Bifurcation Analysis in a Diffusion Mussel-Algae Interaction System with Delays Considering the Half-Saturation Constant

In this paper, the kinetics of a class of delayed reaction-diffusion musselalgae system under Neumann boundary conditions with the half- saturation constant is studied. The global existence and priori bounds as well as the existence conditions of positive equilibrium are obtained. The half-saturation constant affect the stability of the system and may result in Turing instability. When the half-saturation constant exceeds a certain critical value, the boundary equilibrium is globally asymptotically stable which means that the larger half-saturation constant forces the mussel population toward extinction. By analyzing the distribution of the roots of the characteristic equation with two delays, the stability conditions of the positive equilibrium in the parameter space are obtained. The stability of the positive equilibrium can be changed by steady-state bifurcation, Hopf bifurcation, Hopf-Hopf bifurcation or Hopf-steady state bifurcation, which can be verified by some numerical simulations. Among parameters, the half-saturation constant and two delays drive the complexity of the system dynamics.