(a) β1 = 0.45 and β2 = 0.35, (b) β1 = β2 = 0.4. Other parameters correspond to those for Fig. 2.8.37Figure 2.10: Diagram representation of direct (solid arrows) and indirect (dotted arrows) interactionsbetween a plant and its associated mutualist guild. Species 1 is more mutualistic than species 2, andspecies 2 is more mutualistic than species 3. Arrows indicate the direction of each interaction, with thesender at the tail of the arrow and the receiver at the head. Green arrows represent positive effects (‘+’),while red arrows represent negative effects on growth (‘−’). Arrows thickness qualitatively represents theinteraction strength. A ‘+’/‘−’ indicates that the sender has a positive/negative effect on the growthof the receiver. A mutualistic interaction is one in which the two parallel interactions are denoted as(‘+’, ‘+’), while a parasitic interaction is one in which one of the parallel interactions is positive and theother is negative (‘+’,‘−’). A competitive interaction is one in which each species negatively impacts theother one, and so it would be labeled (‘−’,‘−’). Interactions between guild members are not competitive(‘−’,‘−’), but parasitic (‘+’,‘−’).2.4 Discussion2.4.1 Modelling of mutualistic interactionsI present a model that allows for stable coexistence of multiple multualists sharing aresource supplied by a single partner, in the absence of niche-based mechanisms, spatialstructure, or preferential allocation. In the model, multi-species mutualism establishmentdepends on the overall net benefit provided by the guild to the partner. As long asthe plant receives enough nutrients to support its growth, coexistence of all mutualistsis observed. Similarly, in public goods games, coexistence of more and less beneficialmutualists has been shown to occur as long as there are enough good mutualists toproduce the public good (Archetti and Scheuring, 2011, 2013). In my model, I obtainthe additional requirement that establishment is more likely if the variance across guildmembers is small, i.e., that the guild members are similar in mutualistic quality (seeEq. (2.43)).Previous ordinary differential equation models of mutualistic interactions have shownthat stable coexistence in mutualisms involving only two interacting species can be ob-tained by setting an upper limit to the benefit that each species can provide to the other(i.e., the gross benefit (GB) and cost (C) curves in Fig. 2.4 level off for sufficiently largebiomass) (Addicott, 1981, Holland and DeAngelis, 2006, 2009, 2010, Holland et al., 2002,2004, Ingvarsson and Lundberg, 1995, Kot, 2001, May, 1976, Wright, 1989). However,saturation in the benefit makes stable coexistence of multiple mutualists impossible, in the38absence of secondary mechanisms. Inevitably, one of the mutualists will be better thanthe others at obtaining resource from the partner species. Consequently, the biomass ofthis species will increase, leading to the consumption of an increasing proportion of theavailable resource, eventually driving the other mutualists to extinction. On the otherhand, symmetric non-saturating resource exchange (corresponding to linear gross benefit(GB) functional responses) results in unlimited growth of both, the mutualists and thepartner species (Gause and Witt, 1935, Vandermeer and Boucher, 1978). In my model,the benefit increases linearly with the biomass of the species providing the benefit, anddoes not saturate at high biomass of the species giving the benefit. Hence, coexistence ispossible. At the same time, asymmetry in the benefit and cost of mutualism (Eqs. (2.2)and (2.3)) sets a limit to plant growth which is reached when, for the plant, the benefitand cost of the mutualism equalize. The asymmetry, is due to the fact that low plantbiomass sets a limit to the amount of phosphorus provided by the fungus (see Eq. (2.2)),as fungi cannot transfer phosphorus if there is insufficient plant root available for coloni-sation. As plant biomass increases, the availability of root space ceases to be limiting,as shown in studies examining the proportion of root length that is colonised (Hart andReader, 2002, Klironomos and Hart, 2002).Very few studies have attempted to bridge theory and empiricism by experimentallytesting different mathematical forms of benefit and cost functional responses (Geib andGalen, 2012, Kang et al., 2011, Morales, 2011, Morris et al., 2010). As the asymmetry inthe gross benefit and cost functional responses (see Table 2.4 and Fig. 2.4) is an importantpart of the dynamics observed here, further empirical work is needed to determine whethermutualistic interactions can be characterised by the asymmetric resource exchange ratesassumed in the model. In principle, mutualisms other than the AM symbiosis can becharacterised by the same dynamics of multi-species mutualist coexistence, as long as thesame asymmetry in the benefit and cost of mutualism can be assumed.Mutualism-parasitism continuumExperimental work has shown that symbiotic associations are not always necessarily ben-eficial, but can, under certain circumstances, become parasitic or commensalistic. Forexample, the nature of the symbiosis may depend on the genotypes of the interactingspecies or on environmental conditions (Johnson et al., 1997, Lekberg and Koide, 2014).Here, I present a mechanism that causes interactions to change from mutualistic to para-sitic when there is a disproportion in plant and fungal biomass, where the more abundantspecies is exploited by the less abundant one before mutual growth can be observed. Ithas already been hypothesized that depletion in plant growth may be observed when thecarbon cost set by the fungi is not sufficiently compensated by phosphorus transfer (Lek-berg and Koide, 2014, Smith and Smith, 2012). This mechanism could explain a depletion39in plant growth when abundance of AM fungi is low, as observed by Smith and Smith(2012).2.4.2 Indirect parasitism among guild membersIn this chapter, I show that if the main form of interaction between guild members isindirect parasitism, competitive exclusion of more and less beneficial mutualists will notoccur. The presence of a more mutualistic species increases plant biomass, and thereforeresource availability, for other guild members. Each fungal mutualist exploits all of theother species that are more mutualistic fungi than itself, and at the same time is itselfexploited by each of the species that are less mutualistic. The most mutualistic speciesis exploited by all other species, and the least mutualistic one is not exploited by any.The more similar the net benefit provided by the guild members, the more negligible theindirect parasitism and the effect that the mutualist species have on each other’s biomass.Previous work has proposed that stable coexistence of multiple species sharing a lim-iting resource can emerge as a consequence of indirect interactions in multitrophic com-munities (McCann, 2000, Stanton, 2003). The presence of a third species may affect thepopulation size of a superior competitor through an interaction chain, e.g., by predation(Allesina and Levine, 2011, Caswell, 1978, Kerr et al., 2002, May, 1972), or it can alter theeffect that one species has on another through indirect interactions, e.g., by varying theamount of resource available (Abrams, 1983, Levine et al., 2017, Mayfield and Stouffer,2017, Schoener, 1974). In the model presented here, the amount of shared resource avail-able (i.e., plant biomass) is determined by its interaction with the associated mutualists.The fungal species are not directly competing for the resource they are sharing, becausethe amount of resource available increases with increasing fungal abundance.2.4.3 Experimental workExperiments studying variations in the abundance of fungal species grown simultaneouslyon the same plant are rare. Hart et al. (2013) found that the abundance of a less mutual-istic fungal species is increased by the addition of a better mutualist colonizing the samehost plant, and Argu¨ello et al. (2016) showed that a plant receives more phosphorus froma less mutualistic species in the presence of a better mutualist. Jansa et al. (2008) showedthat plant and fungal growth in the presence of multiple mutualists were mostly withinthe range of what was observed when the plant was grown separately with the most andless mutualistic species. All of these results are in agreement with model predictions.Other observations, however, show that plant growth is higher in the presence ofmultiple species of AM fungi than in the presence of any of the species separately (Alkanet al., 2006, Gianinazzi et al., 2010, Jansa et al., 2008, Maherali and Klironomos, 2007,Van der Heijden et al., 1998). This fact presents a contradiction with model predictions,40as plant biomass is expected to be maximized by the presence of a single, most beneficialspecies (e.g., species 1 in Fig. 2.9). The contraddiction could be explained by the fact thatin the model I assume phosphorus supply to be the only benefit provided to the plantby the fungi. Root colonization with AM fungi has also been shown to increase plantresistance to diseases (Jung et al., 2012), protection against parasites (Parniske, 2008),and plant tolerance to abiotic stress (Latef et al., 2016). The presence of fungal speciesproviding complementary benefits can therefore help to maximize plant growth (Afkhamiet al., 2014). Further monitoring of plant and fungal growth over time in monoculturesof AM fungi and in cultures containing different fungal species is needed for experimentalverification of the hypotheses presented here.2.4.4 Future directionsWhile much can be learned from the model presented here, there are certain features of thesystem that may be important to include in future modelling efforts. In particular, I donot distinguish between plant structure above or below ground, although plants are knownto differentially allocate carbon depending on plant size, environmental conditions or AMfungal abundance (Andersen and Rygiewicz, 1991, Rygiewicz and Andersen, 1994). Also,AM fungi have been shown to contribute to nutrient uptake only after having built a fungalnetwork inside the plant root cells (Dickson et al., 1999). There should therefore exista threshold of fungal biomass below which the fungi do not contribute to plant growth.Future models may regard plant and fungal biomass as a system of multiple coupleddifferential equations, to distinguish plant biomass above- and below- ground and intra-and extraradical fungal biomass. In my work, I also do not consider plant-soil feedback(Miki, 2012, Van der Putten et al., 2013). These types of feedback can indeed influencecoexistence patterns (Johnson and Bronstein, 2019). A simple study of the dynamicsof these feedbacks could be carried out with the model presented here by consideringvariations in the intrinsic growth rate of the plant (characterised by parameters rp andµp), as a function of different soils. My model could also be used as a building block inlarger community models which are at present conceptual (Van der Putten et al., 2013)or based on very simple mass action kinetics (Miki, 2012). Additionally, my model couldbe used to investigate how different plant-mutualist guild combinations can be more orless evolutionary successful, where the total mutualist quality of the guild depends on theresource exchange capacities of each mutualist in the guild. Such a study would thereforeoffer an opportunity to couple the physiological qualities of the fungal species present(i.e., their resource exchange capacities) with evolutionary processes (Lekberg and Koide,2014).It is important to note that the model presented in this chapter is not spatially explicit,and allows for an indefinite number of mutualists to be associated with the same host41plant. Space should become a limiting factor, however, at large fungal density (Engelmoeret al., 2014, Hepper et al., 1988, Koch et al., 2011, Mummey et al., 2009). Extension of themodel to explicitly consider competition between fungal species is presented in chapter 3.42Chapter 3Predicting the consequences of theintroduction of an alien mutualistinto a guild3.1 Background informationRecent decades have seen a drastic increase in biological invasion by alien organisms,for instance due to climate change, which allows species to extend their range to newareas, or to anthropogenic disturbance, which facilitates the transportation of speciesfrom one location to another (Hulme, 2009, Lockwood et al., 2013). These invasions haveled to significant ecological and economic damage throughout the world (Cardinale et al.,2012, Vitousek et al., 1997), and it has thus become increasingly important to betterunderstand the mechanisms behind the invasion process, in order to identify the causalfactors promoting or hindering invasion success (Lowry et al., 2013).Invasion is a three stage process (Williamson and Griffiths, 1996), involving (1) theintroduction of an alien species, (2) its local establishment (which may cause differentscales of biodiversity loss within the recipient community), and eventually (3) its regionalspread. Factors making biological invasion more likely to occur have been identified inthe recipient ecosystem (Lonsdale, 1999, Olyarnik et al., 2009), in the introduced species(Fournier et al., 2019, Kolar and Lodge, 2001), and in the interaction between the two(Lowry et al., 2013, Shea and Chesson, 2002). Studies agree that propagule pressure (i.e.the number of individuals introduced in a potential invasion site) plays a decisive rolein determining the establishment of an alien species (Lockwood et al., 2005, Simberloff,2009). Resource availability is also a main factor contributing to successful establishment(Davis et al., 2000, Sher and Hyatt, 1999), as well as the capacity of the alien speciesto make an effective use of the resources acquired (Funk and Vitousek, 2007, Pysˇek andRichardson, 2008). Resource availability is strongly influenced by abiotic factors (Dethier43and Hacker, 2005, Menke and Holway, 2006), by the interactions of the organisms and theenvironment (Byers, 2002, Williamson and Harrison, 2002), or by the direct and indirectinteractions between the alien species and the existing community (Olyarnik et al., 2009,Seabloom et al., 2003, White et al., 2006). Additionally, the establishment of the alienspecies can in turn have an impact on resource availability and affect the invasion dynamics(Didham et al., 2007, Goldstein and Suding, 2014). Thus, biological invasion is a complexprocess, and although individual ecological factors determining invasibility have beenidentified, it remains challenging to understand how their interplay determines invasionsuccess (Byun et al., 2015, Gerhardt and Collinge, 2007, Gurevitch et al., 2011).Ecological modeling can illuminate the investigation of mechanisms driving invasion,as models can disentangle the roles played by different factors in the invasion process (Huiand Richardson, 2017, Lewis et al., 2016). Numerous mathematical approaches and simu-lation model have been developed to analyse the processes involved in biological invasions(Lonsdale, 1999, Olyarnik et al., 2009). Many results focus on the spreading speed of theinvasion (Lewis et al., 2016) (step (3) of the invasion process), while establishment successhas received little attention. In particular, the invasibility of mutualistic communities byan alien species has remained under-explored theoretically (Bray et al., 2019, Minoariveloand Hui, 2016a,b, Traveset and Richardson, 2014).In nature, mutualistic interactions often involve multiple species of mutualists (the mu-tualist guild) sharing a resource supplied by one or more partner (or host) species (Herreet al., 1999, Hoeksema and Bruna, 2000, Palmer et al., 2003). A single plant, for exam-ple, can associate with dozens of species of beneficial mycorrhizal fungi that coexist in theplant roots and surrounding soil and compete between each other for access to the plantresource (Douds Jr and Millner, 1999, Verbruggen et al., 2012). Similarly, different com-peting animal pollinators can coexist on the same floral resources (Button and Elle, 2014,Sardin˜as and Kremen, 2015). Understanding of the mechanisms behind the persistenceof diversity among mutualists and the factors that can threaten this stability has becomecrucial not only from an ecological point of view, but also for our socio-economical wellbe-ing (Bronstein, 2015). Indeed pollination and below-ground mutualisms, two of the mostwidespread earth-mutualisms, are essential drivers of agricultural productivity (Hannaet al., 2013, Ings et al., 2010, Menz et al., 2011). Two recent meta-analyses of experi-mental studies of plant-pollinator relationships (Charlebois and Sargent, 2017, Moralesand Traveset, 2009), underline how challenging it is to untangle these relationships withexperimental tools alone.In this chapter, I will use a deterministic ordinary differential equation model to studythe second stage of the invasion process, i.e. alien species establishment, in a communityof mutualists sharing a resource supplied by the same host. Additionally, I will explorethe ability of the guild to prevent establishment of the alien species. I am interested in thecontext wherein a native community is threatened with invasion by a foreign species. I thus44use the terms ‘mutualist guild, ‘native community’, and ‘established community’ to referto the assemblage of species that was initially present (not necessarily at equilibrium). Iuse the terms ‘alien mutualist’, ‘introduced species’, and ‘introduced mutualist’ to indicatean organism that can associate with the same partner as the existing guild members. Iassume that the alien species is subjected to the same environmental pressures, but thatit might differ in its mutualist quality and/or competitive ability. Note that my modelincludes direct competition between the mutualists present in the native community, aswell as competition with the alien species when it is introduced.More specifically, I will study the relative effect of (i) invasor traits, such as the abilityto exchange resources with the plant, (ii) propagule pressure, and (iii) composition ofthe recipient community, in predicting the growth dynamics of the introduced mutualist.Finally, I will discuss the impact of the introduction of an alien species on the diversityof the native guild and on resource availability, i.e. the biomass of the partner species. Inmutualistic communities, partner species biomass is often associated with productivity (forexample in terms of crop yield) and used as one of the measures of ecosystem functioning(Xu and Mage, 2001), and so the same metric will be used in my analysis.I show that the presence of multiple species in a guild promotes species coexistence andreduces the risk of invasion by an alien mutualist. My case study will be the mutualisticinteractions between arbuscular mycorrhizal (AM) fungi and their host plant (Smith andRead, 2010), although my conclusions apply to a broad range of mutualisms.3.2 Model and methods3.2.1 Formulating a model for competing mutualistsTo study the dynamics of a guild of competing mutualists sharing a resource supplied bythe same partner, I use the model presented in chapter 2, simulating the mutual growthof a plant and its associated mutualist guild of AM fungi. The only interactions directlymodeled in the original system are mutualistic, and so in this chapter I consider the effectof adding direct competition between mutualists. This new model therefore includes bothmutualistic (plant to fungi) and competitive (fungi to fungi) interactions.Mutualists compete for access to a common resource, whose abundance in the modeluniquely corresponds to the biomass of the partner species. Resource availability is in turnaffected by the benefit that the partner species receives from its associated mutualists,and hence by the guild composition (Bever, 1999, 2002). The