Abstract
We study a model of competition for resource through a chemostat-type model where species consume the common resource that is constantly supplied. We assume that the species and resources are characterized by a continuous trait. As already proved, this model, although more complicated than the usual Lotka–Volterra direct competition model, describes competitive interactions leading to concentrated distributions of species in continuous trait space. Here we assume a very fast dynamics for the supply of the resource and a fast dynamics for death and uptake rates. In this regime we show that factors that are independent of the resource competition become as important as the competition efficiency and that the direct competition model is a good approximation of the chemostat. Assuming these two timescales allows us to establish a mathematically rigorous proof showing that our resource-competition model with continuous traits converges to a direct competition model. We also show that the two timescales assumption is required to mathematically justify the corresponding classic result on a model consisting of only finite number of species and resources (MacArthur in, Theor Popul Biol 1:1–11, 1970). This is performed through asymptotic analysis, introducing different scales for the resource renewal rate and the uptake rate. The mathematical difficulty relies in a possible initial layer for the resource dynamics. The chemostat model comes with a global convex Lyapunov functional. We show that the particular form of the competition kernel derived from the uptake kernel, satisfies a positivity property which is known to be necessary for the direct competition model to enjoy the related Lyapunov functional.
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Acknowledgments
The three authors thank Readilab, LIA 197 CNRS, for supporting their collaborations and also Meiji University and UPMC. J.Y.W. is partly supported by Glocal COE program “Formation and Development of Mathematical Sciences Based on Modeling and Analysis”. S. M. benefits from a 2 year “Fondation Mathématique Jacques Hadamard” (FMJH) postdoc scholarship. She would like to thank Ecole Polytechnique for its hospitality. The authors would like also to thank Géza Meszéna for careful review of this paper, valuable comments, several references and interpretations. We borrowed some sentences from the report.
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Mirrahimi, S., Perthame, B. & Wakano, J.Y. Direct competition results from strong competition for limited resource. J. Math. Biol. 68, 931–949 (2014). https://doi.org/10.1007/s00285-013-0659-5
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DOI: https://doi.org/10.1007/s00285-013-0659-5
Keywords
- Ecological competition for resource
- Direct competition
- Multiscale analysis
- Lyapunov functional
- Adaptive dynamics