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Environmental Brownian noise suppresses explosions in population dynamics. (English) Zbl 1058.60046
It is well known that by adding a noise term to the right hand side of an ordinary differential equation it is often possible to change completely stability properties of solutions. In particular, solutions to a stochastic differential equation may exist globally while solutions to the corresponding deterministic problem blow up in finite time. Such phenomena are studied for stochastic Lotka-Volterra systems. Let $$W$$ be an $$n$$-dimensional Wiener process, and let $$b\in \mathbb R^ {n}$$, $$A\in \mathbb R^ {n}\otimes \mathbb R^ {n}$$ be arbitrary. Let a matrix $$\sigma \in \mathbb R^ {n}\otimes \mathbb R^ {n}$$ be such that $$\sigma _ {ii}>0$$ if $$1\leq i\leq n$$, $$\sigma _ {ij}\geq 0$$ if $$i\neq j$$. The main theorem of the paper states that for any initial condition $$x_ 0\in \mathbb R^ {n}_ {+}$$ there exists a unique nonnegative global solution to the system $dx(t) = \text{diag} (x_ 1(t),\dots ,x_ {n}(t))[(b+Ax(t))\,dt + \sigma x(t) \,dW(t)].$

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 92D25 Population dynamics (general)
##### Keywords:
stochastic differential equations; explosion; boundedness
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