guild and resource dynamicsare therefore intrinsically coupled. I will first investigate the effect of competition betweenmutualists on the guild dynamics. I will then investigate the response of the guild to aninvader.To derive the model, I use the context of the mycorrhizal symbiosis. AM fungi facilitate45the plant’s absorption of nutrients (phosphorus in particular) that are limiting to plantgrowth (Jeffries et al., 2003, Smith and Read, 2010). In exchange, the plant provides fixedcarbon to the fungi (Smith and Read, 2010). The model equations describe the evolutionin time of the biomass of the plant (p) and the biomass of a guild of fungal species(m1,m2, ...,mN) as a function of the exchange of these critical nutrients. Mathematically,the model is written as the following system of ordinary differential equations:dpdt=plantfitness︷ ︸︸ ︷rp(p) +∑j[ phosphorusreceived︷ ︸︸ ︷αj fhp(p,mj)−carbonsupplied︷ ︸︸ ︷βj fcp(p,mj)C(mj,mi)], (3.1a)dmjdt= rmj(mj)︸ ︷︷ ︸fungalmaintenance+ βjfcmj(p,mj)C(mj,mi)︸ ︷︷ ︸competition︸ ︷︷ ︸carbonreceived−αjfhmj(p,mj)︸ ︷︷ ︸phosphorussupplied, j = 1..N. (3.1b)The components of the model are explained below.Mutualistic interactions between the plant and the AM fungi: In the absence ofAM fungi, plant fitness is given by the function rp(p), which takes into account the intrinsicgrowth of the plant as well as maintenance costs, such as respiration, or energy costsrelated to nutrient absorption. I assume that plant growth in the absence of the fungi islogistic. In the presence of fungi, the plant receives phosphorus from each fungal mutualist(fhp(p,mj)), and supplies carbon in return (fcp(p,mj)). Fungal biomass increases due tothe carbon received by the plant (fcmj(p,mj)), and decreases due to the phosphorussupplied to the plant (fhmj(p,mj)), as well as due to costs related to the maintenanceof the existing fungal biomass (rmj(mj)). AM fungi are obligate mutualists and can notsurvive in the absence of a host plant, therefore no intrinsic growth term is present in theequation describing fungal growth. I assume that fungal exchange ability for phosphorus(αj) and carbon (βj) may differ among species. The complete forms of the r(·, ·) andf(·, ·) functions are given in Eq. (3.6).The choice of the functional forms of the f(·, ·) functions, describing nutrients transfer,is tied to the biology and fully explained in chapter 2 (see Eqs. (2.2) and (2.3)). Herewe present a brief summary. Phosphorus transfer is proportional to fungal and plantbiomass, when plant biomass is small, and to fungal biomass only, when plant biomass islarge enough. We writefhp(p,mj), fhmj(p,mj) ∝ mjpd+ p. (3.2)Carbon transfer is proportional to both, plant and fungal biomass. Hencefcmj(p,mj), fcp(p,mj) ∝ mj p . (3.3)46Competitive interactions between AM fungi: Competition between fungal speciesreduces the amount of carbon received/supplied in a way that depends on the specificcommunity composition, whereC(mj,mi) = 1−∑i 6=jmiaj +∑i 6=jmi. (3.4)When only one fungal species is present, C(mj,mi) is equal to 1 and Eq. (3.1) reducesto the original model of chapter 2. When two or more species are present, competitionbetween fungi reduces the carbon uptake capacity of each of the fungal species. Thevalue of aj determines how the presence of other fungal mutualists in the communityinfluences the carbon uptake capacity, and therefore the growth, of species j. If aj is large,competition between species j and other fungal species in the community is minimal. Ifaj is small, competition between species j and the rest of the fungal community is strong,and the presence of other mutualists can significantly limit the growth of species j.The literature shows that competition between fungi, for example for access to plantroots, can limit fungal growth in a way that depends on both the species present andtheir abundance (Engelmoer et al., 2014, Hepper et al., 1988, Mummey et al., 2009). Todetermine aj, I assume that each of the fungal species present has a direct negative effecton the growth rate of other fungi that depends on its identity, determining the strengthof the competitive interaction between those two species, and depending on the massproportion occupied by the competitor. Hence, aj is defined as the mean competitionstrength experienced by species j, where the competitive strength of each paired interac-tion is weighted by the proportion, in terms of biomass, that each species occupies withinthe competing community. Mathematically:aj =∑i 6=jaijmi∑i 6=jmi. (3.5)The aij parameter determines how much the biomass of species i affects the carbonuptake capacity of species j. Competition between species i and j constitutes tworeciprocal interactions quantified by aij and aji. If aij and aji are large, only a largequantity of species i influences the growth of species j (and viceversa), and competitionbetween species i and j is minimal. This situation corresponds for example to the casewhere fungal species are functionally complementary, or have low niche overlap, and tendtherefore to compete only weakly between each other. For small aij and aji, competitionbetween species i and j is strong, and the two AM fungal species can be regarded asbeing functionally similar, or as having a large niche overlap. If aij is large and aji issmall, species i has a strong negative influence on the growth of species j, but j haslittle or no influence on the growth of i. Species i is therefore a stronger competitor withrespect to species j.473.2.2 Investigating the effect of competition on mutualist guilddynamicsI begin by studying the mutualist-competitive system in comparison to the uniquely mu-tualistic system studied in chapter 2. In particular I will determine whether competitionbetween mutualists promotes or reduces guild diversity. I will consider multiple scenarios(see section 3.3.1 and Table 3.1 for a summary), but only the most ecologically relevantcases will be discussed. These cases fall into two categories, described below.Mutualism and coexistence among competitorsI first consider the dynamics of a plant in association with two fungal species. I theninvestigate the dynamics of a plant in association with three fungal species. I am partic-ularly interested in the scenario where two of the fungal species are strong competitorsand the third species is a weak competitor (see Fig. 3.5).Alien species introduction and invasion successI will investigate how invasion success of an alien mutualist introduced inside the commu-nity is affected by (i) invasor traits, such as competition strength and resource exchangeability, (ii) propagule pressure, expressed in terms of initial biomass of the introducedspecies, and (iii) composition of the recipient community, in terms of diversity and initialbiomass.The effect of competition will be studied by varying parameters acw and awc, repre-senting the strength of the competitive interactions between the existing guild (mwj) andthe potential invasor (mc). The introduced species, either a weak or a strong competitor,will be added to an existing guild of coexisting mutualists, as shown in the diagrams ofFig. 3.6. The effect of propagule pressure will be modelled by varying the initial biomassof the introduced species mc. Additionally, I will discuss how the mutualist quality of theintroduced species can affect the invasion dynamics, by varying the ratio of phosphorus(parameter α) to carbon (parameter β) exchange capacities of mc. I will refer to ‘cheaters’to species that have a low phosphorus to carbon ratio (α/β). Such species provide lowphosphorus in return for the carbon received.3.2.3 Investigating the impact of invasion on resource availabil-ityI am interested in understanding the impact of alien species invasion on the biomass ofthe partner species. Resource availability, in this case corresponding to plant biomass, is48directly affected by guild composition. I will determine how plant growth varies (in termsof growth rate and final biomass reached) as a consequence of the introduction of a newspecies (mc) in a guild of weakly competing mutualists (mwj). Eq. (3.1) will be solvednumerically to simulate plant growth over time for different combinations of the initialbiomass of the introduced species (propagule pressure) and of the existing community.3.3 Results3.3.1 Analytical criteria for stability of the mutualist guildIn this section I present the analysis of the model given in Eq. (3.1), to understand whatare the conditions promoting coexistence or competitive exclusion among a communityof mutualists. The whole model of Eq. (3.1) can be written asdpdt=Intrinsicgrowth︷ ︸︸ ︷qhprpp+qhpphosphorus received︷ ︸︸ ︷(∑jαjmj)pd+ p−qcpcarbon supplied︷ ︸︸ ︷(∑jβjmj[1−∑i 6=jmiaj +∑i6=jmi])p−maintenance︷︸︸︷µpp2 , (3.6a)dmjdt= qcmj βjpmj[1−∑i 6=jmiaj +∑i 6=jmi]︸ ︷︷ ︸Reduction in carbon uptakedue to competition︸ ︷︷ ︸carbon received−qhmj αjpd+ pmj︸ ︷︷ ︸phosphorus supplied− µmjm2j︸ ︷︷ ︸maintenance. (3.6b)Eq. (3.6) resembles the model developed in chapter 2, except for the term indicating thereduction in carbon uptake due to competition of a fungal species with the surroundingcommunity. When only one fungal species is present, Eq. (3.1) reduces to the originalmodel of chapter 2. When two or more species are present, competition among fungireduces the carbon uptake capacity of each of the fungal species.Parameters αj and βj are respectively the phosphorus and carbon exchange abilityof species j, µp and µmj are the biomass dependent maintenance rates of the plant andof fungal species j, the q−parameters are conversion factors for the amount of resource(phosphorus h or carbon c) into plant biomass (p) or biomass of fungal species j (mj), rpis the intrinsic growth rate of the plant, and d is the half-saturation constant. The valueof aj is defined in Eq. (3.5) and determines how the presence of other fungal mutualists inthe community influences the carbon uptake capacity, and therefore the growth, of speciesj.A linear analysis will be conducted on the 2 fungal species case (j = 2), on the 3species case (j = 3) and on a (N + 1)−species case, where a new species of AM fungiis introduced in a community of N coexisting mutualists. These scenarios correspond tocases A-C of Table 3.1. I am interested in the situation in which the establishment of theplant-fungi mutualism is possible. The conditions for the establishment of the mutualism49correspond to Eqs. (2.37), (2.39) and (2.43) derived from the analysis of chapter 2.In order to give more insights on the effect of the competition, I will explore numericallythe impact of the competition parameters on different characteristic of the mutalist guild,such as total AM fungal biomass, productivity of the plant, and ratio of plant biomassand AM fungal biomass. Numerical computations have been performed with the built-inode45 solver of the software Matlab R2017a.Case A: One plant and 2 AM fungal speciesI consider a system of a plant and 2 fungal mutualists m1 and m2. Note that when onlytwo AM fungal species are present, the value of a1 and a2 reduce to the constants a21and a12 respectively (see Eq. (3.5)). These parameters are a measure for the amount ofbiomass of species 1 that limits the carbon absorption of species 2, and viceversa, and areexpected to be large when competition between the two fungal species is weak, and smallif competition between the two species is strong.To facilitate computations I assume that the plant biomass has reached steady state(p∗). The value of the plant steady state may assume different values depending on thespecific fungal community composition one is looking at. With no loss of generality, I willuse the single symbol p∗ to refer to all these values, without distinction. I further assumethat the two fungal species have identical parameters α, β, qcm, qhm and µm. Eq. (3.6)can be rewritten asdm1dt= qcmβp∗m1a21a21 +m2− qhmα p∗d+ p∗m1 − µmm21 ,dm2dt= qcmβp∗m2a12a12 +m1− qhmα p∗d+ p∗m2 − µmm22 .(3.7a)(3.7b)I assess the dynamics of Eq. (3.7) through linear stability analysis and phase-planeportraits (Case, 1999, Edelstein-Keshet, 1988). Nullclines can be found by settingEqs. (3.7a) and (3.7b) to zero and solving for m1 and m2. One obtainsdm1dt= 0 ⇐⇒ m1 = 0 or m2 = f1(m1) = a21(qcmβp∗qhmαp∗d+p∗ + µmm1− 1), (3.8a)dm2dt= 0 ⇐⇒ m2 = 0 or m2 = f2(m1) = 1µm(qcmβp∗ a12a12 +m1− qhmα p∗d+ p∗). (3.8b)A graphical representation of the nullclines is shown in Fig. 3.1. The first equationsin (3.8a) and in (3.8b) correspond to the vertical and horizontal axes in the phase-plane.The nullclines f1(m1) and f2(m1) are hyperbolae, with vertical asymptote in the negativeregion of the phase-plane. As long as p is large enough, both curves intersect the positivehorizontal and vertical axes at one point, and the competitive exclusion steady states50(m∗1, 0) and (0,m∗2) do always exist (provided, as mentioned above, that conditions (2.37),(2.39) and (2.43) are satisfied and the plant-fungi mutualism can establish).In order for a coexistence steady state to exist, f1(m1) and f2(m1) must intersect. Thiscondition is satisfied if f1(m1) intersect the vertical axis at a larger value than f2(m1) andthe horizontal axis at a lower value (and viceversa). This condition translates into theequationsf1(0) > f2(0) & a < b , where f1(m1=a) = f2(m1=b) = 0 , or (3.9a)f1(0) < f2(0) & a > b , where f1(m1=a) = f2(m1=b) = 0 . (3.9b)Conditions (3.9) simplify intoa12, a21 >qhm p∗ α(d+ p∗)µmor (3.10a)a12, a21 <qhm p∗ α(d+ p∗)µm. (3.10b)Equations (3.10) indicate that in order for a coexistence steady state to exist, competi-tion parameters a12 and a21 need to be either both small or both large. In the symmetriccase a12 = a21, one of the conditions in Eq. (3.10) is always satisfied and the steady state(m∗1,m∗2) does always exist. Note that mathematical equality in Eq. (3.10) correspondsto the case in which the two nullclines exactly overlap and each point on the nullcline isa stable equilibrium. When competition among two species is asymmetric and conditions(3.10) are not satisfied, only the competitive exclusion steady states (m∗1, 0) and (0,m∗2)exist.Stability of the exclusion steady states (m∗1, 0) and (0,m∗2):To understand under which conditions the coexistence and competitive exclusionsteady states are stable, I linearize the model around the steady states. I start by con-sidering the competitive exclusion steady state (m∗1, 0). The Jacobian around the steadystate (m∗1, 0) is given byJ(m∗1,0) =−µmm∗1 − qcm β p∗m∗1a210 qcm β p∗ a12a12+m∗1− qhm α p∗d+p∗ , (3.11)51with eigenvaluesλ1 = −µmm∗1 , (3.12a)λ2 = qcm β p∗ a12a12 +m∗1− qhm α p∗d+ p∗. (3.12b)In order for the (m∗1, 0) to be stable both eigenvalues should be negative. One can directlysee that λ1 < 0. The eigenvalue λ2 is negative ifa12a12 +m∗1<qhmαqcmβ(d+ p∗). (3.13)The values of the competitive exclusion steady state m∗1 and of p∗ do not dependon a12, hence one can directly conclude that condition (3.13) is satisfied if a12 is smallenough. That is, if m1 is a strong competitor, the competitive exclusion steady state(m∗1, 0) is locally stable and species 1 and species 2 will not coexist. The same calculationcan be applied to (0,m∗2) to show that if m2 is a strong competitor, m1 can be drivento extinction. In sum, if a12 and a21 are large enough, the competitive exclusion steadystates are unstable, while if competition between species 1 and 2 is strong, both steadystates (m∗1, 0) and (0,m∗2) are stable. If competition is asymmetric, e.g. a12 is small whilea21 is large, then only the steady state (m∗1, 0), corresponding to competitive exclusion ofthe weak competitor (in this case species 2), is stable.An important factor determining which species is competitively superior whencompetition between species 1 and 2 is strong, is the initial biomass. Fungal parameterscan also shape the size of the basin of attraction of one species or the other. For theanalysis I assumed species 1 and 2 to have identical parameters. If fungal species differby other traits, such as phosphorus absorption capacity (αi), carbon sink strength (βi),nutrient conversion efficiency (qcmi and qhmi) or maintenance cost (µm), the basin ofattraction of the species with the higher resource acquisition efficiency will increase insize, while the basin of attraction of the less competitive species will decrease.Stability of coexistence steady state (m∗1,m∗2):I want to derive the conditions leading to stability of the coexistence steady state(m∗1,m∗2). For this purpose I compute the JacobianJ(m∗1,m∗2) =−µmm∗1 − qcmβp∗a21m∗1(m∗2+a21)2− qcmβp∗a12m∗2(m∗1+a12)2−µmm∗2 . (3.14)The steady state (m∗1,m∗2) is stable if Tr(J)< 0 and det(J) > 0. The trace of the Jacobian52in (3.14) is always negative. The stability of the coexistence steady state depends thereforeon the sign of the determinant of the Jacobian in Eq. (3.14). By setting det(J) > 0 andsolving for a12 and a21 one obtains the following condition for stability of the coexistencesteady state:a12 a21(a12 +m∗1)2 (a21 +m∗2)2<(µmqcmβp∗)2. (3.15)As the values of fungal and plant biomass at steady state are coupled, it is unclear howvariation in parameters a12 and a21 affects the values of m∗1, m∗2 and p∗. For large a’sthe dynamics approaches the zero competition case discussed in chapter 2. One expectstherefore that for a12 and a21 →∞ the values of m∗1, m∗2 and p∗ tends to those obtainedin the absence of species 2 and 1 respectively. Fig. 3.2 shows a graphical representationof the dependence of the plant and steady state on the competition parameter a. As thevalue of the steady states are bounded, and so is the ratio m∗/p∗, one can conclude thatif a21 and a12 are large enough, i.e. competitive interactions between m1 and m2 areweak, Eq. (3.15) is satisfied and the coexistence steady state is stable. There may existmultiple coexistence steady states, however as long as the competitive exclusion steadystate is unstable, one can assume that coexistence should occur.In sum, coexistence of two mutualists sharing a resource supplied by the same plantis observed if competition is weak, while competitive exclusion occurs when competitionbetween the two species is strong. When both species are strong competitors, either theone or the other survives, depending on their competitive ability. When only one of thetwo species is a strong competitor, the weak competitor is driven to extinction. Table 3.1summarizes this information. This scenario is similar to the output of the classic Lotkaand Volterra model of competition (Volterra, 1928), which predicts coexistence of weakcompetitors and competitive exclusion of strong competitors.Case B: One plant and 3 AM fungal speciesI consider a system of a plant and 3 fungal mutualists m1, m2 and m3. This systemcorresponds to the model given by Eq. (3.1) for j = 3. In order to capture the stabilityof the different steady states (coexistence or exclusion states) I assume parameters of allthree species to be identical. I further assume that plant biomass has reached steadystate, and is therefore constant. Again, the value of p∗ may be different depending on thespecific fungal composition we are looking at. With no loss of generality, I will use the53symbol p∗ for all these values. Eq. (3.1) can be rewritten as follow:dm1dt= qcmβp∗m1[a21m2 + a31m3a21m2 + a31m3 + (m2 +m3)2]︸ ︷︷ ︸f(m2,m3)−qhmα p∗d+ p∗m1 − µmm21 ,dm2dt= qcmβp∗m2[a12m1 + a32m3a12m1 + a32m3 + (m1 +m3)2]− qhmα p∗d+ p∗m2 − µmm22 ,dm3dt= qcmβp∗m3[a13m1 + a23m2a13m1 + a23m2 + (m1 +m2)2]− qhmα p∗d+ p∗m3 − µmm23 .