We present a new formula that makes it possible to get sharp global stability results for one-dimensional discrete-time models in an easy way. In particular, it allows to show that the local asymptotic stability of a positive equilibrium implies its global asymptotic stability for a new family of difference equations that finds many applications in population dynamics, economic models, and also in physiological processes governed by delay differential equations. The main ingredients to prove our results are the Schwarzian derivative and some dominance arguments.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
For this reason we use the name gamma-models for (1.1).
Allwright, D.J.: Hypergraphic functions and bifurcations in recurrence relations. SIAM J. Appl. Math. 34, 687–691 (1978)
Bellows, T.S.: The descriptive properties of some models for density dependence. J. Anim. Ecol. 50, 139–156 (1981)
Buedo-Fernández, S., Liz, E.: On the stability properties of a delay differential neoclassical model of economic growth. Electron J. Qual. Theory Differ. Equ. 43, 1–14 (2018)
Coppel, W.A.: The solution of equations by iteration. Proc. Camb. Philos. Soc. 51, 41–43 (1955)
Cull, P.: Population models: stability in one dimension. Bull. Math. Biol. 69, 989–1017 (2007)
Day, R.H.: Irregular growth cycles. Am. Econ. Rev. 72, 406–414 (1982)
El-Morshedy, H.A., Jiménez López, V.: Global attractors for difference equations dominated by one-dimensional maps. J. Differ. Equ. Appl. 14, 391–410 (2008)
Gopalsamy, K., Trofimchuk, S.I., Bantsur, N.R.: A note on global attractivity in models of hematopoiesis. Ukr. Math. J. 50, 3–12 (1998)
Ivanov, A.F., Sharkovsky, A.N.: Oscillations in singularly perturbed delay equations. Dyn. Rep. Expos. Dyn. Syst. (N.S.) 1, 164–224 (1992)
Jiménez López, V., Parreño, E.: L.A.S and negative Schwarzian derivative do not imply G.A.S. in Clark’s equation. J. Dyn. Differ. Equ. 28, 339–374 (2016)
Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, New York (2001)
Levin, S.A., May, R.M.: A note on difference delay equations. Theor. Popul. Biol. 9, 178–187 (1976)
Liz, E.: Local stability implies global stability in some one-dimensional discrete single-species models. Discrete Contin. Dyn. Syst. Ser. B 7, 191–199 (2007)
Liz, E.: A global picture of the gamma-Ricker map: a flexible discrete-time model with factors of positive and negative density dependence. Bull. Math. Biol. 80, 417–434 (2018)
Liz, E.: A new flexible discrete-time model for stable populations. Discrete Contin. Dyn. Syst. B 23, 2487–2498 (2018)
Liz, E., Pinto, M., Robledo, G., Trofimchuk, S., Tkachenko, V.: Wright type delay differential equations with negative Schwarzian. Discrete Contin. Dyn. Syst. 9, 309–321 (2003)
Liz, E., Röst, G.: Global dynamics in a commodity market model. J. Math. Anal. Appl. 398, 707–714 (2013)
Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control system. Science 197, 287–289 (1977)
Mallet-Paret, J., Nussbaum, R.D.: A differential-delay equation arising in optics and physiology. SIAM J. Math. Anal. 20, 249–292 (1989)
May, R.M., Oster, G.F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–599 (1976)
Maynard Smith, J., Slatkin, M.: The stability of predator–prey systems. Ecology 54, 384–391 (1973)
Quinn, T.J., Deriso, R.B.: Quantitative Fish Dynamics. Oxford University Press, New York (1999)
Sedaghat, H.: Nonlinear Difference Equations: Theory with Applications to Social Science Models, Mathematical Modelling: Theory and Applications, vol. 15. Kluwer Academic Publishers, Dordrecht (2003)
Sharkovsky, A.N., Kolyada, S.F., Sivak, A.G., Fedorenko, V.V.: Dynamics of One-Dimensional Maps, Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1997)
Shepherd, J.G.: A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield resources. J. Conserv. Int. Explor. Mer. 40, 67–75 (1982)
Singer, D.: Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)
Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956)
Zheng, J., Kruse, G.H.: Stock–recruitment relationships for three major Alaskan crab stocks. Fish. Res. 65, 103–121 (2003)
The authors sincerely thank Víctor Jiménez López (Universidad de Murcia, Spain) and Ábel Garab (Alpen-Adria-Universität Klagenfurt, Austria) for useful discussions, encouraging comments and relevant remarks, and an anonymous reviewer for his/her helpful comments. Eduardo Liz acknowledges the support of the research Grant MTM2017-85054-C2-1-P (AEI/FEDER, UE). The research of Sebastián Buedo-Fernández has been partially supported by Ministerio de Educación, Cultura y Deporte of Spain (Grant No. FPU16/04416), Consellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia (Grant Nos. GRC2015/004 and R2016/022), and Agencia Estatal de Investigación of Spain (Grant MTM2016-75140-P, cofunded by European Community fund FEDER).
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Liz, E., Buedo-Fernández, S. A New Formula to Get Sharp Global Stability Criteria for One-Dimensional Discrete-Time Models. Qual. Theory Dyn. Syst. 18, 813–824 (2019). https://doi.org/10.1007/s12346-018-00314-4
- Global stability
- Discrete-time model
- Mackey–Glass equation
- Schwarzian derivative
Mathematics Subject Classification
- Primary 39A10
- Secondary 34K20