Positive solutions to the unstirred chemostat model with Crowley-Martin functional response

Hai-Xia Li; Jian-Hua Wu; Yan-Ling Li; Chun-An Liu

doi:10.3934/dcdsb.2017128

Article Contents

2018, Volume 23, Issue 8: 2951-2966. Doi: 10.3934/dcdsb.2017128

This issue Previous Article Next Article Fractional Navier-Stokes equations

Positive solutions to the unstirred chemostat model with Crowley-Martin functional response

1.
Institute of Mathematics and Information Sciences, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China,
2.
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China,
3.
Institute of Mathematics and Information Sciences, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China

* Corresponding author
* Corresponding author

Received: March 2016

Revised: February 2017

Early access: April 2017

Published: September 2018

The work is supported by the Natural Science Foundation of China (61672021,11401356,11671 243), the Natural Science Basic Research Plan in Shaanxi Province of China (2015JM1008), the Foundations of Shaanxi Educational Committee (16JK1046), the Postdoctoral Science Foundation of China (2016M602767) and the Special Fund of Education Department of Shaanxi Province (16JK1710).

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Abstract

A food-chain model with Crowley-Martin functional response in the unstirred chemostat is considered. First, the global framework of coexistence solutions is discussed by the maximum principle and bifurcation theory. We obtain the sufficient and necessary conditions for coexistence of steady-state. Second, the stability and uniqueness of coexistence solutions are investigated by means of the combination of the perturbation theory and fixed point index theory. Our results indicate that if the magnitude of interference among predator is sufficiently large, the model has only one unique linearly stable coexistence solution when the maximal growth rate of predator belongs to certain range. Finally, some numerical simulations are carried out to verify and complement the theoretical results.

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Mathematics Subject Classification: Primary: 35K57, 35B32, 35B20; Secondary: 92B05.

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Figure 1. Existence and non-existence of coexistence states. In (a), $a=5,b=10.$ In (b) and (c), $a=8,b=5$ and $a=15,b=5$ respectively. In (d), $a=5,b=1.$.

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Figure 2. Different values of the parameters $k_{1},k_{2},m_{1}$. In (a) and (b), $k_{2}=1,m_{1}=2,k_{1}=1,2$ respectively.

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Figure 3. Different values of the parameter $m_{2}$. In (a)-(d), $m_{2}=1,2,200,1000$ respectively. In (e) and (f), $m_{2}=500$.

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