(3.16a)(3.16b)(3.16c)Stability of the exclusion states:I investigate the stability of Eq. (3.16) by linearizing the model around the competitiveexclusion steady state (m∗1, 0, 0). The Jacobian of the linearized model corresponds to thematrixJ(m∗1,0,0) =µmm∗1 qcmβp∗m∗1∂f(m2,m3)∂m2∣∣(0,0)qcmβp∗m∗1∂f(m2,m3)∂m3∣∣(0,0)0 qcmβp∗ a12a12+m∗1− qhmα p∗d+p∗ 00 0 qcmβp∗ a13a13+m∗1− qhmα pd+p∗. (3.17)Eigenvalues of (3.17) areλ1 = −µmm∗1 , (3.18a)λ2 = qcmβp∗ a12a12 +m∗1− qhmα p∗d+ p∗, (3.18b)λ3 = qcmβp∗ a13a13 +m∗1− qhmα p∗d+ p∗. (3.18c)For stability, one require all eigenvalues to be negative. Clearly, λ1 < 0. Eigenvalues λ2and λ3 can assume positive or negative values, depending on the competition parametersa12 and a13. By assuming λ2 < 0 and λ3 < 0 one find the following two conditions forstability:a12a12 +m∗1<qhmαqcmβ(p∗ + d)& (3.19a)a13a13 +m∗1<qhmαqcmβ(p∗ + d). (3.19b)Eq. (3.19) states that as long as m1 is a strong competitor, i.e. a12 and a13 are smallenough, the steady state (m∗1, 0, 0) is locally stable and competitive exclusion will occur.54If a12 and a13 are large enough, the competitive exclusion steady state is unstable. Thesame calculation can be applied the steady states representing competitive exclusionof species 2 or 3, i.e. (0,m∗2, 0) and (0, 0,m∗3). If m2 (or m3) is a strong competitor,m1 and m3 (respectively m1 and m2) can be driven to extinction. One can concludethat if m1, m2 and m3 are all strong competitors, the species with higher biomass willcompetitively exclude the others. If only m2 and m3 are strong competitors, while m1is a weak competitor, than m2 or m3 will competitively exclude the other. If fungalspecies have different fungal parameters α, β, qcm, qhm or µm, the basin of attraction ofthe most competitive species will increase in size, while the basin of attraction of the lesscompetitive species will decrease. This situation correspond to cases B(ii) and B(v) ofTable 3.1.Stability of the steady state including 2 AM fungal species:If fungal m1, m2 and m3 are weak competitors, the competitive exclusion steady statesare unstable and coexistence can occur. To understand whether coexistence of all fungalspecies or of only two fungal species is observed, I investigate the stability of the steadystate (0,m∗2,m∗3). I linearize the model around (0,m∗2,m∗3) and compute the correspondingJacobian:J(0,m∗2 ,m∗3) =qcmβp∗ a21m∗2+a31m∗3a21m∗2+a31m∗3+(m∗2+m∗3)2 − qhmα p∗d+p∗ 0 0qcmβp∗m∗2a12−2a32(a32+m∗3)2−µmm∗2 −qcmβp∗m∗2 a32(a32+m∗3)2qcmβp∗m∗3a13−2a23(a23+m∗2)2−qcmβp∗m∗3 a23(a23+m∗2)2 −µmm∗3.(3.20)Eigenvalues of (3.20) are found by solving(qcmβp∗ a21m∗2 + a31m∗3a21m∗2 + a31m∗3 + (m∗2 +m∗3)2− qhmα p∗d+ p∗− λ)×[(−µmm∗2 − λ)(−µmm∗3 − λ)− (qcmβp∗)2m∗2m∗3a23 a32(a23 +m∗2)2(a32 +m∗3)2]= 0 .(3.21)For stability, one requires λ1, λ2 and λ3 to be all negative. Looking at the first factorof Eq. (3.21), one obtains a first condition for λ1 < 0, that isa21m∗2 + a31m∗3a21m∗2 + a31m∗3 + (m∗2 +m∗3)2<qhmαqcmβ(d+ p∗). (3.22)The values of m∗2 and m∗3 do not depend on parameters a21 and a31, as the biomass of55m1 at steady state is zero. Therefore one can directly conclude that for a21 and a31 smallenough, condition (3.22) is satisfied and λ1 < 0 . Hence, competition between m1 and m2needs to be strong enough in order for the steady state (0,m∗2,m∗3) to be locally stable.To find eigenvalues λ2 and λ3 I consider the second factor of Eq. (3.21). I expand thebracket and obtainλ2 + (µmm∗2 + µmm∗3)︸ ︷︷ ︸:=Bλ+ µ2mm∗2m∗3 −(qcmβp∗)2m∗2m∗3a23 a32(a23 +m∗2)2(a32 +m∗3)2︸ ︷︷ ︸:=C= 0 . (3.23)Eq. (3.23) is a quadratic with B > 0. As long as C > 0, the solutions to Eq. (3.23),corresponding to the eigenvalues λ2 and λ3, have negative real part. I find therefore thefollowing condition for stability of the steady state (0,m∗2,m∗3):a23 a32(a23 +m∗2)2(a32 +m∗3)2<(µmqcmβp∗)2. (3.24)Equation (3.24) resembles the condition of Eq. (3.15) derived for the 2 species case. Asdiscussed above, when a gets large, i.e. competition among mutualists decreases, I expectthe values of m∗2, m∗3 and p∗ to tend to the steady states of the corresponding non-competitive model of chapter 2. Because the values of p∗, m∗2 and m∗3 are bounded (seeFig. 3.2), and so is p∗/m∗ (see Fig. 3.3), condition (3.24) is satisfied for a23 and a32 largeenough, i.e. competition between m2 and m3 is weak.One can conclude that in order for the steady state (0,m∗2,m∗3) to be locally stable,both conditions (3.22) and (3.24) need to be satisfied. These conditions state thatcompetition of m2 toward m1 and of m3 toward m1 needs to be strong, while competitionbetween m2 and m3 needs to be weak.In sum, the steady state (m∗1, 0, 0) is locally stable when m1 is a strong competitorand the steady state (0,m∗2,m∗3) is locally stable when competition of m2 and m3 towardm1 is strong, while competition between m∗2 and m∗3 is weak. When m1 competes stronglywith m2 and m3 but m2 and m3 compete weakly with m1, only the steady state (m∗1, 0, 0)is stable and m2 and m3 will be driven to extinction by m1. These scenario correspond tocases B(iii) and (vi) of Table 3.1. When neither the (m∗1, 0, 0) nor the (0,m∗2,m∗3) steadystates are stable (cases B(i) and B(iii)) we expect the coexistence steady state (m∗1,m∗2,m∗3)to be stable, as the extinction steady state is unstable and numerical simulations showthat fungal growth is asymptotically bounded.Case C: one plant and multiple AM fungal speciesTo understand the impact of the introduction of a new fungal species into a coexistingcommunity I consider Eq. (3.1) for j = N + 1. In the equations I will use the variable mc56to indicate the introduced fungus and mw1,mw2, ...,mwN to indicate the N fungal speciespresent in the native community. To simplify computations, I assume that the competitionparameters acw and awc quantify the competition strength between the introduced speciesand the existing community, are the same for each of the species present. Similarly Iassume that competitive interactions among the existing community are all determinedby parameter aww. Like in the previous analyses, I further assume that all fungal specieshave identical parameters qcm, qhm and µm, and that plant biomass has reached steadystate p∗ (where p∗ may assume different values depending on the fungal community). Idenote αw and βw as the phosphorus and carbon exchange capacity of the wild community,and αc and βc as the phosphorus and carbon exchange capacity of the introduced species.With these adjustments, Eq. (3.1) can be rewritten asdmcdt= qcmβcp∗mc(awcawc +∑Nj mwj)− qhmαc p∗d+ p∗mc − µmm2c ,dmwjdt= qcmβwp∗mwj(acwmc + aww∑Ni 6=jmwiacwmc + aww∑Ni 6=jmwi + (mc +∑Ni 6=jmwi)2)−qhmαw p∗d+ p∗mwj−µmm2wj .(3.25a)(3.25b)Stability of the invasion state (m∗c , 0 . . . , 0):I first investigate the stability of the steady state (m∗c , 0, 0, . . . , 0), corresponding tothe situation in which the introduced species drives the existing community to extinction.The (N+1)×(N+1)−Jacobian obtained by linearising Eq. (3.25) around (m∗c , 0, 0, . . . , 0)is given byJ(m∗c ,0,...,0) =−µmm∗c − qcmβcp∗m∗cawc. . . − qcmβcp∗m∗cawc0 qcmβwp∗acwacw+m∗c− qhmαw p∗d+p∗ . . . 0....... . ....0 0 . . . qcmβwp∗acwacw+m∗c− qhmαw p∗d+p∗, (3.26)with eigenvaluesλ1 = −µmm∗c , (3.27a)λ2 =qcmβwp∗acwacw +m∗c− qhmαw p∗d+ p∗, (3.27b)57where λ2 has multiplicity N . Because λ1 is always negative, the steady state(m∗c , 0, 0, . . . , 0) is locally stable as long as the unique conditionacwacw +m∗c<qhmαwqcmβw(d+ p∗). (3.28)is satisfied. That is, as long as acw is small enough, i.e. the introduced species mccompetes strongly with the existing community, Eq. (3.28) holds true and competitiveexclusion can occur.Stability of the native community:To understand whether the existing community can as well competitively ex-clude the introduced species, I investigate under which conditions the steady state(0,m∗w1,m∗w2, . . . ,m∗wN) is locally stable. The analysis will give information on whetherthe existing community can defend itself from invasion by a strong competitor (in thiscase, the introduced species). I linearize Eq. (3.25) around (0,m∗w1,m∗w2, . . . ,m∗wN), andobtain the JacobianJ(0,m∗w1,..,m∗wN ) =qcmβcp∗awc∑Nj m∗wj+awc− qhmαc p∗d+p∗ 0 . . . 0qcmβwp∗(acw−2aww)m∗w1(aww+∑Ni6=1m∗wi)2 −µmm∗w1 . . . − qcmβwp∗awwm∗w1(aww+∑Ni6=1m∗wi)2....... . ....qcmβwp∗(acw−2aww)m∗wN(aww+∑N−1i m∗wi)2− qcmβwp∗awwm∗wN(aww+∑N−1i m∗wi)2. . . −µmm∗N.(3.29)Eigenvalues of (3.29) are given by the solutions to(qcmβcp∗awc∑Nj m∗wjawc−qhmαc p∗d+ p∗− λ)×∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣−µmm∗w1 − λ − qcmβwp∗awwm∗w1(aww+∑Ni6=1m∗wi)2 . . . − qcmβwp∗awwm∗w1(aww+∑Ni6=1m∗wi)2− qcmβwp∗awwm∗w2(aww+∑Ni6=2m∗wi)2 −µmm∗w2 − λ . . . − qcmβwp∗awwm∗w2(aww+∑Ni6=2m∗wi)2....... . ....− qcmβwp∗awwm∗wN(aww+∑N−1i m∗wi)2− qcmβwp∗awwm∗wN(aww+∑N−1i m∗wi)2. . . −µmm∗wN − λ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0.(3.30)58From Eq. (3.30) one can directly derive the first condition for stability, that isawcawc +∑Nj m∗wj<qhmαcqcmβc(d+ p∗). (3.31)Eq. (3.31) is a generalisation of Eqs. (3.15) and (3.22). Because the values of the plantand fungal steady states and their quotient are bounded (Fig. 3.2), one can conclude thatin order for the existing community to outcompete the introduced species mc, awc needsto be small enough, i.e. the existing community should compete strongly enough withthe introduced species.If the existing community is large enough, stability can be achieved for weaker compe-tition between the community and the introduced species, i.e. for larger awc. This means,that the basin of attraction of (0,m∗w1, . . . ,m∗wN) increases with increasing N , while thebasin of attraction of (m∗c , 0, . . . , 0) decreases. Hence, for the same competition strength,a more diverse community can outcompete an introduced species there where a less di-verse community would be driven to extinction. From the analysis, it emerges thereforethat a community composed of multiple species is less vulnerable to invasion by a strongcompetitor. This outcome is investigated numerically in Fig. 3.7.If the ratio αc/βc is small (i.e. if the introduced species provides low phosphorus tothe plant and takes a large amount of carbon), than stability is achieved for smaller valuesof awc, hence for larger competition strength between the community and the intro-duced species. The basin of attraction of (0,m∗w1, . . . ,m∗wN) decreases therefore when themutualist quality of the introduced species is low. These results are illustrated in Fig. 3.8.To derive the second condition for stability of the steady state (0,m∗w1, . . . ,m∗wN) Ilook at the second factor of Eq. (3.30). I assume that all fungal species in the communityare identical, and therefore m∗w1 = . . . = m∗wN = m∗w, and∑Ni 6=jm∗wi = (N − 1)m∗w. I setb = − qcmβwp∗awwm∗w(aww + (N − 1)m∗w)2(3.32)59and rewrite the determinant of Eq. (3.30) asM =−µmm∗w − λ b . . . bb −µmm∗w − λ . . . b....... . ....b b . . . −µmm∗w − λ. (3.33)Let define the diagonal matrixD = diag(M)− b I , (3.34)where I is the (N ×N)−identity matrix. One can rewriteM = D + b e eT , (3.35)where e is an all-one (N × 1)−vector. According to the matrix determinant lemma (Dingand Zhou, 2007), one obtainsdet(M) = det(D + b eeT ) = det(D)(1 + b eTD−1e). (3.36)We know thatdet(D) = (−µmm∗w − λ− b)N , (3.37)andeTD−1e =N−µmm∗w − λ− b. (3.38)Hence, by using Eq. (3.36), the determinant of M can be computed asdet(M) = (−µmm∗w − b− λ)N−1(−µmm∗w + b(N − 1)− λ) . (3.39)The additional eigenvalues of Eq. (3.29) can be found by setting Eq. (3.39) equal to zeroand solving for λ. By substituting the expression for b (Eq. (3.32)), one obtainsλ2 = −µmm∗w − (N − 1)qcmβwp∗awwm∗w(aww + (N − 1)m∗w)2, (3.40a)λ3 = −µmm∗w +qcmβwp∗awwm∗w(aww + (N − 1)m∗w)2, (3.40b)60where λ3 has multiplicity (N − 1).The eigenvalue λ2 is always negative. As long as λ3 is negative as well, the steadystate (0,m∗w1, . . . ,m∗wN) is locally stable. Hence, for stability one requiresaww(aww + (N − 1)m∗w)2<µmqcmβwp∗. (3.41)Eq. (3.41) is a generalisation of Eqs. (3.15) and (3.24). Because the values of p∗, m∗wand of the quotient p∗/m∗w are bounded (see Figs. 3.2 and 3.3), condition (3.41) holds aslong as aww is large enough. This means, if competitive interactions among the existingcommunity are weak, coexistence is observed. For large N , coexistence is observed evenfor stronger competitive interactions among mutualists (i.e., for a smaller aww). Hence,the presence of higher diversity promotes the coexistence of species that would otherwisecompetitively exclude each others.These findings are summarized in Table 3.1 and in Fig. 3.6. One can conclude thatif the introduced species is a weak competitor (Cases C(ii) and C(iv) of Table 3.1 andscenarios (2) and (4) of Fig. 3.6), condition (3.28) is not satisfied, and the competitiveexclusion steady state (m∗c , 0, . . . , 0) is therefore unstable. If the native community com-petes weakly against the introduced species (Case C(ii) and scenario (2)), the coexistencesteady state (m∗c ,m∗w1, . . . ,m∗wN) is stable, meaning that in this case the introduced speciescan establish and coexist with the existing community. However, if the community com-petes strongly against the introduced species, only the steady state (0,m∗w1, ...,m∗wN) isstable and the community will exclude the introduced species. If the introduced speciesis a strong competitor (Cases C(i) and C(iii) of Table 3.1 and scenarios (1) and (3) ofFig. 3.6), acw is small enough to satisfy condition (3.28), and if the community competesweakly against the introduced species, the introduced species can competitively excludethe mutualist community. However if the existing community competes strong enoughwith the introduced species (i.e., awc is small enough to satisfy condition (3.31)), and com-petitive interactions among the existing community are weak (i.e. aww is large enoughand condition (3.41)) holds), then either the introduced species or the existing commu-nity survives (Case (iii) and scenario (3)). This phenomenon is a consequences of thebistability of the steady states (m∗c , 0, . . . , 0) and (0,m∗w1, . . . ,m∗wN). Thus the invasion ofthe commercial AMF or the survival of the establised community will strongly dependson the initial amount of mc compared to the one of the community. The size of the basinof attraction of the steady state (0,m∗w1, . . . ,m∗wN) is directly related to a larger totalfungal biomass of the community, which is larger when a larger number of coexistingnative species are present. The size of the basin of attraction of (m∗c , 0, . . . , 0) increases ifthe introduced species is a bad mutualist (i.e., the ratio αc/βc is low). These results areexplored numerically in Figs. 3.7 and 3.8.61Note that if the introduced species can be regarded as a complete cheater (i.e., a speciesthat does not provide any benefit to the plant, with αc = 0), the steady state (m∗c , 0, . . . , 0)is unstable. Indeed, if the plant is in pairwise association with a complete cheater theplant-fungus mutualism can not establish, as condition (2.6) is never satisfied for αc = 0.Hence, scenarios (1) and (3) of Fig. 3.6 can not occur. Additionally, also scenario (4) cannot occur, as if αc = 0 Eq. (3.31) is never satisfied, and the steady state (0,m∗w1, . . . ,m∗wN)is always unstable. As long as the plant receives enough resources to support its growthand a multi-species mutualism can establish (i.e., Eq. (2.55) is satisfied), the introductionof a complete cheater results into coexistence of the introduced species with the nativecommunity (scenario (2)), and into extinction of the whole guild otherwise. If coexistenceis possible, the presence of the cheater causes a significant reduction in plant biomass andin the biomass of the native community, without however causing the extinction of otherguild members (see Fig. 3.4).62Table 3.1: Representation of the direct interactions between fungal species (mi) sharing a resourcesupplied by the same host plant (not included in the figure). Arrows indicate competition betweenmutualists, for space or access to plant roots, where competition can be weak (thin arrows) or strong(thick arrows). The last column summarizes the steady states stability resulting from the linear analysis.SCENARIOS Stable steady statesCase A (i) (m∗1,m∗2)(ii) (m∗1, 0) and (0,m∗2)(iii) (m∗1, 0)Case B (i) (m∗1,m∗2,m∗3)(ii) (m∗1, 0, 0), (0,m∗2, 0) and (0, 0,m∗3)(iii) (m∗1,m∗2,m∗3)(iv) (m∗1, 0, 0) and (0,m∗2,m∗3)(v) (0,m∗2, 0) and (0, 0,m∗3)(vi) (m∗1, 0)Case C (i) (m∗c , 0, . . . , 0)(ii) (m∗c ,m∗w1, . . . ,m∗wN)(iii) (m∗c , 0, . . . , 0) and (0,m∗w1, . . . ,m∗wN)(iv) (0,m∗w1, . . . ,m∗wN)63Figure 3.1: Phase-plane portrait of the model of Eq. (3.7). The horizontal and vertical axes representthe biomass of fungal species 1 and 2 respectively. Plant biomass is assumed to be at steady state. Leftpanel: Competition between species 1 and 2 is strong (a12 = a21 = 0.3). In this case the coexistencesteady state is unstable while the competitive exclusion steady states (m∗1, 0) and (0,m∗2) are both stable.Central panel: Competition between species 1 and 2 is weak (a12 = a21 = 2.2) and the coexistencesteady state (m∗1,m∗2) is stable. Right panel: Species 1 is a strong competitor while species 2 is a weakcompetitor. In this case, only the competitive exclusion steady state (m∗1, 0) is stable. Model parametersare qhp = 3, qcm = 2, qhm = qcp = 1, β = α = 0.4, µm = µp = 0.3, rp = 0.06, d = 1.2.64Figure 3.2: Dependence of (a) total fungal biomass at steady state and (b) plant biomass at steadystate on the competition parameter a, when a plant is associating with a community of N -coexistingmutualists. Different curves indicates the value of the steady states for a different number of mutualistspresent in the community (N = 2 . . . 