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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

 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.
Citation: Hua Nie, Feng-Bin Wang. Competition for one nutrient with recycling and allelopathy in an unstirred chemostat. Discrete & 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, (1997).   Google Scholar [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [3] E. N. Dancer, On the indices of fixed points of mappings in cones and applications,, J. Math. Anal. Appl., 91 (1983), 131.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar [4] E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar [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.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar [6] Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model,, Differential Integral Equations, 9 (1996), 1043.   Google Scholar [7] J. P. Grover, Resource Competition,, Chapman and Hall, (1997).  doi: 10.1007/978-1-4615-6397-6.  Google Scholar [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.  doi: 10.1093/plankt/fbq070.  Google Scholar [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988).   Google Scholar [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.  doi: 10.1137/0132030.  Google Scholar [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.  doi: 10.1137/0153051.  Google Scholar [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.  doi: 10.1016/j.amc.2009.05.033.  Google Scholar [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.  doi: 10.1016/j.camwa.2013.02.005.  Google Scholar [14] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar [15] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar [16] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [17] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-5282-5.  Google Scholar [18] H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, $2^{nd}$ edition, (1999).  doi: 10.1007/978-1-4612-1468-7.  Google Scholar [19] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs 41, (1995).   Google Scholar [20] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar [21] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar [22] J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar [23] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar [24] M. X. Wang, Nonlinear Parabolic Equations (in Chinese),, Science Press, (1993).   Google Scholar [25] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, Nonlinear Anal., 39 (2000), 817.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar [26] Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions,, SIAM J. Math. Anal., 21 (1990), 327.  doi: 10.1137/0521018.  Google Scholar [27] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003).  doi: 10.1007/978-0-387-21761-1.  Google Scholar

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##### References:
 [1] F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications,, Addison-Wesley Longman, (1997).   Google Scholar [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [3] E. N. Dancer, On the indices of fixed points of mappings in cones and applications,, J. Math. Anal. Appl., 91 (1983), 131.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar [4] E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar [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.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar [6] Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model,, Differential Integral Equations, 9 (1996), 1043.   Google Scholar [7] J. P. Grover, Resource Competition,, Chapman and Hall, (1997).  doi: 10.1007/978-1-4615-6397-6.  Google Scholar [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.  doi: 10.1093/plankt/fbq070.  Google Scholar [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988).   Google Scholar [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.  doi: 10.1137/0132030.  Google Scholar [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.  doi: 10.1137/0153051.  Google Scholar [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.  doi: 10.1016/j.amc.2009.05.033.  Google Scholar [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.  doi: 10.1016/j.camwa.2013.02.005.  Google Scholar [14] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar [15] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar [16] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [17] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-5282-5.  Google Scholar [18] H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, $2^{nd}$ edition, (1999).  doi: 10.1007/978-1-4612-1468-7.  Google Scholar [19] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs 41, (1995).   Google Scholar [20] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar [21] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar [22] J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar [23] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar [24] M. X. Wang, Nonlinear Parabolic Equations (in Chinese),, Science Press, (1993).   Google Scholar [25] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, Nonlinear Anal., 39 (2000), 817.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar [26] Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions,, SIAM J. Math. Anal., 21 (1990), 327.  doi: 10.1137/0521018.  Google Scholar [27] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003).  doi: 10.1007/978-0-387-21761-1.  Google Scholar

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