On the nonautonomous Lotka-Volterra equation (Q1966088)

Consider the nonautonomous Lotka-Volterra equations, \(n\geq 3\), \[ {du_k \over dt}= u_k\left[r(t)- \sum^{n-1}_{j=0} c_j(t)u_{k+j} \right],\quad k=1, \dots,n; \quad -\infty< t<+\infty, \tag{*} \] where the indices are counted cyclically modulo \(n,r\) and \(c_j\), \(j=0, \dots,n-1\), are continuous bounded functions, and \(u_k\geq 0\), \(k=1, \dots,n\). Let \(\gamma_k (t):=\sum^{n-1}_{j=0} c_j(t) \exp(2\pi ijk/n)\), \(i=\sqrt{-1}\). Suppose \(\inf_{-\infty<t <+\infty} r(t) >0\), \(\nu_k: =\inf_{-\infty <t<+ \infty}\{\text{Re} \gamma_k (t)\}>0\), \(k=0, 2,3, \dots, n-2\), \(\nu_1:= \inf_{-\infty< t<+\infty} \gamma_1(t)\geq 0\) and the extra assumption: \(\gamma_1(t)\neq 0\) for \(-\infty< t<+\infty\) when \(n\neq 4\) and \(\nu_1>0\) when \(n=4\). The authors prove that (*) has a unique solution defined for \(-\infty <t<+ \infty\) with components bounded above and below by positive constants. Furthermore, that solution is globally attractive (for any initial condition with positive components). Also, if \(r\) and \(c_j\), \(j=0, \dots, n-1\), are \(T\)-periodic [almost-periodic], so is that solution.