Bulletin of Mathematical Biology

, Volume 74, Issue 10, pp 2315–2338 | Cite as

Quantifying the Likelihood of Co-existence for Communities with Asymmetric Competition

  • Stuart Nattrass
  • Stephen Baigent
  • David J. Murrell
Original Article


Trade-offs in performance of different ecological functions within a species are commonly offered as an explanation for co-existence in natural communities. Single trade-offs between competitive ability and other life history traits have been shown to support a large number of species, as a result of strong competitive asymmetry. We consider a single competition-fecundity trade-off in a homogeneous environment, and examine the effect of the form of asymmetry on the likelihood of species co-existing. We find conditions that allow co-existence of two species for a general competition function, and show that (1) two species can only co-exist if the competition function is sufficiently steep when the species are similar; (2) when competition is determined by a linear function, no more than two species can co-exist; (3) when the competition between two individuals is determined by a discontinuous step function, this single trade-off can support an arbitrarily large number of species. Further, we show analytically that as the degree of asymmetry in competition increases, the probability of a given number of species co-existing also increases, but note that even in the most favourable conditions, large numbers of species co-existing along a single trade-off is highly unlikely. On this basis, we suggest it is unlikely that single trade-offs are able to support high levels of bio-diversity without interacting other processes.


Lotka–Volterra Trade-offs Community ecology Niche Life-history 


  1. Adler, F., & Mosquera, J. (2000). Is space necessary? Interference competition and limits to biodiversity. Ecology, 81, 3226–3232. CrossRefGoogle Scholar
  2. Amarasekare, P. (2007). Trade-offs, temporal variation, and species coexistence in communities with intraguild predation. Ecology, 88, 2720–2728. CrossRefGoogle Scholar
  3. Bonsall, M., & Mangel, M. (2004). Life-history trade-offs and ecological dynamics in the evolution of longevity. Proceedings of the Royal Society of London. Series B, Biological Sciences, 271, 1143–1150. CrossRefGoogle Scholar
  4. Bowers, R., Boots, M., & Begon, M. (1994). Life-history trade-offs and the evolution of pathogen resistance: competition between host strains. Proceedings of the Royal Society of London. Series B, Biological Sciences, 257, 247–253. CrossRefGoogle Scholar
  5. Cadotte, M. (2007). Competition-colonization trade-offs and disturbance effects at multiple scales. Ecology, 88, 823–829. CrossRefGoogle Scholar
  6. Cadotte, M., Mai, D., Jantz, S., Collins, M., Keele, M., et al. (2006). On testing the competition-colonization trade-off in a multispecies assemblage. The American Naturalist, 168, 704–709. CrossRefGoogle Scholar
  7. Calcagno, V., Mouquet, N., Jarne, P., & David, P. (2006). Coexistence in a metacommunity: the competition–colonization trade-off is not dead. Ecology Letters, 9, 897–907. CrossRefGoogle Scholar
  8. Connolly, J., & Wayne, P. (1996). Asymmetric competition between plant species. Oecologia, 108, 311–320. Google Scholar
  9. Costanzo, K., Mormann, K., & Juliano, S. (2005). Asymmetrical competition and patterns of abundance of Aedes Albopictus and Culex pipiens (Diptera: Culicidae). Journal of Medical Entomology, 42, 559. CrossRefGoogle Scholar
  10. Hastings, A. (1980). Disturbance, coexistence, history, and competition for space. Theoretical Population Biology, 18, 363–373. MathSciNetMATHCrossRefGoogle Scholar
  11. HilleRisLambers, R., & Dieckmann, U. (2003). Competition and predation in simple food webs: intermediately strong trade-offs maximize coexistence. Proceedings of the Royal Society of London. Series B, Biological Sciences, 270, 2591–2598. CrossRefGoogle Scholar
  12. Hofbauer, J., & Sigmund, K. (1998). The theory of evolution and dynamical systems. New York: Cambridge University Press. Google Scholar
  13. Hofbauer, J., & Sigmund, K. (1998). Evolutionary games and population dynamics. Cambridge: Cambridge University Press. MATHCrossRefGoogle Scholar
  14. Keddy, P., Twolan-Strutt, L., & Shipley, B. (1997). Experimental evidence that interspecific competitive asymmetry increases with soil productivity. Oikos, 80, 253–256. CrossRefGoogle Scholar
  15. Kisdi, É. (1999). Evolutionary branching under asymmetric competition. Journal of Theoretical Biology, 197, 149–162. CrossRefGoogle Scholar
