September  2011, 10(5): 1479-1501. doi: 10.3934/cpaa.2011.10.1479

On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis

1. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300

2. 

Department of Mathematics, National Tsing-Hua University, Hsinchu 300, Taiwan

Received  August 2009 Revised  June 2010 Published  April 2011

In this paper we construct a mathematical model of two microbial populations competing for a single-limited nutrient with internal storage in an unstirred chemostat. First we establish the existence and uniqueness of steady-state solutions for the single population. The conditions for the coexistence of steady states are determined. Techniques include the maximum principle, theory of bifurcation and degree theory in cones.
Citation: 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 and Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479
References:
[1]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189-203.

[2]

A. Cunningham and R. M. Nisbet, "Transient and Oscillation in Continuous Culture," in Mathematics in Microbiology, M. J. Bazin, ed., Academic ress, New York, 1983.

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[4]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.

[5]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[6]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Am. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.

[7]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion I. General existence results, Nonlinear Anal., 254 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[8]

Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, Fields Institute Communications, Vol. 48, American Mathematical Society, (2006), 95-135.

[9]

Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey, Journal of Differential Equations, 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Am. Math. Soc., 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.

[11]

L. C. Evans, "Partial Differential Equations," American Mathematical Society, 1998.

[12]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Commun. Part. Diff. Eq., 17 (1992), 339-346. doi: 10.1080/03605309208820844.

[13]

J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409-428.

[14]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.

[15]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.

[16]

S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology, SIAM J. Math. Anal., 14 (1983), 1130-1138. doi: 10.1137/0514087.

[17]

S. B. Hsu, S. Hubbell and P. Waltman, Mathematical theoy for single nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030.

[18]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044. doi: 10.1137/0153051.

[19]

J. Keener, "Principles of Applied Mathematics," Addison-Wesley, Reading, MA, 1987.

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984.

[21]

W. Ruan and W. Feng, On the fixed point index and multiple steady-state solutions of reaction-diffusion systems, Differential Integral Equations, 8 (1995), 371-392.

[22]

H. H. Schaefer, "Topological Vector Spaces," Macmillan, New York, 1966.

[23]

Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.

[24]

H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math., 34 (1994), 1113-1131. doi: 10.1137/S0036139993245344.

[25]

H. L. Smith and P. E. Waltman, "The Theory of the Chemostat," Cambridge Univ. Press, 1995.

[26]

M. X. Wang, "Nonlinear Parabolic Equations," Science Press, Beijing, 1993 (in Chinese).

show all references

References:
[1]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189-203.

[2]

A. Cunningham and R. M. Nisbet, "Transient and Oscillation in Continuous Culture," in Mathematics in Microbiology, M. J. Bazin, ed., Academic ress, New York, 1983.

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[4]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.

[5]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[6]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Am. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.

[7]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion I. General existence results, Nonlinear Anal., 254 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[8]

Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, Fields Institute Communications, Vol. 48, American Mathematical Society, (2006), 95-135.

[9]

Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey, Journal of Differential Equations, 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Am. Math. Soc., 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.

[11]

L. C. Evans, "Partial Differential Equations," American Mathematical Society, 1998.

[12]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Commun. Part. Diff. Eq., 17 (1992), 339-346. doi: 10.1080/03605309208820844.

[13]

J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409-428.

[14]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.

[15]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.

[16]

S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology, SIAM J. Math. Anal., 14 (1983), 1130-1138. doi: 10.1137/0514087.

[17]

S. B. Hsu, S. Hubbell and P. Waltman, Mathematical theoy for single nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030.

[18]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044. doi: 10.1137/0153051.

[19]

J. Keener, "Principles of Applied Mathematics," Addison-Wesley, Reading, MA, 1987.

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984.

[21]

W. Ruan and W. Feng, On the fixed point index and multiple steady-state solutions of reaction-diffusion systems, Differential Integral Equations, 8 (1995), 371-392.

[22]

H. H. Schaefer, "Topological Vector Spaces," Macmillan, New York, 1966.

[23]

Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.

[24]

H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math., 34 (1994), 1113-1131. doi: 10.1137/S0036139993245344.

[25]

H. L. Smith and P. E. Waltman, "The Theory of the Chemostat," Cambridge Univ. Press, 1995.

[26]

M. X. Wang, "Nonlinear Parabolic Equations," Science Press, Beijing, 1993 (in Chinese).

[1]

Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7

[2]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[3]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

[4]

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

[5]

J. F. Toland. Path-connectedness in global bifurcation theory. Electronic Research Archive, 2021, 29 (6) : 4199-4213. doi: 10.3934/era.2021079

[6]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[7]

Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012

[8]

Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321

[9]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499

[10]

Alain Rapaport, Mario Veruete. A new proof of the competitive exclusion principle in the chemostat. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3755-3764. doi: 10.3934/dcdsb.2018314

[11]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571

[12]

Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations and Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143

[13]

Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291

[14]

Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034

[15]

Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3495-3502. doi: 10.3934/dcdss.2020248

[16]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[17]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[18]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[19]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure and Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[20]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]