Asymptotic behavior of the solutions of a Kolmogorov forward system (Q796181)

The author discussed the Kolmogorov system: \(y_ i'=-a_{ii}(t)y_ i+\sum_{j\neq i}a_{ij}(t)y_ j\), \(i,j=1,2,.\). where for \(t\geq 0\), \(a_{ij}(t)\geq 0\), \(a_{ii}(t)=\sum_{j\neq i}a_{ji}(t)\), \(\sup_{i}a_{ii}(t)<\infty\). Under the assumptions that \(a_{ij}(t)=\sum^{M}_{k=1}c_ k^{(ij)}b_ k(t)\), \(c_ k^{(ij)}\geq 0\), \(\sup_{i}\sum^{M}_{k=1}c_ k^{(ij)}<\infty\) and \(b_ k(t)\) are locally integrable, the system can be identified with the differential equation \(y'=A(t)y\), where A(t) is a bounded operator in the space of sequences \(\ell_ 1\). Supposing that there is an integer N such that \(a_{ij}(t)\equiv 0\) for \(i>j+N\), and \(\inf_{i>1} \max_{1\leq k\leq N} \underline{\lim}_{t,\tau \to \infty}t^{- 1}\int^{t+\tau}_{\tau}a_{l-k,i}(u)du>0\), the asymptotic stability of the system in the space \(\ell_ 1\) was established.