Parasitism within mutualist guilds explains the maintenance of diversity in multi-species mutualisms

Abstract

We consider here mutualisms where there are multiple species sharing a resource supplied by the same partner. If, as commonly assumed, there is competition between the species, then only the superior competitor should persist. Nevertheless, coexistence of multiple species sharing the same mutualistic partner is a widespread phenomenon. Regulation of nutrient exchange, where each species receives resources from the partner in proportion to the strength of the mutualism between the two, has been proposed as the main mechanism for coexistence in multi-species mutualisms involving the transfer of nutrients. Significant arguments, however, challenge the importance of partner selection processes. We present a mathematical model, applied to the arbuscular mycorrhizal symbiosis, to propose an alternative explanation for this coexistence. We show that asymmetric resource exchange between the plant and its fungal guild can lead to indirect parasitic interactions between guild members. In our model, the amount of carbon supplied by the plant to the fungi depends on both plant and fungal biomass, while the amount of phosphorus supplied by the fungi to the plant depends on both plant and fungal biomass when the plant is small, and effectively on fungal biomass only when the plant is large. As a consequence of these functional responses, more beneficial mutualists increase resource availability, and are indirectly exploited by less beneficial species that consume the resource and grow larger than they would in the absence of the better mutualists. As guild mutualists are not competing, competitive exclusion does not occur. Hence, the interaction structure can explain the maintenance of diversity within guilds in the absence of spatial structure and niche-related processes.

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References

  1. Abrams PA (1983) Arguments in favor of higher order interactions. The American Naturalist 121(6):887–891

    Google Scholar 

  2. Addicott JF (1981) Stability properties of 2-species models of mutualism: simulation studies. Oecologia 49(1):42–49

    PubMed  Google Scholar 

  3. Afkhami ME, Rudgers JA, Stachowicz JJ (2014) Multiple mutualist effects: conflict and synergy in multispecies mutualisms. Ecol 95(4):833–844

    Google Scholar 

  4. Aldrich-Wolfe L (2007) Distinct mycorrhizal communities on new and established hosts in a transitional tropical plant community. Ecol 88(3):559–566

    Google Scholar 

  5. Alkan N, Gadkar V, Yarden O, Kapulnik Y (2006) Analysis of quantitative interactions between two species of arbuscular mycorrhizal fungi, glomus mosseae and g. intraradices, by real-time pcr. Appl Environ Microbiol 72(6):4192–4199

    CAS  PubMed  PubMed Central  Google Scholar 

  6. Allesina S, Levine JM (2011) A competitive network theory of species diversity. Proceedings of the National Academy of Sciences 108(14):5638–5642

    CAS  Google Scholar 

  7. Andersen CP, Rygiewicz PT (1991) Stress interactions and mycorrhizal plant response: understanding carbon allocation priorities. Environ Pollut 73(3-4):217–244

    CAS  PubMed  Google Scholar 

  8. Archetti M, Scheuring I (2011) Coexistence of cooperation and defection in public goods games. Evolution: International Journal of Organic Evolution 65(4):1140–1148

    Google Scholar 

  9. Archetti M, Scheuring I (2013) Trading public goods stabilizes interspecific mutualism. Journal of theoretical biology 318:58–67

    PubMed  Google Scholar 

  10. Argüello A, O’Brien MJ, van der Heijden MG, Wiemken A, Schmid B, Niklaus PA (2016) Options of partners improve carbon for phosphorus trade in the arbuscular mycorrhizal mutualism. Ecol Lett 19(6):648–656

    PubMed  Google Scholar 

  11. Bachelot B, Lee CT (2018) Dynamic preferential allocation to arbuscular mycorrhizal fungi explains fungal succession and coexistence. Ecol 99(2):372–384

    Google Scholar 

  12. Batstone RT, Carscadden KA, Afkhami ME, Frederickson ME (2018) Using niche breadth theory to explain generalization in mutualisms. Ecol 99(5):1039–1050

    Google Scholar 

  13. Bennett AE, Bever JD (2009) Trade-offs between arbuscular mycorrhizal fungal competitive ability and host growth promotion in plantago lanceolata. Oecologia 160(4):807–816

