Yue, Yuanxi (2025) Wavefronts and speed selection of spatiotemporal biochemical models. Doctoral (PhD) thesis, Memorial University of Newfoundland.
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Abstract
This thesis is devoted to the study of wavefronts and speed selection of spatiotemporal biochemical models. We first consider a model of precursor and differentiated cells, with the appearance of non-isolated equilibria on a line in the phase space. This system exhibits transition properties, shifting from monostable to bistable and from monotone to non-monotone, through changes in the impacting parameter. We establish the existence of traveling waves in the monostable monotone case, the monostable non-monotone case, and the bistable monotone case, and derive some propagation properties of the traveling waves for each case. The critical degenerate case in this precursor and differentiated cells model exhibits fundamentally different and significantly more intricate dynamics. We here develop new ideas to prove the existence of traveling waves by a continuation argument via perturbation in a weighted functional space. To extend the above ideas to more complex models, we investigate the Belousov-Zhabotinsky system with non-local delayed interaction, which has non-isolated equilibria and exhibits a dynamical transition structure from bistable to monostable. The existence and uniqueness of bistable traveling waves, as well as the existence of monostable waves in the degenerate case, are first proved. In addition, we derive some necessary and sufficient conditions for speed selection and show how parameters impact speed selection results under the monostable case. Climate change has great impacts on the survival of biological species. Lastly, we consider a reaction-diffusion equation with a shifting environment, where the reaction function shifts from a favorable to an unfavorable environment. The exponential stability of three types of forced waves in this model is proved. Furthermore, numerical simulations are carried out for each model, which match with our theoretical results.
| Item Type: | Thesis (Doctoral (PhD)) |
|---|---|
| URI: | http://research.library.mun.ca/id/eprint/16969 |
| Item ID: | 16969 |
| Additional Information: | Includes bibliographical references (pages 159-163) -- Restricted until May 30, 2026 |
| Keywords: | traveling wave, non-isolated equilibria, speed selection, degenerate nonlinearity |
| Department(s): | Science, Faculty of > Mathematics and Statistics |
| Date: | May 2025 |
| Date Type: | Submission |
| Library of Congress Subject Heading: | Biological models; Nonlinear theories; Traveling-wave tubes |
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