Chaos and Noise: A Look at Stochastic Difference Equations

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Many biological and physiological processes involve self-regulating mechanisms that prevent too much growth while ensuring against extinction. The rate of growth is often random (``noisy"), possibly affected by fluctuations in the environment. Some questions that we'd like to answer are: what happens to the system in the long term?

Does the system have a well-defined average? How does this long-term average compare to the long-term behavior of the deterministic (not random)

system? What can we say about the distribution of ``survival times", i.e. the distribution of times until the system reaches a particular value?

In this talk we answer these questions for a family of maps on the unit interval that model self-limiting growth. We then look at more complicated systems and make several conjectures.

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