Dynamics of a chemostat system with two patches

Shikun Wang

doi:10.3934/dcdsb.2019138

Article Contents

2019, Volume 24, Issue 11: 6261-6278. Doi: 10.3934/dcdsb.2019138

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Dynamics of a chemostat system with two patches

Shikun Wang^a,b,

a.
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
b.
Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, USA, 77030

Received: July 2018

Revised: January 2019

Early access: July 2019

Published: August 2019

The first author is supported by NSF grant of China (11571382)

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Abstract

This paper studies a diffusion model with two patches, which is derived from experiments and includes exploitable resources. Our aim is to provide theoretical proof for experimental observations and extend previous theory to consumer-resource systems with external resource inputs. First, we exhibit nonnegativeness and boundedness of solutions of the model. For one-patch subsystems, we demonstrate the global dynamics by excluding periodic solutions. For the two-patch system, we exhibit uniform persistence of the system and asymptotic stability of the positive equilibria, while the equilibria converge to a unique positive point as the diffusion tends to infinity. Then we demonstrate that homogeneously distributed resources support higher total population abundance than heterogeneously distributed resources with diffusion, which coincides with empirical observation but refutes previous theory. Meanwhile, we exhibit new conditions under which populations diffusing in heterogeneous environments can reach higher total size than if non-diffusing. A new finding of our study is that these results hold even with source-sink populations, and varying the diffusion rate can result in survival/extinction of the species. Our results are consistent with experimental observations and provide new insights.

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Mathematics Subject Classification: 34C37, 92D25, 37N25.

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Figure 1. Phase-plane diagram of subsystem (3). Stable and unstable equilibria are identified by solid and open circles, respectively. Vector fields are shown by gray arrows. Isoclines of the nutrient and consumer are represented by red and blue lines, respectively. Let $ N_{01} = 0.02, r_1 = k = 0.1, \gamma = m_1 = 0.01 $. All positive solutions of (3) converge to equilibrium $ E^+(0.111, 8.85) $

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