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Global stability in the 2D Ricker equation revisited
1. | Department of Mathematics, California State University Bakersfield, Bakersfield, CA 93311-1022, USA |
2. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA |
We offer two improvements to prior results concerning global stability of the 2D Ricker Equation. We also offer some new methods of approach for the more pathological cases and prove some miscellaneous results including a special nontrivial case in which the mapping is conjugate to the 1D Ricker map along an invariant line and a proof of the non-existence of period-2 points.
References:
[1] |
A.S. Ackleh and P.L. Salceanu,
Competitive exclusion and coexistence in an n-species Ricker model, J Biological Dynamics, 9 (2015), 321-331.
doi: 10.1080/17513758.2015.1020576. |
[2] |
S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, Stephen Baigent, 12 (2015), p8167. Google Scholar |
[3] |
E. Cabral Balreira, S. Elaydi and R. Luis,
Local stability implies global stability for the planar Ricker competition model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 323-351.
doi: 10.3934/dcdsb.2014.19.323. |
[4] |
P. Cull,
Stability of one-dimensional population models, Bull. Math. Biology, 50 (1988), 67-75.
doi: 10.1016/S0092-8240(88)90016-X. |
[5] |
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder Colorado, USA, second edition, 2003. Google Scholar |
[6] |
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. Google Scholar |
[7] |
S. Elaydi, Discrete Chaos, Chapman and Hall, CRC, Boca Raton, USA, 2008. Google Scholar |
[8] |
H. Jiang and T. D. Rogers,
The discrete dynamics of symmetric competition in the plane, J. Math. Biol., 25 (1987), 573-596.
doi: 10.1007/BF00275495. |
[9] |
J. Li,
Simple mathematical models for mosquito populations with genetically altered mosquitos, Math. Bioscience, 189 (2004), 39-59.
doi: 10.1016/j.mbs.2004.01.001. |
[10] |
E. Liz,
Local stability implies global stability in some one-dimensional discrete single-species models, Discrete and Continuous Dynam. Syst.-B, 7 (2007), 191-199.
doi: 10.3934/dcdsb.2007.7.191. |
[11] |
C. Mira, L. Gardini, A. Barugola and J. -C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, volume 20 of Series in Nonlinear Sciences, World Scientific, Tokyo, Japan, 1996. Google Scholar |
[12] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, volume 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010. Google Scholar |
[13] |
W. E. Ricker, Stock and recruitment, J. Fisheries Research Board Canada, 11 (1954), 559-623. Google Scholar |
[14] |
B. Ryals and R. J. Sacker,
Global stability in the 2-D Ricker equation, J. Difference Eq. and Appl., 21 (2015), 1068-1081.
doi: 10.1080/10236198.2015.1065825. |
[15] |
R. J. Sacker,
A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.
doi: 10.1080/10236190601008752. |
[16] |
R. J. Sacker and H. F. von Bremen, Global asymptotic stability in the Jia Li model for genetically altered mosquitos, In Linda J. S. Allen-et. al. , editor, Difference Equations and Discrete Dynamical Systems, Proc. 9th Internat. Conf. on Difference Equations and Appl. (2004), pages 87-100. World Scientific, 2005. Google Scholar |
[17] |
R. J. Sacker and H. F. von Bremen,
Dynamic reduction with applications to mathematical biology and other areas, J. Biological Dynamics, 1 (2007), 437-453.
doi: 10.1080/17513750701605572. |
[18] |
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Federenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers Group, Dordrecht, Netherlands, 1997. Google Scholar |
[19] |
H. Smith,
Planar competitive and cooperative difference equations, J. Difference Eq. and Appl., 3 (1998), 335-357.
doi: 10.1080/10236199708808108. |
show all references
References:
[1] |
A.S. Ackleh and P.L. Salceanu,
Competitive exclusion and coexistence in an n-species Ricker model, J Biological Dynamics, 9 (2015), 321-331.
doi: 10.1080/17513758.2015.1020576. |
[2] |
S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, Stephen Baigent, 12 (2015), p8167. Google Scholar |
[3] |
E. Cabral Balreira, S. Elaydi and R. Luis,
Local stability implies global stability for the planar Ricker competition model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 323-351.
doi: 10.3934/dcdsb.2014.19.323. |
[4] |
P. Cull,
Stability of one-dimensional population models, Bull. Math. Biology, 50 (1988), 67-75.
doi: 10.1016/S0092-8240(88)90016-X. |
[5] |
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder Colorado, USA, second edition, 2003. Google Scholar |
[6] |
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. Google Scholar |
[7] |
S. Elaydi, Discrete Chaos, Chapman and Hall, CRC, Boca Raton, USA, 2008. Google Scholar |
[8] |
H. Jiang and T. D. Rogers,
The discrete dynamics of symmetric competition in the plane, J. Math. Biol., 25 (1987), 573-596.
doi: 10.1007/BF00275495. |
[9] |
J. Li,
Simple mathematical models for mosquito populations with genetically altered mosquitos, Math. Bioscience, 189 (2004), 39-59.
doi: 10.1016/j.mbs.2004.01.001. |
[10] |
E. Liz,
Local stability implies global stability in some one-dimensional discrete single-species models, Discrete and Continuous Dynam. Syst.-B, 7 (2007), 191-199.
doi: 10.3934/dcdsb.2007.7.191. |
[11] |
C. Mira, L. Gardini, A. Barugola and J. -C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, volume 20 of Series in Nonlinear Sciences, World Scientific, Tokyo, Japan, 1996. Google Scholar |
[12] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, volume 2002 of Lecture Notes in Mathematics, Springer, Berlin, 2010. Google Scholar |
[13] |
W. E. Ricker, Stock and recruitment, J. Fisheries Research Board Canada, 11 (1954), 559-623. Google Scholar |
[14] |
B. Ryals and R. J. Sacker,
Global stability in the 2-D Ricker equation, J. Difference Eq. and Appl., 21 (2015), 1068-1081.
doi: 10.1080/10236198.2015.1065825. |
[15] |
R. J. Sacker,
A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.
doi: 10.1080/10236190601008752. |
[16] |
R. J. Sacker and H. F. von Bremen, Global asymptotic stability in the Jia Li model for genetically altered mosquitos, In Linda J. S. Allen-et. al. , editor, Difference Equations and Discrete Dynamical Systems, Proc. 9th Internat. Conf. on Difference Equations and Appl. (2004), pages 87-100. World Scientific, 2005. Google Scholar |
[17] |
R. J. Sacker and H. F. von Bremen,
Dynamic reduction with applications to mathematical biology and other areas, J. Biological Dynamics, 1 (2007), 437-453.
doi: 10.1080/17513750701605572. |
[18] |
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Federenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers Group, Dordrecht, Netherlands, 1997. Google Scholar |
[19] |
H. Smith,
Planar competitive and cooperative difference equations, J. Difference Eq. and Appl., 3 (1998), 335-357.
doi: 10.1080/10236199708808108. |








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