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

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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

Abstract
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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 and 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.
[2]	R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953.
[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]	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.
[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.
[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.
[7]	J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, 1982.
[21]	M. X. Wang, Nonlinear Elliptic Equations, (in Chinese) Science Press, Beijing, 2010.
[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.
[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.
[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.
[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.
[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.

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.
[2]	R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953.
[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]	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.
[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.
[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.
[7]	J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, 1982.
[21]	M. X. Wang, Nonlinear Elliptic Equations, (in Chinese) Science Press, Beijing, 2010.
[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.
[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.
[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.
[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.
[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.

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