    PubMed  Google Scholar 

  14. Bever JD (2015) Preferential allocation, physio-evolutionary feedbacks, and the stability and environmental patterns of mutualism between plants and their root symbionts. New Phytologist 205(4):1503–1514

    CAS  PubMed  Google Scholar 

  15. Bever JD, Richardson SC, Lawrence BM, Holmes J, Watson M (2009) Preferential allocation to beneficial symbiont with spatial structure maintains mycorrhizal mutualism. Ecology letters 12(1):13–21

    PubMed  Google Scholar 

  16. Caswell H (1978) Predator-mediated coexistence: a nonequilibrium model. The American Naturalist 112(983):127–154

    Google Scholar 

  17. Christian N, Bever JD (2018) Carbon allocation and competition maintain variation in plant root mutualisms. Ecology and Evolution 8(11):5792–5800

    PubMed  PubMed Central  Google Scholar 

  18. Dickson S, Smith S, Smith F (1999) Characterization of two arbuscular mycorrhizal fungi in symbiosis with allium porrum: colonization, plant growth and phosphate uptake. The New Phytologist 144(1):163–172

    CAS  Google Scholar 

  19. Douds DD, Pfeffer PE, Shachar-Hill Y (2000) Carbon partitioning, cost, and metabolism of arbuscular mycorrhizas. In: Arbuscular mycorrhizas: physiology and function, pages 107–129. Springer

  20. Drew E, Murray R, Smith S, Jakobsen I (2003) Beyond the rhizosphere:, growth and function of arbuscular mycorrhizal external hyphae in sands of varying pore sizes. Plant and Soil 251(1):105–114

    CAS  Google Scholar 

  21. Engelmoer DJ, Behm JE, Toby Kiers E (2014) Intense competition between arbuscular mycorrhizal mutualists in an in vitro root microbiome negatively affects total fungal abundance. Molecular ecology 23(6):1584–1593

    CAS  PubMed  Google Scholar 

  22. Fitter A (2006) What is the link between carbon and phosphorus fluxes in arbuscular mycorrhizas? a null hypothesis for symbiotic function. New Phytol 172(1):3–6

    CAS  PubMed  Google Scholar 

  23. Gause G, Witt A (1935) Behavior of mixed populations and the problem of natural selection. The American Naturalist 69(725):596–609

    Google Scholar 

  24. Geib JC, Galen C (2012) Tracing impacts of partner abundance in facultative pollination mutualisms: from individuals to populations. Ecol 93(7):1581–1592

    Google Scholar 

  25. Gianinazzi S, Gollotte A, Binet M-N, van Tuinen D, Redecker D, Wipf D (2010) Agroecology: the key role of arbuscular mycorrhizas in ecosystem services. Mycorrhiza 20(8):519–530

    PubMed  Google Scholar 

  26. Graham J (2000) Assessing costs of arbuscular mycorrhizal symbiosis agroecosystems fungi. In: Current advances in mycorrhizae research

  27. Hammer EC, Pallon J, Wallander H, Olsson PA (2011) Tit for tat? a mycorrhizal fungus accumulates phosphorus under low plant carbon availability. FEMS Microbiol Ecol 76(2):236–244

    CAS  PubMed  Google Scholar 

  28. Hardin G (1960) The competitive exclusion principle. Science 131(3409):1292–1297

    CAS  PubMed  Google Scholar 

  29. Hart MM, Forsythe J, Oshowski B, Bücking H., Jansa J, Kiers ET (2013) Hiding in a crowd—does diversity facilitate persistence of a low-quality fungal partner in the mycorrhizal symbiosis? Symbiosis 59(1):47–56

    Google Scholar 

  30. Hart MM, Reader RJ (2002) Taxonomic basis for variation in the colonization strategy of arbuscular mycorrhizal fungi. New Phytol 153(2):335–344

    Google Scholar 

  31. Hassell M (1975) Density-dependence in single-species populations. The Journal of animal ecology 44:283–295

    Google Scholar 

  32. Hepper CM, Azcon-Aguilar C, Rosendahl S, Sen R (1988) Competition between three species of glomus used as spatially separated introduced and indigenous mycorrhizal inocula for leek (allium porrum l.) New Phytol 110(2):207–215

