A competition model with dynamically allocated toxin production in the unstirred chemostat

Hua Nie; Sze-bi Hsu; Jianhua Wu

doi:10.3934/cpaa.2017066

Article Contents

2017, Volume 16, Issue 4: 1373-1404. Doi: 10.3934/cpaa.2017066

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A competition model with dynamically allocated toxin production in the unstirred chemostat

1.
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China
2.
Department of Mathematics and National Center of Theoretical Science, National Tsing-Hua University, Hsinchu 300, Taiwan

^* Corresponding author
^* Corresponding author

Received: July 2015

Revised: July 2016

Published: May 2017

The first author is supported by the NSF of China(11671243), the Shaanxi New-star Plan of Science and Technology(2015KJXX-21), and the Fundamental Research Funds for the Central Universities(GK201701001)

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Abstract

This paper deals with a competition model with dynamically allocated toxin production in the unstirred chemostat. First, the existence and uniqueness of positive steady state solutions of the single population model is attained by the general maximum principle, spectral analysis and degree theory. Second, the existence of positive equilibria of the two-species system is investigated by the degree theory, and the structure and stability of nonnegative equilibria of the two-species system are established by the bifurcation theory. The results show that stable coexistence solution can occur with dynamic toxin production, which cannot occur with constant toxin production. Biologically speaking, it implies that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting. Finally, numerical results illustrate that a wide variety of dynamical behaviors can be achieved for the system with dynamic toxin production, including competition exclusion, bistable attractors, stable positive equilibria and stable limit cycles, which complement the analytic results.

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Mathematics Subject Classification: Primary: 35K57, 35K55, 35B32; Secondary: 35B50, 92D25.

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Figure 1. We further fix $d=0.1, \alpha=0.2, \beta=1, $ and observe the effects of the parameter $c$: coexistence in the form of equilibria is observed in (b)(c) when $c=0.2, 0.3$ respectively; competitive exclusion occurs in (a)(d) when $c=0.01, 0.6$ respectively

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Figure 2. We further fix $\alpha=0.8, \beta=0.001, c=0.2$, take the diffusion rates $d=0.4, 0.6, 0.65, 0.7, 0.9, 1.5$ in (a)-(f), and observe the effects of the diffusion rate $d$: (a) competition exclusion, (b)-(d) stable limit cycles, (e) stable positive equilibrium, (f) washout solution

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Figure 3. We further fix $d=0.1, \alpha=0.2, \beta=1, $ and observe the effects of the parameter $c$: coexistence in the form of equilibria is observed in (b)(c) when $c=0.2, 0.3$ respectively; competitive exclusion occurs in (a)(d) when $c=0.01, 0.6$ respectively

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Figure 4. Taking $d=0.1, \alpha=0.2, \beta=1, $ and $c=0.1$, bistable attractors occur, which are plotted in (a) and (b). Moreover, two positive equilibria appear (see (c) and (d))

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