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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 toexamine the Hopf bifurcation and global dynamics of a predator-prey model ofLeslie type with generalized Holling type III functional response for the twocases: (A) when it has a unique non-degenerate positive equilibrium; (B) whenit has three distinct positive equilibria. For each case, the type andstability of each equilibrium, Hopf bifurcation at each weak focus, and thenumber and distribution of limit cycles in the first quadrant are studied. Itis shown that every equilibrium is not a center. For the case (A), $i$ limitcycle(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 positiveanswer to the open problem proposed by Coleman [Differential equationsmodel,1983]: finding at least two ecologically stable cycles in any natural (orlaboratory) predator-prey system. For the case (B), one positive equilibrium isa saddle and the others are both anti-saddle. If one of the two anti-saddles isweak 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 oforder one. Moreover, one limit cycle can bifurcate from each of themsimultaneously. Numerical simulations demonstrate that there is also a bigstable limit cycle enclosing these two limit cycles. Our results indicate thatthe 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 $2$.

Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response