Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion

In this paper, a delayed biological economic system which considers a plankton system with harvest effort on phytoplankton is proposed. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the plankton dynamics, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. Then by numerical simulations the impact of delay, diffusion and economic interest on plankton system are explored. It is found that delay can cause system into chaos and can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, diffusion and economic profit can also affect the dynamic behavior of the system.