Hopf bifurcation and stability analyses of a neural network model with delay and a predator-prey model with delay

In this thesis, the main aim is to investigate the stability and Hopf bifurcation analyses of delayed differential equation systems. For this, we take two different differential equation systems which belong to population dynamics and artificial neural networks areas. These systems are enhanced by incorporating delay terms where needed and their dynamical behaviours are studied. Bifurcation theory is a research area that observes the changes of dynamical systems that depend on time according to a chosen control parameter. The goal is to study the large time behaviour of the system with respect to a control parameter that we want to observe. Bifurcation is the change of qualitative structure of dynamical systems associated with varying parameter values. There are many different bifurcation types in both difference and differential equations. Also, the names and types of bifurcations change according to the dimension of models. In dynamical systems literature, periodic solutions have an extensive study area. In continuous time dynamical systems, the method of determination of local periodic solutions that depend on a control parameter is Hopf bifurcation analysis. Hopf bifurcation is a type of bifurcation that after a critical parameter value the stability of the equilibrium point changes (the direction may change according to axes which we look) and limit cycles (periodic solutions) occur. Hopf bifurcation can be seen in at least two dimensional differential equation systems. It is very common to choose the delay parameter as a bifurcation parameter. In order to reflect dynamical behaviour of models that depend on the past history of the system, we often incorporate time delays into models. Even it is more difficult to analyse systems with delays, delays always exist in nature. In population dynamics, maturation time to be a prey of a specie or time needed for predation, incubation period of a bacteria, time needed for a response after a neuron is fired can be given for examples. This thesis mainly involves two parts: In the first part, it has been incorporated a discrete delay term into a ratio-dependent differential equation system which has rich content in population dynamics. This part includes Hopf bifurcation and stability analyses of two different systems, that is, one is the system with equal time delays and the other one is the system with different two time delays. In the second part, it has been incorporated a discrete and a distributed delay term into a recurrent neural network system with two neurons. Again, the system is considered in two categories. Both in two systems, the conditions on parameters are determined to guarantee that two systems have Hopf bifurcation and existence of periodic solutions is demonstrated. Theoretical results that have been obtained for four systems supported with numerical examples by using MATLAB program.