Delay driven vegetation patterns of a plankton system on a network

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A two-component delayed reaction–diffusion system is introduced to a complex network to describe the vegetation patterns in plankton system. The positive equilibrium is shown by the linear stability analysis to be asymptotically stable in the absence of time delay, but when the time delay increases beyond a threshold it loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the method of center manifold theory. Our result reveals that the stability of Hopf bifurcation leads to the emergence of vegetation patterns. Numerical calculations are performed to confirm our theoretical results.