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September  2015, 20(7): 2129-2155. doi: 10.3934/dcdsb.2015.20.2129

Competition for one nutrient with recycling and allelopathy in an unstirred chemostat

Hua Nie 1, and  Feng-Bin Wang 2,

1. 

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

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

Received  August 2014 Revised  January 2015 Published  July 2015

In this paper, we study a PDE model of two species competing for a single limiting nutrient resource in a chemostat in which one microbial species excretes a toxin that increases the mortality of another. Our goal is to understand the role of spatial heterogeneity and allelopathy in blooms of harmful algae. We first demonstrate that the two-species system and its single species subsystem satisfy a mass conservation law that plays an important role in our analysis. We investigate the possibilities of bistability and coexistence for the two-species system by appealing to the method of topological degree in cones and the theory of uniform persistence. Numerical simulations confirm the theoretical results.
Keywords: allelopathy, unstirred chemostat., Harmful algae, spatial heterogeneity, nutrient recycling.
Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 92D2.
Citation: Hua Nie, Feng-Bin Wang. Competition for one nutrient with recycling and allelopathy in an unstirred chemostat. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2129-2155. doi: 10.3934/dcdsb.2015.20.2129
References:
[1]

F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications, Addison-Wesley Longman, Harlow, UK, 1997.

[2]

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.

[3]

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.

[4]

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.

[5]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part I, General existence results, Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[6]

Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 9 (1996), 1043-1066.

[7]

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

[8]

J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: A theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227. doi: 10.1093/plankt/fbq070.

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[10]

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), 366-383. doi: 10.1137/0132030.

[11]

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.

[12]

I. P. Martines, H. V. Kojouharov and J. P. Grover, A chemostat model of resource competition and allelopathy, Applied Mathematics and Computation, 215 (2009), 573-582. doi: 10.1016/j.amc.2009.05.033.

[13]

I. P. Martines, H. V. Kojouharov and J. P. Grover, Nutrient recycling and allelopathy in a gradostat, Computers and Mathematics with Applications, 66 (2013), 1613-1626. doi: 10.1016/j.camwa.2013.02.005.

[14]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[15]

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.

[16]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[18]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, $2^{nd}$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1468-7.

[19]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.

[20]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[21]

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.

[22]

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.

[23]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[24]

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

[25]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[26]

Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345. doi: 10.1137/0521018.

[27]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications, Addison-Wesley Longman, Harlow, UK, 1997.

[2]

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.

[3]

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.

[4]

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.

[5]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part I, General existence results, Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[6]

Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 9 (1996), 1043-1066.

[7]

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

[8]

J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: A theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227. doi: 10.1093/plankt/fbq070.

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[10]

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), 366-383. doi: 10.1137/0132030.

[11]

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.

[12]

I. P. Martines, H. V. Kojouharov and J. P. Grover, A chemostat model of resource competition and allelopathy, Applied Mathematics and Computation, 215 (2009), 573-582. doi: 10.1016/j.amc.2009.05.033.

[13]

I. P. Martines, H. V. Kojouharov and J. P. Grover, Nutrient recycling and allelopathy in a gradostat, Computers and Mathematics with Applications, 66 (2013), 1613-1626. doi: 10.1016/j.camwa.2013.02.005.

[14]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[15]

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.

[16]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[18]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, $2^{nd}$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1468-7.

[19]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.

[20]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[21]

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.

[22]

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.

[23]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[24]

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

[25]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[26]

Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345. doi: 10.1137/0521018.

[27]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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