Electronic Thesis and Dissertation Repository

Degree

Master of Science

Program

Applied Mathematics

Supervisor

Yu, Pei.

Abstract

In this thesis, we apply bifurcation theory to study two biological systems. Main attention is focused on complex dynamical behaviors such as stability and bifurcation of limit cycles. Hopf bifurcation is particularly considered to show bistable or even tristable phenomenon which may occur in biological systems. Recurrence is also investigated to show that such complex behavior is common in biological systems.

First we consider a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. Main attention is focused on the sta- bility and bifurcation of equilibria when the prey has a linear growth. Coexistence of different species is shown in the food chain, showing bistable or even tristable phenomenon. Hopf bi- furcation is studied to show complex dynamics due to the existence of multiple limit cycles. In particular, normal form theory is applied to prove that three limit cycles can bifurcate from an equilibrium in the vicinity of a Hopf critical point.

Further investigation is focused on the recurrence behavior in oscillating networks of bio- logically relevant organic reactions. This model has one unique equilibrium solution. Analysis is first given to the stability and bifurcation of the equilibrium. Then, particular attention is fo- cused on recurrence behavior of the system when the equilibrium become unstable. Numerical simulations are compared with the analytical predictions to show a very good agreement.

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