Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]