DIRECTION AND STABILITY OF HOPF BIFURCATION

The dynamics of the spatial competition mathematical model for the invasion, removal of Kappaphycus Algae (KA) in Gulf of Mannar (GoM) with propagation delays is investigated by applying the normal form theory and the center manifold theorem.

In recent years we have witnessed an increasing interest in dynamical systems with time delays, especially in applied mathematics.Stability and direction of the Hopf bifurcation for the predator-prey system have been discussed by using normal form theory and center manifold theory [5,6,10,13,15,16,17]. Direction and stability of the equilibrium for a neural network model with two delays have been investigated [7,12].Bifurcation analysis of the predtor-prey model has been detailed [8].Direction and stability of the equilibrium involving various fields have been discussed [4,9,11,14].We reported the shifting of algal dominated reef ecosystem due to the invasion of KA in Gulf of Mannar [1].Subsequently, the dominance of KA over NA and corals in competing for space has also been reported.KA sexual reproduction by spores in the Gulf of Mannar Marine Biosphere Reserve (GoM) in future, when environmental conditions unanimously favor this alga has been deliberated [2].To simulate the three way competition among corals, KA and NA, we proposed the following system of non-linear ODE's [3].

Direction and Stability of Hopf bifurcation
We assume that the system undergoes a Hobf bifurcation at the positive equilibrium E(0, y * , 0) for τ 1 = τ * 1 and then ±iω denotes the corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E(0, y * , 0).

Without loss of generality, we assume that τ
..Here µ = 0 is the bifurcation parameter and dropping the bars, the system becomes a functional differential equation in where x(t) = (x 11 , x 21 , x 31 ) ∈ R 3 and L µ : C → R 3 , f : R ×C → R 3 are respectively given by φ 2 (−1) and where Q= .
Thus we can take We first compute the coordinate to describe the center manifold C 0 at µ = 0. Let X t be the solution of the system (2.5)when µ = 0. Define z(t) =< q * , X t > On the center manifold C 0 , we have and z and z are local coordinates for center manifold C 0 in the direction of q * and q * .Note that W is real if X t is real.We consider only real solutions.For solution X t ∈ C 0 of Eq.
p 2 , p 3 and p 4 values can be calculaed by using the formula.