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Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data. (English) Zbl 0427.35011

35B35Stability of solutions of PDE
35B40Asymptotic behavior of solutions of PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35K60Nonlinear initial value problems for linear parabolic equations
Full Text: DOI
[1] Amann, H.: On the existence of positive solutions of nonlinear elliptic boundary value problems. Indiana univ. Math. J. 21, 125-146 (1971) · Zbl 0219.35037
[2] Bradford, E.; Philip, J. R.: Stability of steady distributions of asocial populations dispersing in one dimension. J. theoret. Biol. 29, 13-26 (1970)
[3] Casten, R.; Holland, C.: Stability properties of solutions to systems of reaction-diffusion equations. SIAM J. Appl. math. 33, 353-364 (1977) · Zbl 0372.35044
[4] Chueh, K. N.; Conley, C. C.; Smoller, J. A.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana univ. Math. J. 26, 373-392 (1977) · Zbl 0368.35040
[5] D. Clark, On differential inequalities for systems of diffusion-reaction type, to appear.
[6] Conway, E.; Smoller, J.: Diffusion and the predator-prey interaction. SIAM J. Appl. math. 33, 673-686 (1977) · Zbl 0368.35021
[7] Courant, R.; Hilbert, D.: Methods of mathematical physics. (1961) · Zbl 57.0245.01
[8] Hilborn, R.: The effect of spatial heterogeneity on the persistence of predator-prey interactions. Theoret. population biol. 8, 346-355 (1975)
[9] Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’cera, N. N.: Linear and quasilinear equations of parabolic type. Translation of mathematical monographs 23 (1968)
[10] Leung, A.: Limiting behavior for a prey-predator model with diffusion and crowding effects. J. math. Biol. 6, 87-93 (1978) · Zbl 0386.92011
[11] A. Leung and D. Clark, Bifurcations and large-time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, to appear. · Zbl 0427.35014
[12] Levin, S.: Spatial patterning and the structure of ecological communities. Lectures on mathematics in the life sciences 8 (1976) · Zbl 0338.92017
[13] Maynard-Smith, J.: Models in ecology. (1974) · Zbl 0312.92001
[14] Protter, M. H.; Weinberger, H.: Maximum principles in differential equations. (1967) · Zbl 0153.13602
[15] Sattinger, D. H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana univ. Math. J. 21, 979-1000 (1972) · Zbl 0223.35038
[16] Sattinger, D. H.: Topics in stability and bifurcation theory. Lecture notes in mathematics no. 309 (1973) · Zbl 0248.35003
[17] Tsai, L. Y.: Nonlinear boundary value problems for systems of second order elliptic differential equations. Bull. inst. Math. acad. Sinica 5, 157-165 (1977) · Zbl 0356.35029
[18] Williams, S.; Chow, P. L.: Nonlinear reaction-diffusion models for interacting populations. J. math. Anal. appl. 62, 157-169 (1978) · Zbl 0372.35047