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Revising the Role of Species Mobility in Maintaining Biodiversity in Communities with Cyclic Competition

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Abstract

One of the most crucial tasks faced by biologists today is revealing the mechanisms which account for biodiversity, yet we are still far from a full understanding of these mechanisms, and in particular the role of spatially heterogeneous population distributions. Recently, the spatially heterogeneous coexistence seen in cyclic competition models—in which species compete as in the game rock-paper-scissors—has brought them to the fore as a paradigm for biodiversity. Research into cyclic competition has so far been focused almost exclusively on stochastic lattice models with discrete space, which ignore several key dynamical aspects. In particular, such models usually assume that species disperse at the same speed. This paper aims to extend our understanding of cyclic competition by applying a reaction–diffusion Lotka–Volterra scheme to the problem, which allows us to vary the mobility of each species, and lets us take into account cyclic competition with more complex underlying mechanisms. In this paper we reveal an entirely new kind of cyclic competition—‘conditional’ cyclic competition, with a different underlying mechanism to ‘classic’ cyclic competition—and we show that biodiversity in communities with cyclic competition in fact depends heavily on the ratios between the species mobilities. Furthermore, we show that this dependence can be completely different for conditional and classic cyclic competition. We also present a wide range of spatiotemporal patterns which are formed in the system, including spiral and target waves, spiralling patches, and irregular chaotic patches. We show that the previously unknown case of conditional cyclic competition is host to a scenario of patchy co-invasion, where the spread of the population front takes place via the formation, splitting and propagation of patches of high species density. This is also an example of invasional meltdown because one competitor facilitates the invasion of the other, but unlike well-known cases of invasional meltdown the co-invaders in this system are not mutualists but antagonistic competitors, and the overall result mitigates, rather than amplifies, the damage done to the native ecosystem.

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References

  • Alzahrani, E. O., Davidson, F. A., & Dodds, N. (2010). Travelling waves in near-degenerate bistable competition models. Math. Model. Nat. Phenom., 5(5), 13–35.

    Article  MathSciNet  MATH  Google Scholar 

  • Alzahrani, E. O., Davidson, F. A., & Dodds, N. (2011). Reversing invasion in bistable systems. J. Math. Biol. doi:10.1007/s00285-011-0490-9.

    MATH  Google Scholar 

  • Amarasekare, P. (2003). Competitive coexistence in spatially structured environments: A synthesis. Ecol. Lett., 6, 1109–1122.

    Article  Google Scholar 

  • Boerlijst, M. C., & Hogeweg, P. (1995). Spatial gradients enhance persistence of hypercycles. Physica D, 88, 29–39.

    Article  MATH  Google Scholar 

  • Boerlijst, M. C., Lamers, M. E., & Hogeweg, P. (1993). Evolutionary consequences of spiral waves in a host–parasitoid system. Proc. R. Soc. Lond. B, 253, 15–18.

    Article  Google Scholar 

  • Buss, L. W., & Jackson, J. B. (1979). Competitive networks: Nontransitive competitive relationships in cryptic coral reef environments. Am. Nat., 113, 223–234.

    Article  Google Scholar 

  • Cameron, D. D., White, A., & Antonovics, J. (2009). Parasite-grass-forb interactions and rock-paper-scissor dynamics: Predicting the effects of the parasitic plant Rhinanthus minor on host plant communities. J. Ecol., 97, 1311–1319.

    Article  Google Scholar 

  • Cantrell, R. S., & Cosner, C. (2003). Spatial ecology via reaction-diffusion equations. Chichester: Wiley.

    MATH  Google Scholar 

  • Chesson, P. (2000). Mechanisms of maintenance of species diversity. Annu. Rev. Ecol. Syst., 31, 342–366.

    Article  Google Scholar 

  • Crosby, A. W. (1986). Ecological imperialism. The biological expansion of Europe, 900–1900. Cambridge: Cambridge University Press.

    Google Scholar 

  • Davis, M. B., Calcote, R. R., Sugita, S., & Takahara, H. (1998). Patchy invasion and the origin of a hemlock-hardwoods forest mosaic. Ecology, 79, 2641–2659.

    Google Scholar 

  • Ducharme, M. B., Larochelle, J., & Richard, D. (1989). Thermogenic capacity in gray and black morphs of the gray squirrel, Sciurus Carolinensis. Physiol. Zool., 62, 1273–1292.

    Google Scholar 

  • Durrett, R., & Levin, S. (1998). Spatial aspects of interspecific competition. Theor. Popul. Biol., 53, 30–43.

