Graduate Thesis Or Dissertation
Nonlocal Models with Applications to Ecology Public Deposited
Understanding how species move in their environment is a key objective in ecology. In the face of climate change, invasive species, and habitat destruction, understanding primary factors that drive species can help predict how territories change or whether a species survives as the environment changes, another species goes extinct, or a competing species invades. Mechanistic models using only local information and local processes have been used to understand the forces that drive species movement and to predict the persistence or extinction of species. However, many ecological phenomena are deeply affected by nonlocal interactions and reactions, and including these forces into mathematical models can change the dynamics and more accurately describe observed phenomena. We investigate two particular ecological phenomena affected by nonlocal forces.
First, we study a nonlocal mechanistic model describing territory development of social groups to determine the most prominent mechanisms driving movement. While the nonlocal nature of this model leads to a better approximation of physical reality, it also causes numerical and analytical challenges. The focus in this work is to address these issues. Therefore, we employ several strategies to solve the nonlocal system of equations. We are particularly interested in model verification through the connection with data and understanding how the nonlocal terms and parameters in the model influence solutions.
Second, we investigate an integro-differential equation modeling a birth-jump process, where birth and dispersal cannot be decoupled. This has been argued to be a more suitable model for processes such as seed dispersal, cancer cell growth, and fire propagation. We prove global existence and uniqueness of solutions and investigate persistence and extinction of a population subject to various growth terms. We are particularly interested in cases where a population can overcome the Allee effect, where population density affects the fitness of individuals.
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