  16. Kisdi, E., & Geritz, S. (2003). On the coexistence of perennial plants by the competition-colonization trade-off. The American Naturalist, 161, 350–354. CrossRefGoogle Scholar
  17. Klausmeier, C. (1998). Extinction in multispecies and spatially explicit models of habitat destruction. The American Naturalist, 152, 303–310. CrossRefGoogle Scholar
  18. Kubota, Y., & Hara, T. (1995). Tree competition and species coexistence in a sub-boreal forest, northern Japan. Annals of Botany, 76, 503. CrossRefGoogle Scholar
  19. Law, R. (1979). Optimal life histories under age-specific predation. The American Naturalist, 114, 399–417. MathSciNetCrossRefGoogle Scholar
  20. Law, R., & Morton, R. (1996). Permanence and the assembly of ecological communities. Ecology, 77, 762–775. CrossRefGoogle Scholar
  21. Law, R., Marrow, P., & Dieckmann, U. (1997). On evolution under asymmetric competition. Evolutionary Ecology, 11, 485–501. CrossRefGoogle Scholar
  22. Lawton, J., & Hassell, M. (1981). Asymmetrical competition in insects. Nature, 289, 793–795. CrossRefGoogle Scholar
  23. Levin, S., & Paine, R. (1974). Disturbance, patch formation, and community structure. Proceedings of the National Academy of Sciences of the United States of America, 71, 2744. MATHCrossRefGoogle Scholar
  24. Levins, R., & Culver, D. (1971). Regional coexistence of species and competition between rare species. Proceedings of the National Academy of Sciences of the United States of America, 68, 1246–1248. MATHCrossRefGoogle Scholar
  25. May, R., & Leonard, W. (1975). Nonlinear aspects of competition between three species. SIAM Journal on Applied Mathematics, 29, 243–253. MathSciNetMATHCrossRefGoogle Scholar
  26. May, R., & Nowak, M. (1994). Superinfection, metapopulation dynamics, and the evolution of diversity. Journal of Theoretical Biology, 170, 95. CrossRefGoogle Scholar
  27. Meszena, G., Gyllenberg, M., Pasztor, L., & Metz, J. (2006). Competitive exclusion and limiting similarity: a unified theory. Theoretical Population Biology, 69, 68–87. MATHCrossRefGoogle Scholar
  28. Morin, P., & Johnson, E. (1988). Experimental studies of asymmetric competition among anurans. Oikos, 53, 398–407. CrossRefGoogle Scholar
  29. Muller-Landau, H. (2010). The tolerance–fecundity trade-off and the maintenance of diversity in seed size. Proceedings of the National Academy of Sciences of the United States of America, 107, 4242. CrossRefGoogle Scholar
  30. Nowak, M., & May, R. (1994). Superinfection and the evolution of parasite virulence. Proceedings of the Royal Society of London. Series B, Biological Sciences, 255, 81. CrossRefGoogle Scholar
  31. Resetarits, W. Jr (1995). Competitive asymmetry and coexistence in size-structured populations of brook trout and spring salamanders. Oikos, 73, 188–198. CrossRefGoogle Scholar
  32. Robinson, G., Quinn, J., & Stanton, M. (1995). Invasibility of experimental habitat islands in a California winter annual grassland. Ecology, 76, 786–794. CrossRefGoogle Scholar
  33. Schluter, D. (1995). Adaptive radiation in sticklebacks: trade-offs in feeding performance and growth. Ecology, 76, 82–90. CrossRefGoogle Scholar
  34. Strobeck, C. (1973). N species competition. Ecology, 54, 650–654. CrossRefGoogle Scholar
  35. Tilman, D. (1994). Competition and biodiversity in spatially structured habitats. Ecology, 75, 2–16. CrossRefGoogle Scholar
  36. Turnbull, L., Rees, M., & Crawley, M. (1999). Seed mass and the competition/colonization trade-off: a sowing experiment. Journal of Ecology, 87, 899–912. CrossRefGoogle Scholar
  37. Venable, D. (1992). Size-number trade-offs and the variation of seed size with plant resource status. The American Naturalist, 140, 287–304. CrossRefGoogle Scholar
  38. Weiner, J. (1990). Asymmetric competition in plant populations. Trends in Ecology & Evolution, 5, 360–364. CrossRefGoogle Scholar
  39. White, A., & Bowers, R. (2005). Adaptive dynamics of Lotka–Volterra systems with trade-offs: the role of interspecific parameter dependence in branching. Mathematical Biosciences, 193, 101–117. MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Stuart Nattrass
    • 1
  • Stephen Baigent
    • 2
    • 3
  • David J. Murrell
    • 1
    • 3
  1. 1.Department of Genetics, Evolution and EnvironmentUniversity College LondonLondonUK
  2. 2.Department of MathematicsUniversity College LondonLondonUK
  3. 3.CoMPLEX (Centre for Mathematics and Physics in the Life Sciences and Experimental Biology)University College LondonLondonUK

Personalised recommendations