Abstract
We construct a model of competition of three consumers for one single biotic resource ; simulations show that the three species coexist. Using singular perturbations theory we sketch a mathematical proof for this coexistence. The main mathematical tool used is an extension of the Pontryagin-Rodygin theorem on the “slow” motion of a “slow-fast” differential system when the “fast” motion possesses a stable limit cycle. The mathematical analysis is done within the framework of Non Standard Analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Armstrong, R.A., McGehee, R.: Coexistence of species competing for shared resources. Theoretical Pop. Biol. (9), 317–328 (1976)
Butler, G., Waltman, P.: Persistence in dynamical systems. J. Differential Equations (63), 255–263 (1986)
Callot, J.-L., Sari, T.: Stroboscopie et moyennisation dans les systèmes d’équations différentielles à solutions rapidement oscillantes. Mathematical Tools and Models for Control, Systems Analysis and Signal Processing, CNRS Paris (3), 345–353 (1983)
Jost, J.L., Drake, T.J.F., Frederickson, A.G., Tsuchiya, M.: Interaction of Tetrahimen pyriformis, Escherichi coli, Azotobacter vinelandi and Glucose in a Minimal Madium. J. of Bacteriology 113(2), 834–840 (1973)
Lobry, C., Sari, T.: Nonstandard analysis and representation of real world. International Journal on Control 80(3), 171–193 (2007)
McGehee, R., Armstrong, R.A.: Some mathematical problems concerning the ecological principle of competitive exclusion. Journal od Differential Equations (23), 30–52 (1977)
Nelson, E.: Internal Set Theory: a new approach to nonstandard analysis. Bull. Amer. Math. Soc. 83(6), 1165–1198 (1977)
Pontryagin, L.S., Rodygin, L.V.: Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives. Soviet. Math. Dokl. (1), 237–240 (1960)
Reeb, G.: La mathématique non standard vieille de soixante ans ? Troisième Colloque sur les Catégories, dédi é à C. Ehresmann, Amiens, 1980. Cahiers Topologie Géom. Différentielle 22(2), 149–154 (1981)
Robinson, A.: Nonstandard Analysis. American Elsevier, New York (1974)
Sari, T.: Averaging in Ordinary Differential Equations and Functional Differential Equations. In: van den Berg, I., Neves, V. (eds.) The Strength of Nonstandard Analysis, pp. 286–305. Springer, Wien (2007)
Sari, T., Yadi, K.: On Pontryagin–Rodygin’s theorem for convergence of solutions of slow and fast systems. Electron. J. Diff. Eqns (139), 1–17 (2004)
Yadi, K.: Singular perturbations on infinite time interval. Revue Arima 9, 37–560 (2008)
Yadi, K.: Perturbations Singulières: Approximations, Stabilité Pratique et Applications à des Modèles de Compétition. Thèse de doctorat de l’Université de Haute-Alsace de Mulhouse (2008), http://tel.archives-ouvertes.fr/tel-00411503/fr/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lobry, C., Sari, T., Yadi, K. (2010). Coexistence of Three Predators Competing for a Single Biotic Resource. In: Lévine, J., Müllhaupt, P. (eds) Advances in the Theory of Control, Signals and Systems with Physical Modeling. Lecture Notes in Control and Information Sciences, vol 407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16135-3_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-16135-3_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16134-6
Online ISBN: 978-3-642-16135-3
eBook Packages: EngineeringEngineering (R0)