In the context of gene expression, we proposed a nonlinear stochastic delay differential equation as a mathematical model to study the effects of extrinsic noise on a delay induced Hopf bifurcation. We envisaged its direct numerical resolution. Following the example of the noisy oscillator, we first solved a linearized version of the equation, close to the Hopf bifurcation. The numerical scheme used is a combination of a standard algorithm to solve a deterministic delay differential equation and a stochastic Euler scheme. From our calculations we verified that the deterministic behaviour is fully recovered. For the stochastic case, we found that our solution is qualitatively accurate, in the sense that the noise induced shift in the critical value a, follows the trend the known analytic results predict. However, our numerical solution systematically overestimates the value of the shift. This is explained because the accuracy in the numerical estimation of the decay rate of a solution towards the stationary state value is a function of the control parameter a. We believe the mismatch between the numerical solution and the analytic results is due to a lack of convergence of our scheme, rather than to lack of accuracy. As our numerical scheme is an hybrid, the convergence problem can be improved, both at the deterministic and at the stochastic parts of the scheme. In this work we left our numerical results on the nonlinear case out, because before proceeding to the investigation of the nonlinear equation, the convergence must be assured in the linear case.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.101627 |
| Date | January 2006 |
| Creators | Mondraǵon Palomino, Octavio. |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Master of Science (Department of Physics.) |
| Rights | © Octavio Mondraǵon Palomino, 2006 |
| Relation | alephsysno: 002592790, proquestno: AAIMR32755, Theses scanned by UMI/ProQuest. |
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