Abstract
A stability condition for Hopf-bifurcating solutions from the uniform equilibrium of clasical Lotka-Volterra interaction-diffusion equations is presented. Using this condition, it is shown that stable spatio-temporal oscillations exist in the framework of such equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arneodo, A., Coullet, J., Peyraud, J., Tresser, C.: Strange Attractors in Volterra equations for species in competition. J. Math. Biol. 14, 153–157 (1982)
Casten, R. G., Holland, C. J.: Instability results for reaction diffusion equations with Neumann boundary conditions. J. Math. Differential Equations 27, 266–273 (1978)
Chafee, N.: Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Differential Equations 18, 111–134 (1975)
Chow, S. N., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differential Equations 27, 112–159 (1977)
Conway, E., Hopf, D., Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35, 1–16 (1978)
Evans, G. T.: Diffusive structure: Counterexamples to any explanations? J. Theoret. Biology 82, 313–315 (1980)
Fife, P. C.: Private communication (1982)
Fleming, W. H.: A selection-migration model in population genetics. J. Math. Biol. 2, 219–234 (1975)
Hastings, A.: Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol. 6, 163–168 (1978)
Jorne, J., Carmi, S.: Liapunov stability of the diffusive Lotka-Volterra equations. Math. Biosci. 37, 51–61 (1977)
Kishimoto, K.: Instability of non-constant equilibrium solutions of a system of competition-diffusion equations. J. Math. Biol. 13, 105–114 (1981)
Kishimoto, K.: The diffusive Lotka-Volterra system with three species can have a stable non-constant equilibrium solution. J. Math. Biol. 16, 103–112 (1982)
Kishimoto, K., Weinberger, H. F.: The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains. (submitted for publication)
Leung, A.: Limiting behavior for a prey-predator model with dffusion and crowding effects. J. Math. Biol. 6, 87–93 (1978)
Levin, S. A.: Dispersion and population interactions. Amer. Natur. 108, 207–208 (1974)
Matano, H.: Asymptotic behavior and stability of solutions of semi-linear diffusion equations. Publ. RIMS Kyoto Univ. 15, 401–451 (1979)
May, R. M., Leonard, W. J.: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975)
Mimura, M.: Asymptotic behavior of a parabolic system related to a planktonic prey and predator model. SIAM J. Appl. Math. 37, 499–512 (1979)
Mimura, M.: Stationary pattern of some density-dependent diffusive system with competing dynamics. Hiroshima Math. J. 11, 621–635 (1981)
Mimura, M., Nishida, T.: On a certain semilinear parabolic system related to the Lotka-Volterra ecological model. Publ. RIMS Kyoto Univ. 14, 269–282 (1978)
Mimura, M., Nishiura, Y., Yamaguti, M.: Some diffusive prey and predator systems and their bifurcation problems. In: Bifurcation theory and applications in scientific disciplines, pp. 490–510, New York Academy of Science, 1978
Murray, J. D.: Non-existence of wave solutions for the class of reaction-diffusion equations given by the Volterra interaction-population equations with diffusion. J. Theoret. Biology 52, 459–469 (1975)
Nakajima, J.: The stability and periodicity in model ecosystem (in Japanese). Bussei Kenkyu 28, 245–387 (1978)
Okubo, A.: Diffusion and ecological problems: Mathematical models. Berlin-Heidelberg-New York: Springer 1980
Rothe, F.: Convergence to the equilibrium state in the Volterra Lotka diffusion equations. J. Math. Biol. 3, 319–324 (1976)
Segel, L. A., Jackson, J. L.: Dissipative Structure: An explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)
Segel, L. A., Levin, S. A.: Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions, In: Topics in statistical mechanics and biophysics: A memorial to Julius L. Jackson (Riccirelli, R. A., ed.) Proc. AIP Conf. 27, 123–152 (1976)
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979)
Schoener, T. W.: Resource partitioning in ecological communities. Science 185, 27–39 (1974)
Skellam, J. D.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)
Steele, J. H.: Spatial heterogeneity and population stability. Nature 248, 83 (1974)
Williams, S. T., Chow, P. L.: Nonlinear reaction-diffusion models for interacting populations. J. Math. Anal. Appl. 62, 159–169 (1978)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kishimoto, K., Mimura, M. & Yoshida, K. Stable spatio-temporal oscillations of diffusive Lotka-Volterra system with three or more species. J. Math. Biology 18, 213–221 (1983). https://doi.org/10.1007/BF00276088
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276088