Date on Master's Thesis/Doctoral Dissertation

8-2019

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Li, Bingtuan

Committee Co-Chair (if applicable)

Emery, Sarah

Committee Member

Emery, Sarah

Committee Member

Gie, Gung-Min

Committee Member

Hu, Changbing

Committee Member

Swanson, David

Author's Keywords

Overcompensation; reaction-diffusion equation; spreading speed

Abstract

We consider both the nonspatial model and spatial model of a species that gives birth to eggs at the end of the year. It is assumed that the timing of emergence from eggs is controled by phenology, which is density dependent. In general, the solution maps for our models are implicit; When the solution map is explicit, it is extremely complex and it is easier to work with the implicit map. We derive integral conditions for which the nonspatial model exhibits strong Allee effect. We also provide a necessary condition and a sufficient condition for the existence of positive equilibrium solutions. We also numerically explore the complex dynamics of both models. It is shown that varying a parameter can cause an Allee threshold to appear/disappear. We also show that the spatial model can have a growth function with overcompensation, wave solutions, oscillating waves, and nonspreading solutions. It is also shown that the wave solutions can have constant, oscillating, or chaotic spreading speeds. We also provide an example where the solutions to the spatial model are persistent, even though the underlying dynamics of the nonspatial model is essential extinction.

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