Coexistence states and global attractivity for some convective diffusive competing species models
HTML articles powered by AMS MathViewer
- by Julián López-Gómez and José C. Sabina de Lis PDF
- Trans. Amer. Math. Soc. 347 (1995), 3797-3833 Request permission
Abstract:
In this paper we analyze the dynamics of a general competing species model with diffusion and convection. Regarding the interaction coefficients between the species as continuation parameters, we obtain an almost complete description of the structure and stability of the continuum of coexistence states. We show that any asymptotically stable coexistence state lies in a global curve of stable coexistence states and that Hopf bifurcations or secondary bifurcations only may occur from unstable coexistence states. We also characterize whether a semitrivial coexistence state or a coexistence state is a global attractor. The techniques developed in this work can be applied to obtain generic properties of general monotone dynamical systems.References
- Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
- J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21–34. MR 751174, DOI 10.1017/S0308210500031802
- Yi Hong Du and K. J. Brown, Bifurcation and monotonicity in competition reaction-diffusion systems, Nonlinear Anal. 23 (1994), no. 1, 1–13. MR 1288495, DOI 10.1016/0362-546X(94)90248-8
- Robert Stephen Cantrell and Chris Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math. 13 (1987), no. 3, 337–352. MR 916141
- Robert Stephen Cantrell and Chris Cosner, Should a park be an island?, SIAM J. Appl. Math. 53 (1993), no. 1, 219–252. MR 1202850, DOI 10.1137/0153014
- Chris Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math. 44 (1984), no. 6, 1112–1132. MR 766192, DOI 10.1137/0144080
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131–151. MR 688538, DOI 10.1016/0022-247X(83)90098-7
- E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984), no. 2, 729–743. MR 743741, DOI 10.1090/S0002-9947-1984-0743741-4
- E. N. Dancer, On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Amer. Math. Soc. 326 (1991), no. 2, 829–859. MR 1028757, DOI 10.1090/S0002-9947-1991-1028757-9
- J. C. Eilbeck, J. E. Furter, and J. López-Gómez, Coexistence in the competition model with diffusion, J. Differential Equations 107 (1994), no. 1, 96–139. MR 1260851, DOI 10.1006/jdeq.1994.1005
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Jacques E. Furter and Julián López-Gómez, On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatially dependent coefficients, Nonlinear Anal. 25 (1995), no. 4, 363–398. MR 1336979, DOI 10.1016/0362-546X(94)00139-9
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
- Jack K. Hale, Infinite-dimensional dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 379–400. MR 730278, DOI 10.1007/BFb0061425
- Peter Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1100011
- Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999–1030. MR 588690, DOI 10.1080/03605308008820162 T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1975.
- Anthony Leung, Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl. 73 (1980), no. 1, 204–218. MR 560943, DOI 10.1016/0022-247X(80)90028-1
- Anthony W. Leung, Systems of nonlinear partial differential equations, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1989. Applications to biology and engineering. MR 1621827, DOI 10.1007/978-94-015-3937-1 J. López-Gómez, Positive periodic solutions of Lotka-Volterra $R$ - $D$ systems, Differential Integral Equations 5 (1992), 55-72.
- Julián López-Gómez and Marcela Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential Integral Equations 7 (1994), no. 2, 383–398. MR 1255895
- Julián López-Gómez and Rosa Pardo, Existence and uniqueness for some competition models with diffusion, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 13, 933–938 (English, with French summary). MR 1143448
- C. V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl. 83 (1981), no. 1, 54–76. MR 632326, DOI 10.1016/0022-247X(81)90246-8
- C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
- David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624, DOI 10.1007/BFb0060079
- A. Schiaffino and A. Tesei, Competition systems with Dirichlet boundary conditions, J. Math. Biol. 15 (1982), no. 1, 93–105. MR 684781, DOI 10.1007/BF00275791
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3797-3833
- MSC: Primary 35Q80; Secondary 34C99, 35K55, 92D25
- DOI: https://doi.org/10.1090/S0002-9947-1995-1311910-8
- MathSciNet review: 1311910