Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Coexistence states and global attractivity for some convective diffusive competing species models


Authors: Julián López-Gómez and José C. Sabina de Lis
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. MR 0415432 (54:3519)
  • [2] J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Royal Soc. Edinburgh 97A (1984), 21-43. MR 751174 (85k:92053)
  • [3] K. J. Brown and Y. Du, Bifurcation and monotonicity in competition reaction-diffusion systems, Preprint. MR 1288495 (95h:35112)
  • [4] R. Cantrell and C. Cosner, On the steady-state problem for the Lotka-Volterra competition model with diffusion, Houston J. Math. 13 (1987), 337-352. MR 916141 (89d:92052)
  • [5] -, Should a park be an island, SIAM J. Math. Appl. 53 (1993), 219-252. MR 1202850 (94c:92016)
  • [6] C. Cosner and A. C. Lazer, Stable coexistence states in the Lotka Volterra competition model with diffusion, SIAM J. Appl. Math. 44 (1984), 1112-1132. MR 766192 (86j:92020)
  • [7] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. II, Wiley Interscience, 1962. MR 1013360 (90k:35001)
  • [8] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8(1971), 321-340. MR 0288640 (44:5836)
  • [9] -, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), 131-151. MR 688538 (84d:58020)
  • [10] -, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284(1984), 729-743. MR 743741 (85i:35056)
  • [11] -, On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Amer. Math. Soc. 326 (1991), 829-859. MR 1028757 (91k:35122)
  • [12] J. C. Eilbeck, J. E. Furter, and J. López-Gómez, Coexistence in the competition model with diffusion, J. Differential Equations 107 (1994), 96-139. MR 1260851 (95c:35072)
  • [13] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 0181836 (31:6062)
  • [14] J. E. Furter and J. López-Gómez, On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatial dependent coefficients, Nonlinear Anal., TMA (to appear). MR 1336979 (96j:35247)
  • [15] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of the second order, Springer-Verlag, Berlin, 1977. MR 0473443 (57:13109)
  • [16] J. K. Hale, L. T. Magalhães and, W. M. Oliva, An Introduction to infinite dimensional dynamical systems, geometric theory, Springer-Verlag, Berlin, 1984. MR 730278 (85d:58026)
  • [17] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Res. Notes in Math., vol. 247, Pitman, 1991. MR 1100011 (92h:35001)
  • [18] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), 999-1030. MR 588690 (81m:35102)
  • [19] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1975.
  • [20] A. Leung, Equilibria and stabilities for competing-species, reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl. 73 (1980), 204-218. MR 560943 (81e:35067)
  • [21] A. Leung, Systems of nonlinear partial differential equations, Kluwer, Dordrecht-Boston, 1989. MR 1621827 (99m:35245)
  • [22] J. López-Gómez, Positive periodic solutions of Lotka-Volterra $ R$ - $ D$ systems, Differential Integral Equations 5 (1992), 55-72.
  • [23] J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly elliptic systems and some applications, Differential Integral Equations 7 (1994), 383-398. MR 1255895 (94k:35053)
  • [24] J. López-Gómez and R. Pardo, Existence and uniqueness for some competition models with diffusion, C. R. Acad. Sci. Paris Ser. I 313 (1991), 933-938. MR 1143448 (93h:35058)
  • [25] C. V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl. 83 (1981), 54-76. MR 632326 (82m:35077)
  • [26] C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084 (94c:35002)
  • [27] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N.J., 1967. MR 0219861 (36:2935)
  • [28] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. MR 0301587 (46:745)
  • [29] D. H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math., vol. 309, Springer-Verlag, 1973. MR 0463624 (57:3569)
  • [30] A. Schiaffino and A. Tesei, Competition systems with Dirichlet boundary conditions, J. Math. Biol. 15 (1982), 93-105. MR 684781 (84k:92029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35Q80, 34C99, 35K55, 92D25

Retrieve articles in all journals with MSC: 35Q80, 34C99, 35K55, 92D25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1311910-8
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society