  This paper is concerned with a predator-prey model of Leslie type with
simplified Holling type IV functional response, provided that it has either a
unique non-degenerate positive equilibrium or three distinct positive
equilibria. The type and stability of each equilibrium, Hopf cyclicity of each
weak focus, and the number and distribution of limit cycles in the first
quadrant are studied. It is shown that every equilibrium cannot be a center. If
system has a unique positive equilibrium which is a weak focus, then its order
is at most $2$ and it has Hopf cyclicity $2$. Moreover, some sufficient
conditions for the global stability of the unique equilibrium are established
by applying Dulac's criterion and constructing the Liapunov function. If system
has three distinct positive equilibria, then one of them is a saddle and the
others are both anti-saddles. For two anti-saddles, we prove that the Hopf
cyclicity for positive equilibrium with smaller abscissa (resp. bigger
abscissa) is $2$ (resp. $1$). Furthermore, if both anti-saddle positive
equilibria are weak foci, then they are unstable multiple foci with
multiplicity one. Moreover, one limit cycle can bifurcate from each of them
simultaneously. Numerical simulations demonstrate that there is a big stable
limit cycle enclosing these two small limit cycles.
