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Interactions among mutualism, competition, and predation foster species coexistence in diverse communities

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Abstract

In natural systems, organisms are simultaneously engaged in mutualistic, competitive, and predatory interactions. Theory predicts that species persistence and community stability are feasible when the beneficial effects of mutualisms are balanced by density-dependent negative feedbacks. Enemy-mediated negative feedbacks can foster plant species coexistence in diverse communities, but empirical evidence remains mixed. Disparity between theoretical expectations and empirical results may arise from the effects of mutualistic mycorrhizal fungi. Here, we build a multiprey species/predator model combined with a bidirectional resource exchange system, which simulates mutualistic interactions between plants and fungi. To reach population persistence, (1) the per capita rate of increase of all plant population must exceed the sum of the negative per capita effects of predation, interspecific competition, and costs of mycorrhizal association, and (2) the per capita numerical response of enemies to mycorrhizal plants must exceed the magnitude of the per capita enemy rate of mortality. These conditions reflect the balance between regulation and facilitation in the system. Interactions between plant natural enemies and mycorrhizal fungi lead to shifts in the strength and direction of net mycorrhizal effects on plants over time, with common plant species deriving greater benefits from mycorrhizal associations than rare plant species.

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References

  • Addicott JF (1979) A multispecies aphid–ant association: density dependence and species-specific effects. Can J Zool 57:558–569

    Article  Google Scholar 

  • Addicott JF (1981) Properties of 2-species models of mutualism: simulation studies. Oecologia 49:42–49

    Article  Google Scholar 

  • Addicott JF, Freedman HI (1984) On the structure and stability of mutualistic systems: analysis of predator–prey and competition models as modified by the action of a slow-growing mutualist. Theor Popul Biol 26:320–339

    Article  Google Scholar 

  • Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483:205–208

    Article  CAS  PubMed  Google Scholar 

  • Auge RM, Schekel KA, Wample RL (1987) Leaf water and carbohydrate status of VA mycorrhizal rose exposed to drought stress. Plant Soil 302:291–302

    Article  Google Scholar 

  • Bachelot B, Kobe RK (2013) Rare species advantage? Richness of damage types due to natural enemies increases with species abundance in a wet tropical forest. J Ecol 101:846–856

    Article  Google Scholar 

  • Bagchi R, Gallery RE, Gripenberg S, Gurr SJ, Narayan L, Addis CE, Freckleton RP, Lewis OT (2014) Pathogens and insect herbivores drive rainforest plant diversity and composition. Nature 506:85–90

    Article  CAS  PubMed  Google Scholar 

  • Barber NA, Adler LS, Theis N, Hazzard RV, Kiers ET (2012) Herbivory reduces plant interactions with above- and belowground antagonists and mutualists. Ecology 93:1560–1570

    Article  PubMed  Google Scholar 

  • Bardgett RD, Wardle DA (2010) Aboveground-belowground linkages. Oxford University Press, Oxford

    Google Scholar 

  • Bever JD (2003) Soil community feedback and the coexistence of competitors: conceptual frameworks and empirical tests. New Phytol 157:465–473

    Article  Google Scholar 

  • Bever JD, Westover KM, Antonovics J (1997) Incorporating the soil community into plant population dynamics: the utility of the feedback approach. J Ecol 85:561–573

    Article  Google Scholar 

  • Bi HH, Song YY, Zeng RS (2007) Biochemical and molecular responses of host plants to mycorrhizal infection and their roles in plant defence. Allelopathy J 20:15–27

    Google Scholar 

  • Booth MG, Hoeksema JD (2010) Mycorrhizal networks counteract competitive effects of canopy trees on seedling survival. Ecology 91:2294–2302

    Article  PubMed  Google Scholar 

  • Bronstein JL (2001a) The exploitation of mutualisms. Ecol Lett 4:277–287

    Article  Google Scholar 

  • Bronstein JL (2001b) The costs of mutualism. Am Zool 41:825–839

    Google Scholar 

  • Bronstein J, Wilson W, Morris W (2003) Ecological dynamics of mutualist/antagonist communities. Am Nat 162:S24–S39