10). To simplify computations, I assume that all model parametersare equal for the two fungal species and that aij = aji = a, and thus m∗j = m∗ and∑Nj m∗j = Nm∗. Theparameters used for the simulations correspond to those for Fig. 3.1.65Figure 3.3: Ratio of plant biomass at steady state (p∗) to total the fungal biomass at steady state of acommunity of N -coexisting mutualists (∑jm∗j ), as a function of the competition parameter a. Differentcurves corresponds to a different number of mutualists present in the community (N = 2..10). Similarlyto Fig. 3.2, to simplify computations, all model parameters are assumed to be equal for the two fungalspecies and that aij = aji = a, and therefore that m∗j = m∗ and∑Nj m∗j = Nm∗. The parameters usedfor the simulations correspond to those for Fig. 3.1.Figure 3.4: Plant biomass (p) and biomass of native fungal species (mw) over time in the absence (leftpanel) and presence (right panel) of a complete cheater (mc), i.e., a fungal species that does not provideany benefit to the plant (αc = 0). Competition parameters are aww = 2.2, awc = 2.2, and acw = 0.3.Qualitatively similar results are observed for any choice of the competition parameters. Other modelparameters correspond to those of Fig. 3.1.3.3.2 Effect of competition on mutualist guild dynamicsThe analysis of section 3.3.1 provides the mathematical foundations and quantitativecriteria on which are based the results presented in this section. Below, I present the key66ecological insights emerging from the mathematical results.Mutualism promotes coexistence among competitorsCoexistence of two mutualists (m1 and m2) is observed if competition is weak, whilecompetitive exclusion occurs when competition between the two species is strong. Thelatter case is shown in Fig. 3.5 (for full details see Case A in Table 3.1). This outcomeis similar to the output of classical competition models (Volterra, 1928), that predictcoexistence of weak competitors and competitive exclusion of strong competitors.The presence of a third species (m3) that competes only weakly with the other twomutualists, can change the exclusion scenario to one of coexistence. More specifically,indirect interactions are created that promote the coexistence of two strong competitorsthat would otherwise competitively exclude each other. The growth of the weak competi-tor (m3) is not significantly reduced by the presence of the other mutualists in the guild,and m3 improves the growth of the associated plant (p). An increase in plant biomasscorresponds to an increase in resource availability for all mutualists present, with a con-sequent reduction in competition strength between m1 and m2, allowing their coexistence(indeed a1 and a2 in Eq. (3.5) increase in the presence of m3).Alien species introduction and invasion successThe different possible outcomes, following the introduction of an alien mutualist in a guild,are summarized in Fig. 3.6. Four possible scenarios can be observed: (1) The alien speciesdisplaces the native community, (2) the alien species coexists with the native community,(3) either the alien or the native community is competitively excluded, depending on theirinitial biomasses, or (4) establishment of the alien species is prevented by the presence ofa native community. Which scenario occurs depends on the competition strength betweenthe alien mutualist and the native community, corresponding to the horizontal and verticalaxes of Fig. 3.6 (parameters awc and acw). Small a parameters indicate strong competitiveinteractions, while large a parameters indicate weak competition. The values a∗wc and a∗cwcan be quantified analytically, and determine the thresholds above and below which thedifferent scenarios (1)-(4) occur.If a weak competitor is added to a guild (Fig. 3.6, right column), two outcomes arepossible: Either the new species establishes and coexists with the community (scenario(2)), or it is competitively excluded by the native community (scenario (4)). Coexistenceis achieved when competition between the alien mutualist and the community is weak(i.e., acw > a∗cw and awc > a∗wc). Exclusion is the outcome when the community competesstrongly with the introduced mutualist (i.e., acw > a∗cw and awc < a∗wc). If competitionbetween the introduced species and the community is strong in both directions (i.e.,acw < a∗cw and awc < a∗wc), the existing community is displaced by the strong competitor67or the community persists and the new species is driven to extinction (scenario (3)). Theoutcome depends strongly on the the initial biomass of the introduced species (propagulepressure) and of the biomass and diversity of the existing fungal community (see Fig. 3.7).Finally, if the introduced species is a strong competitor but the community competesonly weakly against it (i.e., acw < a∗cw and awc > a∗wc), the introduced mutualist willcompetitively exclude all of the other guild members (scenario (1) in Fig. 3.6), independentof its initial biomass.The minimal competition strength needed in the community to overcome an invasionof the alien species, i.e. the value of a∗wc determining whether scenario (1) or scenario(3) occurs, depends on the diversity of the existing community. Diversity increases theresilience of the guild against invasion (see Fig. 3.7). Additionally establishment of thealien mutualist also depends on the mutualist quality of the introduced species (i.e. theratio αc/βc), with cheaters being more likely to invade (see Fig. 3.8).One can see the effect of diversity and mutualist quality in the expression for a∗wc:a∗wc =ρwαcβc1− ρwαcβcN∑j=1m∗wj , (3.42)where m∗wj represents the biomass of the native community, αc/βc is the ratio of phospho-rus to carbon exchange capacity of the alien species and ρw is a constant that depends onthe plant biomass and on the characteristics of the native community (explicitly stated inEq. (3.31)). As long as awc < a∗wc, extinction of the native community due to invasion ofthe alien mutualist is unlikely to happen (c.f. scenarios (3) and (4)). A large a∗wc indicates,therefore, a more stable community.From Eq. (3.42) it can be directly seen that if the total biomass of the native com-munity (i.e.∑jmwj) is large, the threashold value of a∗wc increases and invasion can beprevented for weaker competition between the community and the introduced species, i.e.for larger awc. One can see from Fig. 3.2a that a species rich community can reach alarger total biomass, and hence, for the same competition strength a diverse communitycan outcompete an introduced species where a less diverse community would be drivento extinction. If, on the other hand, the introduced species is a cheater (i.e. the ratioof αc/βc is small), the threshold a∗wc decreases, indicating that invasion can occur evenif the community competes strongly with the introduced species (i.e. awc is small). Theintroduction of a cheater is therefore more likely to result in invasion and displacement ofthe existing guild.683.3.3 The impact of invasion on resource availability (i.e. plantbiomass)To understand how plant biomass is affected by the introduction of an alien mutualist,I look at the evolution dynamics of an ecosystem composed of a plant, a native guild oftwo coexisting fungal species, and an introduced species, where competition between theintroduced species and the native fungi is either strong (scenario (3) of Fig. 3.6) or weak(scenario (2)).The introduction of a weak competitor in the guild results in the establishment of theintroduced species and its coexistence with the rest of the community (scenario (2)). Inthis case, the addition of a new species increases both plant growth rate and final plantbiomass (see Fig. 3.2). If the introduced species is a strong competitor (scenario (3)),the consequences on plant growth (and therefore on resource availability) depend largelyon the initial biomass of the introduced species and on the total biomass of the nativecommunity (see Fig. 3.9). If initially the biomass of the community is low, the additionof a new species speeds up plant growth, and at the same time reduces the growth rateof the community of mutualists (see Fig. 3.9, top two rows). When the initial biomass ofthe introduced species is not large enough to guarantee persistence, the community willdisplace the strong competitor, with no consequences on final plant and fungal biomass inthe long term (top row). When the introduced species can persist, the native communitywill be displaced with negative consequences on the plant final biomass (middle row). Ifinitially the total biomass of the community is large, the effect of the introduced species onthe plant and fungal growth rates is minimal (bottom row). In this case, the introducedspecies can not establish, therefore not affecting the final biomass reached by the plantor by the fungal community.3.4 Discussion3.4.1 Competition and mutualist guild dynamicsMutualism promotes coexistence among competitorsI suggest that the presence of a good mutualist within a guild can indirectly facilitatethe coexistence of species that would otherwise competitively exclude each other. If allguild members depend on the resource provided by a single plant, the presence of a weakcompetitor within the guild increases plant biomass and therefore resource availability.Increasing the amount of resource available leads to a consequent reduction in competitionamong all of the mutualists present.It has been proposed that the presence of particular species in a community can en-hance resource availability and provide habitat to the establishment of other organisms69that could not have otherwise survived (Floerl and Inglis, 2005, Schwindt and Iribarne,2000). More specifically, the role of mutualism in mediating competition and enhancingdiversity has also been noted experimentally (Aschehoug and Callaway, 2015, Schmittand Holbrook, 2003, Siefert et al., 2019). However, very few models have dealt with thisissue (Bever, 1999). In the context of mutualist guilds, the interdependence of commu-nity composition and host growth, determining resource availability, makes the dynamicscomplex to analyse (Bever, 2002). My model provides a mathematical tool to study thedifferent facets of these coupled dynamics. I found that the removal of a key species, suchas a weak competitor, from a community of strongly competing mutualists (e.g. speciesm3 in Fig. 3.5) may cause a drastic reduction in the diversity of the guild.Predicting the invasion success of an alien mutualistIn my results, I formulate testable predictions on biological invasion in mutualist guilds. Idisentangle the effect of different factors, such as niche availability, propagule pressure andtraits of the alien mutualist and the recipient community, in determining the establishmentof an introduced species.Niche opportunities have already been identified as important drivers of invasion suc-cess (Shea and Chesson, 2002, Vall-llosera et al., 2016). In agreement with these ob-servations, I found that if competition between the introduced species and the existingcommunity is weak, e.g. due for example to functional complementarity among species,the alien species will establish and coexist with the rest of the community (scenario (2)in Fig. 3.6).I show that if competition between the alien species and the existing community isstrong, coexistence is not possible (scenario (3) in Fig. 3.6). The introduced species willtherefore either competitively exclude the whole community, or fail to persist. Simula-tions show that when the native community is already established and has reached alarge biomass, establishment of the alien species is unlikely. However, when the biomassof the native community is low, establishment of the alien species occurs when its initialbiomass is large enough. Hence, the biomass of the native community at the time ofintroduction of the alien species and the propagule pressure of the alien species are keyfactors determining invasion success. The literature shows that early arrivals can estab-lish and colonize available resources, and prevent their exploitation by late arrivals. Thisphenomenon has been observed for competitors in general (Fraser et al., 2015, Schantzet al., 2015, Wainwright et al., 2012), as well as in the context of mutualistic commu-nities (Kennedy and Bruns, 2005, Peay, 2018, Werner and Kiers, 2015). Priority effectsand timing can therefore play a fundamental role in creating invasion opportunities forcompetitors. My model supports this finding, as early arrivals have time to increase theirbiomass, and by doing so, to gain a competitive advantage over an introduced speciesI found that the most worrisome scenario occurs when the competitive ability of the70introduced species is largely superior than that of the existing community (scenario (1) inFig. 3.6). In this case, the alien species invades and displaces the existing guild, indepen-dent of its propagule pressure. The literature has reported cases where the introductionof a strong competing alien mutualist, i.e., an alien pollinator sharing nesting and floralresources with native species, causes the decline or extinction of native pollinators (Ingset al., 2006, Morales et al., 2013). I show that this scenario is more likely to occur whenthe alien species has low mutualist quality, and when diversity in the existing communityis low.Whether diversity promotes resilience of a community has been frequent matter ofdebate in recent decades (McCann, 2000). Generally, species rich systems are reputedto be more stable in the face of disturbance (Downing et al., 2012), mainly through theinsurance effect (Yachi and Loreau, 1999), i.e. where the presence of multiple speciesperforming the same function makes an ecosystem less vulnerable to species loss, andthrough functional complementarity, i.e., where the existing community exhibits a betterexploitation of resources (Booth and Hoeksema, 2010, Cardinale, 2011, Maherali andKlironomos, 2007, Naeem et al., 1996). This second mechanism predicts that a better useof the available resources is directly related to an increase in fitness of the community,and therefore higher resistance to disturbance.Although there is copious theoretical support for a negative relationship between di-versity and invasion success (Elton, 2000, Hobbs and Huenneke, 1992, Tilman, 2004), em-pirical observations have been inconsistent (Dickerson et al., 1984, Fargione and Tilman,2005, Levine, 2000, Planty-Tabacchi et al., 1996, Robinson et al., 1995, Wiser et al., 1998).In particular, the role of species identity in promoting or opposing invasion is still underinvestigation (Zheng et al., 2018). My work adds further theoretical evidence in supportof diversity promoting resilience, this time in the specific context of a community of mu-tualists. I suggests that a diverse community of mutualists can efficiently monopolize theavailable resource, in a way that makes it more resilient to invasion by a strong com-petitor. In the model, weak competitors occupying different niches do not significantlyhinder each others’ growth. Hence, the biomass of a group of weak competitors increasesmuch faster than the biomass of a single species, even when the species in question is astrong competitor. A larger biomass provides a competitive advantage to the coexistingcommunity, by preventing access by the introduced species to the common resource. Thelarger the number of functionally different species, the faster the growth in terms of totalbiomass, and the easier it becomes to outcompete a strong competitor. A large commu-nity can therefore prevent the invasion of a strong competitor in cases where a smallercommunity would be competitively excluded.713.4.2 The impact of invasion on resource availabilityI show that the addition of an alien mutualist to an existing guild leads to an increasein the growth rate of the host plant, independent of introduced species persistence (seeFig. 3.9). The increase in the plant growth rate can be more or less pronounced, dependingon the abundance of native fungi in the community. In the presence of a less abundantinitial community, plant growth can be significantly enhanced by the introduction of analien species, while the effect is minimal when abundance of native fungi is already high.Although the short term effect of the introduction of an alien species on plant growthcan be extremely positive, the situation changes when looking at its long term conse-quences. Indeed, I show that the introduction of a strongly competitive species and itspermanent establishment may result in an initial increase in the plant growth rate, butalso in the subsequent displacement of native species, with a consequent decrease in finalplant size. While the introduction of a mutualist that is a weak competitor (e.g. a specieswhose function is complementary to the native community) can result in an overall posi-tive effect on both plant growth rate and final size, the introduction of a strong competitormay have detrimental long term consequences on plant biomass.Although studies generally agree that high functional diversity leads to more produc-tive and stable communities (Garibaldi et al., 2014, Hoeksema et al., 2010, Male´zieuxet al., 2009, Ratnadass et al., 2012, Yachi and Loreau, 1999), it is not clearly understoodwhether the short term positive effects on productivity are related to diversity by itself,or are rather due to a general increase in the abundance of mutualists (Freeman et al.,2013, Pellegrino et al., 2011). My predictions suggest that abundance, and not diversity, ispositively related to plant growth rate on a short time scale. Long term productivity andresilience are increased by diversity in the mutualist community. In order to disentanglethe short and long term consequences of invasion, experimental and field studies need totake into account the abundance and diversity of the existing fungal community.3.4.3 Future directionsThe model presented in this chapter sheds light on the mechanisms behind the stability ofmutualistic communities. My results have direct implications for conservation biology, byproviding insights into the possible consequences of species manipulation among a group ofmutualists depending on the same resource, such as below-ground microbial communitiesor pollinators (Hanna et al., 2013, Ings et al., 2010, Menz et al., 2011). Because themodel is based on a consumer-resource framework for mutualistic interactions (Hollandand DeAngelis, 2010), simulations predicting the output of competition within a guildare easily testable in an experiment. Plant and fungal parameter estimation could bepossibly achieved by monitoring plant and fungal growth, and resource exchange rates inmonocultures. Pairwise competition parameters may be determined by measuring growth72depletion of a species in the presence of another fungus.In nature, fungal species or pollinators can often associate simultaneously with mul-tiple different plants. To better simulate this real world scenario, our model should beextended to include not only a single host, but multiple possible partners. Consideringassociations with multiple different plants could, for example, answer questions relatedto the impact of plant diversity on pollination services and productivity (Isbell et al.,2017), or give insighs into community assemblage and invasion dynamics in forests oragroecosystems (Horn et al., 2017, Potts et al., 2003).Spatial factors may play an important role in invasion dynamics, or in creating het-erogeneous patterns of guild and host persistence (Kerr et al., 2002, Melbourne et al.,2007, Westphal et al., 2006, Yu and Wilson, 2001). In chapter 4, I develop and analyse ofa spatially explicit version of the model presented here. The consideration of the spatialcomponent will provide interesting insights about the spread and speed of invasion of theintroduced species.Figure 3.5: Diagram representing the direct interactions of a plant (p) in association with two or threemutualists (m1, m2 and m3). Interactions can be mutualistic (‘+’,‘+’) or competitive (‘−’,‘−’). Arrowthickness represents the strength of the interactions. (a) When only two strongly competiting mutualists(m1 and m2) are in association with the plant, coexistence is not possible and one of the two specieswill drive the other to extinction. (b) When a third weakly competing mutualist is present, coexistenceis achieved. The presence of m3 increases plant biomass (i.e. resource availability), reduces competitionbetween m1 and m2 and promotes their mutualistic interactions with the plant.73Figure 3.6: Representation of the possible outcomes following the introduction of an alien species intoa guild of coexisting mutualists. The diagrams represent the direct interactions between the plant (p),its mutualist guild (mw1, ...,mwN ) and the introduced mutualist (mc). Interactions between the plantand the fungal species are mutualistic, while interactions among fungal species are competitive. Arrowthickness represents the strength of the interactions. In the figure, competition between the introducedmutualist and the guild (parameters awc and acw) is varied along a gradient of strong competition (a’sparameters are small) to weak competition (a’s parameters are large). The critical values a∗wc and a∗cwdefine the boundary between the occurrence of the different scenarios, and depend on diversity in thenative community and on the mutualist quality of the introduced species.74Figure 3.7: Outcome of competition between a strong competitor (mc) and a community of mutualists(mwj), for different initial biomass of the strong competitor (the alien species) and of the community.The solid curves represent the boundaries between establishment of the alien species (area above eachline) and establishment of the community (area below each line) for the situation illustrated in caseC(iii) of Table 3.1 and scenario (3) of Fig. 3.6. The horizontal axis represents the total initial biomassof the community, while the vertical axis represents the initial biomass of the alien species (or propagulebiomass). Different curves correspond to a different number of species present in the community, rangingfrom 1 to 9, where the number of species is indicated on each curve. Increasing the number of speciesin the community increases resilience of the community against invasion, in the sense that for the sametotal biomass of the community, a higher propagule biomass is needed in order to displace the commu-nity. Competition parameters used for the simulations are awc = acw = 0.3, aww = 2.2. Other modelparameters are qhp = 3, qcmj = 2, qhmj = qcp = 1, βj = αj = 0.4, µp = µmj = 0.3, rp = 0.02, d = 1.2,N = 2..75Figure 3.8: Outcome of competition between a strong competitor (mc) and a community of weakcompeting mutualists (mwj), corresponding to case C(iii) in Table. 