    Article  MATH  Google Scholar 

  • Frachebourg, L., Krapivsky, P. L., & Ben-Naim, E. (1996). Spatial organization in cyclic Lotka–Volterra systems. Phys. Rev. E, 54, 6186–6200.

    Article  Google Scholar 

  • Frean, M., & Abraham, R. (2001). Rock-scissors-paper and the survival of the weakest. Proc. R. Soc. Lond. B, 268, 1323–1327.

    Article  Google Scholar 

  • Gause, G. F. (1934). The struggle for existence. New York: Hafner Publishing Company.

    Book  Google Scholar 

  • Gilpin, M. E. (1975). Limit cycles in competition communities. Am. Nat., 109, 51–60.

    Article  Google Scholar 

  • Gurney, W. S. C., Veitch, A. R., Cruickshank, I., & McGeachin, G. (1998). Circles and spirals: Population persistence in a spatially explicit predator–prey model. Ecology, 79, 2516–2530.

    Google Scholar 

  • Hanski, I. (1994). Spatial scale, patchiness and population dynamics on land. Philos. Trans. R. Soc. Lond. B, 343, 19–25.

    Article  Google Scholar 

  • Hardin, G. (1960). The competitive exclusion principle. Science, 131, 1292–1298.

    Article  Google Scholar 

  • Hauert, C., De Monte, S., Hofbauer, J., & Sigmund, K. (2002). Replicator dynamics for optional public good games. J. Theor. Biol., 218, 187–194.

    Article  Google Scholar 

  • Hosono, Y. (1998). The minimal speed of traveling fronts for a diffusive Lotka–Volterra competition model. Bull. Math. Biol., 60, 435–448.

    Article  MATH  Google Scholar 

  • Huisman, J., Johansson, A. M., Folmer, E. O., & Weissing, F. J. (2001). Towards a solution of the plankton paradox: The importance of physiology and life history. Ecol. Lett., 4, 408–411.

    Article  Google Scholar 

  • Hutchinson, G. E. (1961). The paradox of the plankton. Am. Nat., 95, 137–145.

    Article  Google Scholar 

  • Ikeda, H. (2001). Multiple travelling wave solutions of three-component systems with competition and diffusion. Methods Appl. Anal., 8(3), 479–496.

    MathSciNet  MATH  Google Scholar 

  • Ives, A. R., & Carpenter, S. R. (2007). Stability and diversity of ecosystems. Science, 317, 58.

    Article  Google Scholar 

  • Jackson, J. B. C., & Buss, L. W. (1975). Allelopathy and spatial competition among coral reef invertebrates. Proc. Natl. Acad. Sci. USA, 72, 5160–5163.

    Article  Google Scholar 

  • Jiang, L., Zhou, T., Perc, M., & Wang, B. (2011). Effects of competition on pattern formation in the rock-paper-scissors game. Phys. Rev. E, 84, 021912.

    Article  Google Scholar 

  • Károlyi, G., Neufeld, Z., & Scheuring, I. (2005). Rock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms. J. Theor. Biol., 236, 12–20.

    Article  Google Scholar 

  • Kerr, B., Riley, M. A., Feldman, M. W., & Bohannan, J. M. (2002). Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors. Nature, 418, 171–174.

    Article  Google Scholar 

  • Kim, P.J., Jeong, H. (2005). Spatio-temporal dynamics in the origin of genetic information. Physica D, 203, 88–99.

    Article  MathSciNet  Google Scholar 

  • Kirkup, B. C., & Riley, M. A. (2004). Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo. Nature, 428, 412–414.

    Article  Google Scholar 

  • Kolb, A., & Alpert, P. (2003). Effects of nitrogen and salinity on growth and competition between a native grass and an invasive congener. Biol. Invasions, 5, 229–238.

    Article  Google Scholar 

  • Laird, R. A., & Schamp, B. S. (2008). Does local competition increase the coexistence of species in intransitive networks? Ecology, 89, 237–247.

    Article  Google Scholar 

  • Levin, P. S., Coyer, J. A., Petrik, R., & Good, T. P. (2002). Community-wide effects of nonindigenous species on temperate rocky reefs. Ecology, 83, 3182–3193.

    Article  Google Scholar 

  • Levine, J. M., & Hille Ris Lambers, J. (2010). The importance of niches for the maintenance of species diversity. Nature, 461, 254–257.

