# American Institute of Mathematical Sciences

July  2020, 40(7): 4427-4451. doi: 10.3934/dcds.2020185

## 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

Received  August 2019 Revised  December 2019 Published  April 2020

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.

Citation: Xiaoqing He, Sze-Bi Hsu, Feng-Bin Wang. A periodic-parabolic Droop model for two species competition in an unstirred chemostat. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4427-4451. doi: 10.3934/dcds.2020185
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