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

Previous Article
Long time dynamics of a multidimensional nonlinear lattice with memory
DCDS-B Home
This Issue
Next Article
Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays

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
Related Papers

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.

[1]	Robert Stephen Cantrell, King-Yeung Lam. Competitive exclusion in phytoplankton communities in a eutrophic water column. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1783-1795. doi: 10.3934/dcdsb.2020361
[2]	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
[3]	J. F. Toland. Path-connectedness in global bifurcation theory. Electronic Research Archive, 2021, 29 (6) : 4199-4213. doi: 10.3934/era.2021079
[4]	Linfeng Mei, Sze-Bi Hsu, Feng-Bin Wang. Growth of single phytoplankton species with internal storage in a water column. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 607-620. doi: 10.3934/dcdsb.2016.21.607
[5]	Danfeng Pang, Hua Nie, Jianhua Wu. Single phytoplankton species growth with light and crowding effect in a water column. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 41-74. doi: 10.3934/dcds.2019003
[6]	Linfeng Mei, Feng-Bin Wang. Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2115-2132. doi: 10.3934/dcdsb.2020359
[7]	Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[8]	Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070
[9]	Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669
[10]	Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190
[11]	Jingli Ren, Gail S. K. Wolkowicz. Preface: Recent advances in bifurcation theory and application. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : i-ii. doi: 10.3934/dcdss.2020417
[12]	Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163
[13]	Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control and Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315
[14]	Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213
[15]	Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[16]	Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[17]	Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013
[18]	Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212
[19]	Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089
[20]	Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Hua Nie Sze-Bi Hsu Jianhua Wu
on Google Scholar
Hua Nie Sze-Bi Hsu Jianhua Wu

[Back to Top]