Hopf bifurcation analysis in the diffusive nutrient-microorganism model with time delay

In this paper, the diffusive nutrient-microorganism model with time delay subject to Neumann boundary conditions is considered. The effect of time delay on the Hopf bifurcation and stability of positive constant equilibrium in the partial differential equation (PDE) and ordinary differential equation (ODE) models are analyzed. By choosing the time delay as the bifurcation parameter, it is found that the positive equilibrium is asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than this value. In addition, it is also indicated that Hopf bifurcation occurs when the delay passes through a family of critical values. And then, the formulae to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by applying the normal form theory and center manifold theorem for partial differential equation are derived. Finally, to support our analytical results obtained for the ODE and PDE model with time delay, some numerical simulations are also included.