American Institute of Mathematical Sciences

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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), 895-915. 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), 55-66. 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), 1205-1222. 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), 1214-1227. 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), 399-422. doi: 10.1016/j.cam.2004.02.001.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  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), 1537-1551. 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), 516-539. 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), 582-595. 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), 588-597. doi: 10.1016/j.jmaa.2007.08.014.  Google Scholar

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

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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.  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), 11-28. 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), 1751-1758. 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), 419-428. 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), 2139-2154. 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), 405-418. 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), 277-289. 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), 9 pp. 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), 95-110. 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), 17-27. 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), 2675-2685. 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), 433-451. 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, Solitons & Fractals, 26 (2005), 507-517. 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), 1155-1179. 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), 895-915. 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), 55-66. 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), 1205-1222. 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), 1214-1227. 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), 399-422. 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), 1141-1151. 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., USA 70 (1973), 3590-3593. 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), 275-279. 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), 1537-1551. 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), 516-539. 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), 582-595. 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), 588-597. 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), 580-583. 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), 11-28. 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), 1751-1758. 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), 419-428. 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), 2139-2154. 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), 405-418. 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), 277-289. 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), 9 pp. 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), 95-110. 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), 17-27. 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), 2675-2685. 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), 433-451. 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, Solitons & Fractals, 26 (2005), 507-517. 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), 1155-1179. doi: 10.1137/060649343.  Google Scholar

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