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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 n , A n n be arbitrary. Let a matrix σ n n be such that σ ii >0 if 1in, σ ij 0 if ij. The main theorem of the paper states that for any initial condition x 0 + n there exists a unique nonnegative global solution to the system

dx(t)=diag(x 1 (t),,x n (t))[(b+Ax(t))dt+σx(t)dW(t)]·

60H10Stochastic ordinary differential equations
92D25Population dynamics (general)