    Google Scholar 

  33. Herre EA, Knowlton N, Mueller UG, Rehner SA (1999) The evolution of mutualisms: exploring the paths between conflict and cooperation. Trends in Ecology & Evolution 14(2):49–53

    CAS  Google Scholar 

  34. Hoeksema JD, Bruna EM (2000) Pursuing the big questions about interspecific mutualism: a review of theoretical approaches. Oecologia 125(3):321–330

    PubMed  Google Scholar 

  35. Hoeksema JD, Kummel M (2003) Ecological persistence of the plant-mycorrhizal mutualism:, a hypothesis from species coexistence theory. The american naturalist 162(S4):S40–S50

    PubMed  Google Scholar 

  36. Holland J, DeAngelis D (2006) Interspecific population regulation and the stability of mutualism: fruit abortion and density-dependent mortality of pollinating seed-eating insects. Oikos 113(3):563–571

    Google Scholar 

  37. Holland JN, DeAngelis DL (2009) Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions. Ecol Lett 12(12):1357–1366

    PubMed  Google Scholar 

  38. Holland JN, DeAngelis DL (2010) A consumer–resource approach to the density-dependent population dynamics of mutualism. Ecol 91(5):1286–1295

    Google Scholar 

  39. Holland JN, DeAngelis DL, Bronstein JL (2002) Population dynamics and mutualism: functional responses of benefits and costs. The American Naturalist 159(3):231–244

    PubMed  Google Scholar 

  40. Holland JN, DeAngelis DL, Schultz ST (1550) Evolutionary stability of mutualism: interspecific population regulation as an evolutionarily stable strategy. Proceedings of the Royal Society of London B: Biological Sciences 271:1807–1814

    Google Scholar 

  41. Husband R, Herre EA, Turner S, Gallery R, Young J (2002) Molecular diversity of arbuscular mycorrhizal fungi and patterns of host association over time and space in a tropical forest. Mol Ecol 11 (12):2669–2678

    CAS  PubMed  Google Scholar 

  42. Ingvarsson PK, Lundberg S (1995) Pollinator functional response and plant population dynamics: pollinators as a limiting resource. Evol Ecol 9(4):421–428

    Google Scholar 

  43. Jansa J, Smith FA, Smith SE (2008) Are there benefits of simultaneous root colonization by different arbuscular mycorrhizal fungi? New Phytol 177(3):779–789

    CAS  PubMed  Google Scholar 

  44. Ji B, Bever JD (2016) Plant preferential allocation and fungal reward decline with soil phosphorus: implications for mycorrhizal mutualism. Ecosphere 7(5):e01256

    Google Scholar 

  45. Johnson CA, Bronstein JL (2019) Coexistence and competitive exclusion in mutualism. Ecol 100(6):e02708

    Google Scholar 

  46. Johnson NC, Graham J-H, Smith FA (1997) Functioning of mycorrhizal associations along the mutualism–parasitism continuum. New phytologist 135(4):575–585

    Google Scholar 

  47. Jung SC, Martinez-Medina A, Lopez-Raez JA, Pozo MJ (2012) Mycorrhiza-induced resistance and priming of plant defenses. Journal of chemical ecology 38(6):651–664

    CAS  PubMed  Google Scholar 

  48. Kang Y, Clark R, Makiyama M, Fewell J (2011) Mathematical modeling on obligate mutualism: Interactions between leaf-cutter ants and their fungus garden. Journal of theoretical biology 289:116–127

    PubMed  Google Scholar 

  49. Kerr B, Riley MA, Feldman MW, Bohannan BJ (2002) Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors. Nature 418(6894):171

    CAS  PubMed  Google Scholar 

  50. Kiers ET, Duhamel M, Beesetty Y, Mensah JA, Franken O, Verbruggen E, Fellbaum CR, Kowalchuk GA, Hart MM, Bago A, et al. (2011) Reciprocal rewards stabilize cooperation in the mycorrhizal symbiosis. Science 333(6044):880–882

    CAS  PubMed  Google Scholar 

  51. Kiers ET, Van Der Heijden MG (2006) Mutualistic stability in the arbuscular mycorrhizal symbiosis: exploring hypotheses of evolutionary cooperation. Ecol 87(7):1627–1636

