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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 & 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, (1996).  doi: 10.1142/9789812798732.  Google Scholar

[2]

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

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer, (1982).   Google Scholar

[4]

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

[5]

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

[6]

S. Elaydi, Discrete Chaos: With Applications in Science and Engineering., Chapman and Hall/CRC, (2008).   Google Scholar

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1007/BF01077471.  Google Scholar

[11]

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

[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.  doi: 10.1080/10236198.2010.504377.  Google Scholar

[13]

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

[14]

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

[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, (1992).   Google Scholar

[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.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[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.  doi: 10.1080/17513758.2011.581764.  Google Scholar

[18]

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

[19]

M. Martelli, Global stability of stationary states of discrete dynamical systems,, Ann. Sci. Math. Québec, 22 (1998), 201.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

C. Mira, Chaotic Dynamics,, World Scientific, (1987).   Google Scholar

[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).   Google Scholar

[24]

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

[25]

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

[26]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Springer, (1990).   Google Scholar

[27]

S. Willard, General Topology,, Dover Publications, (2004).   Google Scholar

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, (1996).  doi: 10.1142/9789812798732.  Google Scholar

[2]

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

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer, (1982).   Google Scholar

[4]

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

[5]

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

[6]

S. Elaydi, Discrete Chaos: With Applications in Science and Engineering., Chapman and Hall/CRC, (2008).   Google Scholar

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1007/BF01077471.  Google Scholar

[11]

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

[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.  doi: 10.1080/10236198.2010.504377.  Google Scholar

[13]

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

[14]

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

[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, (1992).   Google Scholar

[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.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[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.  doi: 10.1080/17513758.2011.581764.  Google Scholar

[18]

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

[19]

M. Martelli, Global stability of stationary states of discrete dynamical systems,, Ann. Sci. Math. Québec, 22 (1998), 201.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[22]

C. Mira, Chaotic Dynamics,, World Scientific, (1987).   Google Scholar

[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).   Google Scholar

[24]

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

[25]

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

[26]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Springer, (1990).   Google Scholar

[27]

S. Willard, General Topology,, Dover Publications, (2004).   Google Scholar

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