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Resource Competition: A Bifurcation Theory Approach

Published online by Cambridge University Press:  28 November 2013

B. W. Kooi ,
P. S. Dutta  and
U. Feudel
B. W. Kooi*
Affiliation:
Department of Theoretical Biology, VU University de Boelelaan 1085, 1081 HV Amsterdam, The Netherlands
P. S. Dutta
Affiliation:
Theoretical Physics/Complex Systems, ICBM Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg, Germany Department of Mathematics, Indian Institute of Technology Ropar Rupnagar-140001, Punjab, India
U. Feudel
Affiliation:
Theoretical Physics/Complex Systems, ICBM Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg, Germany
*
⋆ Corresponding author. E-mail: bob.kooi@vu.nl
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Abstract

We develop a framework for analysing the outcome of resource competition based on bifurcation theory. We elaborate our methodology by readdressing the problem of competition of two species for two resources in a chemostat environment. In the case of perfect-essential resources it has been extensively discussed using Tilman’s representation of resource quarter plane plots. Our mathematically rigorous analysis yields bifurcation diagrams with a striking similarity to Tilman’s method including the interpretation of the consumption vector and the resource supply vector. However, our approach is not restricted to a particular class of models but also works with other trophic interaction formulations. This is illustrated by the analysis of a model considering interactively-essential or complementary resources instead of prefect-essential resources. Additionally, our approach can also be used for other ecosystem compositions: multiple resources–multiple species communities with equilibrium or oscillatory dynamics. Hence, it gives not only a new interpretation of Tilman’s graphical approach, but it constitutes an extension of competition analyses to communities with many species as well as non-equilibrium dynamics.

Keywords

bifurcation analysiscompetitive exclusioncomplementary resourcesresource competitionresource quarter plane plot
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 6: Ecology , 2013 , pp. 165 - 185
Copyright
© EDP Sciences, 2013

