Citation: Ting Kang, Yanyan Du, Ming Ye, Qimin Zhang. Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6702-6719. doi: 10.3934/mbe.2020349
[1] | L. Duan, Q. Lu, Z. Yang, L. Chen, Effects of diffusion on a stage-structured population in a polluted environment, Appl. Math. Comput., 02 (2004), 347-359. |
[2] | A. J. Shaw, Ecological genetics of plant populations in polluted environment. Ecological Genetics and Air Pollution, Springer New York, 1991. |
[3] | G. P. Samanta, A. Maiti, Dynamical model of a single-species system in a polluted environment, J. Appl. Mathe. Comput., 16 (2004), 231-242. |
[4] | T. G. Hallam, C. E. Clark, R. R. Lassiter, Effects of toxicants on populations: a qualitative approach I. Equilibrium environmental exposure, Ecol. Model., 18 (1983), 291-304. |
[5] | T. G. Hallam, C. E. Clar, G. S. Jordan, Effects of toxicant on population: a qualitative approach II. First Order Kinetics, J. Math. Biol., 109 (1983), 411-429. |
[6] | D. Mukherjee, Persistence and global stability of a population in a polluted environment with delay, J. Biol. Syst., 10 (2008), 225-232. |
[7] | Z. Ma, G. Cui, W. Wang, Persistence and extinction of a population in a polluted environment, Math. Biosci., 101 (2004), 75-97. |
[8] | T. G. Hallam, Z. Ma, Persistence in Population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. |
[9] | J. Pan, Z. Jin, Z. Ma, Thresholds of survival for an n-dimensional Volterra mutualistic system in a polluted environment, J. Math. Anal. Appl., 252 (2000), 519-531. |
[10] | Z. Ma, B. J. Song, T. G. Hallam, The threshold of survival for systems in a fluctuating environment, Bull. Math. Biol., 51 (1989), 311-323. |
[11] | M. Liu, K. Wang, Persistence and extinction of a stochastic single-species population model in a polluted environment with impulsive toxicant input, Electron. J. Differ. Equ., 230 (2013), 823-840. |
[12] | X. Yu, S. Yuan, T. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 359-374. |
[13] | F. Wei, S. A. H. Geritz, J. Cai, A stochastic single-species population model with partial pollution tolerance in a polluted environment, Appl. Math. Lett., 63 (2017), 130-136. |
[14] | M. Liu, K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment. J. Theor. Biol., 267 (2010), 283-291. |
[15] | M. Liu, K. Wang, Survival analysis of a stochastic single-species population model with jumps in a polluted environment, Int. J. Biomath., 09 (2016), 1-15. |
[16] | Y. Zhao, S. Yuan, Q. Zhang, The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment, Appl. Math. Model., 40 (2016), 7583-7600. |
[17] | T. C. Gard, Stability for multi-species population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. |
[18] | J. Tong, Z. Zhang, J. Bao, The stationary distribution of the facultative population model with a degenerate noise, Discrete Continuous Dyn. Syst., 83 (2013), 655-664. |
[19] | M. Liu, K. Wang, Y. Wang, Long term behaviors of stochastic single-species growth models in a polluted environment, Appl. Math. Model., 35 (2011), 4438-4448. |
[20] | C. Yuan, X. Mao, Stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching and their convergence, J. Differ. Equ. Appl., 11 (2005), 29-48. |
[21] | G. Yin, X. Mao, K. Yin, Numerical approximation of invariant measures for hybrid diffusion systems, IEEE Trans. Automat. Contr., 50 (2005), 934-946. |
[22] | J. Bao, J. Shao, C. Yuan, Approximation of invariant measures for regime-switching diffusions, Potential Anal., 44 (2016), 707-727. |
[23] | H. Yang, X. Li, Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons, J. Differ. Equ., 265 (2018), 2921-2967. |
[24] | X. Mao, C. Yuan, Stochastic differential equations with Markovian switching, Imperial College Press, 2006. |
[25] | G. Dhariwal, A. Jungel, N. Zamponi, Global martingale solutions for a stochastic population crossdiffusion system, Stoch. Process. their Appl., 129 (2019), 3792-3820. |
[26] | G. Da Prato, J. Zabczyk, Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge, 1996. |
[27] | G. Yin, C. Zhu, Hybrid swithching diffusion: Properties and Applications, Springer, 2010. |
[28] | Y. Zhao, S. Yuan, Q. Zhang, Numerical solution of a fuzzy stochastic single-species age-structure model in a polluted environment, Appl. Math. Comput., 260 (2015), 385-396. |
[29] | W J. Anderson, Continuous-time Markov chains, Springer, Berlin, 1991. |
[30] | J. Tan, A. Rathinasamy, Y. Pei, Convergence of the split-step θ-method for stochastic agedependent population equations with Poisson jumps, Elsevier Science Inc, 254 (2015), 305-317. |