# American Institute of Mathematical Sciences

November  2015, 20(9): 3255-3266. doi: 10.3934/dcdsb.2015.20.3255

## Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model

 1 School of Mathematics and Physics, Changzhou University, Changzhou, 213164, China 2 School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

Received  May 2014 Revised  June 2015 Published  September 2015

We shall obtain the parameter region that ensures the global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. The parameter region can be illustrated graphically and examples of such regions are presented. Our result partially answers an open problem proposed by Elaydi and Luís [3] and complements the very recent work by Balreira, Elaydi and Luís [1].
Citation: 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
##### References:
 [1] E. C. Balreira, S. Elaydi and R. Luís, Local stability implies global stability for the planar Ricker competition model,, Discrete and Continuous Dynamical Systems series B, 19 (2014), 323. doi: 10.3934/dcdsb.2014.19.323. [2] Y. M. Chen and Z. Zhou, Stable peoriodic solution of a discrete periodic Lotka-Volterra competition system,, J. Math. Anal. Appl., 277 (2003), 358. doi: 10.1016/S0022-247X(02)00611-X. [3] S. Elaydi and R. Luís, Open problems in some competition models,, J. Diff. Equ. Appl., 17 (2011), 1873. doi: 10.1080/10236198.2011.559468. [4] J. Hofbauer, R. Kon and Y. Saito, Qualitative permanence of Lotka-Volterra equations,, J. Math. Biol., 57 (2008), 863. doi: 10.1007/s00285-008-0192-0. [5] A. N. W. Hone, M. V. Irle and G. W. Thurura, On the Neimark-Sacker bifurcation in a discrete predator-prey system,, J. Biol. Dyna., 4 (2010), 594. doi: 10.1080/17513750903528192. [6] Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model,, J. Biol. Dyna., 2 (2008), 89. doi: 10.1080/17513750801956313. [7] M. R. S. Kulenović, Invariants and related Liapunov functions for difference equations,, Appl. Math. Lett., 13 (2000), 1. doi: 10.1016/S0893-9659(00)00068-9. [8] Z. Lu and W. Wang, Permanence and global attractivity for Lotka-Volterra difference systems,, J. Math. Biol., 39 (1999), 269. doi: 10.1007/s002850050171. [9] Z. Lu and Y. Zhou, Advances in Mathematical Biology,, Science Press, (2006). [10] R. Luís, S. Elaydi and H. Oliveira, Stability of a Ricker-type competition model and the competitive exclusion principle,, J. Biol. Dyna., 5 (2011), 636. doi: 10.1080/17513758.2011.581764. [11] R. M. May, Biological populations with nonoverlapping generations: stable points, stable cycles and chaos,, Science, 186 (1974), 645. doi: 10.1126/science.186.4164.645. [12] Y. Saito, W. Ma and T. Hara, A necessary and sufficient condition for permanence of a Lotka-Voltera discrete system with delays,, J. Math. Anal. Appl., 256 (2001), 162. doi: 10.1006/jmaa.2000.7303. [13] H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperatiive Systems,, American Mathematical Society, (1995). [14] W. Wang and Z. Lu, Global stability of discrete models of Lotka-Voltera type,, Nonl. Anal. RWA, 35 (1999), 1019. doi: 10.1016/S0362-546X(98)00112-6. [15] L. Wang and M. Q. Wang, Ordinary Difference Equations,, Xinjiang University Press, (1991). [16] C. Wu, Permanence and stable periodic solution for a discrete competitive system with multidelays,, Advances in Difference Equations, (2009). doi: 10.1155/2009/375486.

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##### References:
 [1] E. C. Balreira, S. Elaydi and R. Luís, Local stability implies global stability for the planar Ricker competition model,, Discrete and Continuous Dynamical Systems series B, 19 (2014), 323. doi: 10.3934/dcdsb.2014.19.323. [2] Y. M. Chen and Z. Zhou, Stable peoriodic solution of a discrete periodic Lotka-Volterra competition system,, J. Math. Anal. Appl., 277 (2003), 358. doi: 10.1016/S0022-247X(02)00611-X. [3] S. Elaydi and R. Luís, Open problems in some competition models,, J. Diff. Equ. Appl., 17 (2011), 1873. doi: 10.1080/10236198.2011.559468. [4] J. Hofbauer, R. Kon and Y. Saito, Qualitative permanence of Lotka-Volterra equations,, J. Math. Biol., 57 (2008), 863. doi: 10.1007/s00285-008-0192-0. [5] A. N. W. Hone, M. V. Irle and G. W. Thurura, On the Neimark-Sacker bifurcation in a discrete predator-prey system,, J. Biol. Dyna., 4 (2010), 594. doi: 10.1080/17513750903528192. [6] Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model,, J. Biol. Dyna., 2 (2008), 89. doi: 10.1080/17513750801956313. [7] M. R. S. Kulenović, Invariants and related Liapunov functions for difference equations,, Appl. Math. Lett., 13 (2000), 1. doi: 10.1016/S0893-9659(00)00068-9. [8] Z. Lu and W. Wang, Permanence and global attractivity for Lotka-Volterra difference systems,, J. Math. Biol., 39 (1999), 269. doi: 10.1007/s002850050171. [9] Z. Lu and Y. Zhou, Advances in Mathematical Biology,, Science Press, (2006). [10] R. Luís, S. Elaydi and H. Oliveira, Stability of a Ricker-type competition model and the competitive exclusion principle,, J. Biol. Dyna., 5 (2011), 636. doi: 10.1080/17513758.2011.581764. [11] R. M. May, Biological populations with nonoverlapping generations: stable points, stable cycles and chaos,, Science, 186 (1974), 645. doi: 10.1126/science.186.4164.645. [12] Y. Saito, W. Ma and T. Hara, A necessary and sufficient condition for permanence of a Lotka-Voltera discrete system with delays,, J. Math. Anal. Appl., 256 (2001), 162. doi: 10.1006/jmaa.2000.7303. [13] H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperatiive Systems,, American Mathematical Society, (1995). [14] W. Wang and Z. Lu, Global stability of discrete models of Lotka-Voltera type,, Nonl. Anal. RWA, 35 (1999), 1019. doi: 10.1016/S0362-546X(98)00112-6. [15] L. Wang and M. Q. Wang, Ordinary Difference Equations,, Xinjiang University Press, (1991). [16] C. Wu, Permanence and stable periodic solution for a discrete competitive system with multidelays,, Advances in Difference Equations, (2009). doi: 10.1155/2009/375486.
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