3.1 and scenario (3) in Fig. 3.6,for different ratios of the phosphorus and carbon exchange ability of the alien species (parameters αcand βc respectively). The solid curves represent the boundaries between establishment of the alienspecies (area above each line) and establishment of the community (area below each line). The curvelabeled with a ‘1’ corresponds to the case in which the native community and the alien species havethe same resource exchange ability (i.e. αc/βc = αw/βw). Other curves correspond to situations wherethe alien species has a higher/lower resource exchange capacity with respect to the native community(i.e. αc/βc = xαw/βw, where x = 0.5, 0.75, 1.25, 1.5). The plot shows that establishment of an alienis more likely if the introduced species has lower mutualist quality. Exact values are chosen for theresource exchange parameters are: αw = βw = 0.4 for all curves, while αc = 0.3, 0.35, 0.4, 0.45, 0.5 andβc = 0.6, 0.467, 0.4, 0.36, 0.333 respectively. Other parameters correspond to those of Fig. 3.7.76Figure 3.9: Plant biomass (left panels, solid black lines) and fungal biomass over time (right panels, solidblue lines) in response to the addition of an alien species that is a strong competing mutualist (dashed redlines) to the native community, for the situation corresponding to scenario (3) of Fig. 3.6. The dotted linesshow plant growth (left panels) and fungal growth (right panels) in the absence of the introduced species.Simulations were run with a guild of two native species (i.e., N = 2). In the top panels, the biomassof the introduced species is too low to guarantee persistence (mc0 = 0.2, 2mwj = 0.04). In the middlepanels, the alien species establishes and displaces the native community (mc(0) = 0.3, 2mwj(0) = 0.04).In the bottom panels, the initial biomass of native fungi is much larger than the alien propagule biomass,and species introduction has a very little effect on plant and fungal growth (mc(0) = 0.2, 2mwj(0) =0.8). The plant initial biomass used for the simulations is p(0) = 0.15, competition parameters are aww= 2.2, acw = awc = 0.3. Other parameters correspond to those for Fig. 3.8.77Chapter 4Investigating the spread of an alienmutualist in a field4.1 Background informationThe study of mutualistic interactions is gaining increasing attention in the scientific litera-ture (Bronstein, 2015). Indeed pollination and plant-microbe symbioses, two of the mostcommon earth mutualisms, play a key role in agricultural productivity. In particular,the presence in the soil of arbuscular mycorrhizal (AM) fungi, beneficial fungi living inassociation with the roots of the vast majority of plants, has been found to significantlyincrease crop yield, mainly by facilitating plant access to limiting nutrients in the soil(Jeffries et al., 2003, Smith and Read, 2010). Additional benefits provided to the plant byAM fungi are pathogen protection (Pozo and Azco´n-Aguilar, 2007) and resistance againstabiotic stress, such as drought or salinity (Latef et al., 2016). The plant, in exchange,provides the fungi with fixed carbon (Smith and Read, 2010).Because of their positive influence on plant growth and fitness, in recent decades AMfungi have been commercialised, and are widely used as organic fertilizers (Gianinazzi andVosa´tka, 2004). Despite the potential benefits of field inoculation with commercial AMfungi, studies investigating the effective establishment of the mycorrhizal inoculum haveled to contradicting results (Baum et al., 2015, Berruti et al., 2016, Verbruggen et al.,2013). Some observations show that the inoculants have successfully established in thefield and improved crop yield (Bender et al., 2019, Buysens et al., 2016, Ko¨hl et al., 2016),while other studies showed poor establishment of the inoculated species and no effect onproductivity (Emam, 2016, Loja´n et al., 2017, Ryan and Graham, 2002).Moreover, it is still unclear how native fungal communities might respond to theintroduction of a new fungal species (Hart et al., 2018, Thomsen and Hart, 2018). In somestudies, inoculation caused little or no impact on the native fungal community (Loja´net al., 2017, Sy`korova´ et al., 2012), while other observations show that native species78were displaced by the introduction of the inoculants (Bender et al., 2019, Koch et al.,2011, Pellegrino et al., 2012, Symanczik et al., 2015). These last results in particular raiseconcerns as to whether the inoculated fungi could possibly spread over non-target areacausing a loss in soil biodiversity and, ultimately, negative effects on plant productivity(Hart et al., 2017, Ricciardi et al., 2017, Schwartz et al., 2006). It is therefore importantto assess the conditions under which the use of commercial inoculants is beneficial, andunder which it is harmful.In this chapter, I use a partial differential equation model to formulate testable pre-dictions on the establishment and spread of fungal inoculants in the field. I analyse howthe field community dynamics and invasion speed are affected by the interactions of thecommercial inoculants with the native fungal community and with the host plant. Forthis purpose, I use the model of chapter 3, considering the consequences of the introduc-tion of an alien mutualist (in this case the commercial inoculant) in a guild of coexistingmutualists (the native fungal community), and add a spatial component to the model tocharacterise the regional spread of the inoculants. In particular, I will discuss how fungalspread affects resource availability (in this case determined by the biomass density of thehost plant), and influences in turn the invasion dynamics of the inoculated fungi (Didhamet al., 2007, Goldstein and Suding, 2014).The replacement of native species by an invasive species has already been investigatedusing Lotka-Volterra competition models (Hosono, 1998, Murray, 2007, Okubo et al.,1989). However, the determination of the spreeading speed has been obtained only re-cently (Holzer and Scheel, 2012), even for homogeneous environment. Other studies havedescribed the spread of competitors in a spatially heterogeneous environment (Goldsteinet al., 2019, Neubert and Caswell, 2000, Real and Biek, 2007). In my work I derive con-crete quantitative criteria to determine the invasion speed of the inoculants in the fieldin heterogeneous environment, where heterogeneity may evolve with the spread of thespecies. Indeed, in my model the resource (i.e., the plant) interacts with the species incompetition (i.e., the fungi).Recently Girardin (2016) has provided analytical formulae for the spreading speedusing non-cooperative system of reaction-diffusion equations. Similarly to these systems,my model involves interactions that are competitive (fungi to fungi), but also mutualistic(plant to fungi). However, Girardin (2016) assumed that all species may grow in theabsence of the other, while in my study fungi are obligate mutualists, and can not survivein the absence of the host plant. Additionally, I assume that interaction are non-linear,conversely to the linear system investigated by Girardin (2016).My results elucidate some of the mechanisms responsible for the inconsistent exper-imental observations on inoculum success reported in the literature, and help the for-mulation of concrete quantitative criteria to determine the risks and benefits associatedwith the introduction of mycorrhizal inoculants. Additionally my work provide useful79information for the selection of inoculants that are likely to promote plant growth whilehelping the restoration of a flourishing native fungal community. Mathematically, mywork provides new insights on traveling waves solutions in systems involving competitiveand mutualistic interactions.4.2 Model and methods4.2.1 A model for investigating the persistence and spread offungal inoculantsAM fungi are obligate symbionts, and their growth depends on their mutualistic interac-tions with a plant host (p) and on the strength of the competitive interactions amongthe different fungal mutualists present (Palmer et al., 2003, Smith and Read, 2010).Plant growth, in turn, is enhanced by the presence the mutualists in a way that de-pends on the specific fungal community composition (see chapters 2 and 3). In themodel, plant and fungal coupled growth is therefore determined by the density depen-dent functions fp(p,mc,mwj), for plant growth, fmc(p,mc,mwj), for inoculum growth, andfmwj (p,mc,mwj), for the growth of the native fungal community.Fungi can spread in the soil through fungal hyphae (Friese and Allen, 1991), or throughshort or long range dispersal of spores by different agents (Warner et al., 1987). Tomodel fungal spread, I will consider only short range dispersal of fungal biomass, eitherthrough hyphae or spores (not distinguishable in the model). I assume therefore that fungidisperse by random diffusion, and quantity the dispersal ability of each fungal species bythe diffusion coefficients Dc (for the inoculated fungi) and Dwj (for each of the nativefungal species).To investigate the persistence and spread of the introduction of commercial fungalinoculants (mc) into a field inhabited by a native fungal community (mwj , where j =1 .. N), I extend the model of chapter 3 to explicitly consider random fungal dispersal.Plant and fungal biomass densities at each time and location are given by the solutionsto ∂t p(x, t) = fp(p,mc,mwj) ,∂tmc(x, t) = Dc ∂2xmc + fmc(p,mc,mwj) ,∂tmwj(x, t) = Dwj∂2xmwj + fmj(p,mc,mwj) ,(4.1a)(4.1b)(4.1c)where Dc and Dwj are the diffusion coefficients of the inoculants and of the native fungalspecies respectively. The functions fp, fmc and fmwj are defined in chapter 3 and can be80written asfp =plantfitness︷ ︸︸ ︷rp(p) +∑i=c,wjN∑j=1[ phosphorusreceived︷ ︸︸ ︷αi fhp(p,mi)−carbonsupplied︷ ︸︸ ︷βi fcp(p,mi)C(mi,mwj)], (4.2)fmc = rmc (mc) + βc fcmc(p,mi)C(mc,mwj) − αc fhmc(p,mi), (4.3)fmwj = rmwj (mwj)︸ ︷︷ ︸fungalmaintenance+ βwjfcmwj (p,mi)C(mwj ,mc)︸ ︷︷ ︸competition︸ ︷︷ ︸carbonreceived−αwjfhmwj (p,mi)︸ ︷︷ ︸phosphorussupplied. (4.4)where α-parameters represent the fungal exchange ability for phosphorus, while β-parameters represent the fungal exchange ability for carbon.Modeling competition between fungi: Competition between fungal species is de-fined asC(mj,mi) = 1−∑i 6=jmiaj +∑i 6=jmi, (4.5)where aj determines how the presence of other fungal mutualists in the community in-fluences the carbon uptake capacity of species j. The value of aj and depends on thepairwise competition strength between species j and each of the fungal species present(aij), weighted by the proportion, in terms of biomass, occupies by each fungal specieswithin the competing community (see chapter 3, Eqs. (3.4) and (3.5)). We writeaj =∑i 6=jaijmi∑i 6=jmi. (4.6)I will use awjc to indicate the competition strength of native fungal species j toward theinoculant, parameter acwj to indicate the competition strength of the inoculant towardnative fungal species j, and awiwj to indicate the competition strength between two nativefungal species.81The complete model: The model of Eq. (4.1) can be written as∂tp = qhprpp+ qhp(αcmc +∑jαwjmwj)pd+ p− qcp[βcmc( ∑j awjcmwj∑j awjcmwj + (∑jmwj)2)+∑jβwjmwj(acwjmc +∑i 6=j awiwjmwiacwmc +∑i6=j awiwjmwi +(mc +∑i 6=jmiw)2)]p− µpp2 ,∂tmc = Dc∂2xmc + qcmcβcpmc( ∑j awjcmwj∑j awjcmwj + (∑jmwj)2)− qhmcαcpd+ pmc − µmcm2c ,∂tmwj = Dwj∂2xmwj + qcmwjβwjpmwj(acwjmc +∑i 6=j awiwjmwiacwmc +∑i 6=j awiwjmwi +(mc +∑i 6=jmwi)2)−qhmwjαwjpd+ pmwj − µmwjm2wj .(4.7a)(4.7b)(4.7c)where model parameters are given in Table 4.1.For the analysis, I will assume that the competition strength between the commercialfungus mc and individuals of the wild community mwj and between individuals of the wildcommunity are the same for all fungi (i.e. awjc = awc, acwj = acw and awiwj = aww). I willfurther assume that all fungal parameters are the same, except for the phosphorus andcarbon exchange abilities (parameters α and β). With these simplifications, the model ofEq. (4.7) becomes∂tp = qhprpp+qhp(αcmc +Nαwjmwj)pd+ p− qcp[βcmcawcawc +Nmwj+N(βwjmwjacwmc + aww(N − 1)mwjacwmc + aww(N − 1)mwj + (mc + (N − 1)mwj)2)]p− µpp2 ,∂tmc = Dc∂2xmc + qcmβcpmc(awcawc +Nmwj)− qhmαc pd+ pmc − µmm2c ,∂tmwj = Dw∂2xmwj + qcmβwjpmwj(acwmc + aww(N − 1)mwjacwmc + aww(N − 1)mwj + (mc + (N − 1)mwj)2)−qhmαwj pd+ pmwj − µmm2wj .(4.8a)(4.8b)(4.8c)Model outputs will be analysed through traveling wave analysis and numerical simu-lations (see section 4.3.1). Numerical simulations are run in Matlab R2017a. Solutionsare found by using a semi-implicit scheme, where the reaction term is solved explicitlyand the diffusion is solved implicitly. The discretization used for the space is dx = 0.05,and the discretion used for the time is dt = 0.05.824.2.2 Investigating inoculum persistence and spreadI will use Eq. (4.1) to study how the competition strength between the inoculant andnative fungal species affect fungal spread. I will consider a community of coexistingweakly competing fungi (i.e., aww is large), competing with the inoculated fungal speciesmc. The scenarios considered are sketched in Table. 4.2. In case (1), competition betweenthe inoculants and the native community is weak (i.e., awc and acw are large). In cases(2) and (3) competition between the inoculants and the native community is asymmetric,where either the inoculated species or the native community is a superior competitor (i.e.,acw < awc or acw > awc). In case (4) competition between the inoculants and the nativefungal community is strong (i.e., awc and acw are small).Additionally, I will determine how the spread speed is affected by the diffusion abilityof the introduced fungi (intended as parameter Dc), by their mutualist quality (intendedas the ratio of the phosphorus and carbon ability of the inoculants, i.e., αc/βc), and bythe number of fungal species present in the native community.4.2.3 Insights for agroecosystem managementNeighboring fields: I will study how the spread of inoculants from a field to anothercan affect fungal community composition and plant productivity. I will consider a land-scape consisting of two fields next to each other. I will assume that the field on the leftcontains inoculants only, while the field on the right contains an established native fungalcommunity. I will look at the spread of the inoculants from the left to the right field,when competition between the inoculants and the native community is weak (case (1)) orstrong (case (4)). I will assume that strongly competing inoculants have lower resourceexchange capacity with respect to native species (i.e., lower α/β ratio), as commercialinoculants have been shown to invest more in their own reproduction at the expensesof their mutualistic relationship with the plant (Kokkoris et al., 2019). I will considerthe situation in which the native community consists of multiple fungal species, and thesituation in which only one native fungal species is present.Inoculation and productivity: I will study how productivity (in terms of plantbiomass density) is affected by the introduction of fungal inoculants. As seen in chapter3, the introduction of inoculants has little or no effect on productivity in the presenceof a well-established native fungal community. Here I will therefore focus on the effectof inoculation in depleted soils, i.e., when the density of native fungal species is low, tounderstand how the introduction of inoculants can promote the regeneration of a nativefungal community and boost productivity. I will assume inoculation occurs at the leftedge of a landscape and I will simulate plant and fungal growth and spread over time.834.3 Results4.3.1 Traveling wave analysisI am interested in understanding whether the introduction of a new fungal species (mc)in the presence of an existing fungal community (mwj) will result in a spread of thenew species into the field. To tackle this problem, I will look at travelling wave solutionof Eq. (4.1), which are particular solutions describing the invasion of one steady stateby an other one at constant speed c. More precisely, I assume that the solution ofEq. (4.1) are of the form p(t, x) = P (x − ct) = P (z), mc(t, x) = Mc(x − ct) = Mc(z)and mwj(t, x) = Mwj(x− ct) = Mwj(z) for an unknown speed c ∈ R. By replacing theseexpressions into Eq. (4.1) one obtains−cdPdz= FP (P,Mc,Mwj) ,−cdMcdz= Dcd2Mcdz2+ FMc(P,Mc,Mwj) ,−cdMwjdz= Dwd2Mwjdz2+ FMwj(P,Mc,Mwj) .(4.9a)(4.9b)(4.9c)In order to lighten the notations, I define the vector U(x − ct) = U(z) = (P,Mc,Mwj),the diffusion matrix D = diag(0, Dc, Dwj) and the interaction function F(U(z)) =(FP , FMc , FMwj). Then Eq. (4.9) can be written asDU′′(z) + cU′(z) + F(U(z)) = 0 . (4.10)I will focus on three particular steady states which stability have been already investigatedin chapter 3:- Community only: U∗w = (P∗w, 0,M∗wj) ,- Coexistence: U∗∗cw = (P∗∗cw,M∗∗c ,M∗∗wj) ,- Inoculants only: U∗c = (P∗c ,M∗c , 0) .The stability of these steady states depends on the strength of the competitive interac-tions between the fungi. As shown in Table 4.2, in case (1), competition between theinoculants and the native community is weak. In cases (2) and (3) competition betweenthe inoculants and the native community is asymmetric, where either the inoculants orthe native community is a superior competitor. In case (4) competition between the in-oculants and the native fungal community is strong. From the analysis of the non-spatialmodel, as investigated in chapter 3, we know that• Case (1): U∗∗cw stable, U∗w unstable,84• Case (2): U∗c unstable, U∗w stable,• Case (3): U∗c stable, U∗w unstable,• Case (4): U∗c stable, U∗w stable.I will consider on the right of the domain a steady state where the inoculant biomassdensity is zero (i.e. Mc(∞) = 0) and only the native community and the plant are present.On the left I will assume that the steady state has a positive density of commercialinoculants (Mc(−∞) > 0), plant biomass density is positive and the biomass densityof the native community can be zero or positive depending on which scenario has beeninvestigated. This situation corresponds to investigating whether the steady state U∗w canbe invaded by a steady state with a positive density of inoculants (i.e. either U∗∗cw or U∗c).Thus the boundary conditions can be expressed asU(−∞) =U∗∗cw Case (1)U∗c Cases (2)− (4) and U(+∞) = U∗w (4.11)The solutions to Eq. (4.10) satify boundary conditions for P , Mwj and Mc. The stabilityof the steady states involved in the boundary conditions, may help us to describe thespeed at which the solution moves.When one stable steady state and one unstable steady state are present (cases (1)-(3))the stable state will invade the unstable state at constant speed c. In this case, there isa monostable traveling waves and one may expect an estimate of the minimal speed ofpropagation using the linearized problem around the unstable steady state. In case (4),a bistable traveling wave connects two stable steady state. In this case it is difficult todetermine the actual speed of invasion, however, there are some techniques to estimate thesign of the propagation speed, to determine which of steady states will invade the other.I will first analyse the three monostable cases, and then investigate the bistable one. Inorder to gain more insight into the results of the analysis, I will explore numerically howthe invasion speed varies as a function of diversity of the native community, as well as afunction of different characteristic of the inoculated fungi, such as dispersal ability (Dc)and mutualist quality (αc/βc).85Table 4.1: Description of variables and parameters of the model given in (4.7), with respective mea-surement units.Symbol Description Unitp Plant plant biomass densitymc Fungal inoculant fungal biomass densitymwj Native fungal species fungal biomass densityDc Diffusion coefficient of mcspacetimeDwj Diffusion coefficient of mwjspacetimeawjc Competition parameter (mwj toward mc) fungal biomass densityacwj Competition parameter (mc toward mwj) fungal biomass densityawiwj Competition parameter (mwi toward mwj) fungal biomass densityd Half-saturation constant plant biomass densityrp Intrinsic growth ratephosphorusplant biomass density × timeα Phosphorus exchange ability phosphorus(time)×(fungal biomass density)β Carbon exchange ability carbon(time)×(fungal biomass density)×(plant biomass density)µp Plant maintenance rate1(time)×(plant biomass density)µm Fungal maintenance rate1(time)×(fungal biomass density)qhm Conversion phosphorus to fungal biomass fungal biomass density/ phosphorusqhp Conversion phosphorus to plant biomass plant biomass density/ phosphorusqcm Conversion carbon to fungal biomass fungal biomass density/ carbonqcp Conversion carbon to plant biomass plant biomass density/ carbon86Table 4.2: Cases 1-4 indicate the four possible outputs following the introduction of a fungal inoculant (mc) in a field inhabited by a native fungal community(mwj ), where all fungi share a resource supplied by the same plant. Interactions between the fungi and the plant are mutualistic, while interactions between fungalspecies are competitive. In the top row, arrow thickness represents the strength of the interactions. The second row summarises the predictions on inoculumestablishment discussed in chapter 3. The third and fourth rows illustrate the predictions on inoculum spread in each of the four scenarios. The big arrows in thebottom row indicate the spread direction and on the left of each arrow are the plant, inoculants and native community densities expected after inoculation.Case (1) (2) (3) (4)InteractionsStablesteady states(no diffusion)Coexistence(p˜∗, m˜∗c , m˜∗w1, . . . , m˜∗wN)Community only(p∗, 0,m∗w1, . . . ,m∗wN)Inoculants only(˜˜p∗,m∗c , 0, . . . , 0)Bistability(˜˜p∗,m∗c , 0, . . . , 0) and(p∗, 0,m∗w1, . . . ,m∗wN)Spreadfrom (p∗, 0, m˜∗w1, . . . , m˜∗wN)to (p˜∗, m˜∗c , m˜∗w1, . . . , m˜∗wN)from (˜˜p∗,m∗c , 0, . . . , 0)to (p∗, 0,m∗w1, . . . ,m∗wN)from (p∗, 0,m∗w1, . . . ,m∗wN)to (˜˜p∗,m∗c , 0, . . . , 0)from (˜˜p∗,m∗c , 0, . . . , 0)to (p∗, 0,m∗w1, . . . ,m∗wN)Sketchof the spread87Monostable traveling wavesCases (1) and (3): In these cases, the unstable steady state U∗w will be invaded by asteady state that contains commercial inoculants, where either coexistence of the inocu-lants with the native community is observed (U∗∗cw, case (1)) or the native community isdisplaced by the inoculants (U∗c , case (3)). To find an estimate for the minimal propaga-tion speed c, I linearize Eq. (4.10) around the unstable steady state U∗w and obtain− cP ′ = FP∣∣∣∣U∗w+∂FP∂P∣∣∣∣U∗wP +∂FP∂Mc∣∣∣∣U∗wMc +∂FP∂Mwj∣∣∣∣U∗wMwj− cM ′c = DcM ′′c + FMc∣∣∣∣U∗w+∂FMc∂P∣∣∣∣U∗wP +∂FMc∂Mc∣∣∣∣U∗wMc +∂FMc∂Mwj∣∣∣∣U∗wMwj− cM ′wj = DwM ′′wj + FMwj∣∣∣∣U∗w+∂FMwj∂P∣∣∣∣U∗wP +∂FMwj∂Mc∣∣∣∣U∗wMc +∂FMwj∂Mwj∣∣∣∣U∗wMwj(4.12a)(4.12b)(4.12c)Because FMc|U∗w = 0 and∂FMc∂P|U∗w =∂FMc∂Mwj|U∗w = 0, I can observe that the componentMc of the linearized problem of Eq. (4.12) satisfiesDcM′′c + cM′c + F∗cMc = 0 , (4.13)whereF ∗c = P∗w(qcmβcawcawc +NM∗w− qhmαcP ∗w + d). (4.14)From the analysis of the non spatial version of the model carried on in chapter 3 (seeEq. (3.31)), we known that F ∗c > 0 for cases (1) and (3). Hence, one can conclude that alower bound for the propagation speed c∗c ≤ c is given by (Murray, 1989)c∗c = 2√DcF ∗c = 2√DcP ∗w(qcmβcawcawc +NM∗w− qhmαcP ∗w + d). (4.15)Thus in cases (1) and (3) the commercial inoculants invade a field inhabitated by a nativecommunity at a minimal speed that increases with increasing diffusion ability of thecommercial fungus and decreases with increasing diversity of the native fungal community.Additionally, the minimal propagation speed increases if the commercial fungus can beregarded as a cheater, i.e. when αc is small and βc is large. These findings are supportedby numerical simulations. In Fig. 4.1, the theoretically predicted linear speed of Eq. (4.15)is compared with the speed resulting from numerical simulations, for different dispersalability and mutualist quality of the inoculants, and for increasing diversity of the nativecommunity. The speed resulting from the numerical simulations does not depend onwhether the native community is displaced (case (3)) or the inoculants coexist with thecommunity (case (1)), as anticipated by the traveling wave analysis.88Case (2): In this case, the native community competes strongly against the inoculants.The steady state U∗w is stable while the unstable steady state is now U∗c . I linearizetherefore Eq. (4.10) around U∗c . Because FMwj |U∗c = 0 and∂FMwj∂P|U∗c =∂FMwj∂Mwj|U∗c = 0, allcomponents Mwj of the linearized problem satisfyDwM′′w + cM′w + F∗wMw = 0 (4.16)whereF ∗w = P∗c(qcmβacwacw +M∗c− qhmαwP ∗c + d)(4.17)It is known that F ∗w > 0 for case (2) (see Eq. (3.28) of chapter 3). A lower bound for thepropagation speed c∗w ≤ c is thus given byc∗w = 2√DwF ∗w = 2√DwP ∗c(qcmβacwacw +M∗c− qhmαwP ∗c + d). (4.18)One can conclude that in case (2) commercial inoculants are displaced by the nativecommunity at a minimal speed c∗w. The speed increases with increasing mutualist qualityof the inoculants (i.e., with larger αc/βc ratio, determining the value of P∗c and M∗c ) andwith decreasing competitive strength of the inoculants (acw), and it is independent onthe dispersal ability of the inoculants. The linear speed of Eq. (4.18) seems to be a goodapproximation when the competitive ability of the inoculants is small (i.e., acw is large,see Fig. 4.2c), and when the mutualist quality of the inoculants is low (i.e., the ratioαc/βc is small, see Fig. 4.2b). However the speed c∗w appears to be non-linear for highercompetitive ability and mutualist quality of the inoculants.Bistable traveling waveCase (4), when N > 1: In this case competition between the inoculants and the nativecommunity is strong, and the steady state U∗w and U∗c are both stable. I will first discussthe case where N > 1, i.e., the native community consists of two or more fungal species.To understand which steady state will invade the other, I will determine the sign of thepropagation speed c. For this purpose, I multiply Eq. (4.9b) by M ′c and integrate overthe whole z-domain. I obtain the equation∫ ∞−∞[DcM′′cM′c + cM′2c + FMcM′c]dz = 0 . (4.19)We know that M ′c = 0 when z → ±∞. Hence∫∞−∞M′′cM′cdz =∫∞−∞ 1/2((M′c)2)′dz = 089and Eq. (4.19) can be rewritten asc∫ ∞−∞(M ′c)2dz︸ ︷︷ ︸>0= −∫ ∞−∞FMcM′cdz . (4.20)The sign of the wave speed c depends therefore on the sign of the integral on the right-handside of Eq. (4.20). One can observe that−∫ ∞−∞FMcM′cdz = −∫ ∞−∞FMcMcM ′cMcdz = −∫ ∞−∞FMc2Mc(M2c )′dz. (4.21)According to the boundary conditions of Eq. (4.11),FMcMc(−∞) = FMcMc(U∗c) = 0, andFMcMc(+∞) = FMcMc(U∗w) = F∗c . (4.22)From chapter 3 we know that, for case (4), F ∗c < 0. Additionally, from numerical sim-ulations one can notice that the slope of Mc is really steep (see Table 4.2) and thus theintegral of Eq. (4.21) can be approximated by−∫ ∞−∞FMc2Mc(M2c )′dz =12(FMcMc(∞)− FMcMc(−∞))=12F ∗c < 0 . (4.23)Hence, the integral on the right side of Eq. (4.21) is always negative. I conclude thatthe propagation speed c is negative, and therefore the stable steady state U∗w is expectedto invade the state U∗c . In other words, the commercial inoculants are displaced by thenative fungal community.To gain more insights into the speed at which commercial inoculants are displaced,I explore the dependence of the speed on diversity of the native community, and on thedispersal ability and mutualist quality of the inoculants numerically (see Fig. 4.3). I findthat the speed at which inoculants are displaced increases with increasing diversity inthe native community, and decreases with increased dispersal ability and with decreasingmutualist quality of the inoculants.Case (4), when N = 1: I consider now the case where a strong competing inoculantmc is introduced in the field is inhabited by a single native fungal species mw, that isa strong competitor as well. In this case, the competitive exclusion steady states U∗w(only mw) and U∗c (only mc) are both stable. If mc and mw have different parameters,the strongest competitor will invade the other. A competitive advantage can be given bylower mutualist quality (i.e., smaller α/β ratio) or lower maintenance cost (i.e., smallerµm).If the two fungi are identical, none of the species has a competitive advantage over90the other and stationary solutions will be observed. To explore this scenario numerically,I consider on the left of the domain a steady state where mw is zero, and on the righta steady state a steady state where mc is zero. In this case, neither mw nor mc willprevail over the other. At the boundary between the two domains, where the two fungalspecies meet, we observe an increase in plant biomass density due to the partial presenceof both species (see Fig. 4.4). The increase in plant biomass density is less pronouncedfor stronger competition between the fungi.91Figure 4.1: Speed of invasion of the inoculants in a weak competing native community (cases (1) and (3))as a function of (a) the diffusion constant of the inoculants, (b) the mutualist quality of the inoculants, and(c) the number of species present in the native community. The solid lines are the theoretically predictedlower bounds (see Eq. (4.15)) while the scattered plots are the results of the numerical simulations. Thespeed of invasion does not depend on whether the inoculants coexist with the native community (case(1)) or displace the native community (case (3)). In figures (a) and (b) competition parameters areaww = awc = 2.2, acw = 0.3, αw = βw = 0.4. In figure (c) aww = awc = 3.0, acw = 0.1, αw = 0.7 and βw= 0.3. Other parameter values are qhp = 3, qcm = 2, qhm = qcp = 1, µp = µm = 0.3, d = 1.2, rp = 0.02,Dw = 0.3.92Figure 4.2: Numerical simulations representing the speed at which weakly competing inoculants aredisplaced by the native community (case (2)). The solid lines are the theoretically predicted lower bounds(see Eq. (4.18)) while the scattered plots are the results of the numerical simulations. The speed is plotas a function of (a) the diffusion constant of the inoculants, (b) the mutualist quality of the inoculants,and (c) the competitive strength of the inoculants. Competition parameters are aww = 2.2, awc = 0.3.Other parameters correspond to those of Fig. 4.1a.93Figure 4.3: Numerical simulations representing the speed at which competitive inoculants are displacedby a strongly competing native community (case (4), N > 1), as a function of (a) the diffusion constantof the inoculants, (b) the mutualist quality of the inoculants, and (c) the number of species present inthe native community. Competition parameters are aww = 2.2, awc = acw = 0.3. Other parameterscorrespond to those of Fig. 4.1a.94Figure 4.4: Numerical simulation showing the plant (solid lines) and fungal biomass density distributionresulting from the encounter of two strongly competing fungi (mc, scattered line, and mw, dotted line),for different competition strengths. Competition parameters acw and awc have the same value and theyare varies from 0.1 (strongest competition observed, black curves) to 0.7 (weakest competition observed,green curves) in steps of 0.2. Other parameters correspond to those of Fig. 4.1.4.3.2 Investigating inoculum persistence and spreadWhen varying the strength of competition between the community and the inoculants oneencounter four different possible scenarios, summarised in Table 4.2. The mathematicalfoundations on which my results are based are presented in the analysis.Case (1): Inoculants and native community coexistence: When competitionbetween the inoculated species and the existing community is low, coexistence is possibleand the inoculants can establish and spread in the field. The community will experiencea reduction in biomass density due to the introduction of the inoculants. This reductionhowever should not threaten the community survival. The spread speed increases withincreasing diffusion ability and with decreasing mutualist quality of the inoculants (seeEq. (4.15) and Figs. 4.1a and 4.1b). Additionally, the spread speed of the inoculants islower in the presence of a diverse native community (see Fig. 4.1c).Case (2): Inoculants do not persist: If the inoculated species is competitively in-ferior than the native fungal species present, the community prevents the persistence ofthe inoculants. The community will displace the commercial fungi at a speed that in-creases with increasing mutualist quality and with decreasing competitive ability of theintroduced fungus (Figs. 4.2b and 4.2c), and it is independent on the dispersal ability ofthe inoculants (Fig. 4.2).95Case (3): Invasion of the inoculants and displacement of the native com-munity: If the inoculated species is competitively superior than the native community,fungal inoculants establish, displace the native population and spread throughout the fieldcausing a regional biodiversity loss. Diversity in the native community can contribute toreduce the invasion speed of the inoculants (see Eq. (4.15) and Fig. 4.1c). Higher dispersalability and low mutualist quality of the inoculants coincide with a higher invasion speed(see Eq. (4.15) and Figs. 4.1a and 4.1b).Case (4): Inoculants establish locally but fail to persist: When competitionbetween the inoculated species and the existing community is high, coexistence is not likelyto occur. The initial establishment of the inoculants depends mainly on propagule densityand on the initial biomass density of the native community (as discussed in chapter 3).Establishment is not possible when the biomass density of the native fungal community ishigh, while it is likely to occur when the density of native fungi is low and propagule densityis high (see Fig. 3.7). For a combination of low native fungal density and high propagulepressure, the spatial dynamics follows a multi-step process illustrated in Fig. 4.6, that is:• The inoculants can temporarily establish and spread from their initial location.This initial spread prevents the growth of native fungi in areas where the density offungal inoculants is high, but at the same time the massive presence of inoculantsaccelerates plant growth (Fig. 4.6, T = 20).• Beyond the dispersal range of the inoculants, the density of native fungi increasesand so does plant productivity, eventually reaching a higher density than when onlythe inoculants are present (Fig. 4.6, T = 30 and T = 40).• A higher plant density determines higher resource availability, what gives a compet-itive advantage to the native community with respect to the inoculants. The nativecommunity is able to recolonize the portion of field taken by the inoculants leadingthe inoculated population to extinction (Fig. 4.6, T = 60 and T = 100).The speed at which the inoculants are displaced increases with increasing diversity of thenative fungal community, and decreases with increasing dispersal ability of the inoculantand with decreasing mutualist quality (see Fig. 4.3).4.3.3 Insights for agroecosystem managementNeighboring fields: The spread of weakly competing inoculants in a field causes adecline in native fungal biomass density (without, however, causing the extinction ofnative species), and an increase in plant biomass density. The decline in native fungiand increase in productivity are more pronounced when diversity in the native fungal96community is low. Strongly competing inoculants do not spread into a field where nativefungal diversity is high. When diversity is low, strongly competing inoculants are able toinvade the field, displace native fungi and cause a reduction in plant productivity. Theseresults are illustrated in Fig. 4.5.Inoculation and productivity: If the biomass density of the native fungal communityis low, inoculation increases the growth rate of the plant, independently of the inoculantsidentity and on their persistence. Generally, an increase in fungal abundance automati-cally corresponds to an increase in the plant growth rate, for any of the four cases discussedabove.In Fig. 4.7 I simulate plant and fungal growth over a landscape, following the inoc-ulation with weakly competing inoculants (case (2)). Although the inoculants can notestablish, they significantly enhance plant and native fungal growth in the whole field.The growth is even more enhanced if the inoculants can establish and coexist with thenative community (case (1), results not shown). Inoculation with strongly competingspecies leads as well to a short-term enhancement of plant growth (compare Fig. 4.7 withthe first two subfigures of Fig. 4.6). However in this case the introduction of the inocu-lants leads to a displacement of native fungi, and to negative long term consequences forplant productivity.4.4 Discussion4.4.1 Mathematical insightsPrevious work investigating traveling wave solutions in competitive systems have foundthat the speed of propagation can be linear or non-linear (Girardin, 2016, Huang, 2010,Lewis et al., 2002, Okubo et al., 1989), depending for example on interspecific competition(Huang and Han, 2011, Roques et al., 2015), or on the dispersal ability of the invasor(Holzer and Scheel, 2012). My analysis shows that in our specific case, when consideringthe spread of competitors in a heterogeneous interactive medium, the linear speed appearsto be a valid approximation to two of the monostable cases (Fig. 4.1). However, for highmutualist quality of the inoculants (Fig. 4.2b) and for high competitive ability of theinoculants (Fig. 4.2c) the speed appears to be non-linear. Similarly, Roques et al. (2015)found non-linearity when interspecific competition is high, and linearity when interspecificcompetition is low.In the bistable case, I show that a strong competitor is displaced by a native com-munity consisting of two or more species. Similarly, Guo et al. (2019) found that in abistable system two weaker competitors can displace a strongest competitor, under certainparametric conditions. As the competition system studied by Guo et al. (2019) is similar97to the one discussed in this chapter, we could expect to gain more analytical insights intothe bistable case by adapting Guo et al. (2019) analysis to our specific case. However,this task will be left to future work.4.4.2 Investigating inoculum persistence and spreadCase (1); Inoculants and native community coexistence: My results suggest thatif competition between the introduced species and the existing community is weak, e.g.,due to small niche overlap between the species present, the inoculants will coexist withthe rest of the community (case (1), Table 4.2). For example, if the introduced speciescomes to occupy an empty niche spaces available in the soil (Herbold and Moyle, 1986,Lekevicˇius, 2009, MacArthur and Levins, 1967), the inoculants will establish, persist andspread in the field.Experiments, indicate that inoculum establishment in the presence of a native AMfungal community is possible, and niche availability is one of the factors determininginoculation success (del Mar Alguacil et al., 2011, Farmer et al., 2007, Mummey et al.,2009, Sy`korova´ et al., 2012, Verbruggen et al., 2013). Mummey et al. (2009) showed thatthe identity of the fungal species present before inoculation is a strong determinant ofinoculation establishment, and Farmer et al. (2007) found that establishment was moresuccessful when the inoculants had a different functionality than the indigeneous fungalspecies present in the field.Case (2); Inoculants do not persist: If the competitive ability of the inoculants islow and the native community competes strongly with the inoculants (e.g., if no nichespace is available in the soil), persistence of the inoculants will not occur. Assumingthat the availability of empty niches decreases with increasing diversity and abundanceof AM fungal species, inoculum persistence becomes less likely in the presence of a wellestablished fungal community. Indeed, experiments have reported a negative relationshipbetween inoculum establishment and abundance of native AM fungi (Bender et al., 2019,Niwa et al., 2018, Thomsen and Hart, 2018).Case (3); Invasion of the inoculants and displacement of the native community:My results show that highly competitive fungi are likely to become invaders and cause abiodiversity loss in the native fungal community. Plenty of concerns about invasiveness ofcommercial inoculants and its detrimental consequences on ecosystem functionality havebeen raised by the scientific community (Hart et al., 2017, Ricciardi et al., 2017, Schwartzet al., 2006, Thomsen and Hart, 2018). Commercial fungi are mostly grown in-vitro andsubjected to intense breeding pressure (Gianinazzi and Vosa´tka, 2004), what gives them ahighly competitive functionality with respect to wild fungi, such as increased sporulationrate and a propensity to invest in their own reproduction, rather then in the mutualistic98relationship with the plant (Calvet et al., 2013, Jin et al., 2013, Kokkoris et al., 2019). Myanalysis clearly indicates that the risk of invasion from the inoculants is effectively present,where competitive fungi are likely to become invasors and high dispersal ability and lowermutualist quality can contribute to increasing the invasion speed of the inoculants (seeFig. 