Stability and Hopf Bifurcation Analysis on a Numerical Discretization of the Distributed Delay Equation

A kind of discrete logistic model with distributed delays obtained by the Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, it is found that the stability of the positive equilibrium and Hopf occurs when τ crosses some critical value. Then the explicit formulae which determine the stability, direction and other properties of the bifurcating periodic solution are derived by using the theory of normal form and center manifold. Finally, numerical simulations are performed to verify and illustrate the analytical results.