    Google Scholar 

  52. Klironomos JN, Hart MM (2002) Colonization of roots by arbuscular mycorrhizal fungi using different sources of inoculum. Mycorrhiza 12(4):181–184

    PubMed  Google Scholar 

  53. Knowlton N, Rohwer F (2003) Multispecies microbial mutualisms on coral reefs:, the host as a habitat. The american naturalist 162(S4):S51–S62

    PubMed  Google Scholar 

  54. Koch AM, Antunes PM, Barto EK, Cipollini D, Mummey DL, Klironomos JN (2011) The effects of arbuscular mycorrhizal (am) fungal and garlic mustard introductions on native am fungal diversity. Biol Invasions 13(7):1627–1639

    Google Scholar 

  55. Kot M (2001) Elements of mathematical ecology. Cambridge University Press

  56. Křivan V, Revilla TA (2019) Plant coexistence mediated by adaptive foraging preferences of exploiters or mutualists. Journal of theoretical biology 480:112–128

    PubMed  Google Scholar 

  57. Latef AAHA, Hashem A, Rasool S, Abd allah EF, Alqarawi A, Egamberdieva D, Jan S, Anjum NA, Ahmad P (2016) Arbuscular mycorrhizal symbiosis and abiotic stress in plants: A review. Journal of plant biology 59(5):407–426

    Google Scholar 

  58. Lekberg Y, Koide RT (2014) Integrating physiological, community, and evolutionary perspectives on the arbuscular mycorrhizal symbiosis. Botany 92(4):241–251

    CAS  Google Scholar 

  59. Levine JM, Bascompte J, Adler PB, Allesina S (2017) Beyond pairwise mechanisms of species coexistence in complex communities. Nature 546(7656):56

    CAS  PubMed  Google Scholar 

  60. Maherali H, Klironomos JN (2007) Influence of phylogeny on fungal community assembly and ecosystem functioning. Science 316(5832):1746–1748

    CAS  PubMed  Google Scholar 

  61. Martignoni MM, Hart MM, Tyson RC, Garnier J (2020) Diversity within mutualist guilds promotes coexistence and reduces the risk of invasion from an alien mutualist. Proceedings of the Royal Society B 287(1923):20192312

    PubMed  Google Scholar 

  62. May RM (1972) Will a large complex system be stable? Nature 238(5364):413

    CAS  PubMed  Google Scholar 

  63. May RM (1976) Models for two interacting populations. In: May RM (ed) Theoretical ecology: principles and applications, Saunders, Philadelphia. pp 49–71

  64. Mayfield MM, Stouffer DB (2017) Higher-order interactions capture unexplained complexity in diverse communities. Nature ecology & evolution 1(3):0062

    Google Scholar 

  65. McCann KS (2000) The diversity–stability debate. Nature 405(6783):228

    CAS  PubMed  Google Scholar 

  66. Miki T (2012) Microbe-mediated plant–soil feedback and its roles in a changing world. Ecol Res 27(3):509–520

    CAS  Google Scholar 

  67. Moeller HV, Neubert MG (2016) Multiple friends with benefits: an optimal mutualist management strategy? The American Naturalist 187(1):E1–E12

    PubMed  Google Scholar 

  68. Molbo D, Machado CA, Sevenster JG, Keller L, Herre EA (2003) Cryptic species of fig-pollinating wasps: implications for the evolution of the fig–wasp mutualism, sex allocation, and precision of adaptation. Proceedings of the National Academy of Sciences 100(10):5867–5872

    CAS  Google Scholar 

  69. Morales MA (2011) Model selection analysis of temporal variation in benefit for an ant-tended treehopper. Ecol 92(3):709–719

    Google Scholar 

  70. Morris WF, Vázquez DP, Chacoff NP (2010) Benefit and cost curves for typical pollination mutualisms. Ecol 91(5):1276–1285

    Google Scholar 

  71. Mummey DL, Antunes PM, Rillig MC (2009) Arbuscular mycorrhizal fungi pre-inoculant identity determines community composition in roots. Soil Biol Biochem 41(6):1173–1179