    Article  Google Scholar 

  • Lewis, M. A. (2000). Spread rate for a nonlinear stochastic invasion. J. Math. Biol., 41, 430–454.

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis, M. A., & Pacala, S. (2000). Modeling and analysis of stochastic invasion processes. J. Math. Biol., 41, 387–429.

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis, M., Li, B., & Weinberger, H. F. (2002). Spreading speed and linear determinacy for two-species competition models. J. Math. Biol., 45, 219–233.

    Article  MathSciNet  MATH  Google Scholar 

  • MacArthur, R., & Levins, R. (1967). The limiting similarity, convergence, and divergence of coexisting species. Am. Nat., 101(921), 377–385.

    Article  Google Scholar 

  • May, R. M., & Leonard, W. J. (1975). Nonlinear aspects of competition between three species. SIAM J. Appl. Math., 29, 243–253.

    Article  MathSciNet  MATH  Google Scholar 

  • Merino, S. (1996). Cyclic competition of three species in the time periodic and diffusive case. J. Math. Biol., 34, 789–809.

    MathSciNet  MATH  Google Scholar 

  • Metz, J. A. J., Nisbet, R. M., & Geritz, S. A. H. (1992). How should we define fitness for general ecological scenarios. Trends Ecol. Evol., 7, 198–202.

    Article  Google Scholar 

  • Moen, J. (1989). Diffuse competition: A diffuse concept. Oikos, 54, 260–263.

    Article  Google Scholar 

  • Morozov, A.Yu., & Li, B.-L. (2007). On the importance of dimensionality of space in models of space-mediated population persistence. Theor. Popul. Biol., 71, 278–289.

    Article  MATH  Google Scholar 

  • Morozov, A. Yu., Petrovskii, S. V., & Li, B. (2006). Spatiotemporal complexity of the patchy invasion in a predator–prey system with the Allee effect. J. Theor. Biol., 238, 18–35.

    Article  MathSciNet  Google Scholar 

  • Müller, A. P. O., & Gallas, J. A. C. (2010). How community size affects survival chances in cyclic competition games that microorganisms play. Phys. Rev. E, 82, 052901.

    Article  Google Scholar 

  • Murray, J.D. (1989). Mathematical biology. Berlin: Springer.

    MATH  Google Scholar 

  • Ni, X., Yang, R., Wang, W.-X., Lai, Y.-C., & Grebogi, C. (2010). Cyclic competition of mobile species on continuous space: pattern formation and coexistence. Phys. Rev. E, 82, 066211.

    Article  MathSciNet  Google Scholar 

  • O’Dowd, D. J., Green, P. T., & Lake, P. S. (2003). Invasional ‘meltdown’ on an oceanic island. Ecol. Lett., 6, 812–817.

    Article  Google Scholar 

  • Okubo, A., Maini, P., Williamson, M. H., & Murray, J. D. (1989). On the spatial spread of the grey squirrel in Britain. Proc. R. Soc. Lond. B, 238, 113–125.

    Article  Google Scholar 

  • Paquin, C. E., & Adams, J. (1983). Relative fitness can decrease in evolving asexual populations of S. cerevisiae. Nature, 306, 368–371.

    Article  Google Scholar 

  • Petrovskii, S. V., Kawasaki, K., Takasu, F., & Shigesada, N. (2001). Diffusive waves, dynamical stabilization and spatiotemporal chaos in a community of three competitive species. Jpn. J. Ind. Appl. Math., 18, 459–481.

    Article  MathSciNet  MATH  Google Scholar 

  • Petrovskii, S. V., Morozov, A. Y., & Venturino, E. (2002). Allee effect makes possible patchy invasion in a predator–prey system. Ecol. Lett., 5, 345–352.

    Article  Google Scholar 

  • Prado, F., & Kerr, B. (2008). The evolution of restraint in bacterial biofilms under nontransitive competition. Evolution, 62, 538–548.

    Article  Google Scholar 

  • Reichenbach, T., Mobilia, M., & Frey, E. (2007). Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games. Nature, 448, 1046–1049.

    Article  Google Scholar 

  • Richardson, D. M., Allsopp, N., D’Antonio, C. M., Milton, S. J., & Rejmanek, M. (2000). Plant invasions—the role of mutualisms. Biol. Rev., 75, 65–93.

    Article  Google Scholar 

  • Rohani, P., Lewis, T. J., Grünbaum, D., & Ruxton, G. D. (1997). Spatial self-organization in ecology: Pretty patterns or robust reality? Trends Evol. Ecol., 12(2), 70–74.

