考虑一类随机泊松系统，这类系统来自于对一类Lotka-Volterra系统进行Stratonovich型白噪声扰动。给出该系统的解几乎处处存在(全局不爆发)且唯一的充分条件，并进一步证明在这个条件下，解是几乎处处正的和有界的。数值实验对结论进行了验证。
In this paper, a class of stochastic Poisson systems, arising from randomly perturbing a type of Lotka-Volterra systems by certain Stratonovich white noise, are considered. We give the sufficient conditions for the almost sure existence (global non-explosion) and uniqueness of the solution of the system, and further prove that the solution is positive and bounded almost surely under the proposed conditions. Numeraical experiments are performed to verify the results.
[1] Hairer E, Lubich C, Wanner G. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations[M]. 2nd ed. Berlin: Springer Berlin Heidelberg, 2002: 256-257.
[2] Plank M. Hamiltonian structures for the n-dimensional Lotka-Volterra equations[J]. Journal of Mathematical Physics, 1995, 36(7): 3520-3534.DOI:10.1063/1.530978.
[3] Arnold L, Horsthemke W, Stucki J W. The influence of external real and white noise on the Lotka-Volterra model[J]. Biometrical Journal, 1979, 21(5): 451-471.DOI:10.1002/bimj.4710210507.
[4] Mao X R, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics[J]. Stochastic Processes and Their Applications, 2002, 97(1): 95-110.DOI:10.1016/S0304-4149(01)00126-0.
[5] Mao X R, Sabanis S, Renshaw E. Asymptotic behavior of the stochastic Lotka-Volterra model[J]. Journal of Mathematical Analysis and Applications, 2003, 287(1): 141-156.DOI:10.1016/S0022-247X(03)00539-0.
[6] Rudnicki R, Pichór K. Influence of stochastic perturbation on prey-predator systems[J]. Mathematical Biosciences, 2007, 206(1): 108-119.DOI：10.1016/j.mbs.2006.03.006
[7] Khasminskii R Z, Klebaner F C. Long term behavior of solutions of the Lotka-Volterra system under small random perturbations[J]. The Annals of Applied Probability, 2001, 11(3): 952-963.DOI:10.1214/aoap/1015345354.
[8] Cohen D, Dujardin G. Energy-preserving integrators for stochastic Poisson systems[J]. Communications in Mathematical Sciences, 2014, 12(8): 1523-1539.DOI:10.4310/cms.2014.v12.n8.a7.
[9] Hong J L, Ruan J L, Sun L Y, et al. Structure-preserving numerical methods for stochastic Poisson systems[J]. Communications in Computational Physics, 2021, 29(3):802-830.DOI:10.4208/cicp.oa-2019-0084.
[10] Li X Y, Ma Q, Ding X H. High-order energy-preserving methods for stochastic Poisson systems[J]. East Asian Journal on Applied Mathematics, 2019, 9(3): 465-484.DOI:10.4208/eajam.290518.310718.
[11] Wang P J, Wang L J. Stochastic Poisson integrators based on Padé approximations for linear stochastic Poisson systems[J]. Journal of University of Chinese Academy of Sciences, 2021,38(2):160-170.DOI:10.7523/j.issn.2095-6134.2021.02.002.
[12] Arnold L. Stochastic differential equations: theory and applications[M]. New York: Wiley, 1974: 112-113.
[13] Mao X R. Stochastic differential equations and applications[M]. 2nd ed. Cambridge: Woodhead Publishing, 2007: 56-58.