    CAS  Google Scholar 

  72. Murray JD (2007) Mathematical biology: I. An introduction, volume 17. Springer Science & Business Media

  73. Öpik M, Moora M, Liira J, Zobel M (2006) Composition of root-colonizing arbuscular mycorrhizal fungal communities in different ecosystems around the globe. Journal of Ecology 94(4):778–790

    Google Scholar 

  74. Palmer TM, Stanton ML, Young TP (2003) Competition and coexistence: Exploring mechanisms that restrict and maintain diversity within mutualist guilds. The american naturalist 162(S4):S63–S79

    PubMed  Google Scholar 

  75. Parniske M (2008) Arbuscular mycorrhiza: the mother of plant root endosymbioses. Nat Rev Microbiol 6(10):763

    CAS  PubMed  Google Scholar 

  76. Pearson J, Jakobsen I (1993) Symbiotic exchange of carbon and phosphorus between cucumber and three arbuscular mycorrhizal fungi. New phytologist 124(3):481–488

    CAS  Google Scholar 

  77. Pellmyr O (1999) Systematic revision of the yucca moths in the tegeticula yuccasella complex (lepidoptera: Prodoxidae) north of mexico. Syst Entomol 24(3):243–271

    Google Scholar 

  78. Peterson RL, Guinel FC (2000) The use of plant mutants to study regulation of colonization by am fungi. In: Arbuscular mycorrhizas: physiology and function, pages 147–171. Springer

  79. Powell JR, Bennett AE (2016) Unpredictable assembly of arbuscular mycorrhizal fungal communities. Pedobiologia 59(1-2):11– 15

    Google Scholar 

  80. Ravnskov S, Jakobsen I (1995) Functional compatibility in arbuscular mycorrhizas measured as hyphal p transport to the plant. New Phytol 129(4):611–618

    Google Scholar 

  81. Rygiewicz PT, Andersen CP (1994) Mycorrhizae alter quality and quantity of carbon allocated below ground. Nature 369(6475):58

    Google Scholar 

  82. Sawers RJ, Svane SF, Quan C, Grønlund M, Wozniak B, Gebreselassie M-N, González-Muñoz E, Montes RAC, Baxter I, Goudet J, et al. (2017) Phosphorusacquisition efficiency in arbuscular mycorrhizal maize is correlated with the abundance of root-external hyphae and the accumulation of transcripts encoding pht1 phosphate transporters. New Phytol 214(2):632–643

    CAS  PubMed  Google Scholar 

  83. Schoener TW (1974) Some methods for calculating competition coefficients from resource-utilization spectra. The American Naturalist 108(961):332–340

    PubMed  Google Scholar 

  84. Smith GR, Steidinger BS, Bruns TD, Peay KG (2018) Competition–colonization tradeoffs structure fungal diversity. The ISME journal 12(7):1758

    CAS  PubMed  PubMed Central  Google Scholar 

  85. Smith SE, Read DJ (2010) Mycorrhizal symbiosis. Academic press

  86. Smith SE, Smith FA (2012) Fresh perspectives on the roles of arbuscular mycorrhizal fungi in plant nutrition and growth. Mycologia 104(1):1–13

    PubMed  Google Scholar 

  87. Smith SE, Smith FA, Jakobsen I (2003) Mycorrhizal fungi can dominate phosphate supply to plants irrespective of growth responses. Plant physiology 133(1):16–20

    CAS  PubMed  PubMed Central  Google Scholar 

  88. Stanton ML (2003) Interacting guilds: moving beyond the pairwise perspective on mutualisms. The American Naturalist 162(S4):S10–S23

    PubMed  Google Scholar 

  89. Sugiyama A, Ueda Y, Zushi T, Takase H, Yazaki K (2014) Changes in the bacterial community of soybean rhizospheres during growth in the field. PloS one 9(6):e100709

    PubMed  PubMed Central  Google Scholar 

  90. Sylvia DM, Williams SE (1992) Vesicular-arbuscular mycorrhizae and environmental stress. Mycorrhizae in sustainable agriculture, (mycorrhizaeinsu), pp 101–124

