次线性框架下的随机Lotka-Volterra多种群互惠系统
Stochastic Cooperative Lotka-Volterra Systems under a Sublinear Expectation Framework
摘要:
众所周知，Lotka-Volterra系统描述了某个群落中n(n≥2)个种群的相互作用关系。本文主要讨论由G-布朗运动驱动的随机Lotka-Volterra多种群互惠系统。在次线性期望框架下，我们证明了系统正解的存在唯一性，另外，通过构造合适的Lyapunov函数，我们得到系统存在平稳分布，且具有遍历性。
Abstract:
As we all know, the Lotka-Volterra system describes the interaction relationship between popula-tions in a community. This paper mainly discusses the stochastic cooperative Lotka-Volterra system driven by G-Brownian motion. Under the framework of sub-linear expectations, we prove the existence and uniqueness of the positive solution of the system. In addition, by constructing a suitable Lyapunov function, we obtain that the system has a stable distribution and ergodicity.
参考文献
|
[1]
|
Hofbauer, J. and Sigmund, K. (1998) The Theory of Evolution and Dynamical Systems.Mathematical Aspects of Se-lection. Cambridge University Press, New York.
|
|
[2]
|
Goh, B.S. (1979) Stability in Models of Mutualism. American Naturalist, 113, 261-275. [Google Scholar] [CrossRef]
|
|
[3]
|
Peng, S. (2010) Nonlinear Expectations and Stochastic Calculus under Un-certainty. arXiv:1002.4546 [math.PR]
|
|
[4]
|
Peng, S. (2007) G-Expectation, G-Brownian Motion and Related Sto-chastic Calculus of Types. In: Benth, F.E., Di Nunno, G., Lindstrøm, T., Øksendal, B. and Zhang, T., Eds., Sto-chastic Analysis and Applications. Abel Symposia, Vol. 2. Springer, Berlin, Heidelberg, 541-567. [Google Scholar] [CrossRef]
|
|
[5]
|
Denis, L., Hu, M. and Peng, S. (2001) Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths. Potential Analysis, 34, 139-161. [Google Scholar] [CrossRef]
|
|
[6]
|
Arnold, L. (1972) Stochastic Differential Equations: Theory and Applications. Wiley, New York.
|
|
[7]
|
Friedman, A. (1976) Stochastic Differential Equations and Their Applications. Academic Press, New York. [Google Scholar] [CrossRef]
|
|
[8]
|
Mao, X.R. (1997) Stochastic Differential Equations and Applications. Horwood, New York.
|
|
[9]
|
Khas’minskii, R.Z. (2012) Stochastic Stability of Differential Equations. 2nd Edition, Springer-Verlag, Berlin Heidelberg.
|