• 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:
[1]

S. AbbasM. Banerjee and N. Hungerbuhler, Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model, J. Math. Anal. Appl., 367 (2010), 249-259.  doi: 10.1016/j.jmaa.2010.01.024.  Google Scholar

[2]

M. AnD. L. LiuI. R. Johnson and J. V. Lovett, Mathematical modelling of allelopathy: Ⅱ. The dynamics of allelochemicals from living plants in the environment, Ecol. Modell., 161 (2003), 53-66.  doi: 10.1016/S0304-3800(02)00289-2.  Google Scholar

[3]

B. L. Bassler, How bacteria talk to each other: regulation of gene expression by quorum sensing, Curr. Opinion Microbiol., 2 (1999), 582-587.  doi: 10.1016/S1369-5274(99)00025-9.  Google Scholar

[4]

F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications, Addison-Wesley Longman, Harlow, UK, 1997.  Google Scholar

[5]

J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Math. Biosci., 173 (2001), 55-84.  doi: 10.1016/S0025-5564(01)00078-5.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.   Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearixed stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[8]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.   Google Scholar

[9]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.   Google Scholar

[10]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part Ⅰ, General existence results, Nonlinear Anal., 24 (1995), 337-357.   Google Scholar

[11]

P. FergolaE. Beretta and M. Cerasuolo, Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production, Nonliear Anal. Real World Appl., 7 (2006), 1081-1095.  doi: 10.1016/j.nonrwa.2005.10.001.  Google Scholar

[12]

P. FergolaM. CerasuoloA. Pollio and G. Pinto, Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecol. Modell., 208 (2007), 205-214.  doi: 10.1016/j.ecolmodel.2007.05.024.  Google Scholar

[13]

P. FergolaJ. Li and Z. Ma, On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ric. Mat., 60 (2011), 313-332.  doi: 10.1007/s11587-011-0108-y.  Google Scholar

[14]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. Google Scholar

[15]

S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490.  doi: 10.1007/BF03167323.  Google Scholar

[16]

F. D. HulotP. J. Morin and M. Loreau, Interactions between algae and the microbial loop in experimental microcosms, Oikos, 95 (2001), 231-238.  doi: 10.1034/j.1600-0706.2001.950205.x.  Google Scholar

[17]

J. López-Gómez and R. Parda, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case, Differential Integral Equations, 6 (1993), 1025-1031.   Google Scholar

[18]

H. NieN. Liu and J. H. Wu, Coexistence solutions and their stability of an unstirred chemostat model with toxins, Nonlinear Analysis: Real World Appl., 20 (2014), 36-51.  doi: 10.1016/j.nonrwa.2014.04.002.  Google Scholar

[19]

H. Nie and J. H. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, International J. Bifurcation and Chaos, 16 (2006), 989-1009.  doi: 10.1142/S0218127406015246.  Google Scholar

[20]

H. Nie and J. H. Wu, Asymptotic behavior on a competition model arising from an unstirred chemostat, Acta Mathematicae Applicatae Sinica, English Series, 22 (2006), 257-264.  doi: 10.1007/s10255-006-0301-z.  Google Scholar

[21]

H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat, Discrete Contin. Dyn. Syst., 32 (2012), 303-329.  doi: 10.3934/dcds.2012.32.303.  Google Scholar

[22]

H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin, European J. Appl. Math., 25 (2014), 481-510.  doi: 10.1017/S0956792514000096.  Google Scholar

[23]

H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, Appl. Anal., 89 (2010), 1141-1159.  doi: 10.1080/00036811003717954.  Google Scholar

[24]

G. PintoA. PollioM. D. Greca and R. Ligrone, Linoleic acid-a potential allelochemical released by Eichhornia crassipes (Mart.) Solms in a continuous trapping apparatus, Allelopathy J., 2 (1995), 169-178.   Google Scholar

[25] E. L. Rice, Allelopathy, 2nd ed, Academic Press, Orlando, FL, .   Google Scholar
[26]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.   Google Scholar

[27] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[28]

T. F. Thingstad and B. Pengerud, Fate and effect of allochthonus organic material in aquatic microbial ecosystems: an analysis based on chemostat theory, Mar. Ecol. Prog. Ser., 21 (1985), 47-62.   Google Scholar

[29] M. X. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010.   Google Scholar
[30]

J. H. WuH. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM J. Appl. Math., 65 (2004), 209-229.  doi: 10.1137/S0036139903423285.  Google Scholar

[31]

J. H. WuH. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885.  doi: 10.1137/050627514.  Google Scholar

[32]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

show all references

References:
[1]

S. AbbasM. Banerjee and N. Hungerbuhler, Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model, J. Math. Anal. Appl., 367 (2010), 249-259.  doi: 10.1016/j.jmaa.2010.01.024.  Google Scholar

[2]

M. AnD. L. LiuI. R. Johnson and J. V. Lovett, Mathematical modelling of allelopathy: Ⅱ. The dynamics of allelochemicals from living plants in the environment, Ecol. Modell., 161 (2003), 53-66.  doi: 10.1016/S0304-3800(02)00289-2.  Google Scholar