  91. Thomson B, Robson A, Abbott L (1990) Mycorrhizas formed by gigaspora calospora and glomus fasciculatum on subterranean clover in relation to soluble carbohydrate concentrations in roots. New Phytol 114 (2):217–225

    CAS  Google Scholar 

  92. Treseder KK (2013) The extent of mycorrhizal colonization of roots and its influence on plant growth and phosphorus content. Plant Soil 371(1-2):1–13

    CAS  Google Scholar 

  93. Treseder KK, Cross A (2006) Global distributions of arbuscular mycorrhizal fungi. Ecosystems 9(2):305–316

    Google Scholar 

  94. Valdovinos FS, Moisset de Espanés P, Flores JD, Ramos-Jiliberto R (2013) Adaptive foraging allows the maintenance of biodiversity of pollination networks. Oikos 122(6):907–917

    Google Scholar 

  95. van Aarle IM, Olsson PA (2003) Fungal lipid accumulation and development of mycelial structures by two arbuscular mycorrhizal fungi. Applied and Environmental Microbiology 69(11):6762–6767

    PubMed  PubMed Central  Google Scholar 

  96. Van der Heijden MG, Klironomos JN, 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(6706):69

    Google Scholar 

  97. Van der Putten WH, Bardgett RD, Bever JD, Bezemer TM, Casper BB, Fukami T, Kardol P, Klironomos JN, Kulmatiski A, Schweitzer JA, et al. (2013) Plant–soil feedbacks: the past, the present and future challenges. J Ecol 101(2):265–276

    Google Scholar 

  98. Vandermeer JH, Boucher DH (1978) Varieties of mutualistic interaction in population models. J Theor Biol 74(4):549–558

    CAS  PubMed  Google Scholar 

  99. Vierheilig H, Bago B, Lerat S, Piché Y. (2002) Shoot-produced, light-dependent factors are partially involved in the expression of the arbuscular mycorrhizal (am) status of am host and non-host plants. J Plant Nutr Soil Sci 165(1):21–25

    CAS  Google Scholar 

  100. Walder F, Niemann H, Mathimaran N, Lehmann MF, Boller T, Wiemken A (2012) Mycorrhizal networks: Common goods of plants shared under unequal terms of trade. Plant physiology 159:pp–112

    Google Scholar 

  101. Walder F, van der Heijden MG (2015) Regulation of resource exchange in the arbuscular mycorrhizal symbiosis. Nature Plants 1(11):15159

    CAS  PubMed  Google Scholar 

  102. Wang W, Shi J, Xie Q, Jiang Y, Yu N, Wang E (2017) Nutrient exchange and regulation in arbuscular mycorrhizal symbiosis. Molecular plant 10(9):1147–1158

    CAS  PubMed  Google Scholar 

  103. Wilson W, Morris W, Bronstein JL (2003) Coexistence of mutualists and exploiters on spatial landscapes. Ecological monographs 73(3):397–413

    Google Scholar 

  104. Wright DH (1989) A simple, stable model of mutualism incorporating handling time. The American Naturalist 134(4):664–667

    Google Scholar 

  105. Zhu Y-G, Miller RM (2003) Carbon cycling by arbuscular mycorrhizal fungi in soil–plant systems. Trends in plant science 8(9):407–409

    CAS  PubMed  Google Scholar 

Download references

Funding

MMH received NSERC Discovery Grant RGPIN-2018-237774. JG received NONLOCAL project (ANR-14-CE25-0013), GLOBNETS project (ANR-16- CE02-0009), and the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No 639638, MesoProbio). RCT received NSERC Discovery Grant RGPIN-2016-05277 and the “Make our planet great again (MOPGA)” grant.

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Correspondence to Maria M. Martignoni.

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Martignoni, M.M., Hart, M.M., Garnier, J. et al. Parasitism within mutualist guilds explains the maintenance of diversity in multi-species mutualisms. Theor Ecol (2020). https://doi.org/10.1007/s12080-020-00472-9

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Keywords

  • Mutualism
  • Mutualist guilds
  • Coexistence
  • Arbuscular mycorrhizal fungi
  • Indirect interactions
  • Mathematical model
  • Ordinary differential equations
  • Functional response
  • Density-dependent resource exchange