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Local stability implies global stability for the planar Ricker competition model
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 |
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] |
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] |
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