In this thesis, we study a two-component reaction diffusion system in one and two spatial dimensions, both numerically and asymptotically. The system is related to a nonlocal reaction diffusion equation which has been proposed as a model for a single species that competes with itself for a common resource. In one spatial dimension, we find that this system admits traveling wave solutions that connect the two homogeneous steady states. We also analyse the long-time behaviour of the solutions. Although there exists a lower bound on the speed of travelling wave solutions, we observe that numerical solutions in some regions of parameter space exhibit travelling waves that propagate for an asymptotically long time with speeds below this lower bound. We use asymptotic methods to both verify these numerical results and explain the dynamics of the problem, which include steady, unsteady, spike-periodic travelling and gap-periodic travelling waves. In two spatial dimensions, the numerical solutions of the axisymmetric form of the system are presented. We also establish the existence of a steady axisymmetric solution which takes a form of a circular patch. We then carry out a linear stability analysis of the system. Finally, we perform bifurcation analysis of these patch solutions via a numerical continuation technique and show how these solutions change with respect to variation of one bifurcation parameter.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:698041 |
| Date | January 2016 |
| Creators | Barwari Bala, Farhad |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://eprints.nottingham.ac.uk/37231/ |
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