Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay

A phytoplankton-zooplankton system with the delay and reaction-diffusion term is investigated. Firstly, existence and priori bound of solution without delay system are given. The stability of the axial steady state solution with delay system are analyzed by using the comparison arguments and modifying the coupled lower-upper solution pairs. By considering the effects of delay and diffusion, the stability and Hopf bifurcation of the positive steady state solution is investigated. When the delay does not exist, the diffusion cannot vary the stability of the steady state solutions, that is, the Turing instability cannot occur. When the delay exists, the effects of big and small diffusions to Hopf bifurcation are investigated, under certain conditions, the space inhomogeneous periodic solutions may produce. Furthermore, the algorithm determining the properties of bifurcation periodic solutions is given. At last, some numerical simulations are carried out to confirm the correctness of theoretical analyses.