#### Publication Date

2015-07-31

#### Availability

Open access

#### Embargo Period

2015-07-31

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PHD)

#### Department

Mathematics (Arts and Sciences)

#### Date of Defense

2015-06-30

#### First Committee Member

Ilie Grigorescu

#### Second Committee Member

George C. Cosner

#### Third Committee Member

Victor C. Pestien

#### Fourth Committee Member

Burton J. Rosenberg

#### Abstract

We derive a fluid limit for a multi-type urn model, also known as a hydrodynamic limit, in the sense that random trajectories of the microscopic process are shown to follow, as the scaling factor L, proportional with the initial population, approaches infinity, a unique trajectory characterized as the strong global solution of a specific dynamical system (the macroscopic equation). The result is a weak Law of Large Numbers of random variables with values in the Skorokhod space of right continuous with left limits functions with values in R^k , where k is the number of types. We obtain that while the macroscopic process does not vanish in finite time, the microscopic process has a probability of extinction no larger than O(1/L) . A similar limit is proven for the normalized vector of population proportions, together with qualitative results on its asymptotic behavior. Both limits are in probability, uniformly in time for any fixed time interval. The model and the scaling studied is inspired by earlier work by Schreiber, Benaïm et al. and generalizes the well known Replicator model with applications in mathematical ecology and genome population dynamics.

#### Keywords

hydrodynamic limit; scaling; urn model; replicator model

#### Recommended Citation

Zhang, Zhe, "Scaling Limit of a Generalized Pólya Urn Model" (2015). *Open Access Dissertations*. 1490.

https://scholarlyrepository.miami.edu/oa_dissertations/1490