American Institute of Mathematical Sciences

• Previous Article
Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property
• CPAA Home
• This Issue
• Next Article
Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat
July  2017, 16(4): 1373-1404. doi: 10.3934/cpaa.2017066

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

Received  July 2015 Revised  July 2016 Published  February 2017

Fund Project: 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).

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.

Citation: Hua Nie, Sze-bi Hsu, Jianhua Wu. A competition model with dynamically allocated toxin production in the unstirred chemostat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1373-1404. doi: 10.3934/cpaa.2017066
References:

show all references

References:
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
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
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
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))
 [1] Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1279-1297. doi: 10.3934/cpaa.2013.12.1279 [2] Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621 [3] Hua Nie, Feng-Bin Wang. Competition for one nutrient with recycling and allelopathy in an unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2129-2155. doi: 10.3934/dcdsb.2015.20.2129 [4] Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169 [5] Hai-Xia Li, Jian-Hua Wu, Yan-Ling Li, Chun-An Liu. Positive solutions to the unstirred chemostat model with Crowley-Martin functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2951-2966. doi: 10.3934/dcdsb.2017128 [6] 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, 2020, 40 (7) : 4427-4451. doi: 10.3934/dcds.2020185 [7] Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591 [8] Hua Nie, Jianhua Wu. The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 303-329. doi: 10.3934/dcds.2012.32.303 [9] Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669 [10] Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391 [11] Haixiang Yao, Zhongfei Li, Yongzeng Lai. Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate. Journal of Industrial & Management Optimization, 2016, 12 (1) : 187-209. doi: 10.3934/jimo.2016.12.187 [12] Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 875-891. doi: 10.3934/dcdsb.2009.11.875 [13] Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013 [14] Jingli Ren, Gail S. K. Wolkowicz. Preface: Recent advances in bifurcation theory and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : i-ii. doi: 10.3934/dcdss.2020417 [15] Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387 [16] Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163 [17] Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 [18] Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009 [19] Kunquan Lan, Wei Lin. Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 849-861. doi: 10.3934/dcdsb.2016.21.849 [20] Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591

2019 Impact Factor: 1.105