American Institute of Mathematical Sciences

Advanced
March  2014, 19(2): 323-351. doi: 10.3934/dcdsb.2014.19.323

Local stability implies global stability for the planar Ricker competition model

E. Cabral Balreira 1, , Saber Elaydi 1, and  Rafael Luís 2,

1. 

Department of Mathematics, Trinity University, San Antonio, Texas, United States, United States
2. 

Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Portugal

Received  March 2013 Revised  July 2013 Published  February 2014

Under certain analytic and geometric assumptions we show that local stability of the coexistence (positive) fixed point of the planar Ricker competition model implies global stability with respect to the interior of the positive quadrant. This result is a confluence of ideas from Dynamical Systems, Geometry, and Topology that provides a framework to the study of global stability for other planar competition models.
Keywords: compact invariant set, critical curves, fold, cusp., global stability, Competition model, principal preimage function, local stability.
Mathematics Subject Classification: Primary: 39A30, 30A60; Secondary: 57R7.
Citation: E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323
References:
[1]

A. Barugola, C. Mira, L. Gardini and J. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, Nonlinear Sciences Series A. World Scientific, Singapore, 1996. doi: 10.1142/9789812798732.

[2]

M. Chamberland, Dynamics of maps with nilpotent Jacobians, J. Difference Equ. Appl., 12 (2006), 49-56. doi: 10.1080/10236190500267970.

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, 1982.

[4]

P. Cull, Stability of discrete one-dimensional population models, Bull. Math. Biol., 50 (1988), 67-75. doi: 10.1016/S0092-8240(88)90016-X.

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd Edition,, 2003., (). 

[6]

S. Elaydi, Discrete Chaos: With Applications in Science and Engineering. Chapman and Hall/CRC, second edition, 2008.

[7]

S. Elaydi and R. Luís, Open problems in some competition models, Journal of Difference Equations and Applications, 17 (2011), 1873-1877. doi: 10.1080/10236198.2011.559468.

[8]

R. Feşler, A proof of the two-dimensional markus-yamabe stability conjecture and a generalization, Ann. Polon. Math., 62 (1995), 45-74.

[9]

L. Gardini, Some global bifurcations of two-dimensional endomorphisms by use of critical lines, Nonlinear Analysis, 18 (1992), 361-399. doi: 10.1016/0362-546X(92)90152-5.

[10]

A. A. Glutsyuk, The asymptotic stability of the linearization of a vector field on the plane with a singular point implies global stability, Funktsional. Anal. i Prilozhen., 29 (1995), 17-30. doi: 10.1007/BF01077471.

[11]

C. Gutierrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 12 (1995), 627-671.

[12]

M. Guzowska, R. Luís and S. Elaydi, Bifurcation and invariant manifolds of the logistic competition model, Journal of Difference Equations and Applications, 17 (2011), 1851-1872. doi: 10.1080/10236198.2010.504377.

[13]

H. Kestelman, Mappings with non-vanishing jacobian, The American Mathematical Monthly, 78 (1971), 662-663. doi: 10.2307/2316581.

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Applied Mathematical Sciences), Springer-Verlag, New York, 2004.

[15]

J. Cathala, L. Gardini and C. Mira, Contact bifurcation of absorbing areas and chaotic areas in two-dimensional endomorphisms, In Procedings of the European Conference on Iteration Theory, Austria, 1992.

[16]

E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete and Continuous Dynamical Systems - Series B, 7 (2007), 191-199. doi: 10.3934/dcdsb.2007.7.191.

[17]

R. Luís, S. Elaydi and H. Oliveira, Stability of a Ricker-type competition model and the competitive exclusion principle, Journal of Biological Dynamics, 5 (2011), 636-660. doi: 10.1080/17513758.2011.581764.

[18]

L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J., 12 (1960), 305-317.

[19]

M. Martelli, Global stability of stationary states of discrete dynamical systems, Ann. Sci. Math. Québec, 22 (1998), 201-212. Dedicated to the memory of Gilles Fournier (Sherbrooke, PQ, 1997).

[20]

C. Mira, Détermination pratique du dumaine de stabilité d'un point d'une récurrence non-lineaire du deuxiéme ordre à variables réelles, C. R. Acad. Sc. Paris, 261 (1964), 5314-5317. Groupe 2.

[21]

C. Mira, Sur quelques propriétés de la frontiére de stabilité d'un point double d'une récurrence et sur un cas de bifurcation de cette frontiére, C. R. Acad. Sc. Paris, 262 (1966), 951-954. Sér. A.

[22]

C. Mira, Chaotic Dynamics, World Scientific, Singapore, 1987.

[23]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, volume 407 of Mathematics and its Applications, Kluwer Academic Publishers Group, 1997.

[24]

H. Smith, Planar competitive and cooperative difference equations, Journal of Difference Equations and Applications, 3 (1998), 335-357. doi: 10.1080/10236199708808108.

[25]

H. Whitney, On singularities of mappings of euclidean spaces. mappings of the plane into the plane, Annals of Mathematics, 62 (1955), 374-410. doi: 10.2307/1970070.

[26]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 1990.

