  The Turing-Hopf type spatiotemporal patterns in a diffusive Holling-Tanner
model with discrete time delay is considered. A global Turing bifurcation
theorem for $\tau=0$ and a local Turing bifurcation theorem for $\tau>0$ are
given by the method of eigenvalue analysis and prior estimation. Further
considering the degenerated situation, the existence of Bogdanov-Takens
bifurcation and Turing-Hopf bifurcation are obtained. The normal form method is
used to study the explicit dynamics near the Turing-Hopf singularity, and we
establish the existence of various self-organized spatiotemporal patterns, such
as two non-constant steady states (stripe patterns) coexist and two spatially
inhomogeneous periodic solutions (spot patterns) coexist. Moreover, the
Turing-Turing-Hopf type spatiotemporal patterns, that is a subharmonic
phenomenon with two spatial wave numbers and one temporal frequency, are also
found and theoretically explained, when there is another Turing bifurcation
curve which is relatively closed to the studied one.
