Ergodicity and loss of capacity for a random family of concave maps

Random fluctuations of an environment are common in ecological and economical settings. We consider a family of concave quadratic polynomials on the unit interval that model a self-limiting growth behavior. The maps are parametrized by an independent, identically distributed random parameter. What in the deterministic setting would be a single fixed point is now replaced by an invariant measure, which is a fixed point of the Perron-Frobenius operator. In select cases, we show the existence of a unique invariant ergodic measure of the resulting random dynamical system. Moreover, there is an attenuation of the mean of the state variable compared to the constant environment with the averaged parameter. This is joint work with Ami Radunskaya (Pomona College, Pomona, CA).