On the existence and uniqueness of positive solutions for competing species models with diffusion
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- by E. N. Dancer PDF
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Abstract:
In this paper, we consider strictly positive solutions of competing species systems with diffusion under Dirichlet boundary conditions. We obtain a good understanding of when strictly positive solutions exist, obtain new nonuniqueness results and a number of other results, showing how complicated these equations can be. In particular, we consider how the shape of the underlying domain affects the behaviour of the equations.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
- H. Berestycki and P.-L. Lions, Some applications of the method of super and subsolutions, Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978) Lecture Notes in Math., vol. 782, Springer, Berlin, 1980, pp. 16–41. MR 572249
- Peter N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980), no. 1, 22–37. MR 559078, DOI 10.1137/0138002
- 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
- E. D. Conway, Diffusion and the predator-prey interaction: steady states with flux at the boundaries, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 215–234. MR 706101
- 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
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
- 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 positive solutions of some pairs of differential equations. II, J. Differential Equations 60 (1985), no. 2, 236–258. MR 810554, DOI 10.1016/0022-0396(85)90115-9
- E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math. 371 (1986), 46–66. MR 859319, DOI 10.1515/crll.1986.371.46
- 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 the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc. (3) 53 (1986), no. 3, 429–452. MR 868453, DOI 10.1112/plms/s3-53.3.429
- E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann. 272 (1985), no. 3, 421–440. MR 799671, DOI 10.1007/BF01455568
- E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), no. 1, 120–156. MR 949628, DOI 10.1016/0022-0396(88)90021-6
- E. N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc. 20 (1988), no. 6, 600–602. MR 980763, DOI 10.1112/blms/20.6.600
- E. N. Dancer and P. Hess, On stable solutions of quasilinear periodic-parabolic problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 1, 123–141. MR 937539
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- 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
- Andrzej Granas, Points fixes pour les applications compactes: espaces de Lefschetz et la théorie de l’indice, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 68, Presses de l’Université de Montréal, Montreal, Que., 1980 (French). With an appendix, “Infinite-dimensional cohomology and bifurcation theory”, by Kazimierz Gęba. MR 569745
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
- Morris W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 267–285. MR 706104
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- Philip Korman and Anthony Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Appl. Anal. 26 (1987), no. 2, 145–160. MR 921723, DOI 10.1080/00036818708839706
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- Hiroshi Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 645–673. MR 731522
- Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 326 (1991), 829-859
- MSC: Primary 35K57; Secondary 47H15, 92D25
- DOI: https://doi.org/10.1090/S0002-9947-1991-1028757-9
- MathSciNet review: 1028757