2014, 19(10): 3169-3189. doi: 10.3934/dcdsb.2014.19.3169

Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage

1. 

Department of Mathematics and The National Center for Theoretical Science, National Tsing-Hua University, Hsinchu 30013

2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795

3. 

Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333

Received  June 2013 Revised  August 2013 Published  October 2014

The dynamics of a reaction-diffusion system for two species of microorganism in an unstirred chemostat with internal storage is studied. It is shown that the diffusion coefficient is a key parameter of determining the asymptotic dynamics, and there exists a threshold diffusion coefficient above which both species become extinct. On the other hand, for diffusion coefficient below the threshold, either one species or both species persist, and in the asymptotic limit, a steady state showing competition exclusion or coexistence is reached.
Citation: 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
References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion,, American Naturalist, 115 (1980), 151. doi: 10.1086/283553.

[2]

A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous culture,, Mathematics in Microbiology, (1983), 77.

[3]

M. R. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis lutheri,, J. Mar. Biol. Assoc. UK, 48 (1968), 689.

[4]

M. R. Droop, Some thoughts on nutrient limitation in algae,, Journal of Phycology, 9 (1973), 264.

[5]

J. P. Grover, Dynamics of competition among microalgae in variable environments: experimental tests of alternative models,, Oikos, 62 (1991), 231.

[6]

J. P. Grover, Resource competition in a variable environment: Phytoplankton growing according to the variable-internal-stores model,, American Naturalist, 138 (1991), 811.

[7]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability,, American Naturalist, 178 (2011), 124.

[8]

J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation,, J. Math. Biol., 64 (2012), 713. doi: 10.1007/s00285-011-0426-4.

[9]

S.-B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760. doi: 10.1137/0134064.

[10]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage,, SIAM J. Appl. Math., 68 (2008), 1600. doi: 10.1137/070700784.

[11]

S.-B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms,, SIAM J. Appl. Math., 32 (1977), 366. doi: 10.1137/0132030.

[12]

S.-B. Hsu, J.-F. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat,, J. Differential Equations, 248 (2010), 2470. doi: 10.1016/j.jde.2009.12.014.

[13]

S.-B. Hsu, J.-F. Jiang and F.-B. Wang, Reaction-diffusion equations of two species competing for two complementary resources with internal storage,, J. Differential Equations, 251 (2011), 918. doi: 10.1016/j.jde.2011.05.003.

[14]

S.-B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces,, Trans. Amer. Math. Soc., 348 (1996), 4083. doi: 10.1090/S0002-9947-96-01724-2.

[15]

S.-B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat,, SIAM J. Appl. Math., 53 (1993), 1026. doi: 10.1137/0153051.

[16]

S.-B. Hsu and F.-B. Wang, On a mathematical model arising from competition of phytoplankton species for a single nutrient with internal storage: steady state analysis,, Commun. Pure Appl. Anal., 10 (2011), 1479. doi: 10.3934/cpaa.2011.10.1479.

[17]

C. A. Klausmeier, E. Litchman, T. Daufresne and S. A. Levin, Phytoplankton stoichiometry,, Ecological Research, 23 (2008), 479.

[18]

B. S. Leadbeater, The 'Droop Equation'-Michael Droop and the legacy of the 'Cell-Quota Model' of phytoplankton growth,, Protist, 157 (2006), 345.

[19]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995).

[20]

H. L. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat,, J. Math. Biol., 35 (1997), 545. doi: 10.1007/s002850050065.

[21]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model,, SIAM J. Appl. Math., 54 (1994), 1113. doi: 10.1137/S0036139993245344.

[22]

M. C. White and X.-Q. Zhao, A periodic Droop model for two species competition in a chemostat,, Bull. Math. Biol., 71 (2009), 145. doi: 10.1007/s11538-008-9357-7.

show all references

References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion,, American Naturalist, 115 (1980), 151. doi: 10.1086/283553.

[2]

A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous culture,, Mathematics in Microbiology, (1983), 77.

[3]

M. R. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis lutheri,, J. Mar. Biol. Assoc. UK, 48 (1968), 689.

[4]

M. R. Droop, Some thoughts on nutrient limitation in algae,, Journal of Phycology, 9 (1973), 264.

[5]

J. P. Grover, Dynamics of competition among microalgae in variable environments: experimental tests of alternative models,, Oikos, 62 (1991), 231.

[6]

J. P. Grover, Resource competition in a variable environment: Phytoplankton growing according to the variable-internal-stores model,, American Naturalist, 138 (1991), 811.

[7]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability,, American Naturalist, 178 (2011), 124.

[8]

J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation,, J. Math. Biol., 64 (2012), 713. doi: 10.1007/s00285-011-0426-4.

[9]

S.-B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760. doi: 10.1137/0134064.

[10]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage,, SIAM J. Appl. Math., 68 (2008), 1600. doi: 10.1137/070700784.

[11]

S.-B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms,, SIAM J. Appl. Math., 32 (1977), 366. doi: 10.1137/0132030.

[12]

S.-B. Hsu, J.-F. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat,, J. Differential Equations, 248 (2010), 2470. doi: 10.1016/j.jde.2009.12.014.

[13]

S.-B. Hsu, J.-F. Jiang and F.-B. Wang, Reaction-diffusion equations of two species competing for two complementary resources with internal storage,, J. Differential Equations, 251 (2011), 918. doi: 10.1016/j.jde.2011.05.003.

[14]

S.-B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces,, Trans. Amer. Math. Soc., 348 (1996), 4083. doi: 10.1090/S0002-9947-96-01724-2.

[15]

S.-B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat,, SIAM J. Appl. Math., 53 (1993), 1026. doi: 10.1137/0153051.

[16]

S.-B. Hsu and F.-B. Wang, On a mathematical model arising from competition of phytoplankton species for a single nutrient with internal storage: steady state analysis,, Commun. Pure Appl. Anal., 10 (2011), 1479. doi: 10.3934/cpaa.2011.10.1479.

[17]

C. A. Klausmeier, E. Litchman, T. Daufresne and S. A. Levin, Phytoplankton stoichiometry,, Ecological Research, 23 (2008), 479.

[18]

B. S. Leadbeater, The 'Droop Equation'-Michael Droop and the legacy of the 'Cell-Quota Model' of phytoplankton growth,, Protist, 157 (2006), 345.

[19]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995).

[20]

H. L. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat,, J. Math. Biol., 35 (1997), 545. doi: 10.1007/s002850050065.

[21]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model,, SIAM J. Appl. Math., 54 (1994), 1113. doi: 10.1137/S0036139993245344.

[22]

M. C. White and X.-Q. Zhao, A periodic Droop model for two species competition in a chemostat,, Bull. Math. Biol., 71 (2009), 145. doi: 10.1007/s11538-008-9357-7.

[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]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[3]

Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053

[4]

Sze-Bi Hsu, Feng-Bin Wang. On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479

[5]

Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373

[6]

Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805

[7]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587

[8]

Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667

[9]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[10]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[11]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[12]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[13]

Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227

[14]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893

[15]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299

[16]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

[17]

Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673

[18]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817

[19]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85

[20]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]