##### Department of Mathematics,

University of California San Diego

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### Colloquium sponsored by Microsoft Research

## Ami Radunskaya

#### Pomona College

## Random dynamical systems: is noisy growth better?

##### Abstract:

\indent Many biological and physiological processes involve self-regulating mechanisms that prevent too much growth while ensuring against extinction: the rate of growth is somewhat random (``noisy"), but the distribution depends on the current state of the system. Cancer growth and neurological control mechanisms are just a few examples. In finance, as well, markets self-regulate since people want to "buy low" and "sell high". \indent Some questions that we'd like to answer are: does the system have a well-defined average? In more technical terms, we want to know if the system is ergodic. 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? \indent In this talk we will look at (and listen to) a simple example of a noisy, discrete dynamical system with parametric noise and explore ways to answer these questions analytically. We prove ergodicity for a class of growth models, and show that the randomness is harmful to the population in the sense that the long-term average is decreased by the presence of noise. When systems obeying noisy growth laws are connected together as a coupled lattice, the long-term effects of the noise can have damaging effects on the organism as a whole, even though local interactions might favor growth in a particular area. We will present simulations that highlight the effect of both the noise and the local coupling on the survival of the organism.

Hosts: Microsoft Research/UCSD

### November 4, 2011

### 4:00 PM

### AP&M 6402

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