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We show that two dynamical systems exhibiting very different deterministicbehaviours possess very similar stationary distributions when stabilized by amultiplicative Gaussian white noise. We also discuss practical aspects ofnumerically simulating these systems. We show that there exists a noise levelthat is optimal in the sense that the interval during which discrete-timeversions of the systems remain physical is maximized. Analytical results areillustrated by numerical examples.

Population explosion suppressed by noise: stationary distributions and how to simulate them