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Title: A study of systems of two cross-interacting species
Author: Schmidtchen, Markus
ISNI:       0000 0004 7963 8217
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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This dissertation is dedicated to studying certain systems of two different species that interact with each other both locally and nonlocally. The systems under consideration are given as a combination of three contributions - nonlinear and nonlocal drifts that model the self-interactions and cross-interactions, re- spectively, between the agents of both species, cross-diffusion terms that de- scribe the dispersal of both species either due to an intrinsic urge to avoid over- crowding or due to size-exclusion effects, and cross-reaction terms that govern the birth and death processes within both subpopulations. These systems typ- ically arise in biomedical or physical contexts and display a variety of fascinat- ing behaviours such as self-sorting phenomena with a rich variety of patterns exhibiting regimes of segregation or mixing. While their one-species counter- parts are well-understood, coupled systems tend to have a higher complexity as many techniques that work for single equations fail in the case of coupled systems. With the results of this dissertation we add new insights to the current discourse of cross-interacting systems. For a certain choice of interaction potentials we provide explicit stationary states and travelling pulse solutions and we propose a numerical scheme whose solutions agree well with the analytical solutions. For a related model, the finite volume scheme is proven to converge. Besides, we prove and generalise an existence result for a certain class of reaction-cross- diffusion systems. Last we provide an existence result for a purely nonlocal interaction system with Newtonian interactions.
Supervisor: Carrillo de la Plata, José Antonio Sponsor: Imperial College London
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral