Abstract
In this paper, we consider a discrete-time prey–predator model with a strong Allee effect. At first, applying the center manifold reduction, we discuss the topological structure of orbits near each fixed point, including hyperbolic one and non-hyperbolic one. Then, we apply the center manifold reduction to prove that the model undergoes fold bifurcation and transcritical bifurcation. At last, by the numerical analysis, we illustrate our results and find that the model possesses heteroclinic orbits in the first quadrant. Furthermore, we present a stable 7-periodic orbit and a stable 8-periodic orbit on the invariant cycle produced from the Neimark–Sacker bifurcation.
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Acknowledgements
The authors thank Professor Weinian Zhang for his supervision with patience and thank Professor Wenmeng Zhang for many helpful discussions during this research. The authors also thank three anonymous referees for their valuable comments and suggestions.
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This work was supported in part by the National Natural Science Foundation of China under Grants 11371314 and 11771197, in part by the High-Level Talent Project of Colleges and Universities in Guangdong Province under Grant QBS201501, in part by the Guangdong Natural Science Foundation of China under Grant 2017A030313030 and by the Startup Foundation for Doctors of Lingnan Normal University under Grant ZL1605.
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Zhong, J., Yu, Z. Qualitative properties and bifurcations of Mistro–Rodrigues–Petrovskii model. Nonlinear Dyn 91, 2063–2075 (2018). https://doi.org/10.1007/s11071-017-3932-0
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DOI: https://doi.org/10.1007/s11071-017-3932-0