    Article  PubMed  Google Scholar 

  • Bruno J, Stachowicz J, Bertness M (2003) Inclusion of facilitation into ecological theory. Trends Ecol Evol 18:119–125

    Article  Google Scholar 

  • Cantrell RS, Cosner C (2001) On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J Math Anal Appl 257:206–222

    Article  Google Scholar 

  • Chesson P (2000) Mechanisms of maintenance of species diversity. Annu Rev Ecol Syst 31:343–366

    Article  Google Scholar 

  • Chesson P, Kuang JJ (2008) The interaction between predation and competition. Nature 456:235–238

    Article  CAS  PubMed  Google Scholar 

  • Christiansen FB, Fenchel T (1977) Theories of populations in biological communities. Springer

  • Connell JH (1971) On the role of natural enemies in preventing competitive exclusion in some marine animals and in rain forest trees. Pages 298–312. In: Boer P J, Graadwell G R (eds) Dynamics of numbers in populations

  • Connell JH, Tracey JG, Webb LJ (1984) Compensatory recruitment, growth, and mortality as factors maintaining rain forest. Ecol Monogr 54:141–164

    Article  Google Scholar 

  • Diez JM, Dickie I, Edwards G, Hulme PE, Sullivan JJ, Duncan RP (2010) Negative soil feedbacks accumulate over time for non-native plant species. Ecol Lett 13:803–809

    Article  PubMed  Google Scholar 

  • Eppstein MJ, Molofsky J (2007) Invasiveness in plant communities with feedbacks. Ecol Lett 10:253–263

    Article  PubMed  Google Scholar 

  • Eppstein MJ, Bever JD, Molofsky J (2006) Spatio-temporal community dynamics induced by frequency dependent interactions. Ecol Model 197:133–147

    Article  Google Scholar 

  • Fitter A (1991) Costs and benefits of mycorrhizas: implications for functioning under natural conditions. Experientia 47:350–355

    Article  Google Scholar 

  • Fontaine C, Guimarães PR, Kéfi S, Loeuille N, Memmott J, van der Putten WH, van Veen FJF, Thébault E (2011) The ecological and evolutionary implications of merging different types of networks. Ecol Lett 14:1170–1181

    Article  PubMed  Google Scholar 

  • Fox L, Morrow P (1981) Specialization: species property or local phenomenon? Science 211:887–893

    Article  CAS  PubMed  Google Scholar 

  • Freedman HI, Addicott JF, Rai B (1987) Obligate mutualism with a predator: stability and persistence of three-species models. Theor Popul Biol 342:157–175

    Article  Google Scholar 

  • Gange A, West HM (1994) Interactions between arbuscular mycorrhizal fungi and foliar-feeding insects in Plantago lanceolata L. New Phytol 128:79–87

    Article  Google Scholar 

  • Gause G, Witt A (1935) Behavior of mixed populations and the problem of natural selection. Am Nat 69:596–609

    Article  Google Scholar 

  • Gehring CA, Whitham TG (1994) Comparisons of ectomycorrhizae on pinyon pines (Pinus edulis; Pinaceae) across extremes of soil type and herbivory. Ecology 81:1509–1516

    Google Scholar 

  • Georgelin E, Loeuille N (2014) Dynamics of coupled mutualistic and antagonistic interactions, and their implications for ecosystem management. J Theor Biol 346:67–74

    Article  CAS  PubMed  Google Scholar 

  • Hierro JL, Maron JL, Callaway RM (2005) A biogeographical approach to plant invasions: the importance of studying exotics in their introduced and native range. Ecology 93:5–15

    Article  Google Scholar 

  • Hoeksema JD, Chaudhary VB, Gehring CA, Johnson NC, Karst J, Koide RT, Umbanhowar J (2010) A meta-analysis of context-dependency in plant response to inoculation with mycorrhizal fungi. Ecol Lett 13:394–407

