The Stochastic Logistic Map: Theory and Applications — Kimberly Ayers (CSU San Marcos)

Abstract: The logistic map is one of the most famous examples in dynamical systems. Originally introduced as a simple model of population growth, it exhibits a surprisingly rich range of behavior: as a single parameter varies, stable equilibria give way to periodic orbits and eventually to chaos. But in real populations, quantities such as reproductive and death rates are unlikely to remain constant over time. What happens when we allow this parameter to vary randomly from one generation to the next?

In this talk, we explore a stochastic version of the logistic map in which the parameter is chosen randomly at each iteration. Introducing randomness changes the questions we ask: rather than studying individual fixed points or periodic orbits, we investigate probability distributions that describe the long-term behavior of the system. We will see how tools from dynamical systems and Markov processes can be used to show that, under appropriate conditions, the stochastic system settles into a unique long-term statistical behavior that is independent of its initial state. We conclude with simulations illustrating this invariant distribution and discuss several questions about how the system’s behavior changes as the amount and type of randomness vary.

For questions about the event, please contact your CCMS Colloquium chairs:

Prof. Robert Cass (Robert.Cass@ClaremontMcKenna.edu) or Prof. Qidi Peng (Qidi.Peng@cgu.edu)

For logistical questions, contact Maxwell Kooiker (Maxwell.Kooiker@cgu.edu), the CCMS Colloquium Assistant.