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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 $b\in \Bbb R\sp {n}$, $A\in \Bbb R\sp {n}\otimes \Bbb R\sp {n}$ be arbitrary. Let a matrix $\sigma \in \Bbb R\sp {n}\otimes \Bbb R\sp {n}$ be such that $\sigma \sb {ii}>0$ if $1\le i\le n$, $\sigma \sb {ij}\ge 0$ if $i\neq j$. The main theorem of the paper states that for any initial condition $x\sb 0\in \Bbb R\sp {n}\sb {+}$ there exists a unique nonnegative global solution to the system $$dx(t) = \text{diag} (x\sb 1(t),\dots ,x\sb {n}(t))[(b+Ax(t))\,dt + \sigma x(t) \,dW(t)].$$

MSC:
60H10Stochastic ordinary differential equations
92D25Population dynamics (general)
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