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References

A. Ajbar, K. Alhumaizi. Dynamics of the Chemostat: A Bifurcation Theory Approach. Boca Raton FL, Taylor & Francis Group, CRC Press, 2012.
Baltzis, B.C., Fredrickson, A.G.. Limitation of growth rate by two complementary nutrients: Some elementary but neglected considerations. Biotechnol. Bioeng., 31 (1988), 75–86. CrossRefGoogle Scholar
A. Cunningham, R.M. Nisbet. Transients and oscillations in continuous culture. In M.J. Bazin (ed), Mathematics in Microbiology, pages 77–103, London, Academic Press, 1983.
Dhooge, A., Govaerts, W., Kuznetsov, Yu.A.. Matcont: A MATLAB package for numerical bifurcation analysis of ODEs. ACM T. Math. Software, 29 (2003), 141–164. CrossRefGoogle Scholar
E.J. Doedel, B. Oldeman. Auto 07p: Continuation and bifurcation software for ordinary differential equations. Technical report, Concordia University, Montreal, Canada, 2009.
G.F. Gause. The Struggle for Existence. Hafner Publishing, New York, 1969.
Geritz, S.A.H., Kisdi, É., Meszéna, G., Metz, J.A.J.. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol., 12 (1998), 35–57. CrossRefGoogle Scholar
J.P. Grover. Resource Competition. Population and Community Biology series. Chapman & Hall, London, 1997.
J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 2 edition, 1985.
Hardin, G.. The competitive exclusion principle. Science, 131:3409 (1960), 1292–1297. CrossRefGoogle ScholarPubMed
Harpole, W.S., Ngai, J.T., Cleland, E.E., Seabloom, E.W., Borer, E.T., Bracken, M.E.S., Elser, J.J., Gruner, D.S., Hillebrand, H., Shurin, J.B., Smith, J.E.. Nutrient co-limitation of primary producer communities. Ecol. Lett., 125 (2011), 852–862. CrossRefGoogle Scholar
Hsu, S.-B., Cheng, K.-S., Hubbell, S.P.. Exploitative competition of microorganisms for two complementary nutrients in continuous cultures. SIAM J. Appl. Math., 41:3 (1981), 422–444 . CrossRefGoogle Scholar
Huisman, J., Weissing, F.J.. Biological conditions for oscillations and chaos generated by multispecies competition. Ecology, 82:10 (2001), 2682–2695 . CrossRefGoogle Scholar
Hutchinson, G. E.. The paradox of the plankton. Am. Nat., 95:882 (1961), 137–145. CrossRefGoogle Scholar
Kooi, B.W., Boer, M.P., Kooijman, S.A.L.M.. Resistance of a food chain to invasion by a top predator. Math. Biosci., 157 (1999), 217–236. CrossRefGoogle ScholarPubMed
Kooi, B.W.. Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment. Acta Biotheor., 51:3 (2003), 189-222. CrossRefGoogle Scholar
Kooi, B.W., Kuijper, L.D.J., Kooijman, S.A.L.M.. Consequence of symbiosis for food web dynamics. J. Math. Biol., 49:3 (2004), 227-271. CrossRefGoogle Scholar
S.A.L.M. Kooijman. Dynamic Energy Budget theory for metabolic organisation. Cambridge University Press, Cambridge, 2010.
Kooijman, S.A.L.M., Dijkstra, H.A., Kooi, B.W.. Light-induced mass turnover in a mono-species community of mixotrophs. J. Theor. Biol., 214 (2002), 233–254. CrossRefGoogle Scholar
Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. volume 112 of Applied Mathematical Sciences. Springer-Verlag, New York, 3 edition, 2004.
Miller, T.E., Burns, J.H., Munguia, P., Walters, E.L., Kneitel, J.M., Richards, P. M., Mouquet, N., Buckley, H. L.. A critical review of Twenty Years’ use of the Resource-Ratio Theory. Am. Nat., 165:4 (2005), 439-448. Google ScholarPubMed
Morton, R.D., Law, R., Pimm, S.L., Drake, J.A.. On models for assembling ecological communities. Oikos, 75:3 (1996), 493-499. CrossRefGoogle Scholar
Nattrass, S., Baigent, S., Murrell, D.J.. Quantifying the likelihood of co-existence for communities with asymmetric competition. B. Math. Biol., 74:10 (2012), 2315-2338. CrossRefGoogle ScholarPubMed
O’Neill, R.V., DeAngelis, D.L., Pastor, J.J., Jackson, B.J., Post, W.M.. Multiple nutrient limitations in ecological models. Ecol. Model., 46 (1989), 147–163. CrossRefGoogle Scholar
Schipper, P., Verschoor, A.M., Vos, M., Mooij, W.M.. Does “supersaturated coexistence” resolve the “paradox of the plankton”? Ecol. Lett., 4 (2001), 404–407. CrossRefGoogle Scholar
Sperfeld, E., Martin-Creuzburg, D., Wacker, A.. Multiple resource limitation theory applied to herbivorous consumers: Liebig’s minimum rule vs. interactive co-limitation. Ecol. Lett., 15 (2012) 142–150. CrossRefGoogle ScholarPubMed
H.L. Smith, P. Waltman. The Theory of the Chemostat. Cambridge University Press, Cambridge, 1994.
Troost, T.A., Kooi, B.W., Kooijman, S.A.L.M.. Bifurcation analysis can unify ecological and evolutionary aspects of ecosystems. Ecol. Model., 204 (2007), 253-268. CrossRefGoogle Scholar
Tilman, D.. Resource competition between planktonic algae: an experimental and theoretical approach. Ecology, 58 (1977), 338–348. CrossRefGoogle Scholar
Tilman, D.. Resources: A graphical-mechanistic approach to competition and predation. Am. Nat., 116 (1980), 363–393. CrossRefGoogle Scholar
D. Tilman. Resource competition and community structure. Princeton University Press, Princeton, 1982.
Tilman, D.. The resource-ratio hypothesis of plant succession. Am. Nat., 125:6 (1985), 827-852. CrossRefGoogle Scholar
D. Tilman. Plant strategies and the Dynamics and Structure of Plant Communities. Princeton University Press, Princeton, 1988.
Wilson, J.B., Spijkerman, E., Huisman, J.. Is there really insufficient support for Tilman’s R* concept? A comment on Miller et al.. Am. Nat., 169:5 (2007), 700-706. Google ScholarPubMed