4.1a and 4.1b).The presence of a diverse native community can mitigate the invasion risk by reducingthe invasion speed (see Fig. 4.1a), or by making establishment of an invasor less likely tooccur (see Eq. (3.42) and Fig. 3.7). Indeed, the results presented in chapter 3 show howthe total biomass of a community of coexisting weak competitors (e.g., species presentinga small niche overlap) increases much faster than the biomass of a single species. A largerbiomass provides a competitive advantage to the community over the introduced species,by preventing access by the introduced species to the common resource. The situationwhere competitive inoculants are added to a strongly competing community correspondsto case (4), discussed below.Case (4); Inoculants establish locally but fail to persist: My model predicts thatif competition between the introduced fungi and the existing community is strong, e.g.due to a niche overlap between the inoculants and the native community, coexistence isnot possible. The introduced species will either establish and locally exclude the wholecommunity, or not establish at all. When a native community is present and has reacheda large biomass density, inoculation will likely be unsuccessful, in a scenario similar towhat discussed in case (2). However, when the biomass density of both, the inoculatedspecies and the community, are low, initial spatial introduction plays a decisive role indetermining the output of the competition. Inoculation when abundance of native fungiis low comports therefore the risk that the introduced species locally displaces the nativefungi and colonizes part of the field, leading to a decrease in belowground biodiversity.Eventually, the native community might be able to recolonize the field (see Fig. 4.6).However, recolonization might occur only on a long-time scale.The importance of founder effects and timing in AM fungal community formation hasbeen recognized empirically (Dickie et al., 2012, Dumbrell et al., 2011, Lekberg et al.,2012, Omirou et al., 2016, Verbruggen et al., 2013) and it is widely agreed that inoculumabundance is a key factors determining inoculum establishment (Bender et al., 2019, Niwaet al., 2018, Streeter, 1994, Sy`korova´ et al., 2012, Verbruggen et al., 2013). Additionally, adecline in native AM fungi following inoculation success has been experimentally observed(Bender et al., 2019, Koch et al., 2011, Pellegrino et al., 2012, Symanczik et al., 2015).All these results are in agreement with my predictions.994.4.3 Insights for agroecosystem managementNeighboring fields: My results underline the presence of an increased invasion riskwhen competitive inoculants are used in conjunction with agricultural management prac-tices that reduce the abundance and diversity of native fungi, such as tillage, fertilizationstretegy or fallow periods (Curaqueo et al., 2011, Karasawa and Takebe, 2012, Sa¨le et al.,2015, Schnoor et al., 2011). Indeed inoculants may be able to spread in a field and dis-place native fungi when native fungal diversity is low, while failing to spread when nativefungal diversity is higher.High competitiveness has been seen as a desirable trait in the inoculants, increasingestablishment success (Farmer et al., 2007, Verbruggen et al., 2013). However, here weshow that the highly competing inoculants are not desirable, as their spread can coincidewith the displacement of the native fungi and a decrease in plant productivity, as alreadysuggested by experimental studied (Bender et al., 2019, Koch et al., 2011, Pellegrino et al.,2012, Symanczik et al., 2015). On the other hand, the use of weakly competing inoculantsis desirable, as their spread increases plant productivity without causing the extinctionof native fungi. The increase is more significant when diversity in the native communityis low.Inoculation and productivity: I show that propagule biomass, and not necessarilythe type of inoculant itself, contributes to a short-term increase in plant productivity indepleted soils (i.e., when fungal biomass is low) (see Figs. 4.6 and 4.7). However, as seenin chapter 3, the positive effect of inoculation on productivity is less significant in thepresence of a well established native community (see Fig. 3.9). Experiments investigatingvariation in crop yield following field inoculation with AM fungi agree with these results,by showing that the immediate consequences of inoculation can range from positive (Ce-ballos et al., 2013, Hijri, 2016, Omirou et al., 2016, Zhang et al., 2019) to no significanteffect (Emam, 2016, Hamel and Smith, 1991, Loja´n et al., 2017, Ryan and Graham, 2002).It has also already been proposed that the short term positive impact on plant growthcan be related to an increase in fungal abundance, rather than to fungal identities (delMar Alguacil et al., 2011, Gosling et al., 2016, Lekberg and Koide, 2005, Pellegrino et al.,2011, Wagg et al., 2015).My results predict that if the introduced fungal species can effectively coexist with thenative community, inoculation will lead to positive, long-term consequences on productiv-ity (case (1)). Additionally, I show that the introduction of weakly competiting fungi canboost plant and native fungal growth and help to accelerate the process of repopulationof a field with a native fungal community, even without an effective long term persistencein the field (case (2)), in agreement with the observations of del Mar Alguacil et al. (2011)However, inoculation with strongly competing species may result into a biodiversity loss,and ultimately in a decrease in productivity (cases (3) and (4)).100Experiments monitoring the consequences of inoculation on fungal diversity and plantproductivity over a long period of time are rather rare. Although there is exhaustive evi-dence for a positive relationship between vegetation diversity and productivity (Male´zieuxet al., 2009, Ratnadass et al., 2012), relatively few studies have focused on the implica-tions of microbial diversity for crop yield and quality (Gosling et al., 2016, Hoeksemaet al., 2010, Lekberg and Koide, 2005, Wagg et al., 2015, 2011). These studies, arguethat diversity helps to maintain a stable and efficient crop production, due to the insur-ance effect and complementary benefits provided by the presence of multiple functionallydifferent species (Booth and Hoeksema, 2010, Maherali and Klironomos, 2007, Yachi andLoreau, 1999). My findings show that a diverse community contributes not only to anincrease plant biomass (see Fig. 3.2), but also to decreasing the invasion risk from a strongcompetitor, by reducing establishment success and invasion speed.My work suggests that it would be best to use weakly competing species, as stronglycompeting inoculants can be harmful for the ecosystem. I suggest the use of regenerativeagriculture, rather then agricultural practises that can reduce below-ground biodiversityand increase the invasion risk. My results show that the presence of a native fungal com-munity plays a critical role in determining establishment success and plant productivity.To confirm my hypotheses, there is an urgent need of empirical studies considering theeffect of the resident community over space and time.101Figure 4.5: Simulation showing the movement of the commercial inoculants in a a field when diversityof native fungi is (a) high or (b) low. The biomass density of strongly competing inoculants is indicatedby the dashed red line, while the biomass density of weakly competing inoculants is shown by thedashed blue line. Competition parameters are awc = acw = 0.3 for strongly competing inoculants, andawc = acw = 2.2 for weakly competing inoculants. αc = 0.35, βc = 0.45. Other parameters correspondto those of Fig. 4.1.102Figure 4.6: Spatio-temporal dynamics observed after inoculation when competition between the inoc-ulants (mc) and the native fungal community (mw) is strong (Case (4) in Table 4.2), where propaguledensity is much larger than the biomass density of the native community. Model parameters used forthe simulation are: qhp = 3, qcm = 2, qhm = qcp = 1, α = β = 0.4, µp = µm = 0.3, d = 1.2, rp = 0.02,aww = 2.2, awc = acw = 0.3, Dc = Dwj = 5, N = 2.103Figure 4.7: Top row: spatio-temporal dynamics observed when the inoculants (mc) are weakly com-peting fungi that are eventually displaced by the native community mw (case (2) in Table 4.2), wherepropagule density is much larger than the biomass density of the native community. The dashed redline represents the inoculants density. The top row represents plant growth in the presence (solid blackline) or absence (dotted black line) of inoculants. The bottom row represents native fungal density in thepresence (solid blue line) or absence (dotted blue line) of inoculants. Model parameters used for the sim-ulation correspond to those for Fig. 4.6, except for the competition parameters that are aww = acw = 2.2,awc = 0.3.104Chapter 5Fungal diversity and plantproductivity5.1 Background informationIn nature, multiple species of AM fungi can coexist in the roots of the same host plant(Verbruggen et al., 2012). It is proposed that a broad level of functional diversity existsamong coexisting fungal species (Feddermann et al., 2010, Lee et al., 2013, Mensah et al.,2015, Powell and Rillig, 2018, Smith et al., 2000). However, the consequences of thisdiversity on the relationship between AM fungi, other organisms and the environment arelargely unexplored (but see Lee et al. (2013), Powell and Rillig (2018)). AM fungi playa key role in determining ecosystem productivity and function (Van der Heijden et al.,1998), and a better understanding of the relationship between diversity in the fungalcommunity composition and plant growth is of particular importance.With the occurrence of functional complementarity among fungal species one wouldexpect plant benefit to be higher in the presence of multiple species of AM fungi than in thepresence of any of the species separately (Gianinazzi et al., 2010, Koide, 2000). However,experimental observations on plant growth are contradictory (Alkan et al., 2006, Daft,1983, Edathil et al., 1996, Gustafson and Casper, 2006, Jansa et al., 2008, Maherali andKlironomos, 2007, Van der Heijden et al., 1998, Violi et al., 2007). Indeed, simultaneouscolonization by multiple AM fungal species does not always lead to a greater benefit forthe plant, and in some cases productivity has been shown to be larger in the presence ofa single, most beneficial fungal species (Daft, 1983, Edathil et al., 1996, Gustafson andCasper, 2006, Jansa et al., 2008, Violi et al., 2007). It is therefore unclear under whatcircumstances the presence of multiple fungal species is beneficial for the plant and whatare the mechanisms behind this synergy.In an experiment, Jansa et al. (2008) compared plant and fungal growth at harvest timeacross different plant-AM fungi combinations. Two plants (medic and leek), and three105fungal species were used (Glomus mossae, Glomus intraradices, and Glomus claroideum).Their results suggest that a plant grown with a mixture of two fungal species receives morephosphorus than when grown in association with either of the two species alone. Jansaet al. (2008) attributed this synergistic effect to complementary phosphorus absorptionstrategies of the fungal species, according to the idea proposed by Koide (2000). Thishypothesis is based on the finding that AM fungal species often present different growthhabits, that affect where lies the phosphorus absorption volume of each species (Smithet al., 2000). Indeed, one of the fungal species used in the Jansa et al. (2008) experimentis known to absorb phosphorus from soil volume close to the roots, while the other oneabsorbs from further away (Jansa et al., 2005). The association with both fungal specieswould therefore allow the plant to receive more phosphorus than when in combinationwith any of the two species alone.Nevertheless, attributing the occurrence of a synergy in plant growth to differences inphosphorus absorption strategies fails to explain key aspects of the Jansa et al. (2008)results. Indeed, the synergy occurred only for a mixture of two of the three species tested,despite the fact that all species differed in their phosphorus absorption strategies. In thischapter, I provide an alternative mechanism for the results of Jansa et al. (2008). I showthat differences in carbon sink strength are likely to induce a synergistic effect in plantgrowth.I use the mathematical model for plant and fungal growth presented in chapter 2 toinvestigate plant and AM fungal growth mechanistically, by disentangling the effects ofphosphorus supply and carbon sink strength on plant and fungal biomass over time. Thethree main questions I wish to answer are the following:Q1 : According to model predictions, how do the (i) phosphorus supply and (ii) carbonsink strength of the fungal mutualists affect plant and AM fungal growth rates andfinal sizes, when the plant is considered in pairwise association with a single AMfungal species.Q2 : What are model predictions in terms of plant and AM fungal growth rates and finalsizes, when the plant is simultaneously colonized by two fungal species presentingdifferences in (i) phosphorus supply and (ii) carbon sink strength?Q3 : Can the results of Jansa et al. (2008) be qualitatively reproduced in a simulationexperiment that involves species of AM fungi with different phosphorus supply andcarbon sink strength?My results outline how the presence of multiple fungal species with complementary carbonsink strengths can contribute to maximize plant productivity.1065.2 Model and methodsTo simulate the coupled growth of multiple species of AM fungi sharing a resource suppliedby the same host plant, I use the ordinary differential equation model presented in chapter2, Eq. (2.1). The system of equations describing the variation over time of plant biomassp and the biomass of all fungal species mi is given bychange inplant biomass︷︸︸︷dpdt= qhpphosphorusfrom AM fungi︷ ︸︸ ︷∑i(αimi)pd+ p− qcpcarbonto AM fungi︷ ︸︸ ︷∑i(βimi) p +intrinsicgrowth︷ ︸︸ ︷qhp rp p−maintenancecost︷︸︸︷µp p2 , (5.1a)dmidt︸︷︷︸change in totalfungal biomass= qcmi βi pmi︸ ︷︷ ︸carbonfrom plant− qhmi αipd+ pmi︸ ︷︷ ︸phosphorusto plant− µmim2i︸ ︷︷ ︸maintenancecost. (5.1b)A description of the model parameters is given in Table 5.1. Model formulation is indetail in chapter 2 (see Eq. (2.1)).To answer the three research questions posed above, I focus on the effect of varyingparameters αi and βi, which quantify the resource exchange between the plant and theAM fungi, while other parameters are kept constant across all species (standard valueschosen for the simulations are given in Table 5.1). Parameter αi is a measure of the abilityof fungal species i to transfer phosphorus to the plant, while parameter βi quantifies theamount of carbon that species i receives from the plant, and consequently its carbonsink strength. Jansa et al. (2008) interpreted their results by relying on differences inphosphorus absorption strategies of the fungi. Here we show that differences in fungalcarbon sink strength (βi), can provide a comprehensive explanation of the results observed.5.2.1 Q1 and Q2: The effect of varying αi and βiTo understand the effect of varying phosphorus supply (αi) and carbon sink strength (βi)on plant and fungal growth rates and final sizes, I compute numerical solutions of themodel given in Eq. (5.1) in a different scenario for each question:Q1 : The plant is considered in pairwise association with two fungal species separately(see Fig. 5.1), left panel.Q2 : The plant is simultaneously colonized by two fungal species (see Fig. 5.1, rightpanel).For both scenarios, I consider the two following cases:(a) Fungal species m1 and m2 differ in their ability to transfer phosphorus to the plant(α1 6= α2) but have the same carbon sink strength (β1 = β2).107(b) Fungal species differ in carbon sink strength (β1 6= β2) but have the same ability totransfer phosphorus to the plant (α1 = α2).As seen in chapter 2, the model predicts bounded growth curves for all species. The finalsizes correspond to plant and fungal biomass at equilibrium. The average growth rate iscalculated by taking the biomass corresponding to 95% of the the final size, and dividingit by the time needed to reach 95% of the final size. Results are shown in Figs. 5.2 and5.3.Figure 5.1: Schematic representation of the scenarios used to address question 1 (left panel) and 2(right panel), through numerical simulation. In the left panel I consider a plant interacting separatelywith two different species of AM fungi, m1 and m2, taken one at a time. In the right panel the plantinteracts with the two fungal species simultaneously. Fungi m1 and m2 share the resource supplied bythe plant and therefore interact indirectly with each other.5.2.2 Q3: Qualitative reproduction of experimental resultsIn order to reproduce Jansa et al. (2008) results I perform the following steps:Step 1 : I qualitatively estimate fungal parameters of the three different species of AM fungiused by Jansa et al. (2008) by identifying general trends in the fungal growth curvesobtained experimentally (see Fig. A.1, reproduced with permission from Jansa et al.(2008)) and comparing these curves to the model output of a plant (medic or leek) inpairwise association with each of the fungal species. I refer to the fungal species as‘M’ for Glomus mossae, ‘I’ for Glomus intraradices and ‘C’ for Glomus claroideum.For simplicity, I differentiate fungal species ‘M’, ‘I’ and ‘C’ only by their parametersα and β, while keeping all other parameters constant across all species.Step 2 : I solve the system given in Eq. (5.1) numerically for each of the array of plant-AM fungi associations studied by Jansa et al (2008). The cases considered are thefollowing:(i) The plant is grown with each of the three fungal species separately (caseslabeled ‘M’, ‘I’ and ‘C’).(ii) The plant is grown with a mixture of two fungal species (cases labeled ‘MI’,‘MC’, ‘CI’)108(iii) The plant is grown with a mixture of all three fungal species (case labeled‘MIC’).In the model, fungal species contribution to plant growth is non additive (see lastterm in Eq. (5.1)b). To be able to compare plant growth in the presence of one,two or three fungal species, I assume the plant p to always be in association withsix AM fungi. The six AM fungi can be all of the same species (case (i)), three ofone species and three of another (case (ii)), or two individuals of each of the threespecies (case (iii)).Step 3 : For all of the cases considered in Step 2, I compute plant biomass and total fungalbiomass at a time that I assume to correspond to the harvesting time of 8 weeks ofJansa et al. (2008). I qualitatively compare simulations of fungal and plant biomassat harvest time to the plots of Fig. A.2, reproduced with permission from Jansaet al. (2008). Observations are limited to the identification of general trends in thegrowth curves of the plant and the AM fungi, rather than a quantitative analysis.The model estimates total fungal biomass, while Jansa et al. (2008) measured hyphallength density, i.e., only a proportion of the total fungal biomass, not including, forexample, the fungal biomass due to spores.5.3 Results5.3.1 Q1 and Q2: The effect of differences in phosphorus supplyand carbon sink strengthThe model predicts that variations in phosphorus supply and in carbon sink strength willaffect growth differently. An increase in phosphorus supply (α) increases both averagegrowth rates and final sizes of the plant and AM fungi (Fig. 5.2a). From the analyticalexpression for plant (p∗) and fungal biomass (m∗) at steady state (see Eqs. (2.11) and(2.13)) we know that p∗ is linear with respect to α, and m∗ increases with α as well.Final sizes and average growth rates respond nonmonotonically to changes in carbon sinkstrength (β), increasing at low β and decreasing at high β (Fig. 5.2b). The values of βmaximizing plant and fungal final sizes are much lower than the β values that maximizesthe average growth rates of the plant and of the fungi. There exists therefore a window ofβ values for which an increase in β leads to a decrease in the final sizes and to an increasein average plant and fungal growth rates.Plant growth in the presence of two species of AM fungi differing in phosphorus supplylies between the growth rate observed when the plant is in association with each of thespecies alone. That is, plant biomass associating with a mixture of two fungal species isconsistently lower than that ultimately obtained when the plant is in pairwise association109with the species with the largest αi, and consistently higher than when the plant is inassociation with the species with the lower αi (Fig. 5.3a, upper figure). The same canbe said for the total fungal biomass: This biomass is an average of that obtained for aplant-AM fungi pairwise association with species 1 and 2 (Fig. 5.3a, bottom figure).In contrast, the situation with AM fungi differing in carbon sink strength is morecomplex. In the model, a higher plant final size can be reached when the carbon sinkstrength βi is low, but at the expense of the plant growth rate, which is lower for smallβi (Fig. 5.2). If the plant is in simultaneous association with two species of AM fungidiffering only in their carbon sink strength, for a temporary period of time the plant willdo better when both species are present than in association with either of the species alone(Fig. 5.3b, upper figure). The combination of two fungal species with different carbonsink strength can therefore lead to a temporary synergistic effect in plant growth. Thesynergy is not necessarily seen in total fungal biomass (Fig. 5.3b, bottom figure) and canbe observed only when initial fungal biomass is low enough (see Fig. 5.4). If initial fungalbiomass is large, the species of AM fungi with the lower carbon sink strength is the mostbeneficial to plant growth.Figure 5.2: Qualitative effect on plant and fungal growth of varying (a) phosphorus supply α to theplant and (b) carbon sink strength β of the fungal species. In (a) parameter β is set constant to 0.5,while α is increased from 0.3 to 0.8 in steps of size 0.01. In (b) α = 0.6 and β is increased from 0.25 to0.75 in steps of size 0.01. Initial biomasses are p(0) = 0.3 and m(0) = 0.15.110Figure 5.3: Plant and total biomass over time is computed when the plant is in association with twofungi of the same species, either species 1 or species 2 (dotted lines) and when the plant is in simultaneousassociation with both species (solid lines, 12). In (a) species 1 and 2 differ only in their ability to transferphosphorus to the plant, where species 1 provides a larger quantity of phosphorus with α1 = 0.6 andα2 = 0.5 while β1 = β2 = 0.4. In (b) species 1 and 2 differ only in their carbon sink strength, wherespecies 1 has stronger carbon sink with β1 = 0.4 and β2 = 0.3, while α1 = α2 = 0.6. Initial biomassesare p(0) = 0.3 and mi(0) = 0.05.111Figure 5.4: Plant biomass over time for different initial fungal biomass (top figure: mi(0) = 0.2, bottomfigure: mi(0) = 1.0). The plant is in association with two fungi of the same species, either species 1 orspecies 2 (dotted lines) or plant is in simultaneous association with both species (solid lines, 12). Species1 and 2 differ only in their carbon sink strength, where species 1 has stronger carbon sink with β1 = 0.4and β2 = 0.3.5.3.2 Q3: Qualitative reproduction of experimental resultsFrom the measurements presented in Jansa et al. (2008) (Fig. 1) one can observe that:(i) Species ‘I’ has a reduced growth rate (especially when in association with leek) butreaches a larger final size with respect to species ‘M’ and ‘C’.(ii) Species ‘M’ has a faster growth rate and reaches a higher final size than ‘C’.In light of the observation above, and the model behaviours presented in section 5.3.1, Iassume that:(i) Species ‘I’ acts has low carbon sink strength, as low carbon sink strength reducesthe growth rate and increase final size of the AM fungi. I consequently chose βI tobe smaller than βM and βC . As the decrease in growth rate and the increase in finalsize are more pronounced when species ‘I’ is in association with leek, with respectto medic, I chose βI(leek) < βI(medic).(ii) Species ‘M’ supplies the plant with more phosphorus than species ‘C’, i.e., αM > αC ,as phosphorus transfer contributes positively to the fungal growth rate and final size.Other parameters are estimated such that growth curves given by the simulation qual-itatively match the growth curves shown in Fig. A.1 (reproduced with permission from112Jansa et al. (2008)). The growth curves of species ‘M’, ‘I’ and ‘C’ produced by the sim-ulation experiment are shown in Fig. 5.5. Plots of plant and total fungal biomass at achosen harvesting time for different plant-AM fungi combinations are shown in Fig. 5.6.Results are qualitatively comparable to Fig. A.2, reproduced with permission from Jansaet al. (2008).Simulation results show that total fungal biomass in the presence of multiple speciesis generally an average of that observed when each of the species is considered in pairwiseassociation with the plant. The same trend can be observed in plant biomass, whenconsidering medic as the host plant. Total plant biomass in leek, however, may be largerwhen the plant is in simultaneous association with a mixture of species of AM fungi, thanwith any of the species alone. These results are coherent with the observations of Jansaet al. (2008).Simulations of Fig. 5.3 show that a synergistic effect in plant growth occurs as a con-sequence of significant differences in carbon sink strength of the fungal species inoculatedsimultaneously (parameter β). Indeed, in the model the carbon sink strength of species‘I’ in leek (βI) is significantly lower than the carbon sink strength of species ‘M’ and ‘C’(βM and βC). The simulation experiment predicts the occurrence of a synergistic effectwhen leek is grown with a mixture of species ‘I’ and ‘C’ or ‘M’. However, experimentallythe synergy was observed only for a mixture of species ‘I’ and ‘C’ (Jansa et al., 2008).A second inconsistency between the theoretical and experimental results is that the totalbiomass of species ‘I’ in leek predicted by the simulation is much lower than that observedby Jansa et al. (2008).Figure 5.5: Experimental simulation obtained by solving Eq. (5.1) of the growth over time of fungalspecies ‘M’, ‘I’ and ‘C’ when regarded in pairwise association with a plant (medic or leek) (compare withFig. A.1). Fungal species are differentiated by parameters α and β only, where αM = 0.56, βM = 0.5, αI= 0.55, βI = 0.42 (medic) and βI = 0.3 (leek) , αC= 0.46, βC = 0.5. Initial conditions are p(0) = 0.1and m(0) = 0.04, and rp = 0.113Figure 5.6: Total fungal biomass (top figures) and plant biomass (bottom figures) computed when theplant (medic or leek) is grown with single fungal species (‘M’, ‘I’ and ‘C’), when a combination of twofungal species are simultaneously grown on the same plant (‘MI’, ‘MC’ and ‘IC’) and when the plant isgrown with a mixture of all three species (‘MIC’) (compare with Fig. A.2). All biomass values correspondto those obtained at time T = 30 in the simulation experiment.5.4 Discussion5.4.1 Q1 and Q2: The effect of differences in phosphorus supplyand carbon sink strengthIt has been proposed that high nutrient supply from the fungi to the plant positivelycorrelates with increased plant biomass (Mensah et al., 2015, Verlinden et al., 2018).However, the effect of differences in carbon sink strength on productivity has remainedexperimentally and theoretically unexplored.My results suggest that differences in carbon sink strength can induce a temporarysynergistic effect in plant growth. Fungal species with high carbon sink strength aremost beneficial when the total fungal biomass is low. In this case, a higher amount of114carbon per biomass intake by the fungi helps the quick establishment of an abundantfungal community, with a consequent increase in the plant growth rate. Fungal speciescharacterised by low carbon sink strength are not able to support plant growth when fungalbiomass is low, but by reducing the carbon cost to the plant they help the plant to reacha higher final size once total fungal biomass is large enough. Hence, to maximize plantproductivity in depleted soils (i.e., where fungal biomass is low), plants should ideally beassociated with multiple fungal species with high and low carbon sink strengths, wherefungi with high carbon sink strength contribute to an initial increase in the plant growthrate, and fungi with low carbon sink strength support a maximal plant final size.I find that differences in phosphorus absorption do not result in any synergistic effectson plant growth. An increase in phosphorus supply always induces an increased plantgrowth rate and final size. The combination of two fungal species that differ in phosphorussupply to the plant does, therefore, result in at each point in time in plant and fungalgrowth rates and final sizes that are an average of that observed when the plant is grownin association with the respective fungal species separately. Plant productivity is thereforemaximized in the presence of fungal species with high phosphorus supply.5.4.2 Q3: Qualitative reproduction of experimental resultsModel predictions present qualitatively similar results to Jansa et al. (2008), with plantand fungal biomasses obtained in a mixture being generally an average of that obtained forthe pairwise plant-AM fungus associations. The combination of two of the three speciesused (‘CI’) was shown to have a synergistic effect on plant growth, coherently with whatobtained experimentally. The model attributes the synergy to differences in AM fungalcarbon sink strength, and not to different phosphorus absorption strategies as suggestedby Jansa et al. (2008). As discussed above, I suggest that species with different carbonsink strength can be complementary in their function, where fungal species with highcarbon sink strength better enhance the plant growth rate when fungal biomass is low,and fungi with low carbon sink strength are more beneficial at large fungal biomass.While the simulation and experimental results show remarkable correspondence, thereare two features of the experiments not reproduced by the model. First, the modelpredicts a synergistic effect in plant growth to occur not only for a mixture of species‘C’ and ‘I’ but also for a mixture of species ‘M’ and ‘I’. Indeed a synergistic effect dueto a combination of species ‘M’ and ‘I’ was theoretically expected to occur, due to thedifferences in carbon sink strength of the two fungi. However, this effect was not detectedexperimentally. A possible explanation can be found in the growth habits of the two fungi.Both, Glomus mossae (species ‘M’) and Glomus intraradice (species ‘I’) have similar rootcolonizatiom strategies (Hart and Reader, 2002, Jansa et al., 2005, Van Tuinen et al.,1998), and experiments show that both species differ in their spatial proliferation when115grown simultaneously on plant roots (Alkan et al., 2006), indicating competition. It istherefore likely that ‘M’ and ‘I’ compete for space and limit each other’s biomass in thisway. The model used to perform the simulation experiment does not take into accountlimitations due to competition for space. Competition for space would therefore explainthe overestimate in total fungal biomass of the ‘MI’ mixture. Second, the model estimatesthe fungal biomass of species ‘I’ to be much lower than what found experimentally. Thisdiscrepancy could be explained by the fact that Jansa et al. (2008) measurements inFig. A.1 considered root length colonized, while measurement in Fig. A.2 considered onlythe hyphal length density of the fungi, and did not take into account the additional fungalbiomass due to spores. However, our findings were given in terms of total fungal biomass.5.4.3 Future workThe similarities found between the Jansa et al. (2008) results and my simulation exper-iment suggest that my model can be an efficient tool in predicting, and explaining themechanisms behind, the response of plant productivity to a fungal community. For moreaccurate predictions, it could be interesting to infer the resource exchange parameters αand β from the experimental data, for instance by using maximum likelihood estimation.I find that fungal species with large phosphorus supply are always beneficial for plantgrowth, while a mixture of fungal species with complementary carbon sink strength canhelp to maximize plant growth at different points in time. For the verification of myhypotheses, systematic measurements of fungal and plant biomass over longer periods oftime are needed, as well as comparison of the grow curves of fungal species differing intheir carbon sink strength and in their phosphorus supply to the plant.116Table 5.1: Standard values of the variables of the model given in Eq. (5.1)Parameter DescriptionStandardvalueµp Plant maintenance rate 0.3µmi Fungal maintenance rate of species i 0.3qhp Conversion phosphorus to plant biomass 3qcmi Conversion carbon to fungal biomass ofspecies i2qcp Conversion carbon to plant biomass 1qhmi Conversion phosphorus to fungal biomassof species i1d Half-saturation constant 1.2rp Plant intrinsic growth rate 0.02αi Phosphorus supply from fungal species i 0.3− 0.8βi Carbon sink strength of fungal species i 0.3− 0.6117Chapter 6ConclusionManaging the conservation of mutualistic communities has become a global priority,due to the fact that two of the most common earth mutualisms, pollination and plant-microbe symbioses, play a key role in agricultural productivity (Ku¨hsel and Blu¨thgen,2015, Manoharan et al., 2017). A diverse mutualist community is typically associatedwith higher productivity and resilience (Ku¨hsel and Blu¨thgen, 2015, Manoharan et al.,2017). However, up to date little is known about the mechanisms behind the persistence ofthis diversity. My work provides a theoretical framework to determine the circumstancesthat can promote or threaten the maintenance of a diverse mutualist community, and theimpact of species gain or loss on plant productivity. In particular, I provide quantitativecriteria to determine the invasion success and spread speed of an alien mutualist in apre-existing community, and the benefits and biodiversity risks associated with speciesintroduction.When regarded in the context of the mycorrhizal symbiosis, these findings help toassess the consequences of the introduction of a commercially-grown fungus (the alienspecies) into a field inhabited by a native fungal community. I show that that inoculationof depleted soils leads to have a positive effect on productivity on a short-term scale.This positive effect is mainly due to an increase in fungal abundance, and it is largelyindependently on the identity of the inoculants.On a long-term scale however, the use of strongly competing fungi constitutes a bio-diversity risk and a consequent productivity loss. The use of weak competitors, such asfungi whose function is complementary to the one of the fungal species already presentin the soil, leads to a positive effect on productivity, by boosting plant growth while sup-porting the restoration of native fungal community. I show that particularly desirable isthe inoculation of depleted soils with a combination of fungal species that provide highphosphorus supply to the plant and differ in their carbon sink strength. Indeed, suchmixtures can induce a synergistic effect on plant growth.My results suggest that the presence of a native fungal community plays a key role indetermine the output of inoculation. Indeed in the presence of a well-established native118community inoculation is of little advantage to productivity, and establishment of theinoculants is less likely to be observed. Additionally the presence of a diverse native fungalcommunity can enhance community resilience, and mitigate the invasion risk following theintroduction of strongly competing inoculants. To confirm my hypothese, there an urgentneed of empirical studies considering the effect of the resident community on plant growthand inoculum establishment. Future work should attempt to estimate the value of theparameters used in the models and test the validity of the quantitative criteria providedto predict inoculum success and spread.Possible extensions of the models presented in this thesis could be used to evaluatethe effect of environmental variation on mutualist communities. For example the explicitconsideration of nutrients dynamics could answer questions related to the effect of soilwashing (Park and Son, 2017) or nutrients stripping (Jones et al., 2013). Mathematically,the nutrient dynamics could be explicitly modelled by considering an heterogeneous distri-bution of nutrients over a landscape, and introducing space-dependent nutrients exchangeparameters. The effect of climate change on the evolution of mutualist community couldalso be modelled by considering time-dependent parameters. Additionally, my modelsconsider only systems including multiple mutualists interacting with a single host or part-ner. Future models could be extended to include multiple possible partners and considerplant diversity. This extension could allow to analyse community assemblage and invasiondynamic in forests or agroecosystems (Horn et al., 2017, Potts et al., 2003).The contribution of my work to our understanding of mutualistic interactions may beimportant not only in the ecological and agricultural contexts, but also for economical(Noe¨, 2016) or medical purposes (Scher et al., 2015, Zhang et al., 2015). For example,the mutualistic interactions between organisms can be compared to the exchange of goodand services between trading partners in economical markets (Noe¨, 2016). 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The figures have been reproduced with permission from Jansa et al. (2008)139Figure A.2: Experimental results from Jansa et al. (2008), reproduced qualitatively in Fig. 5.6. Hyphallength density (top figures) and phosphorus uptake (bottom figures) are measured at harvest time, whenthe plant (medic or leek) is grown in association with different combinations of three fungal species(Glomus mossae (M, circles), Glomus intraradices (I, squares), and Glomus claroideum (C, triangles)).The figures have been reproduces with permission from Jansa et al. (2008)140
Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-ShareAlike 4.0 International
Graduate
Mathematical insights into mechanisms leading to coexistence and competitive exclusion among mutualist guilds
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