Poisson Representations for Spatial Population Models with Competition
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Abstract
Abstract: In this thesis we extend the Kurtz-Rodrigues representation to spatial population models implementing competition between the individuals. The KR-representation is a Poisson representation of the Dawson-Watanabe superprocess meaning that it is a particle model with the superprocess as high density limit . But the KR-representation has the special property, that the genealogical information of the particle does not get lost, when we construct the high-density-limit .
Competition is modeled by an additional death rate for the particles, which may vary depending on the size and the distribution of the population. The considered competition models are extensions of the Bolker-Pacala models and the spatial equivalent of the logistic Feller-diffusion. In the spirit of Evans and Perkins we obtain our Poisson representation by cutting those out from the KR-representation. Therefore we develop an integration theory for the KR-representation based on the ideas of Perkins. This allows us to equip the particles with death markers identifying those particles which suffered from a prematurely death due to competition. Additionally to the construction of the Poisson representations, we also show, how these could be used to study the extinction behavior of Bolker-Pacala models.