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

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

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.
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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, 26 (2005), 507. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0898-1221(00)00228-5.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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, 26 (2005), 507. 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. doi: 10.1137/060649343.

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