  In this paper, we investigate the complex dynamics of a spatial plankton-fish
system with Holling type III functional responses. We have carried out the
analytical study for both one and two dimensional system in details and found
out a condition for diffusive instability of a locally stable equilibrium.
Furthermore, we present a theoretical analysis of processes of pattern
formation that involves organism distribution and their interaction of
spatially distributed population with local diffusion. The results of numerical
simulations reveal that, on increasing the value of the fish predation rates,
the sequences spots $\rightarrow$ spot-stripe mixtures$\rightarrow$
stripes$\rightarrow$ hole-stripe mixtures holes$\rightarrow$ wave pattern is
observed. Our study shows that the spatially extended model system has not only
more complex dynamic patterns in the space, but also has spiral waves.
