A periodic-parabolic Droop model for two species competition in an unstirred chemostat

Xiaoqing He; Sze-Bi Hsu; Feng-Bin Wang

doi:10.3934/dcds.2020185

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2020, Volume 40, Issue 7: 4427-4451. Doi: 10.3934/dcds.2020185

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A periodic-parabolic Droop model for two species competition in an unstirred chemostat

1.
School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
2.
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
3.
Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan
4.
Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

^* Corresponding author: Feng-Bin Wang
^* Corresponding author: Feng-Bin Wang

Received: August 2019

Revised: December 2019

Published: April 2020

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Abstract

We study a periodic-parabolic Droop model of two species competing for a single-limited nutrient in an unstirred chemostat, where the nutrient is added to the culture vessel by way of periodic forcing function in time. For the single species model, we establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear periodic eigenvalue problem. In particular, when diffusion rate is sufficiently small or large, the sign can be determined. We then show that for the competition model, when diffusion rates for both species are small, there exists a coexistence periodic solution.

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Mathematics Subject Classification: Primary: 35B10, 35K51, 37C75, 35K57, 92D25.

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