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Thus, the boundary order-1 limit cycle ((x^{T}(t), 0)) is globally attractive.

Figure 8 Stability of boundary order-1 limit cycle (pmb{(x^{T}(t),0)}).

The boundary order-1 limit cycle ((x^{T}(t), 0)) is unstable for (SC11) and (SC2).

In conclusion, we have the following main results for the boundary order-1 limit cycle.

Thus, the boundary order-1 limit cycle is globally stable if (tau=0) and (A_{h}neq0) in case (SC123).

As (A_{h}) increases and goes beyond zero (i.e. (A_{h}>0)), then the boundary order-1 limit cycles disappear.

The boundary order-1 limit cycle ((x^{T}(t), 0)) is globally asymptotically stable for (SC123), and it is locally asymptotically stable for (SC12).

The local stability of the boundary order-1 limit cycle for (SC12) is obvious due to the domain of the phase set.

If (tau=0) and (A_{0}neq0), then there exists a unique boundary order-1 limit cycle ((x^{T}(t), 0)) for model (2.3).

In [32], Tang et al. presented a semi-dynamic predator-prey model with Holling type-II functional response and studied the global stability of boundary order-1 limit cycle.

The existence, local and global stability of an order-1 limit cycle and obtain sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided.