Article Contents
Article Contents

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

• 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].
Mathematics Subject Classification: Primary: 39A30, 39A60.

 Citation:

•  [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), 323-351.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), 358-366.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), 1873-1877.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), 863-881.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), 594-606.doi: 10.1080/17513750903528192. [6] Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model, J. Biol. Dyna., 2 (2008), 89-101.doi: 10.1080/17513750801956313. [7] M. R. S. Kulenović, Invariants and related Liapunov functions for difference equations, Appl. Math. Lett., 13 (2000), 1-8.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), 269-282.doi: 10.1007/s002850050171. [9] Z. Lu and Y. Zhou, Advances in Mathematical Biology, Science Press, Beijing, 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), 636-660.doi: 10.1080/17513758.2011.581764. [11] R. M. May, Biological populations with nonoverlapping generations: stable points, stable cycles and chaos, Science, 186 (1974), 645-647.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), 162-174.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, Providence, Rhode Island, 1995. [14] W. Wang and Z. Lu, Global stability of discrete models of Lotka-Voltera type, Nonl. Anal. RWA, 35 (1999), 1019-1030.doi: 10.1016/S0362-546X(98)00112-6. [15] L. Wang and M. Q. Wang, Ordinary Difference Equations, Xinjiang University Press, Urumqi, 1991. [16] C. Wu, Permanence and stable periodic solution for a discrete competitive system with multidelays, Advances in Difference Equations, (2009), Article ID 375486, 12 pages.doi: 10.1155/2009/375486.