  This paper, as a complement to the works by Hsu et al [SIAM. J. Appl. Math.
55 (1995)] and Huang et al [J. Differential Equations 257 (2014)], aims to
examine the Hopf bifurcation and global dynamics of a predator-prey model of
Leslie type with generalized Holling type III functional response for the two
cases: (A) when it has a unique non-degenerate positive equilibrium; (B) when
it has three distinct positive equilibria. For each case, the type and
stability of each equilibrium, Hopf bifurcation at each weak focus, and the
number and distribution of limit cycles in the first quadrant are studied. It
is shown that every equilibrium is not a center. For the case (A), $i$ limit
cycle(s) can appear near the unique positive equilibrium, $i=1, \cdots, 4$. For
$i=3$ or $4$, the model has two stable limit cycles, which gives a positive
answer to the open problem proposed by Coleman [Differential equations
model,1983]: finding at least two ecologically stable cycles in any natural (or
laboratory) predator-prey system. For the case (B), one positive equilibrium is
a saddle and the others are both anti-saddle. If one of the two anti-saddles is
weak focus and the other is not, then the order of the weak focus is at most
$3$. If both anti-saddles are weak foci, then they are unstable weak foci of
order one. Moreover, one limit cycle can bifurcate from each of them
simultaneously. Numerical simulations demonstrate that there is also a big
stable limit cycle enclosing these two limit cycles. Our results indicate that
the maximum number of limit cycles in the model of this kind is at least $4$,
which improves the preceding results that this number is at least $2$.
