A note on global stability of three-dimensional Ricker models

ABSTRACT

In the recent paper [E. C. Balreira, S. Elaydi, and R. Luís, J. Differ. Equ. Appl. 23 (2017), pp. 2037–2071], Balreira, Elaydi and Luís established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luís' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.

KEYWORDS:

• Global stability
• Ricker model
• carrying simplex
• global dynamics
• phase portrait

XXXX MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37Cxx
• 37N25
• 92D25

Acknowledgements

The authors are very grateful to Prof. Saber Elaydi, Prof. Janusz Mierczyński and Dr. Stephen Baigent for valuable suggestions and also thank Prof. Saber Elaydi for sending us the manuscript of reference [1].

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11771295, Shanghai Gaofeng Project for University Academic Program Development, and the Academy of Finland.