Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types

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.