Abstract
We study the asymptotic behavior as t→∞ of positive solutions for random and stochastic parabolic equations of Fisher and Kolmogorov type. The following alternatives are established. Either (i) all positive solutions converge to one and the same trivial equilibrium, or (ii) every positive solution is neither bounded away from the trivial equilibria nor converges to them, or (iii) every positive solution is bounded away from the trivial equilibria. Moreover, for the random equation, we provide in case of alternative (iii) a fairly general condition under which every positive solution converges to uniformly positive equilibria. In the stochastic case, it is proved that there is no uniformly positive equilibrium, and under an appropriate condition, (iii) never occurs.
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REFERENCES
Arnold, L. (1998). Random Dynamical Systems, Springer-Verlag, Berlin.
Arnold, L., and Chueshov, I. (1998). Order-preserving random dynamical systems: Equi-libria, attractors, applications. Dynamics and Stability of Systems 13, 265-280.
Arnold, L., and Chueshov, I. (1998). A limit trichotomy for order-preserving random systems, Institut fü r Dynamische Systeme, Universität Bremen, Report 437, to appear in ``Positivity.''
Aronson, D. C., and Weinberger, H. F. (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Lecture Notes in Mathematics, Vol. 446, Springer, New York, pp. 5-49.
Cantrell, R. S., and Cosner, C. (1991). The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29, 315-338.
Cantrell, R. S., and Cosner, C. (1989). Diffusive logistic equations with indefinite weights: Population models in disrupted environments I. Proc. Roy. Soc. Edinburgh, Sect. A 112, 293-318.
Cantrell, R. S., and Cosner, C. (1991). Diffusive logistic equations with indefinite weights: Population models in disrupted environments II. SIAM J. Math. Anal. 22, 1043-1064.
Chueshov, I. D., and Vuillermot, P.-A. (1998). Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovich's case. Probab. Theory Relat. Fields 112, 149-202.
Chueshov, I. D., and Vuillermot, P.-A. (1998). Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients. Ann. Inst. Henri Poincaré 15, 191-232.
Crauel, H., and Flandoli, F. (1994). Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365-393.
Crauel, H., and Flandoli, F. (1998). Additive noise destroys a Pitchfork bifurcation. J. Dynamics and Diff. Eq. 10, 259-274.
Fife, P. C. (1979). Lecture notes in biomathematics, Springer-Verlag.
Fife, P. C., and Peletier, L. A. (1977). Nonlinear diffusion in population genetics. Arch. Rat. Mech. Anal. 64, 93-109.
Fisher, R. A. (1950). Gene frequencies in a cline determined by selection and diffusion. Biometrics 6, 353-361.
Fleming, W. H. (1975). A selection-migration model in population genetics. J. Math. Biol. 2, 219-233.
Henry, D. (1981). Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Vol.840, Springer-Verlag, Berlin.
Hess, P. (1991). Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics, Vol. 247.
Hess, P., and Weinberger, H. (1990). Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitness. J. Math. Biol. 28, 83-98.
Krylov, N. V., and Rozovskii, B. L. (1981). Stochastic evolution equations. J. Sov. Math. 16, 1233-1277.
Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion, CUP, Cambridge.
Shen, W., and Yi, Y. (1998). Convergence in almost periodic Fisher and Kolmogorov models. J. Math. Biol. 37, 84-102.
Vuillermot, P.-A. (1991). Almost periodic attractors for a class of nonautonomous reaction-diffusion equations on ℝN, I. Global stabilization processes. J. Diff. Eq. 94, 228-253.
Vuillermot, P.-A. (1992). Almost periodic attractors for a class of nonautonomous reaction-diffusion equations on ℝN, II. Codimension-one stable manifolds. Diff. Int. Eq. 5, 693-720.
Vuillermot, P.-A. (1994). Almost periodic attractors for a class of nonautonomous reaction-diffusion equations on ℝN, III. Center curves and Liapunov stability. Nonlinear Anal. 22, 533-559.
Wu, J., Zhao, X.-Q., and He, X. (1996). Global asymptotic behavior in almost periodic Kolmogorov equations and chemostat models, preprint.
Zhao, X.-Q., and Hutson, V. (1994). Permanence in Kolmogorov periodic predator-prey models with diffusion. Nonlinear Anal. Theor. Method. Appl. 23, 651-668.
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Hetzer, G., Shen, W. & Zhu, S. Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types. Journal of Dynamics and Differential Equations 14, 139–188 (2002). https://doi.org/10.1023/A:1012932212645
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DOI: https://doi.org/10.1023/A:1012932212645