    Article  PubMed  Google Scholar 

  • Hoffmann D, Vierheilig H, Schausberger P (2011) Mycorrhiza-induced trophic cascade enhances fitness and population growth of an acarine predator. Oecologia 166:141–9

  • Hoffmann D, Vierheilig H, Peneder S, Schausberger P (2011) Mycorrhiza modulates aboveground tri-trophic interactions to the fitness benefit of its host plant. Ecological Entomology 36:574–581

  • Holland JN (2002) Benefits and costs of mutualism: demographic consequences in a pollinating seed-consumer interaction. Proc Biol Sci R Soc 269:1405–1412

    Article  Google Scholar 

  • Holland J, DeAngelis DL (2006) Interspecific population regulation and the stability of mutualism: fruit abortion and density-dependent mortality of pollinating seed-eating insects. Oikos 3:563–571

    Article  Google Scholar 

  • Holland JN, DeAngelis DL (2010) A consumer-resource approach to the density-dependent population dynamics of mutualism. Ecology 91:1286–1295

    Article  PubMed  Google Scholar 

  • Holland J, DeAngelis D, Bronstein J (2002) Population dynamics and mutualism: functional responses of benefits and costs. Am Nat 159:231–244

    Article  PubMed  Google Scholar 

  • Holland JN, DeAngelis DL, Schultz ST (2004) Evolutionary stability of mutualism: interspecific population regulation as an evolutionarily stable strategy. Proc R Soc Biol Sci 271:1807–1814

    Article  Google Scholar 

  • Holland JN, Wang Y, Sun S, DeAngelis DL (2013) Consumer–resource dynamics of indirect interactions in a mutualism–parasitism food web module. Theor Ecol 6:475–493

    Article  Google Scholar 

  • Holling CS (1959) The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can Entomol 91:293–320

    Article  Google Scholar 

  • Jaenike J (1990) Host specialization in phytophageous insects. Annu Rev Ecol Syst 21:243–273

    Article  Google Scholar 

  • Jang SRJ (2002) Dynamics of herbivore-plant-pollinator models. J Math Biol 44:129–149

    Article  CAS  PubMed  Google Scholar 

  • Janzen DH (1970) Herbivores and the number of tree species in tropical forests. Am Nat 104:501–528

    Article  Google Scholar 

  • Jeschke J, Kopp M, Tollrian R (2002) Predator functional responses: discriminating between handling and digesting prey. Ecol Monogr 72:95–112

    Article  Google Scholar 

  • Johnson ANC, Graham JH (2013) The continuum concept remains a useful framework for studying mycorrhizal functioning. Plant Soil 363:411–419

    Article  CAS  Google Scholar 

  • Johnson ANC, Graham JH, Smith FA (1997) Functioning of mycorrhizal associations along the mutualism-parasitism continuum. New Phytol 135:575–586

    Article  Google Scholar 

  • Kardol P, Bezemer TM, Van der Putten WH (2006) Temporal variation in plant-soil feedback controls succession. Ecol Lett 9:1080–1088

    Article  PubMed  Google Scholar 

  • Kiers ET, Duhamel M, Beesetty Y, Mensah JA, Franken O, Verbruggen E, Fellbaum CR, Kowalchuk GA, Hart MM, Bago A, Palmer TM, West SA, Vandenkoornhuyse P, Jansa J, Bücking H (2011) Reciprocal rewards stabilize cooperation in the mycorrhizal symbiosis. Science 333:880–882

    Article  CAS  PubMed  Google Scholar 

  • Klironomos JN (2002) Feedback with soil biota contributes to plant rarity and invasiveness in communities. Nature 417:67–70

    Article  CAS  PubMed  Google Scholar 

  • Klironomos J, McCune J, Moutoglis P (2004) Species of arbuscular mycorrhizal fungi affect mycorrhizal responses to simulate herbivory. Appl Soil Ecol 26:133–141

    Article  Google Scholar 

  • Koide R (1991) Tansley review no. 29: nutrient supply, and nutrient to plant response mycorrhizal infection. New Phytol 117:365–386

    Article  CAS  Google Scholar 

  • Koide R, Elliott G (1989) Cost, benefit and efficiency of the vesicular-arbuscular mycorrhizal symbiosis. Funct Ecol 3:252–255

    Google Scholar 

  • Kondoh M, Kato S, Sakato Y (2010) Food webs are built up with nested subwebs. Ecology 91:3123–3130

    Article  PubMed  Google Scholar 

  • Koricheva J, Gange AM, Jones T (2009) Effects of mycorrhizal fungi on insect herbivores: a meta-analysis. Ecology 90:2088–2097

    Article  PubMed  Google Scholar 

  • Krebs JR (1974) Behavioral aspects of predation. In: Bateson HG, Klopfer P (eds) Perspectives in ethology. Plenum, New York, pp 73–111

    Google Scholar 

  • Lekberg Y, Koide RT (2005) Is plant performance limited by abundance of arbuscular mycorrhizal fungi? A meta-analysis of studies published between 1988 and 2003. New Phytol 168:189–204

    Article  CAS  PubMed  Google Scholar 

  • Leon JA, Tumpson DB (1975) Competition between two species for two complementary or substitutable resources. J Theor Biol 50:185–201

    Article  CAS  PubMed  Google Scholar 

  • Levine JM, Pachepsky E, Kendall BE, Yelenik SG, Lambers JHR (2006) Plant-soil feedbacks and invasive spread. Ecol Lett 9:1005–1014

    Article  PubMed  Google Scholar 

  • Loeuille N (2010) Influence of evolution on the stability of ecological communities. Ecol Lett 13:1536–1545

    Article  PubMed  Google Scholar 

  • May RM (1976) Models for two interacting populations. In: May RM (ed) Theoretical ecology. Saunders, Philadelphia, pp 49–70

    Google Scholar 

  • McGill B (2005) A mechanistic model of a mutualism and its ecological and evolutionary dynamics. Ecol Model 187:413–425

    Article  Google Scholar 

  • Melián CJ, Bascompte J, Jordano P, Krivan V (2009) Diversity in a complex ecological network with two interaction types. Oikos 118:122–130

    Article  Google Scholar 

  • Moora M, Opik M, Sent R, Zobel M (2004) Native arbuscular fungal communities mycorrhizal influence the seedling performance of rare differentially and common Pulsatilla species. Funct Ecol 18:554–562

    Article  Google Scholar 

  • Morales M (2000) Mechanisms and density dependence of benefit in an ant-membracid mutualism. Ecology 81:482–489

    Google Scholar 

  • Mordecai E (2011) Pathogen impacts on plant communities: unifying theory, concepts, and empirical work. Ecol Monogr 81:429–441

    Article  Google Scholar 

  • Mougi A, Kondoh M (2012) Diversity of interaction types and ecological community stability. Science 337:349–351

    Article  CAS  PubMed  Google Scholar 

  • Mougi A, Kondoh M (2014) Stability of competition-antagonism-mutualism hybrid community and the role of community network structure. J Theor Biol 360:54–58

    Article  PubMed  Google Scholar 

  • Murdoch W (1975) Diversity, complexity, stability and pest control. J Appl Ecol 12:795–807

    Article  Google Scholar 

  • Neuhauser C, Fargione JE (2004) A mutualism–parasitism continuum model and its application to plant–mycorrhizae interactions. Ecol Model 177:337–352

    Article  Google Scholar 

  • Newsham KK, Fitter AH, Watkinson AR (1995) Multi-functionality and biodiversity in arbuscular mycorrhizas. Trends Ecol Evol 10:407–411

    Article  CAS  PubMed  Google Scholar 

  • Novotny V, Miller SE, Baje L, Balagawi S, Basset Y, Cizek L, Craft KJ, Dem F, Drew RAI, Hulcr J, Leps J, Lewis OT, Pokon R, Stewart AJA, Samuelson GA, Weiblen GD (2010) Guild-specific patterns of species richness and host specialization in plant–herbivore food webs from a tropical forest. J Anim Ecol 79:1193–1203

    Article  PubMed  Google Scholar 

  • Olsson PA, Rahm J, Aliasgharzad N (2010) Carbon dynamics in mycorrhizal symbioses is linked to carbon costs and phosphorus benefits. FEMS Microbiol Ecol 72:125–131

    Article  PubMed  CAS  Google Scholar 

  • Pacala S, Crawley M (1992) Herbivores and plant diversity. Am Nat 140:243–260

    Article  CAS  PubMed  Google Scholar 

  • Pozo MJ, Azcón-Aguilar C (2007) Unraveling mycorrhiza-induced resistance. Curr Opin Plant Biol 10:393–398

    Article  CAS  PubMed  Google Scholar 

  • Rai B, Freedman H, Addicott LJ (1983) Analysis of three species models of mutualism in predator–prey and competitive systems. Math Biosci 50:13–50

    Article  Google Scholar 

  • Real L (1977) The kinetics of functional response. Am Nat 111:289–300

    Article  Google Scholar 

  • Ringel MS, Hu HH, Anderson G (1996) The stability and persistence of mutualisms embedded in community interactions. Theor Popul Biol 50:281–297

    Article  CAS  PubMed  Google Scholar 

  • Rosenzweig M, MacArthur R (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97:209–223

    Article  Google Scholar 

  • Roughgarden J (1975) Evolution of marine symbiosis—a simple cost-benefit model. Ecology 56:1201–1208

    Article  Google Scholar 

  • Salzer P, Hübner B, Sirrenberg A, Hager A (1997) Differential effect of purified spruce chitinases and beta-1, 3-glucanases on the activity of elicitors from ectomycorrhizal fungi. Plant Physiol 114:957–968

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Saravesi K, Markkola A, Rautio P, Roitto M, Tuomi J (2008) Defoliation causes parallel temporal responses in a host tree and its fungal symbionts. Oecologia 156:117–123

    Article  PubMed  Google Scholar 

  • Schmitt RJ, Holbrook SJ (2003) Mutualism can mediate competition and promote coexistence. Ecol Lett 6:898–902

    Article  Google Scholar 

  • Selosse MA, Richard FHX, Simard SW (2006) Mycorrhizal networks: des liaisons dangereuses? Trends Ecol Evol 21:621–628

    Article  PubMed  Google Scholar 

  • Silvertown JW (1980) The evolutionary ecology of mast seeding in trees. Biol J Linn Soc 14:235–250

    Article  Google Scholar 

  • Simard S, Durall D (2004) Mycorrhizal networks: a review of their extent, function, and importance. Can J Bot 82:1140–1165

    Article  CAS  Google Scholar 

  • Simonsen AK, Stinchcombe JR (2014) Herbivory eliminates fitness costs of mutualism exploiters. New Phytol

  • Smith SE, Read DJ (2008) Mycorrhizal symbiosis, 3rd edn. Academic, London

    Google Scholar 

  • Smout S, Asseburg C, Matthiopoulos J, Fernández C, Redpath S, Thirgood S, Harwood J (2010) The functional response of a generalist predator. PLoS One 5:e10761

    Article  PubMed Central  PubMed  CAS  Google Scholar 

  • Stephens DW, Krebs JR (1986) Foraging theory, vol 1. Princeton University Press, Princeton, pp 1–100

    Google Scholar 

  • Strauss S, Irwin R (2004) Ecological and evolutionary consequences of multispecies plant-animal interactions. Annu Rev Ecol Evol Syst 35:435–466

    Article  Google Scholar 

  • Suweis S, Grilli J, Maritan A (2014) Disentangling the effect of hybrid interactions and of the constant effort hypothesis on ecological community stability. Oikos 123:525–532

    Article  Google Scholar 

  • Tang S, Awar SP, Allesina S (2014) Correlation between interaction strengths drives stability in large ecological networks. Ecol Lett 17:1094–1100

    Article  PubMed  Google Scholar 

  • Thébault E, Fontaine C (2010) Stability of ecological communities and the architecture of mutualistic and trophic networks. Science 329:853–856

