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February  2020, 25(2): 635-650. doi: 10.3934/dcdsb.2019258

Stability for one-dimensional discrete dynamical systems revisited

Daniel Franco 1, , Juan Perán 1,, and  Juan Segura 2,

1. 

Departamento de Matemática Aplicada, ETSI Industriales, Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, 28040, Madrid, Spain
2. 

Departament d'Economia i Empresa, Universitat Pompeu Fabra, c/ Ramón Trías Fargas 25-27, 08005, Barcelona, Spain

* Corresponding author: Juan Perán

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received  January 2019 Revised  May 2019 Published  November 2019

Fund Project: This work was funded by grant MTM2017-85054-C2-2-P (AEI/FEDER, UE) and ETSII-UNED grant 2019-MAT11.

We present a new method to study the stability of one-dimensional discrete-time models, which is based on studying the graph of a certain family of functions. The method is closely related to exponent analysis, which the authors introduced to study the global stability of certain intricate convex combinations of maps. We show that the new strategy presented here complements and extends some existing conditions for the global stability. In particular, we provide a global stability condition improving the condition of negative Schwarzian derivative. Besides, we study the relation between this new method and the enveloping technique.

Keywords: Global stability, discrete dynamical system, enveloping, Schwarzian derivative, population model.
Mathematics Subject Classification: Primary: 37C05, 37C75; Secondary: 92D25.
Citation: Daniel Franco, Juan Perán, Juan Segura. Stability for one-dimensional discrete dynamical systems revisited. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 635-650. doi: 10.3934/dcdsb.2019258
References:
[1]

D. J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math., 34 (1978), 687-691.  doi: 10.1137/0134057.  Google Scholar

[2]

F. A. Bartha, Á. Garab and T. Krisztin, Local stability implies global stability for the 2-dimensional ricker map, J. Difference Equ. Appl., 19 (2013), 2043-2078.  doi: 10.1080/10236198.2013.804916.  Google Scholar

[3]

S. Buedo-Fernández and E. Liz, On the stability properties of a delay differential neoclassical model of economic growth, Electron. J. Qual. Theory of Differ. Equ., 43 (2018), 1-14.  doi: 10.14232/ejqtde.2018.1.43.  Google Scholar

[4]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87.  doi: 10.1016/j.mbs.2013.12.003.  Google Scholar

[5]

P. Cull, Stability of discrete one-dimensional population models, Bull. Math. Biol., 50 (1988), 67-75.  doi: 10.1007/BF02459978.  Google Scholar

[6]

P. Cull, Population models: Stability in one dimension, Bull. Math. Biol., 69 (2007), 989-1017.  doi: 10.1007/s11538-006-9129-1.  Google Scholar

[7]

P. Cull and J. Chaffee, Stability in discrete population models, AIP Conference Proceedings, 517 (2000), 263-276.  doi: 10.1063/1.1291265.  Google Scholar

[8]

M. E. Fisher, B. S. Goh and T. L. Vincent, Some stability conditions for discrete-time single species models, Bull. Math. Biol., 41 (1979), 861-875.  doi: 10.1007/BF02462383.  Google Scholar

[9]

D. Franco, H. Logemann and J. Perán, Global stability of an age-structured population model, Syst. Control Lett., 65 (2014), 30-36.  doi: 10.1016/j.sysconle.2013.11.012.  Google Scholar

[10]

D. Franco, J. Perán and J. Segura, Effect of harvest timing on the dynamics of the Ricker-Seno model, Math. Biosci., 306 (2018), 180-185.  doi: 10.1016/j.mbs.2018.10.002.  Google Scholar

[11]

D. Franco, J. Perán and J. Segura, Global stability of discrete dynamical systems via exponent analysis: Applications to harvesting population models, Electron. J. Qual. Theory Differ. Equ., 101 (2018), 1-22.  doi: 10.14232/ejqtde.2018.1.101.  Google Scholar

[12]

B.-S. Goh, Management and Analysis of Biological Populations, vol. 8, Elsevier, 2012. Google Scholar

[13]

I. Györi and S. I. Trofimchuk, Global attractivity and persistence in a discrete population model, J. Difference Equ. Appl., 6 (2000), 647-665.  doi: 10.1080/10236190008808250.  Google Scholar

[14]

V. Jiménez López and E. Parreño, L.A.S. and negative Schwarzian derivative do not imply G.A.S. in Clark's equation, J. Dynam. Differential Equations, 28 (2016), 339-374.  doi: 10.1007/s10884-016-9525-7.  Google Scholar

[15]

S. A. Kuruklis and G. Ladas, Oscillations and global attractivity in a discrete delay logistic model, Quart. Appl. Math., 50 (1992), 227-233.  doi: 10.1090/qam/1162273.  Google Scholar

[16]

S. A. Levin and R. M. May, A note on difference-delay equations, Theor. Popul. Biol., 9 (1976), 178-187.  doi: 10.1016/0040-5809(76)90043-5.  Google Scholar

[17]

E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[18]

E. Liz and S. Buedo-Fernández, A new formula to get sharp global stability criteria for one-dimensional discrete-time models, Qual. Theory Dyn. Syst., published online, (2019), 1–12. doi: 10.1007/s12346-018-00314-4.  Google Scholar

[19]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.   Google Scholar

[20]

C. J. Pennycuick, R. M. Compton and L. Beckingham, A computer model for simulating the growth of a population, or of two interacting populations, J. Theor. Biol., 18 (1968), 316-329.  doi: 10.1016/0022-5193(68)90081-7.  Google Scholar

[21]

J. Perán and D. Franco, Global convergence of the second order Ricker equation, Appl. Math. Lett., 47 (2015), 47-53.  doi: 10.1016/j.aml.2015.02.022.  Google Scholar

[22]

H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69.  doi: 10.1016/j.mbs.2008.06.004.  Google Scholar

[23]

A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.  doi: 10.1142/S0218127495000934.  Google Scholar

[24]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

[25]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

show all references

References:
[1]

D. J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math., 34 (1978), 687-691.  doi: 10.1137/0134057.  Google Scholar

[2]

F. A. Bartha, Á. Garab and T. Krisztin, Local stability implies global stability for the 2-dimensional ricker map, J. Difference Equ. Appl., 19 (2013), 2043-2078.  doi: 10.1080/10236198.2013.804916.  Google Scholar

[3]

S. Buedo-Fernández and E. Liz, On the stability properties of a delay differential neoclassical model of economic growth, Electron. J. Qual. Theory of Differ. Equ., 43 (2018), 1-14.  doi: 10.14232/ejqtde.2018.1.43.  Google Scholar

[4]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87.  doi: 10.1016/j.mbs.2013.12.003.  Google Scholar

[5]

P. Cull, Stability of discrete one-dimensional population models, Bull. Math. Biol., 50 (1988), 67-75.  doi: 10.1007/BF02459978.  Google Scholar

[6]

P. Cull, Population models: Stability in one dimension, Bull. Math. Biol., 69 (2007), 989-1017.  doi: 10.1007/s11538-006-9129-1.  Google Scholar

[7]

P. Cull and J. Chaffee, Stability in discrete population models, AIP Conference Proceedings, 517 (2000), 263-276.  doi: 10.1063/1.1291265.  Google Scholar

[8]

M. E. Fisher, B. S. Goh and T. L. Vincent, Some stability conditions for discrete-time single species models, Bull. Math. Biol., 41 (1979), 861-875.  doi: 10.1007/BF02462383.  Google Scholar

[9]

D. Franco, H. Logemann and J. Perán, Global stability of an age-structured population model, Syst. Control Lett., 65 (2014), 30-36.  doi: 10.1016/j.sysconle.2013.11.012.  Google Scholar

[10]

D. Franco, J. Perán and J. Segura, Effect of harvest timing on the dynamics of the Ricker-Seno model, Math. Biosci., 306 (2018), 180-185.  doi: 10.1016/j.mbs.2018.10.002.  Google Scholar

[11]

D. Franco, J. Perán and J. Segura, Global stability of discrete dynamical systems via exponent analysis: Applications to harvesting population models, Electron. J. Qual. Theory Differ. Equ., 101 (2018), 1-22.  doi: 10.14232/ejqtde.2018.1.101.  Google Scholar

[12]

B.-S. Goh, Management and Analysis of Biological Populations, vol. 8, Elsevier, 2012. Google Scholar

[13]

I. Györi and S. I. Trofimchuk, Global attractivity and persistence in a discrete population model, J. Difference Equ. Appl., 6 (2000), 647-665.  doi: 10.1080/10236190008808250.  Google Scholar

[14]

V. Jiménez López and E. Parreño, L.A.S. and negative Schwarzian derivative do not imply G.A.S. in Clark's equation, J. Dynam. Differential Equations, 28 (2016), 339-374.  doi: 10.1007/s10884-016-9525-7.  Google Scholar

[15]

S. A. Kuruklis and G. Ladas, Oscillations and global attractivity in a discrete delay logistic model, Quart. Appl. Math., 50 (1992), 227-233.  doi: 10.1090/qam/1162273.  Google Scholar

[16]

S. A. Levin and R. M. May, A note on difference-delay equations, Theor. Popul. Biol., 9 (1976), 178-187.  doi: 10.1016/0040-5809(76)90043-5.  Google Scholar

[17]

E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[18]

E. Liz and S. Buedo-Fernández, A new formula to get sharp global stability criteria for one-dimensional discrete-time models, Qual. Theory Dyn. Syst., published online, (2019), 1–12. doi: 10.1007/s12346-018-00314-4.  Google Scholar

[19]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.   Google Scholar

[20]

C. J. Pennycuick, R. M. Compton and L. Beckingham, A computer model for simulating the growth of a population, or of two interacting populations, J. Theor. Biol., 18 (1968), 316-329.  doi: 10.1016/0022-5193(68)90081-7.  Google Scholar

[21]

J. Perán and D. Franco, Global convergence of the second order Ricker equation, Appl. Math. Lett., 47 (2015), 47-53.  doi: 10.1016/j.aml.2015.02.022.  Google Scholar

[22]

H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69.  doi: 10.1016/j.mbs.2008.06.004.  Google Scholar

[23]

A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.  doi: 10.1142/S0218127495000934.  Google Scholar

[24]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

[25]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

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