    Article  Google Scholar 

  • Roy, S., & Chattopadhyay, J. (2007). Towards a resolution of ‘the paradox of the plankton’: A brief overview of the proposed mechanisms. Ecol. Complex., 4, 26–33.

    Article  Google Scholar 

  • Rushton, S. P., Lurz, P., Gurnell, J., & Fuller, R. (2000). Modelling the spatial dynamics of parapoxvirus disease in red and grey squirrels: A possible cause of the decline in the red squirrel in the UK? J. Appl. Ecol., 37, 997–1012.

    Article  Google Scholar 

  • Ryabov, A. B., & Blasius, B. (2011). A graphical theory of competition on spatial resource gradients. Ecol. Lett., 14, 220–228.

    Article  Google Scholar 

  • Scheffer, M., Rinaldi, S., Huisman, J., & Weissing, F. J. (2003). Why plankton communities have no equilibrium: Solutions to the paradox. Hydrobiologia, 491, 9–18.

    Article  Google Scholar 

  • Sherratt, J. (2001). Periodic travelling waves in cyclic predator–prey systems. Ecol. Lett., 4, 30–37.

    Article  Google Scholar 

  • Sherratt, J., & Smith, M. (2008). Periodic travelling waves in cyclic populations: Field studies and reaction–diffusion models. J. R. Soc. Interface, 6, 483–505.

    Article  Google Scholar 

  • Shigesada, N., & Kawasaki, K. (1997). Biological invasions: Theory and practice. Oxford: Oxford University Press.

    Google Scholar 

  • Silvertown, J., Holtier, S., Johnson, J., & Dale, P. (1992). Cellular automaton models of interspecific competition for space—The effect of pattern on process. J. Ecol., 80, 527–534.

    Article  Google Scholar 

  • Simberloff, D. (2006). Invasional meltdown 6 years later: Important phenomenon, unfortunate metaphor, or both? Ecol. Lett., 9, 912–919.

    Article  Google Scholar 

  • Simberloff, D., & Von Holle, B. (1999). Positive interactions of nonindigenous species: Invasional meltdown? Biol. Invasions, 5, 179–192.

    Article  Google Scholar 

  • Sinervo, B., & Lively, C. M. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature, 380, 240–243.

    Article  Google Scholar 

  • Szabó, G. (2002). Evolutionary prisoner’s dilemma games with voluntary participation. Phys. Rev. E, 66, 062903.

    Article  MathSciNet  Google Scholar 

  • Tainaka, K. (1993). Paradoxical effect in a three-candidate voter model. Phys. Lett. A, 176, 303–306.

    Article  Google Scholar 

  • Tanner, J. E., Hughes, T. P., & Connell, J. H. (1994). Species coexistence, keystone species, and succession: A sensitivity analysis. Ecology, 75, 2204–2219.

    Article  Google Scholar 

  • Thomas, J. (1995). Texts in applied mathematics: Vol. 22. Numerical partial differential equations: finite difference methods. New York: Springer.

    MATH  Google Scholar 

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

    Google Scholar 

  • Tilman, D., May, R. M., Lehman, C. L., & Nowak, M. A. (1994). Habitat destruction and the extinction debt. Nature, 371, 65–66.

    Article  Google Scholar 

  • Tompkins, D. M., White, A. R., & Boots, M. (2003). Ecological replacement of native red squirrels by invasive greys driven by disease. Ecol. Lett., 6, 189–196.

    Article  Google Scholar 

  • With, K. A. (2001). The landscape ecology of invasive spread. Conserv. Biol., 16, 1192–1203.

    Article  Google Scholar 

  • Wonham, M. J., O’Connor, M., & Harley, C. D. G. (2005). Positive effects of a dominant invader on introduced and native mudflat species. Mar. Ecol. Prog. Ser., 289, 109–116.

    Article  Google Scholar 

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Acknowledgements

We are very grateful to Prof. Sergei Petrovskii (University of Leicester, UK) for an informative discussion of our results and for providing valuable comments.

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  1. Department of Mathematics, University of Leicester, Leicester, UK

    M. W. Adamson & A. Y. Morozov

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Adamson, M.W., Morozov, A.Y. Revising the Role of Species Mobility in Maintaining Biodiversity in Communities with Cyclic Competition. Bull Math Biol 74, 2004–2031 (2012). https://doi.org/10.1007/s11538-012-9743-z

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