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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  February 2020 Early access  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 and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

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