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 \[ dx(t) = \text{diag} (x_ 1(t),\dots ,x_ {n}(t))[(b+Ax(t))\,dt + \sigma x(t) \,dW(t)]. \]


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
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