**5. Conclusions**

The dynamics of a Rosenzweig–MacArthur model with continuous threshold harvesting in predator involving the Caputo fractional-order derivative and ABC fractional-order derivative are studied. We prove the existence and uniqueness of solutions of both Caputo and ABC models. Particularly, we completely investigate the dynamics of the Caputo model including the non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. From the biological meanings, the extinction of both populations never occurs since the origin point (*<sup>E</sup>*0) is a saddle point. Some of the situations that might occur are: (1) the predator goes extinct while the prey still survives, which is indicated by the stability of *E*1; (2) both predator and prey co-exist and converge to a constant population density, which happens if the interior point *E* ˆ or *E*∗ are asymptotically stable; and (3) both predator and prey co-exist where both population densities change periodically, namely when a Hopf bifurcation occurs. We show numerically that our model may undergo a forward bifurcation or a Hopf bifurcation. The Hopf bifurcation in models with both Caputo operator and ABC operator can be driven by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. Our numerical simulations show that the Hopf bifurcation point of both models are different.

**Author Contributions:** All authors equally contributed to this article. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Directorate of Research and Community Service, The Directorate General of Strengthening Research and Development, the Ministry of Research, Technology, and Higher Education (Brawijaya University), Indonesia, via Doctoral Dissertation Research, in accordance with the Research Contract No. 037/SP2H/LT/DRPM/2020, dated 9 March 2020.

**Acknowledgments:** The authors would like to thank the reviewers for all useful and helpful comments to improve the manuscript.

**Conflicts of Interest:** All authors report no conflict of interest relevant to this article.
