American Institute of Mathematical Sciences

Advanced
January  2007, 7(1): 191-199. doi: 10.3934/dcdsb.2007.7.191

Local stability implies global stability in some one-dimensional discrete single-species models

Eduardo Liz 1,

1. 

Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, Campus Marcosende, 36280 Vigo

Received  December 2005 Revised  August 2006 Published  October 2006

We prove a criterion for the global stability of the positive equilibrium in discrete-time single-species population models of the form $x_{n+1}=x_nF(x_n)$. This allows us to demonstrate analytically (and easily) the conjecture that local stability implies global stability in some well-known models, including the Ricker difference equation and a combination of the models by Hassel and Maynard Smith. Our approach combines the use of linear fractional functions (Möbius transformations) and the Schwarzian derivative.
Keywords: discrete-time single-species model, Ricker difference equation, Schwarzian derivative., global stability.
Mathematics Subject Classification: Primary: 39A11; Secondary: 92D25, 92D4.
Citation: Eduardo Liz. Local stability implies global stability in some one-dimensional discrete single-species models. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 191-199. doi: 10.3934/dcdsb.2007.7.191
[1]

Abhyudai Singh, Roger M. Nisbet. Variation in risk in single-species discrete-time models. Mathematical Biosciences & Engineering, 2008, 5 (4) : 859-875. doi: 10.3934/mbe.2008.5.859
[2]

Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213
[3]

James Sandefur. A unifying approach to discrete single-species populations models. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 493-508. doi: 10.3934/dcdsb.2017194
[4]

Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123
[5]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323
[6]

Brian Ryals, Robert J. Sacker. Global stability in the 2D Ricker equation revisited. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 585-604. doi: 10.3934/dcdsb.2017028
[7]

Nikodem J. Poplawski, Abbas Shirinifard, Maciej Swat, James A. Glazier. Simulation of single-species bacterial-biofilm growth using the Glazier-Graner-Hogeweg model and the CompuCell3D modeling environment. Mathematical Biosciences & Engineering, 2008, 5 (2) : 355-388. doi: 10.3934/mbe.2008.5.355
[8]

Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734
[9]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
[10]

Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discrete-time SI and SIS epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (4) : 699-710. doi: 10.3934/mbe.2007.4.699
[11]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[12]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[13]

Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066
[14]

Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253
[15]

Benjamin Webb. Dynamics of functions with an eventual negative Schwarzian derivative. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1393-1408. doi: 10.3934/dcds.2009.24.1393
[16]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
[17]

Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315
[18]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[19]

Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015
[20]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences