Asymptotic Behavior of Certain Branching Processes
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2019
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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.
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Beckman, Erin Melissa (2019). Asymptotic Behavior of Certain Branching Processes. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/18756.
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