  Alexander B. Medvinsky \emph{et al} [A. B. Medvinsky, I. A. Tikhonova, R. R.
Aliev, B.-L. Li, Z.-S. Lin, and H. Malchow, Phys. Rev. E \textbf{64}, 021915
(2001)] and Marcus R. Garvie \emph{et al} [M. R. Garvie and C. Trenchea, SIAM
J. Control. Optim. \textbf{46}, 775-791 (2007)] shown that the minimal
spatially extended reaction-diffusion model of phytoplankton-zooplankton can
exhibit both regular, chaotic behavior, and spatiotemporal patterns in a patchy
environment. Based on that, the spatial plankton model is furtherly
investigated by means of computer simulations and theoretical analysis in the
present paper when its parameters would be expected in the case of mixed
Turing-Hopf bifurcation region. Our results show that the spiral waves exist in
that region and the spatiotemporal chaos emerge, which arise from the far-field
breakup of the spiral waves over large ranges of diffusion coefficients of
phytoplankton and zooplankton. Moreover, the spatiotemporal chaos arising from
the far-field breakup of spiral waves does not gradually involve the whole
space within that region. Our results are confirmed by means of computation
spectra and nonlinear bifurcation of wave trains. Finally, we give some
explanations about the spatially structured patterns from the community level.
