On the existence of bounded solutions for Lotka-Volterra equations (Q1582274)

The Lotka-Volterra model of two competing species in a random environment, given by the system \[ \dot{x}= x(a(t)-b(t)x-c(t)y), \quad \dot{y}=y(d(t)-e(t)x-f(t)y), \] with \(a,b,c,d,e,f\) random processes with continuous trajectories, is studied. The existence of solutions bounded above and below by two positive random variables is proved if \[ \limsup_{|t|\to\infty}\frac{a(t)}{b(t)} <\liminf_{|t|\to\infty}\frac{d(t)} {e(t)}\text{ a.s. }\limsup_{|t|\to\infty}\frac{d(t)}{f(t)} <\liminf_{|t|\to\infty}\frac{a(t)} {c(t)}\text{ a.s.}; \] if, in addition, \(\liminf_{t\to\infty}\frac{b(t)}{e(t)} >\limsup_{t\to\infty}\frac{c(t)}{f(t)},\) the solution is unique.