Dynamics of the periodically forced light-limited Droop model

Type Article
Date 2020-08
Language English
Author(s) Martinez Carlos1, Mairet FrancisORCID1, 3, Bernard Olivier1, 2
Affiliation(s) 1 : Universite Cote d'Azur, Inria, INRA, CNRS, UPMC Univ Paris 06, BIOCORE team, France
2 : LOV-UPMC Sorbonne-CNRS, UMR 7093, Station Zoologique, B.P. 28, 06234 Villefranche-sur-mer, France
3 : IFREMER Physiology and Biotechnology of Algae Laboratory, Nantes, France
Source Journal Of Differential Equations (0022-0396) (Elsevier), 2020-08 , Vol. 269 , N. 4 , P. 3890-3913
DOI 10.1016/j.jde.2020.03.020
Keyword(s) Variable quota model, Positive periodic solution, Global stability, Microalgae, Cooperative system
Abstract

The periodically forced light-limited Droop model represents microalgae growth under co-limitation by light and a single substrate, accounting for periodic fluctuations of factors such as light and temperature. In this paper, we describe the global dynamics of this model, considering general monotone growth and uptake rate functions. Our main result gives necessary and sufficient conditions for the existence of a positive periodic solution (i.e. a periodic solution characterized by the presence of microalgae) which is globally attractive. In our approach, we reduce the model to a cooperative planar periodic system. Using results on periodic Kolmogorov equations and on monotone sub-homogeneous dynamical systems, we describe the global dynamics of the reduced system. Then, using the theory of asymptotically periodic semiflows, we extend the results on the reduced system to the original model. To illustrate the applicability of the main result, we include an example considering a standard microalgae population model.

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