[27]

S. Willard, General Topology, Dover Publications, 2004.

show all references

References:
[1]

A. Barugola, C. Mira, L. Gardini and J. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, Nonlinear Sciences Series A. World Scientific, Singapore, 1996. doi: 10.1142/9789812798732.

[2]

M. Chamberland, Dynamics of maps with nilpotent Jacobians, J. Difference Equ. Appl., 12 (2006), 49-56. doi: 10.1080/10236190500267970.

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, 1982.

[4]

P. Cull, Stability of discrete one-dimensional population models, Bull. Math. Biol., 50 (1988), 67-75. doi: 10.1016/S0092-8240(88)90016-X.

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd Edition,, 2003., (). 

[6]

S. Elaydi, Discrete Chaos: With Applications in Science and Engineering. Chapman and Hall/CRC, second edition, 2008.

[7]

S. Elaydi and R. Luís, Open problems in some competition models, Journal of Difference Equations and Applications, 17 (2011), 1873-1877. doi: 10.1080/10236198.2011.559468.

[8]

R. Feşler, A proof of the two-dimensional markus-yamabe stability conjecture and a generalization, Ann. Polon. Math., 62 (1995), 45-74.

[9]

L. Gardini, Some global bifurcations of two-dimensional endomorphisms by use of critical lines, Nonlinear Analysis, 18 (1992), 361-399. doi: 10.1016/0362-546X(92)90152-5.

[10]

A. A. Glutsyuk, The asymptotic stability of the linearization of a vector field on the plane with a singular point implies global stability, Funktsional. Anal. i Prilozhen., 29 (1995), 17-30. doi: 10.1007/BF01077471.

[11]

C. Gutierrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 12 (1995), 627-671.

[12]

M. Guzowska, R. Luís and S. Elaydi, Bifurcation and invariant manifolds of the logistic competition model, Journal of Difference Equations and Applications, 17 (2011), 1851-1872. doi: 10.1080/10236198.2010.504377.

[13]

H. Kestelman, Mappings with non-vanishing jacobian, The American Mathematical Monthly, 78 (1971), 662-663. doi: 10.2307/2316581.

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Applied Mathematical Sciences), Springer-Verlag, New York, 2004.

[15]

J. Cathala, L. Gardini and C. Mira, Contact bifurcation of absorbing areas and chaotic areas in two-dimensional endomorphisms, In Procedings of the European Conference on Iteration Theory, Austria, 1992.

[16]

E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete and Continuous Dynamical Systems - Series B, 7 (2007), 191-199. doi: 10.3934/dcdsb.2007.7.191.

[17]

R. Luís, S. Elaydi and H. Oliveira, Stability of a Ricker-type competition model and the competitive exclusion principle, Journal of Biological Dynamics, 5 (2011), 636-660. doi: 10.1080/17513758.2011.581764.

[18]

L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J., 12 (1960), 305-317.

[19]

M. Martelli, Global stability of stationary states of discrete dynamical systems, Ann. Sci. Math. Québec, 22 (1998), 201-212. Dedicated to the memory of Gilles Fournier (Sherbrooke, PQ, 1997).

[20]

C. Mira, Détermination pratique du dumaine de stabilité d'un point d'une récurrence non-lineaire du deuxiéme ordre à variables réelles, C. R. Acad. Sc. Paris, 261 (1964), 5314-5317. Groupe 2.

[21]

C. Mira, Sur quelques propriétés de la frontiére de stabilité d'un point double d'une récurrence et sur un cas de bifurcation de cette frontiére, C. R. Acad. Sc. Paris, 262 (1966), 951-954. Sér. A.

[22]

C. Mira, Chaotic Dynamics, World Scientific, Singapore, 1987.

[23]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, volume 407 of Mathematics and its Applications, Kluwer Academic Publishers Group, 1997.

[24]

H. Smith, Planar competitive and cooperative difference equations, Journal of Difference Equations and Applications, 3 (1998), 335-357. doi: 10.1080/10236199708808108.

[25]

H. Whitney, On singularities of mappings of euclidean spaces. mappings of the plane into the plane, Annals of Mathematics, 62 (1955), 374-410. doi: 10.2307/1970070.

[26]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 1990.

[27]

S. Willard, General Topology, Dover Publications, 2004.

[1]

Weiyi Zhang, Ling Zhou. Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022062
[2]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[3]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[4]

Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545
[5]

Siniša Slijepčević. Stability of invariant measures. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345
[6]

Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025
[7]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[8]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715
[9]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119
[10]

Attila Dénes, Yoshiaki Muroya, Gergely Röst. Global stability of a multistrain SIS model with superinfection. Mathematical Biosciences & Engineering, 2017, 14 (2) : 421-435. doi: 10.3934/mbe.2017026
[11]

Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449
[12]

E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461
[13]

Ábel Garab, Veronika Kovács, Tibor Krisztin. Global stability of a price model with multiple delays. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6855-6871. doi: 10.3934/dcds.2016098
[14]

Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995
[15]

Eduardo Liz. Local stability implies global stability in some one-dimensional discrete single-species models. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 191-199. doi: 10.3934/dcdsb.2007.7.191
[16]

Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847
[17]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110
[18]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[19]

Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015
[20]

Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (158)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy