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This paper is concerned with a predator-prey model of Leslie type withsimplified Holling type IV functional response, provided that it has either aunique non-degenerate positive equilibrium or three distinct positiveequilibria. The type and stability of each equilibrium, Hopf cyclicity of eachweak focus, and the number and distribution of limit cycles in the firstquadrant are studied. It is shown that every equilibrium cannot be a center. Ifsystem has a unique positive equilibrium which is a weak focus, then its orderis at most $2$ and it has Hopf cyclicity $2$. Moreover, some sufficientconditions for the global stability of the unique equilibrium are establishedby applying Dulac's criterion and constructing the Liapunov function. If systemhas three distinct positive equilibria, then one of them is a saddle and theothers are both anti-saddles. For two anti-saddles, we prove that the Hopfcyclicity for positive equilibrium with smaller abscissa (resp. biggerabscissa) is $2$ (resp. $1$). Furthermore, if both anti-saddle positiveequilibria are weak foci, then they are unstable multiple foci withmultiplicity one. Moreover, one limit cycle can bifurcate from each of themsimultaneously. Numerical simulations demonstrate that there is a big stablelimit cycle enclosing these two small limit cycles.

Hopf cyclicity and global dynamics for a predator-prey system of Leslie type with simplified Holling type IV functional response