Local stability implies global stability for the 2-dimensional Ricker map

Abstract

In this study, we consider the difference equation where is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version of the equation, S. Levin and R. May conjectured in 1976 that local stability of the non-trivial equilibrium implies its global stability. Based on rigorous, computer-aided calculations and analytical tools, we prove the conjecture for d = 1.

Keywords::

• global stability
• rigorous numerics
• Neimark–Sacker bifurcation
• graph representations
• interval analysis
• discrete-time single species model

AMS Subject Classification::

• 39A30
• 39A28
• 65Q10
• 65G40
• 92D25

Acknowledgements

The authors thank Warwick Tucker from the CAPA group and Daniel Wilczak and Tomasz Kapela, members of the CAPD group [3], for useful suggestions and their help. The authors also acknowledge the useful comments of the anonymous referees, which helped them to improve their paper. The computations were performed on the HPC centre provided by the Hungarian National Information Infrastructure Development Institute [25] at the University of Szeged. The first author was supported by Bergens forskningsstiftelse – The Bergen Research Foundation, with project title: ‘Computer-Aided Proofs in Mathematical Analysis’ and project number: 801458. The second and third authors were supported by the Hungarian Scientific Research Fund, Grant No. K 75517. This work was partially supported by the European Union and co-funded by the European Social Fund under the project ‘Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences’ of project number ‘TÁMOP-4.2.2.A-11/1/KONV-2012-0073’. Ábel Garab was also supported by the European Union and co-funded by the European Social Fund. Project title: ‘Broadening the knowledge base and supporting the long-term professional sustainability of the Research University Centre of Excellence at the University of Szeged by ensuring the rising generation of excellent scientists’. Project number: TÁMOP-4.2.2/B-10/1-2010-0012.