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The stationary distribution of competitive Lotka-Volterra population systems with jumps. (English) Zbl 1406.92543

Summary: Dynamics of Lotka-Volterra population with jumps (LVWJ) have recently been established (see [the third author et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6601–6616 (2011; Zbl 1228.93112); the third author and C. Yuan, J. Math. Anal. Appl. 391, No. 2, 363–375 (2012; Zbl 1316.92063)]). They provided some useful criteria on the existence of stationary distribution and some asymptotic properties for LVWJ. However, the uniqueness of stationary distribution for \(n\geq2\) and asymptotic pathwise estimation \(\lim_{t\rightarrow+\infty}(1/t)\int_0^t|X(s)|^pds\) \((p>0)\) are still unknown for LVWJ. One of our aims in this paper is to show the uniqueness of stationary distribution and asymptotic pathwise estimation for LVWJ. Moreover, some characterizations for stationary distribution are provided.

MSC:

92D25 Population dynamics (general)
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References:

[1] Bao, J.; Mao, X.; Yin, G.; Yuan, C., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Analysis: Theory, Methods & Applications, 74, 17, 6601-6616 (2011) · Zbl 1228.93112 · doi:10.1016/j.na.2011.06.043
[2] Bao, J.; Yuan, C., Stochastic population dynamics driven by Lévy noise, Journal of Mathematical Analysis and Applications, 391, 2, 363-375 (2012) · Zbl 1316.92063 · doi:10.1016/j.jmaa.2012.02.043
[3] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus. Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, xxiv+470 (1991), New York, NY, USA: Springer, New York, NY, USA · Zbl 0734.60060 · doi:10.1007/978-1-4612-0949-2
[4] Bhattacharya, R. N.; Waymire, E. C., Stochastic Processes with Applications. Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, xvi+672 (1990), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0744.60032
[5] Tong, J.; Zhang, Z.; Bao, J., The stationary distribution of the facultative population model with a degenerate noise, Statistics & Probability Letters, 83, 2, 655-664 (2013) · Zbl 1402.92370 · doi:10.1016/j.spl.2012.11.003
[6] Has’minskiĭ, R. Z., Stochastic Stability of Differential Equations. Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, xvi+344 (1980), Alphen aan den Rijn, The Netherlands: Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands · Zbl 0441.60060
[7] Mao, X., Stationary distribution of stochastic population systems, Systems & Control Letters, 60, 6, 398-405 (2011) · Zbl 1387.60107 · doi:10.1016/j.sysconle.2011.02.013
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