Geodesic
Resource Competition: A Bifurcation Theory Approach
B. W. Kooi1 ; P. S. Dutta2, 3 ; U. Feudel2
1 Department of Theoretical Biology, VU University de Boelelaan 1085, 1081 HV Amsterdam, The Netherlands
2 Theoretical Physics/Complex Systems, ICBM Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg, Germany
3 Department of Mathematics, Indian Institute of Technology Ropar Rupnagar-140001, Punjab, India
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 6, pp. 165-185 Cet article a éte moissonné depuis la source EDP Sciences

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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.
DOI : 10.1051/mmnp/20138611

B. W. Kooi 1 ; P. S. Dutta 2, 3 ; U. Feudel 2

1 Department of Theoretical Biology, VU University de Boelelaan 1085, 1081 HV Amsterdam, The Netherlands
2 Theoretical Physics/Complex Systems, ICBM Carl von Ossietzky Universität, PF 2503, 26111 Oldenburg, Germany
3 Department of Mathematics, Indian Institute of Technology Ropar Rupnagar-140001, Punjab, India 
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B. W. Kooi; P. S. Dutta; U. Feudel. Resource Competition: A Bifurcation Theory Approach. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 6, pp. 165-185. doi: 10.1051/mmnp/20138611

[1] A. Ajbar, K. Alhumaizi. Dynamics of the Chemostat: A Bifurcation Theory Approach. Boca Raton FL, Taylor Francis Group, CRC Press, 2012.

[2] B.C. Baltzis, A.G. Fredrickson Biotechnol. Bioeng. 1988 75 86

[3] 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.

[4] A. Dhooge, W. Govaerts, Yu.A. Kuznetsov ACM T. Math. Software 2003 141 164

[5] E.J. Doedel, B. Oldeman. Auto 07p: Continuation and bifurcation software for ordinary differential equations. Technical report, Concordia University, Montreal, Canada, 2009.

[6] G.F. Gause. The Struggle for Existence. Hafner Publishing, New York, 1969.

[7] S.A.H. Geritz, É. Kisdi, G. Meszéna, J.A.J. Metz Evol. Ecol. 1998 35 57

[8] J.P. Grover. Resource Competition. Population and Community Biology series. Chapman Hall, London, 1997.

[9] 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.

[10] G. Hardin Science 1960 1292 1297

[11] W.S. Harpole, J.T. Ngai, E.E. Cleland, E.W. Seabloom, E.T. Borer, M.E.S. Bracken, J.J. Elser, D.S. Gruner, H. Hillebrand, J.B. Shurin, J.E. Smith Ecol. Lett. 2011 852 862

[12] S.-B. Hsu, K.-S. Cheng, S.P. Hubbell SIAM J. Appl. Math. 1981 422 444

[13] J. Huisman, F.J. Weissing Ecology 2001 2682 2695

[14] G. E. Hutchinson Am. Nat. 1961 137 145

[15] B.W. Kooi, M.P. Boer, S.A.L.M. Kooijman Math. Biosci. 1999 217 236

[16] B.W. Kooi Acta Biotheor. 2003 189 222

[17] B.W. Kooi, L.D.J. Kuijper, S.A.L.M. Kooijman J. Math. Biol. 2004 227 271

[18] S.A.L.M. Kooijman. Dynamic Energy Budget theory for metabolic organisation. Cambridge University Press, Cambridge, 2010.

[19] S.A.L.M. Kooijman, H.A. Dijkstra, B.W. Kooi J. Theor. Biol. 2002 233 254

[20] Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. volume 112 of Applied Mathematical Sciences. Springer-Verlag, New York, 3 edition, 2004.

[21] T.E. Miller, J.H. Burns, P. Munguia, E.L. Walters, J.M. Kneitel, P. M. Richards, N. Mouquet, H. L. Buckley Am. Nat. 2005 439 448

[22] R.D. Morton, R. Law, S.L. Pimm, J.A. Drake Oikos 1996 493 499

[23] S. Nattrass, S. Baigent, D.J. Murrell B. Math. Biol. 2012 2315 2338

[24] R.V. O’Neill, D.L. Deangelis, J.J. Pastor, B.J. Jackson, W.M. Post Ecol. Model. 1989 147 163

[25] P. Schipper, A.M. Verschoor, M. Vos, W.M. Mooij Ecol. Lett. 2001 404 407

[26] E. Sperfeld, D. Martin-Creuzburg, A. Wacker Ecol. Lett. 2012 142 150

[27] H.L. Smith, P. Waltman. The Theory of the Chemostat. Cambridge University Press, Cambridge, 1994.

[28] T.A. Troost, B.W. Kooi, S.A.L.M. Kooijman Ecol. Model. 2007 253 268

[29] D. Tilman Ecology 1977 338 348

[30] D. Tilman Am. Nat. 1980 363 393

[31] D. Tilman. Resource competition and community structure. Princeton University Press, Princeton, 1982.

[32] D. Tilman Am. Nat. 1985 827 852

[33] D. Tilman. Plant strategies and the Dynamics and Structure of Plant Communities. Princeton University Press, Princeton, 1988.

[34] J.B. Wilson, E. Spijkerman, J. Huisman Am. Nat. 2007 700 706

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