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Zbl 1062.92055
Mao, Xuerong; Yuan, Chenggui; Zou, Jiezhong
Stochastic differential delay equations of population dynamics.
(English)
[J] J. Math. Anal. Appl. 304, No. 1, 296-320 (2005). ISSN 0022-247X

The authors start with the deterministic $n$-dimensional delay Lotka-Volterra equation $$dx(t)/dt= \text {diag}(x_1(t),\ldots,x_n(t))\big[b+ Ax(t)+ Bx(t-\tau)\big],\tag1$$ where $x$ and $b$ are $n$-dimensional vectors and $A$ and $B$ are $n\times n$ matrices. Equation (1) can be seen as a basic model for the dynamical behaviour of a population of $n$ interacting species. The authors assume that the vector $b$, which represents the intrinsic growth rates of the $n$ species, is subject to noise. This gives rise to a stochastic delay Lotka-Volterra system with multiplicative noise. The drift and diffusion coefficients of this stochastic differential system are locally Lipschitz-continuous but do not satisfy a linear growth condition. In standard arguments the latter conditions ensures that a solution does not blow-up in finite time. Thus, the authors first consider several conditions that guarantee the global existence of a unique solution, which, in addition, stays positive almost surely. Further, several asymptotic properties of the solutions are discussed. In particular, conditions for persistence with probability 1, asymptotic stability with probability 1 and stochastic ultimate boundedness are given. In the last section, an example of a $3$-dimensional stochastic Lotka-Volterra food chain is considered and, as an illustration, specific conditions for asymptotic stability with probability 1 are given.
[Evelyn Buckwar (Berlin)]
MSC 2000:
*92D25 Population dynamics
34K50 Stochastic delay equations
60K99 Special processes
34K60 Applications of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
60H20 Stochastic integral equations
93D99 Stability of control systems
92D40 Ecology

Keywords: stochastic delay differential equation; multiplicative noise; Brownian motion; stochastic Lotka-Volterra system; stochastic population dynamics; persistence; asymptotic stability; ultimate boundedness

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