American Institute of Mathematical Sciences

Advanced
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

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 & 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.  doi: 10.1016/j.nonrwa.2005.04.007.  Google Scholar

[2]

F. Chen, Permanence of a delayed non-autonomous Gilpin-Ayala competition model,, Appl. Math. Comput., 179 (2006), 55.  doi: 10.1016/j.amc.2005.11.079.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2005.11.003.  Google Scholar

[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.  doi: 10.1016/j.camwa.2006.12.015.  Google Scholar

[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.  doi: 10.1016/j.cam.2004.02.001.  Google Scholar

[6]

M. Fan and K. Wang, Global periodic solutions of a generalized $n$-species Gilpin-Ayala competition model,, Comput. Math. Appl., 40 (2000), 1141.  doi: 10.1016/S0898-1221(00)00228-5.  Google Scholar

[7]

M. Gilpin and F. Ayala, Global models of growth and competition,, Proc. Natl. Acad. Sci., 70 (1973), 3590.  doi: 10.1073/pnas.70.12.3590.  Google Scholar

[8]

B. Goh and T. Agnew, Stability in Gilpin and Ayala's models of competition,, J. Math. Biol., 4 (1977), 275.  doi: 10.1007/BF00280977.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.03.007.  Google Scholar

[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.  doi: 10.1006/jmaa.1998.6532.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.12.049.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.08.014.  Google Scholar

[13]

R. Khasminskii, Stochastic Stability of Differential Equations,, Berlin: Springer-Verlag, (2012).  doi: 10.1007/978-3-642-23280-0.  Google Scholar

[14]

D. Li, The stationary distribution and ergodicity of a stochastic generalized logistic system,, Stat. Probab. Lett., 83 (2013), 580.  doi: 10.1016/j.spl.2012.11.006.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.10.053.  Google Scholar

[16]

X. Liao and J. Li, Stability in Gilpin-Ayala competition models with diffusion,, Nonlinear Anal., 28 (1997), 1751.  doi: 10.1016/0362-546X(95)00242-N.  Google Scholar

[17]

B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,, J. Math. Anal. Appl., 339 (2008), 419.  doi: 10.1016/j.jmaa.2007.06.058.  Google Scholar

[18]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching,, Math. Comput. Modelling, 54 (2011), 2139.  doi: 10.1016/j.mcm.2011.05.023.  Google Scholar

[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.  doi: 10.1016/j.mcm.2011.08.019.  Google Scholar

[20]

M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay,, J. Nonlinear Sci., 25 (2015), 277.  doi: 10.1007/s00332-014-9229-2.  Google Scholar

[21]

M. Liu and L. Yu, Stability of a stochastic logistic model under regime switching,, Adv. Difference Equ., 2015 (2015).  doi: 10.1186/s13662-015-0666-5.  Google Scholar

[22]

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics,, Stochastic Process. Appl., 97 (2002), 95.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar

[23]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, London: Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[24]

A. Settati and A. Lahrouz, On stochastic Gilpin-Ayala population model with Markovian switching,, Biosystems, 130 (2015), 17.  doi: 10.1016/j.biosystems.2015.01.004.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.09.015.  Google Scholar

[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.  doi: 10.1007/s12591-014-0229-3.  Google Scholar

[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.  Google Scholar

[28]

S. Zhang, D. Tan and L. Chen, The periodic $n$-species Gilpin-Ayala competition system with impulsive effect,, Chaos, 26 (2005), 507.  doi: 10.1016/j.chaos.2005.01.020.  Google Scholar

[29]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems,, SIAM J. Control Optim., 46 (2007), 1155.  doi: 10.1137/060649343.  Google Scholar

show all references

References:
[1]

F. Chen, Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model,, Nonlinear Anal., 7 (2006), 895.  doi: 10.1016/j.nonrwa.2005.04.007.  Google Scholar

[2]

F. Chen, Permanence of a delayed non-autonomous Gilpin-Ayala competition model,, Appl. Math. Comput., 179 (2006), 55.  doi: 10.1016/j.amc.2005.11.079.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2005.11.003.  Google Scholar

[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.  doi: 10.1016/j.camwa.2006.12.015.  Google Scholar

[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.  doi: 10.1016/j.cam.2004.02.001.  Google Scholar

[6]

