American Institute of Mathematical Sciences

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October  2015, 20(8): 2691-2714. doi: 10.3934/dcdsb.2015.20.2691

Coexistence solutions of a competition model with two species in a water column

Hua Nie 1, , Sze-Bi Hsu 2, and  Jianhua Wu 1,

1. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China
2. 

Department of Mathematics, National Tsing Hua University, National Center of Theoretical Science, Hsinchu 300

Received  October 2014 Revised  March 2015 Published  August 2015

Competition between species for resources is a fundamental ecological process, which can be modeled by the mathematical models in the chemostat culture or in the water column. The chemostat-type models for resource competition have been extensively analyzed. However, the study on the competition for resources in the water column has been relatively neglected as a result of some technical difficulties. We consider a resource competition model with two species in the water column. Firstly, the global existence and $L^\infty$ boundedness of solutions to the model are established by inequality estimates. Secondly, the uniqueness of positive steady state solutions and some dynamical behavior of the single population model are attained by degree theory and uniform persistence theory. Finally, the structure of the coexistence solutions of the two-species system is investigated by the global bifurcation theory.
Keywords: degree theory, coexistence solution, global bifurcation., Water column, uniqueness.
Mathematics Subject Classification: Primary: 35K57, 35B32; Secondary: 35B5.
Citation: Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691
References:
[1]

M. Ballyk, L. Dung, D. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596. doi: 10.1137/S0036139997325345.  Google Scholar

[2]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953.  Google Scholar

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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.  Google Scholar

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J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.  Google Scholar

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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.  Google Scholar

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S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974. doi: 10.1137/100782358.  Google Scholar

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

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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

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P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.  Google Scholar

[13]

J. P. Mellard, K. Yoshiyama, E. Litchman and C. A. Klausmeier, The vertical distribution of phytoplankton in stratified water columns, J. Theoret. Biol., 269 (2011), 16-30. doi: 10.1016/j.jtbi.2010.09.041.  Google Scholar

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H. Nie and J. Wu, Multiplicity results for the unstirred chemostat model with general response functions, Sci. China Math., 56 (2013), 2035-2050. doi: 10.1007/s11425-012-4550-4.  Google Scholar

[15]

H. Nie and J. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat, J. Math. Anal. Appl., 355 (2009), 231-242. doi: 10.1016/j.jmaa.2009.01.045.  Google Scholar

[16]

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

[17]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[18]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

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J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, 1982. Google Scholar

[21]

M. X. Wang, Nonlinear Elliptic Equations, (in Chinese) Science Press, Beijing, 2010. Google Scholar

[22]

J. Wu, H. 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

[23]

J. Wu, H. 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

[24]

J. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332. doi: 10.1006/jdeq.2000.3870.  Google Scholar

[25]

K. Yoshiyama and H. Nakajima, Catastrophic transition in vertical distributions of phytoplankton: alternative equilibria in a water column, J. Theoret. Biol., 216 (2002), 397-408. doi: 10.1006/jtbi.2002.3007.  Google Scholar

[26]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Am. Nat., 174 (2009), 190-203. doi: 10.1086/600113.  Google Scholar

show all references

References:
[1]

M. Ballyk, L. Dung, D. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596. doi: 10.1137/S0036139997325345.  Google Scholar

[2]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953.  Google Scholar

[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.  Google Scholar

[4]

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.  Google Scholar

[5]

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

[6]

Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349. doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[7]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.  Google Scholar

[8]

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.  Google Scholar

[9]

S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974. doi: 10.1137/100782358.  Google Scholar

[10]

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

[11]

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

[12]

P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.  Google Scholar

[13]

J. P. Mellard, K. Yoshiyama, E. Litchman and C. A. Klausmeier, The vertical distribution of phytoplankton in stratified water columns, J. Theoret. Biol., 269 (2011), 16-30. doi: 10.1016/j.jtbi.2010.09.041.  Google Scholar

[14]

H. Nie and J. Wu, Multiplicity results for the unstirred chemostat model with general response functions, Sci. China Math., 56 (2013), 2035-2050. doi: 10.1007/s11425-012-4550-4.  Google Scholar

[15]

H. Nie and J. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat, J. Math. Anal. Appl., 355 (2009), 231-242. doi: 10.1016/j.jmaa.2009.01.045.  Google Scholar

[16]

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

[17]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[18]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[20]

D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, 1982. Google Scholar

[21]

M. X. Wang, Nonlinear Elliptic Equations, (in Chinese) Science Press, Beijing, 2010. Google Scholar

[22]

J. Wu, H. 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

[23]

J. Wu, H. 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

[24]

J. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332. doi: 10.1006/jdeq.2000.3870.  Google Scholar

[25]

K. Yoshiyama and H. Nakajima, Catastrophic transition in vertical distributions of phytoplankton: alternative equilibria in a water column, J. Theoret. Biol., 216 (2002), 397-408. doi: 10.1006/jtbi.2002.3007.  Google Scholar

[26]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Am. Nat., 174 (2009), 190-203. doi: 10.1086/600113.  Google Scholar

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