Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types
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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.
- Arnold, L. (1998) Random Dynamical Systems. Springer-Verlag, Berlin
- Arnold, L., Chueshov, I. (1998) Order-preserving random dynamical systems: Equi-libria, attractors, applications. Dynamics and Stability of Systems 13: pp. 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., Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Springer, New York
- Cantrell, R. S., Cosner, C. (1991) The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29: pp. 315-338
- Cantrell, R. S., Cosner, C. (1989) Diffusive logistic equations with indefinite weights: Population models in disrupted environments I. Proc. Roy. Soc. Edinburgh, Sect. A 112: pp. 293-318
- Cantrell, R. S., Cosner, C. (1991) Diffusive logistic equations with indefinite weights: Population models in disrupted environments II. SIAM J. Math. Anal. 22: pp. 1043-1064
- Chueshov, I. D., 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: pp. 149-202
- Chueshov, I. D., Vuillermot, P.-A. (1998) Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients. Ann. Inst. Henri Poincaré 15: pp. 191-232
- Crauel, H., Flandoli, F. (1994) Attractors for random dynamical systems. Probab. Theory Relat. Fields 100: pp. 365-393
- Crauel, H., Flandoli, F. (1998) Additive noise destroys a Pitchfork bifurcation. J. Dynamics and Diff. Eq. 10: pp. 259-274
- Fife, P. C. (1979). Lecture notes in biomathematics, Springer-Verlag.
- Fife, P. C., Peletier, L. A. (1977) Nonlinear diffusion in population genetics. Arch. Rat. Mech. Anal. 64: pp. 93-109
- Fisher, R. A. (1950) Gene frequencies in a cline determined by selection and diffusion. Biometrics 6: pp. 353-361
- Fleming, W. H. (1975) A selection-migration model in population genetics. J. Math. Biol. 2: pp. 219-233
- Henry, D. (1981) Geometric theory of semilinear parabolic equations. Springer-Verlag, Berlin
- Hess, P. (1991). Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics, Vol. 247.
- Hess, P., Weinberger, H. (1990) Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitness. J. Math. Biol. 28: pp. 83-98
- Krylov, N. V., Rozovskii, B. L. (1981) Stochastic evolution equations. J. Sov. Math. 16: pp. 1233-1277
- Pinsky, R. G. (1995) Positive Harmonic Functions and Diffusion. CUP, Cambridge
- Shen, W., Yi, Y. (1998) Convergence in almost periodic Fisher and Kolmogorov models. J. Math. Biol. 37: pp. 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: pp. 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: pp. 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: pp. 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., Hutson, V. (1994) Permanence in Kolmogorov periodic predator-prey models with diffusion. Nonlinear Anal. Theor. Method. Appl. 23: pp. 651-668
- Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types
Journal of Dynamics and Differential Equations
Volume 14, Issue 1 , pp 139-188
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