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Abstract
This thesis is devoted to study some qualitative properties like as existence, uniqueness, boundedness, persistence, stability and bifurcation of solutions of the vector ordinary differential equations
x’ = f(x), x Rn (1)
and delayed differential equation of the form
x’ = F(t,x(t), x(t-nr), x(t-2nr)), r > 0, n=1,2,3,… (2)
By choosing one of the delayed as a bifurcation parameter we show that equation (2) exhibits Hopf bifurcation, saddle-node bifurcation.
As an application of (1) we consider some physical and biological systems. We first consider a simple food chain model with one prey x(t) and two predators y(t) and z(t) in a chemostat with distinct removal rates. In this model, the pry consumes the nutrient and the predator y(t) consumes the prey but the predator y(t) does not consume the nutrient. Also, the predator z(t) consumes the predator y(t) but the predator z(t) does not consume the nutrient or the prey. We use Liapunov functions in studying the global stability of the equilibria. Mathematical analysis of the model equations with regard to invariance of non-negativity, boundedness of solutions, dissipativity and persistent are studied. Hopf bifurcation theory is applied.