NDLI: Direct competition results from strong competition for limited resource

Content Provider	Springer Nature Link
Author	Waka, Joe Yuichiro Perthame, Beît Mirrahimi, Sepideh
Copyright Year	2013
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.
Ending Page	949
Page Count	19
Starting Page	931
File Format	PDF
ISSN	03036812
e-ISSN	14321416
Journal	Journal of Mathematical Biology
Issue Number	4
Volume Number	68
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-02-26
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Lyapunov functional Adaptive dynamics Integro-differential operators Mathematical and Computational Biology Multiscale analysis Direct competition Ecological competition for resource Ecology Reaction-diffusion equations Applications of Mathematics Multiple scale methods Problems related to evolution
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Agricultural and Biological Sciences Modeling and Simulation

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