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Asymptotic properties of stochastic population dynamics

Pang, S. and Deng, F. and Mao, X. and , EPSRC (U.K.) (Funder) and , National Natural Science Foundation of China (Funder) and , Key Programs of Science and Technology of Guangzhou (Funder) and , Jinan University of China (Funder) (2008) Asymptotic properties of stochastic population dynamics. Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis, 15 (5a). pp. 603-620. ISSN 1201-3390

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In this paper we stochastically perturb the classical Lotka{Volterra model x_ (t) = diag(x1(t); ; xn(t))[b + Ax(t)] into the stochastic dierential equation dx(t) = diag(x1(t); ; xn(t))[(b + Ax(t))dt + dw(t)]: The main aim is to study the asymptotic properties of the solution. It is known (see e.g. [3, 20]) if the noise is too large then the population may become extinct with probability one. Our main aim here is to nd out what happens if the noise is relatively small. In this paper we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss the limit of the average in time of the sample paths.