    Article  PubMed  CAS  Google Scholar 

  • Tilman D (1982) Resource competition and community structure. Princeton University Press, Princeton

    Google Scholar 

  • Toju H, Sato H, Yamamoto S, Kadowaki K, Tanabe AS, Yazawa S, Nishimura O, Agata K (2013) How are plant and fungal communities linked to each other in belowground ecosystems? A massively parallel pyrosequencing analysis of the association specificity of root-associated fungi and their host plants. Ecol Evol 3:1–13

    Google Scholar 

  • Van der Heijden MGA, Horton TR (2009) Socialism in soil? The importance of mycorrhizal fungal networks for facilitation in natural ecosystems. J Ecol 97:1139–1150

    Article  Google Scholar 

  • Van der Heijden MGA, Klironomos J, Ursic M, Moutoglis P, Streitwolf-Engel R, Boller T, Wiemken A, Sanders IR (1998) Mycorrhizal fungal diversity determines plant biodiversity, ecosystem variability and productivity. Nature 396:69–72

    Article  CAS  Google Scholar 

  • Van der Putten WH, Bardgett RD, de Ruiter PC, Hol WHG, Meyer KM, Bezemer TM, Bradford MA, Christensen S, Eppinga MB, Fukami T, Hemerik L, Molofsky J, Schädler M, Scherber C, Strauss SY, Vos M, Wardle DA (2009) Empirical and theoretical challenges in aboveground-belowground ecology. Oecologia 161:1–14

    Article  PubMed Central  PubMed  Google Scholar 

  • Vandermeer JH, Boucher DH (1978) Varieties of mutualistic interaction in population models. J Theor Biol 74:549–558

    Article  CAS  PubMed  Google Scholar 

  • Vannette R, Hunter MD (2011) Plant defense theory re-examined: nonlinear expectations based on the costs and benefits of resource mutualisms. J Ecol 99:66–76

    Article  Google Scholar 

  • Wangersky PJ (1978) Lotka-Volterra population models. Annu Rev Ecol Syst 9:189–218

    Article  Google Scholar 

  • Wardle DA (2002) Communities and ecosystems: linking the aboveground and belowground components. Princeton University Press, Princeton

    Google Scholar 

  • Wolfram Research, Inc. (2008) Mathematica, Version 7.0, Champaign

  • Wootton J (1994) The nature and consequences of indirect effects in ecological communities. Annu Rev Ecol Syst 25:443–466

    Article  Google Scholar 

  • Wright SJ (2002) Plant diversity in tropical forests: a review of mechanisms of species coexistence. Oecologia 130:1–14

    Article  Google Scholar 

  • Zhang Q, Xu L, Tang J, Bai M, Chen X (2011) Arbuscular mycorrhizal mediation of biomass-density relationship of Medicago sativa L. under two water conditions in a field experiment. Mycorrhiza 21:269–277

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Acknowledgments

Benedicte Bachelot was partially supported with funds from National Science Foundation DEB-524989 to MU. We are grateful to Dr. Agnes Bachelot, Dr. Alain Bachelot, Dr. Duncan Menge, Dr. Jason Hoeksema, Dr. Stefano Allesina, Dr. Samir Suweis, Dr. Charlotte Lee, Rachael Eaton, Benton Taylor, Bob Muscarella, Naomi Schwartz, and anonymous reviewers for useful comments on the manuscript. In particular, we are grateful to Axios Review that greatly helped us improve our manuscript.

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  1. Department of Ecology, Evolution and Environmental Biology, Columbia University, 1200 Amsterdam Avenue, New York, NY, 10027, USA

    Benedicte Bachelot & María Uriarte

  2. Department of Biology, Barnard College, Columbia University, New York, NY, 10027, USA

    Krista McGuire

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  1. Benedicte Bachelot
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Correspondence to Benedicte Bachelot.

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Bachelot, B., Uriarte, M. & McGuire, K. Interactions among mutualism, competition, and predation foster species coexistence in diverse communities. Theor Ecol 8, 297–312 (2015). https://doi.org/10.1007/s12080-015-0251-2

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