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Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching
1. | School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China |
2. | Center for Mathematics and Interdisciplinary Sciences, School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China |
3. | Department of Mathematics, Wayne State University, Detroit, Michigan 48202 |
References:
[1] |
F. Chen, Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model, Nonlinear Anal., 7 (2006), 895-915.
doi: 10.1016/j.nonrwa.2005.04.007. |
[2] |
F. Chen, Permanence of a delayed non-autonomous Gilpin-Ayala competition model, Appl. Math. Comput., 179 (2006), 55-66.
doi: 10.1016/j.amc.2005.11.079. |
[3] |
F. Chen, Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays, Nonlinear Anal., 7 (2006), 1205-1222.
doi: 10.1016/j.nonrwa.2005.11.003. |
[4] |
F. Chen, L. Wu and Z. Li, Permanence and global attractivity of the discrete Gilpin-Ayala type population model, Comput. Math. Appl., 53 (2007), 1214-1227.
doi: 10.1016/j.camwa.2006.12.015. |
[5] |
N. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422.
doi: 10.1016/j.cam.2004.02.001. |
[6] |
M. Fan and K. Wang, Global periodic solutions of a generalized $n$-species Gilpin-Ayala competition model, Comput. Math. Appl., 40 (2000), 1141-1151.
doi: 10.1016/S0898-1221(00)00228-5. |
[7] |
M. Gilpin and F. Ayala, Global models of growth and competition, Proc. Natl. Acad. Sci., USA 70 (1973), 3590-3593.
doi: 10.1073/pnas.70.12.3590. |
[8] |
B. Goh and T. Agnew, Stability in Gilpin and Ayala's models of competition, J. Math. Biol., 4 (1977), 275-279.
doi: 10.1007/BF00280977. |
[9] |
M. He, Z. Li and F. Chen, Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses, Nonlinear Anal., 11 (2010), 1537-1551.
doi: 10.1016/j.nonrwa.2009.03.007. |
[10] |
A. Il'in, R. Khasminskii and G. Yin, Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: rapid switchings, J. Math. Anal. Appl., 238 (1999), 516-539.
doi: 10.1006/jmaa.1998.6532. |
[11] |
D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.
doi: 10.1016/j.jmaa.2011.12.049. |
[12] |
D. Jiang, N. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597.
doi: 10.1016/j.jmaa.2007.08.014. |
[13] |
R. Khasminskii, Stochastic Stability of Differential Equations, Berlin: Springer-Verlag, 2012.
doi: 10.1007/978-3-642-23280-0. |
[14] |
D. Li, The stationary distribution and ergodicity of a stochastic generalized logistic system, Stat. Probab. Lett., 83 (2013), 580-583.
doi: 10.1016/j.spl.2012.11.006. |
[15] |
X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28.
doi: 10.1016/j.jmaa.2010.10.053. |
[16] |
X. Liao and J. Li, Stability in Gilpin-Ayala competition models with diffusion, Nonlinear Anal., 28 (1997), 1751-1758.
doi: 10.1016/0362-546X(95)00242-N. |
[17] |
B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428.
doi: 10.1016/j.jmaa.2007.06.058. |
[18] |
M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Modelling, 54 (2011), 2139-2154.
doi: 10.1016/j.mcm.2011.05.023. |
[19] |
M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching II, Math. Comput. Modelling, 55 (2012), 405-418.
doi: 10.1016/j.mcm.2011.08.019. |
[20] |
M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277-289.
doi: 10.1007/s00332-014-9229-2. |
[21] |
M. Liu and L. Yu, Stability of a stochastic logistic model under regime switching, Adv. Difference Equ., 2015 (2015), 9 pp.
doi: 10.1186/s13662-015-0666-5. |
[22] |
X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110.
doi: 10.1016/S0304-4149(01)00126-0. |
[23] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, London: Imperial College Press, 2006.
doi: 10.1142/p473. |
[24] |
A. Settati and A. Lahrouz, On stochastic Gilpin-Ayala population model with Markovian switching, Biosystems, 130 (2015), 17-27.
doi: 10.1016/j.biosystems.2015.01.004. |
[25] |
Q. Wang, M. Ding, Z. Wang and H. Zhang, Existence and attractivity of a periodic solution for an $N$-species Gilpin-Ayala impulsive competition system, Nonlinear Anal., 11 (2010), 2675-2685.
doi: 10.1016/j.nonrwa.2009.09.015. |
[26] |
H. Yang, Q. Zhang and J. Feng, Numerical simulations based on POD for stochastic age-dependent system of two species, Differ. Equ. Dyn. Syst., 23 (2015), 433-451.
doi: 10.1007/s12591-014-0229-3. |
[27] |
G. Yin and C. Zhu, Hybrid Switching Diffusions Properties and Applications, New York: Springer-Verlag, 2010.
doi: 10.1007/978-1-4419-1105-6. |
[28] |
S. Zhang, D. Tan and L. Chen, The periodic $n$-species Gilpin-Ayala competition system with impulsive effect, Chaos, Solitons & Fractals, 26 (2005), 507-517.
