We continue a study of the forest ecosystem model due to Kuzunetsov et al.~\cite{KuAnBiAp} 
in which the Dirichlet conditions are imposed.
In this paper, we introduce three kinds of $\omega$-limit sets, namely, 
$\omega(U_0) \subset L^2\text{-}\omega(U_0) \subset \text{w}^*\text{-}\omega(U_0)$, 
for each point $U_0$ of the dynamical system which has been constructed in our preceding paper \cite{ShiChuYa}. 
Using a Lyapunov function, we will then investigate basic properties of the these $\omega$-limit sets. 
Especially, it shall be shown that $L^2\text{-}\omega(U_0)$ consists of equilibria alone. 
These results are then a modification of those obtained in \cite{ChuTsuYa} 
from the Neumann condition case to the Dirichlet condition case.