Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-18T19:50:12.534Z Has data issue: false hasContentIssue false
  • English
  • Français

Population dynamics driven by truncated stable processes with Markovian switching

Part of: Mathematical biology in general Stochastic processes Mathematical economics Markov processes

Published online by Cambridge University Press:  23 June 2021

Zhenzhong Zhang ,
Jinying Tong ,
Qingting Meng  and
You Liang
Zhenzhong Zhang*
Affiliation:
Donghua University
Jinying Tong*
Affiliation:
Donghua University
Qingting Meng*
Affiliation:
Donghua University
You Liang*
Affiliation:
Ryerson University
*
**Email address: zzzhang@dhu.edu.cn
**Email address: zzzhang@dhu.edu.cn
**Email address: zzzhang@dhu.edu.cn
*Postal address: College of Science, Donghua University, 2999 People’s Road, Songjiang, Shanghai, 201620, China.
Get access Rights & Permissions [Opens in a new window]

Abstract

We focus on the population dynamics driven by two classes of truncated $\alpha$-stable processes with Markovian switching. Almost necessary and sufficient conditions for the ergodicity of the proposed models are provided. Also, these results illustrate the impact on ergodicity and extinct conditions as the parameter $\alpha$ tends to 2.

Keywords

Ergodicityα-stable processesMarkovian switchingstationary distributionextinction

MSC classification

Primary: 60G52: Stable processes 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 92B05: General biology and biomathematics 91B70: Stochastic models
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 505 - 522
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arapostathis, A, Biswas, A. and Caffarelli, L. (2016). The Dirichlet problem for stable like operators and related probabilistic representations. Commun. Part. Diff. Eq. 41, 1472–1511.10.1080/03605302.2016.1207084CrossRefGoogle Scholar
Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Bao, J., Mao, X., Yin, G. and Yuan, C. (2011). Competitive Lotka–Volterra population dynamics with jumps. Nonlin. Anal. 74, 6601–6616.CrossRefGoogle Scholar
Bhattacharya, R. and Waynire, E. (2009). Stochastic Processes with Applications. SIAM, Universal City, PA.10.1137/1.9780898718997CrossRefGoogle Scholar
Chen, X., Chen, Z. Q., Tran, K. and Yin, G. (2019). Recurrence and ergodicity for a class of regime-switching jump diffusions. Appl. Math. Opt. 80, 415–445.CrossRefGoogle Scholar
Chen, X., Chen, Z. Q., Tran, K. and Yin, G. (2019). Properties of switching jump diffusions: maximum principles and Harnack inequalities, Bernoulli 25, 1045–1075.CrossRefGoogle Scholar
Carson, M. and Langer, J. S. (1989). Mechanical model of an earthquake fault. Phys. Rev. A. 40, 6470–6484.CrossRefGoogle Scholar
Cederman, L. E. (2003). Modeling the size of wars: from billiard balls to sandpiles. Am. Polit. Sci. Rev. 97, 135–150.10.1017/S0003055403000571CrossRefGoogle Scholar
Davis, D. R. and Weinstein, D. E. (2002). Bones, bombs, and break points: the geography of economic activity. Amer. Econom. Rev. 92, 1269–1289.CrossRefGoogle Scholar
Gutenberg, B. and Richter, C. F. (1944). Frequency of earthquakes in California. Bull. Seismol. Soc. Amer. 34, 185–188.10.1785/BSSA0340040185CrossRefGoogle Scholar
Luo, Q. and Mao, X. (2007). Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334, 69–84.CrossRefGoogle Scholar
Luo, Q. and Mao, X. (2009). Stochastic population dynamics under regime switching II. J. Math. Anal. Appl. 355, 577–593.10.1016/j.jmaa.2009.02.010CrossRefGoogle Scholar
Mao, X. (2011). Stationary distribution of stochastic population systems. Syst. Control Lett. 60, 398–405.CrossRefGoogle Scholar
Nolan, J. (2009). Stable Distribution: Models for Heavy Tailed Data. Birkhauser, Boston.Google Scholar
Richardson, L. (1948). Variation of the frequency of fatal quarrels with magnitude. J. Amer. Statist. Assoc. 43, 523–546.CrossRefGoogle Scholar
Yin, G. and Zhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. Springer, New York.10.1007/978-1-4419-1105-6CrossRefGoogle Scholar
Khasminskii, R. (2011). Stochastic Stability of Differential Equations. Springer, Berlin.Google Scholar
Tong, J., Ma, X. and Zhang, Z. (2018). Ergodicity for population dynamics driven by stable processes with Markovian switching. Commun. Statist. Theory Meth. 48, 2446–2458.10.1080/03610926.2018.1465089CrossRefGoogle Scholar
Zhang, Z., Tong, J. and Hu, L. (2016). Long-term behavior of stochastic interest rate models with Markov switching. Insurance Math. Econom. 70, 320–326.CrossRefGoogle Scholar
Zhang, Z., Zhang, X. and Tong, J. (2017). Exponential ergodicity for population dynamics driven by $\alpha$-stable processes. Statist. Prob. Lett. 125, 149–159.CrossRefGoogle Scholar