TY - JOUR
T1 - Population dynamical behavior of Lotka-Volterra system under regime switching
AU - Li, Xiaoyue
AU - Jiang, Daqing
AU - Mao, Xuerong
AU - National Natural Science Foundation of China (Funder)
AU - Royal Society of Edinburgh (Funder)
PY - 2009
Y1 - 2009
N2 - In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.
AB - In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.
KW - brownian motion
KW - stochastic differential equation
KW - generalized It^o's formula
KW - markov chain
KW - stochastic permanence.
UR - http://www.sciencedirect.com/science/journal/03770427
U2 - 10.1016/j.cam.2009.06.021
DO - 10.1016/j.cam.2009.06.021
M3 - Article
SN - 0377-0427
VL - 232
SP - 427
EP - 448
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 2
ER -