Abstract
We discuss a competition-diffusion system to study coexistence problems of two competing species in a homogeneous environment. In particular, by using invariant manifold theory, effects of domain-shape are considered on this problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad S., Lazer, A. C.: Asymptotic behavior of solutions of periodic competition diffusion system. Nonlinear Anal. 13, 263–284 (1989)
Carr, J.: Applications of centre manifold theory. (Appl. Math. Sci, vol. 35) Berlin Heidelberg New York: Springer 1981
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)
Conway, E., Hoff, D., Smoller, J.: Large time behaviors of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35, 1–16 (1978)
Ei, S.-I.: Two-timing methods with applications to heterogeneous reaction-diffusion systems. Hiroshima Math. J. 18, 127–160 (1988)
Ei, S.-I., Mimura, M.: Pattern formation in heterogeneous reaction-diffusion-advection system with an application to population dynamics. SIAM J. Appl. Math. 21, 346–361 (1990)
Fang, Q.: Asymptotic behavior and domain-dependency of solutions to a class of reaction-diffusion systems with large diffusion coefficients. To appear in Hiroshima Math. J.
Hale, J. K., Vegas, J.: A nonlinear parabolic equation with varying domain. Arch. Rat. Mech. Anal. 86, 99–123 (1984)
Henry, D.: Geometric theory of semilinear parabolic equations. Lecture notes in Math. 840, 1981
Hirsch, M. W.: Differential equations and convergence almost everywhere of strongly monotone semiflows. PAM Technical Report, University of California, Berkeley (1982)
Horn, H., MacArthur, R. H.: Competition among fugitive species in a harlequin environment, Ecology 53, 749–752 (1972)
Jimbo, S.: Singular perturbation of domains and semilinear elliptic equation. J. Fac. Sci. Univ. Tokyo 35, 27–76 (1988)
Jimbo, S.: Perturbed equilibrium solutions in the singularly perturbed domain: L ±(Ω(ζ)) -formulation and elaborate characterization. Preprint
Kishimoto, K., Weinberger, H. F.: The spatial homogeneity of stable equilibria of some reaction-diffusion system on convex domains. J. Differ. Equations 58, 15–21 (1985)
Levin, S. A.: Dispersion and population interactions. Am. Natur. 108, 207–228 (1974)
Matano, H., Mimura, M.: Pattern formation in competition-diffusion systems in nonconvex domains. Publ. RIMS, Kyoto Univ. 19, 1049–1079 (1983)
Mimura, M.: Spatial distribution of competing species. (Lect. Notes, Biomath. vol. 54, pp. 492–501) Berlin Heidelberg New York: Springer 1984
Mimura, M., Kawasaki, K.: Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9, 49–64 (1980)
Mimura, M., Fife, P. C.: A 3-component system of competition and diffusion. Hiroshima Math. J. 16, 189–207 (1986)
Morita, Y.: Reaction-diffusion systems in nonconvex domains: invariant manifold and reduced form. J. Dynamics Differ. Equations 2, 69–115 (1990)
Vegas, J. M.: Bifurcation caused by perturbing the domain in an elliptic equation. J. Differ. Equations 48, 189–226 (1983)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mimura, M., Ei, S.I. & Fang, Q. Effect of domain-shape on coexistence problems in a competition-diffusion system. J. Math. Biol. 29, 219–237 (1991). https://doi.org/10.1007/BF00160536
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00160536