[3]

B. L. Bassler, How bacteria talk to each other: regulation of gene expression by quorum sensing, Curr. Opinion Microbiol., 2 (1999), 582-587.  doi: 10.1016/S1369-5274(99)00025-9.  Google Scholar

[4]

F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications, Addison-Wesley Longman, Harlow, UK, 1997.  Google Scholar

[5]

J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Math. Biosci., 173 (2001), 55-84.  doi: 10.1016/S0025-5564(01)00078-5.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.   Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearixed stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[8]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.   Google Scholar

[9]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.   Google Scholar

[10]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part Ⅰ, General existence results, Nonlinear Anal., 24 (1995), 337-357.   Google Scholar

[11]

P. FergolaE. Beretta and M. Cerasuolo, Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production, Nonliear Anal. Real World Appl., 7 (2006), 1081-1095.  doi: 10.1016/j.nonrwa.2005.10.001.  Google Scholar

[12]

P. FergolaM. CerasuoloA. Pollio and G. Pinto, Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecol. Modell., 208 (2007), 205-214.  doi: 10.1016/j.ecolmodel.2007.05.024.  Google Scholar

[13]

P. FergolaJ. Li and Z. Ma, On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ric. Mat., 60 (2011), 313-332.  doi: 10.1007/s11587-011-0108-y.  Google Scholar

[14]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. Google Scholar

[15]

S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490.  doi: 10.1007/BF03167323.  Google Scholar

[16]

F. D. HulotP. J. Morin and M. Loreau, Interactions between algae and the microbial loop in experimental microcosms, Oikos, 95 (2001), 231-238.  doi: 10.1034/j.1600-0706.2001.950205.x.  Google Scholar

[17]

J. López-Gómez and R. Parda, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case, Differential Integral Equations, 6 (1993), 1025-1031.   Google Scholar

[18]

H. NieN. Liu and J. H. Wu, Coexistence solutions and their stability of an unstirred chemostat model with toxins, Nonlinear Analysis: Real World Appl., 20 (2014), 36-51.  doi: 10.1016/j.nonrwa.2014.04.002.  Google Scholar

[19]

H. Nie and J. H. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, International J. Bifurcation and Chaos, 16 (2006), 989-1009.  doi: 10.1142/S0218127406015246.  Google Scholar

[20]

H. Nie and J. H. Wu, Asymptotic behavior on a competition model arising from an unstirred chemostat, Acta Mathematicae Applicatae Sinica, English Series, 22 (2006), 257-264.  doi: 10.1007/s10255-006-0301-z.  Google Scholar

[21]

H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat, Discrete Contin. Dyn. Syst., 32 (2012), 303-329.  doi: 10.3934/dcds.2012.32.303.  Google Scholar

[22]

H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin, European J. Appl. Math., 25 (2014), 481-510.  doi: 10.1017/S0956792514000096.  Google Scholar

[23]

H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, Appl. Anal., 89 (2010), 1141-1159.  doi: 10.1080/00036811003717954.  Google Scholar

[24]

G. PintoA. PollioM. D. Greca and R. Ligrone, Linoleic acid-a potential allelochemical released by Eichhornia crassipes (Mart.) Solms in a continuous trapping apparatus, Allelopathy J., 2 (1995), 169-178.   Google Scholar

[25] E. L. Rice, Allelopathy, 2nd ed, Academic Press, Orlando, FL, .   Google Scholar
[26]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.   Google Scholar

[27] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[28]

T. F. Thingstad and B. Pengerud, Fate and effect of allochthonus organic material in aquatic microbial ecosystems: an analysis based on chemostat theory, Mar. Ecol. Prog. Ser., 21 (1985), 47-62.   Google Scholar

[29] M. X. Wang, Nonlinear Elliptic Equations, Science Press, Beijing, 2010.   Google Scholar
[30]

J. H. WuH. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM J. Appl. Math., 65 (2004), 209-229.  doi: 10.1137/S0036139903423285.  Google Scholar

[31]

J. H. WuH. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885.  doi: 10.1137/050627514.  Google Scholar

[32]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

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
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
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
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))
[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]

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

[7]

Hua Nie, Jianhua Wu. The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 303-329. doi: 10.3934/dcds.2012.32.303

[8]

Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387

[14]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591

[15]

Xiao-Ping Wang, Xianmin Xu. A dynamic theory for contact angle hysteresis on chemically rough boundary. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1061-1073. doi: 10.3934/dcds.2017044

[16]

Changjun Yu, Honglei Xu, Kok Lay Teo. Preface: Advances in theory and real world applications of control and dynamic optimization. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020094

[17]

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

[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 - A, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009

[19]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701

[20]

Klaus Reiner Schenk-Hoppé. Random attractors--general properties, existence and applications to stochastic bifurcation theory. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 99-130. doi: 10.3934/dcds.1998.4.99

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (24)
  • HTML views (12)
  • Cited by (1)

Other articles
by authors

[Back to Top]