Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching

Previous Article
Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case
DCDS-B Home
This Issue
Next Article
Traveling waves in an SEIR epidemic model with the variable total population

December 2016, 21(10): 3743-3766. doi: 10.3934/dcdsb.2016119

Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching

Hongfu Yang ^1, , Xiaoyue Li ^2, and George Yin ^3,

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

Received November 2015 Revised June 2016 Published November 2016

Abstract
Related Papers

This work is concerned with permanence and ergodicity of stochastic Gilpin-Ayala models involve continuous states as well as discrete events. A distinct feature is that the Gilpin-Ayala parameter and its corresponding perturbation parameter are allowed to be varying randomly in accordance with a random switching process. Necessary and sufficient conditions of the stochastic permanence and extinction are established, which are much weaker than the previous results. The existence of the unique stationary distribution is also established. Our approach treats much wider class of systems, uses much weaker conditions, and substantially generalizes previous results. It is shown that regime switching can suppress the impermanence. Furthermore, several examples and simulations are given to illustrate our main results.

Keywords: positive recurrence, Markov chain, Stochastic permanence, Gilpin-Ayala model, stationary distribution..

Mathematics Subject Classification: Primary: 60H10; Secondary: 92B0.

Citation: Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119

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.

[1]	Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051
[2]	Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
[3]	Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[4]	Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201
[5]	Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[6]	Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control and Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007
[7]	Kazuhiko Kuraya, Hiroyuki Masuyama, Shoji Kasahara. Load distribution performance of super-node based peer-to-peer communication networks: A nonstationary Markov chain approach. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 593-610. doi: 10.3934/naco.2011.1.593
[8]	Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control and Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012
[9]	Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371
[10]	Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[11]	Ralf Banisch, Carsten Hartmann. Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems". Mathematical Control and Related Fields, 2017, 7 (4) : 623-623. doi: 10.3934/mcrf.2017023
[12]	Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257
[13]	Lin Xu, Rongming Wang. Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate. Journal of Industrial and Management Optimization, 2006, 2 (2) : 165-175. doi: 10.3934/jimo.2006.2.165
[14]	Maedeh Agahgolnezhad Gerdrodbari, Fatemeh Harsej, Mahboubeh Sadeghpour, Mohammad Molani Aghdam. A robust multi-objective model for managing the distribution of perishable products within a green closed-loop supply chain. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021107
[15]	Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004
[16]	Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[17]	Jia Shu, Jie Sun. Designing the distribution network for an integrated supply chain. Journal of Industrial and Management Optimization, 2006, 2 (3) : 339-349. doi: 10.3934/jimo.2006.2.339
[18]	Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005
[19]	Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010
[20]	Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks and Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Hongfu Yang Xiaoyue Li George Yin
on Google Scholar
Hongfu Yang Xiaoyue Li George Yin

[Back to Top]