M. Fan and K. Wang, Global periodic solutions of a generalized $n$-species Gilpin-Ayala competition model,, Comput. Math. Appl., 40 (2000), 1141.  doi: 10.1016/S0898-1221(00)00228-5.  Google Scholar

[7]

M. Gilpin and F. Ayala, Global models of growth and competition,, Proc. Natl. Acad. Sci., 70 (1973), 3590.  doi: 10.1073/pnas.70.12.3590.  Google Scholar

[8]

B. Goh and T. Agnew, Stability in Gilpin and Ayala's models of competition,, J. Math. Biol., 4 (1977), 275.  doi: 10.1007/BF00280977.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.03.007.  Google Scholar

[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.  doi: 10.1006/jmaa.1998.6532.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.12.049.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.08.014.  Google Scholar

[13]

R. Khasminskii, Stochastic Stability of Differential Equations,, Berlin: Springer-Verlag, (2012).  doi: 10.1007/978-3-642-23280-0.  Google Scholar

[14]

D. Li, The stationary distribution and ergodicity of a stochastic generalized logistic system,, Stat. Probab. Lett., 83 (2013), 580.  doi: 10.1016/j.spl.2012.11.006.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.10.053.  Google Scholar

[16]

X. Liao and J. Li, Stability in Gilpin-Ayala competition models with diffusion,, Nonlinear Anal., 28 (1997), 1751.  doi: 10.1016/0362-546X(95)00242-N.  Google Scholar

[17]

B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,, J. Math. Anal. Appl., 339 (2008), 419.  doi: 10.1016/j.jmaa.2007.06.058.  Google Scholar

[18]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching,, Math. Comput. Modelling, 54 (2011), 2139.  doi: 10.1016/j.mcm.2011.05.023.  Google Scholar

[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.  doi: 10.1016/j.mcm.2011.08.019.  Google Scholar

[20]

M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay,, J. Nonlinear Sci., 25 (2015), 277.  doi: 10.1007/s00332-014-9229-2.  Google Scholar

[21]

M. Liu and L. Yu, Stability of a stochastic logistic model under regime switching,, Adv. Difference Equ., 2015 (2015).  doi: 10.1186/s13662-015-0666-5.  Google Scholar

[22]

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics,, Stochastic Process. Appl., 97 (2002), 95.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar

[23]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, London: Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[24]

A. Settati and A. Lahrouz, On stochastic Gilpin-Ayala population model with Markovian switching,, Biosystems, 130 (2015), 17.  doi: 10.1016/j.biosystems.2015.01.004.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.09.015.  Google Scholar

[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.  doi: 10.1007/s12591-014-0229-3.  Google Scholar

[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.  Google Scholar

[28]

S. Zhang, D. Tan and L. Chen, The periodic $n$-species Gilpin-Ayala competition system with impulsive effect,, Chaos, 26 (2005), 507.  doi: 10.1016/j.chaos.2005.01.020.  Google Scholar

[29]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems,, SIAM J. Control Optim., 46 (2007), 1155.  doi: 10.1137/060649343.  Google Scholar

[1]

Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete & 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 & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
[3]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[4]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007
[5]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[6]

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 & Optimization, 2011, 1 (4) : 593-610. doi: 10.3934/naco.2011.1.593
[7]

Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control & Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012
[8]

Ralf Banisch, Carsten Hartmann. Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems". Mathematical Control & Related Fields, 2017, 7 (4) : 623-623. doi: 10.3934/mcrf.2017023
[9]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[10]

Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257
[11]

Lin Xu, Rongming Wang. Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate. Journal of Industrial & Management Optimization, 2006, 2 (2) : 165-175. doi: 10.3934/jimo.2006.2.165
[12]

Jia Shu, Jie Sun. Designing the distribution network for an integrated supply chain. Journal of Industrial & Management Optimization, 2006, 2 (3) : 339-349. doi: 10.3934/jimo.2006.2.339
[13]

Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks & Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437
[14]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[15]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[16]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121
[17]

Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667
[18]

H.Thomas Banks, Shuhua Hu. Nonlinear stochastic Markov processes and modeling uncertainty in populations. Mathematical Biosciences & Engineering, 2012, 9 (1) : 1-25. doi: 10.3934/mbe.2012.9.1
[19]

Felix X.-F. Ye, Yue Wang, Hong Qian. Stochastic dynamics: Markov chains and random transformations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2337-2361. doi: 10.3934/dcdsb.2016050
[20]

Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences