Asymptotic Behavior of Certain Branching Processes
This dissertation examines the asymptotic behavior of three branching processes. The first is a branching process with selection; the selection is dictated by a fitness function which is the sum of a linear part and a periodic part. It is shown that the system has an asymptotic speed and that there is a stationary distribution in an appropriate moving frame. This is done through an examination of tightness of the process and application of an ergodic theorem. The second process studied is a branching process with selection driven by a symmetric function with a single local maximum at the origin and which monotonically decreases away from the origin. For this process, a large particle limit of the system is proven and related to the solution to a free boundary partial differential equation. Finally, a branching process is studied in which the branch rate of particles is a function of the empirical measure. Weak convergence to the solution of a non-local partial differential equation is proven. Tightness is proven first, and then the limit object is characterized by its behavior when applied to test functions.
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.
Rights for Collection: Duke Dissertations