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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model,, J. Biol. Dyna., 2 (2008), 89. doi: 10.1080/17513750801956313. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

Z. Lu and Y. Zhou, Advances in Mathematical Biology,, Science Press, (2006). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperatiive Systems,, American Mathematical Society, (1995). Google Scholar

[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. Google Scholar

[15]

L. Wang and M. Q. Wang, Ordinary Difference Equations,, Xinjiang University Press, (1991). Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

Y. Kang, D. Armbruster and Y. Kuang, Dynamics of a plant-herbivore model,, J. Biol. Dyna., 2 (2008), 89. doi: 10.1080/17513750801956313. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

Z. Lu and Y. Zhou, Advances in Mathematical Biology,, Science Press, (2006). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperatiive Systems,, American Mathematical Society, (1995). Google Scholar

[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. Google Scholar

[15]

L. Wang and M. Q. Wang, Ordinary Difference Equations,, Xinjiang University Press, (1991). Google Scholar

[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. Google Scholar

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