doi: 10.1016/j.chaos.2005.01.020. |
[29] |
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
show all references
References:
[1] |
F. Chen, Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model, Nonlinear Anal., 7 (2006), 895-915.
doi: 10.1016/j.nonrwa.2005.04.007. |
[2] |
F. Chen, Permanence of a delayed non-autonomous Gilpin-Ayala competition model, Appl. Math. Comput., 179 (2006), 55-66.
doi: 10.1016/j.amc.2005.11.079. |
[3] |
F. Chen, Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays, Nonlinear Anal., 7 (2006), 1205-1222.
doi: 10.1016/j.nonrwa.2005.11.003. |
[4] |
F. Chen, L. Wu and Z. Li, Permanence and global attractivity of the discrete Gilpin-Ayala type population model, Comput. Math. Appl., 53 (2007), 1214-1227.
doi: 10.1016/j.camwa.2006.12.015. |
[5] |
N. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422.
doi: 10.1016/j.cam.2004.02.001. |
[6] |
M. Fan and K. Wang, Global periodic solutions of a generalized $n$-species Gilpin-Ayala competition model, Comput. Math. Appl., 40 (2000), 1141-1151.
doi: 10.1016/S0898-1221(00)00228-5. |
[7] |
M. Gilpin and F. Ayala, Global models of growth and competition, Proc. Natl. Acad. Sci., USA 70 (1973), 3590-3593.
doi: 10.1073/pnas.70.12.3590. |
[8] |
B. Goh and T. Agnew, Stability in Gilpin and Ayala's models of competition, J. Math. Biol., 4 (1977), 275-279.
doi: 10.1007/BF00280977. |
[9] |
M. He, Z. Li and F. Chen, Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses, Nonlinear Anal., 11 (2010), 1537-1551.
doi: 10.1016/j.nonrwa.2009.03.007. |
[10] |
A. Il'in, R. Khasminskii and G. Yin, Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: rapid switchings, J. Math. Anal. Appl., 238 (1999), 516-539.
doi: 10.1006/jmaa.1998.6532. |
[11] |
D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.
doi: 10.1016/j.jmaa.2011.12.049. |
[12] |
D. Jiang, N. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597.
doi: 10.1016/j.jmaa.2007.08.014. |
[13] |
R. Khasminskii, Stochastic Stability of Differential Equations, Berlin: Springer-Verlag, 2012.
doi: 10.1007/978-3-642-23280-0. |
[14] |
D. Li, The stationary distribution and ergodicity of a stochastic generalized logistic system, Stat. Probab. Lett., 83 (2013), 580-583.
doi: 10.1016/j.spl.2012.11.006. |
[15] |
X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28.
doi: 10.1016/j.jmaa.2010.10.053. |
[16] |
X. Liao and J. Li, Stability in Gilpin-Ayala competition models with diffusion, Nonlinear Anal., 28 (1997), 1751-1758.
doi: 10.1016/0362-546X(95)00242-N. |
[17] |
B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428.
doi: 10.1016/j.jmaa.2007.06.058. |
[18] |
M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Modelling, 54 (2011), 2139-2154.
doi: 10.1016/j.mcm.2011.05.023. |
[19] |
M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching II, Math. Comput. Modelling, 55 (2012), 405-418.
doi: 10.1016/j.mcm.2011.08.019. |
[20] |
M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277-289.
doi: 10.1007/s00332-014-9229-2. |
[21] |
M. Liu and L. Yu, Stability of a stochastic logistic model under regime switching, Adv. Difference Equ., 2015 (2015), 9 pp.
doi: 10.1186/s13662-015-0666-5. |
[22] |
X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110.
doi: 10.1016/S0304-4149(01)00126-0. |
[23] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, London: Imperial College Press, 2006.
doi: 10.1142/p473. |
[24] |
A. Settati and A. Lahrouz, On stochastic Gilpin-Ayala population model with Markovian switching, Biosystems, 130 (2015), 17-27.
doi: 10.1016/j.biosystems.2015.01.004. |
[25] |
Q. Wang, M. Ding, Z. Wang and H. Zhang, Existence and attractivity of a periodic solution for an $N$-species Gilpin-Ayala impulsive competition system, Nonlinear Anal., 11 (2010), 2675-2685.
doi: 10.1016/j.nonrwa.2009.09.015. |
[26] |
H. Yang, Q. Zhang and J. Feng, Numerical simulations based on POD for stochastic age-dependent system of two species, Differ. Equ. Dyn. Syst., 23 (2015), 433-451.
doi: 10.1007/s12591-014-0229-3. |
[27] |
G. Yin and C. Zhu, Hybrid Switching Diffusions Properties and Applications, New York: Springer-Verlag, 2010.
doi: 10.1007/978-1-4419-1105-6. |
[28] |
S. Zhang, D. Tan and L. Chen, The periodic $n$-species Gilpin-Ayala competition system with impulsive effect, Chaos, Solitons & Fractals, 26 (2005), 507-517.
doi: 10.1016/j.chaos.2005.01.020. |
[29] |
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
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