We are concerned with a forest kinetic model equipped with the Dirichlet boundary conditions 
which has been presented by Kuzunetsov et al.~\cite{KuAnBiAp}. We construct global solutions
and construct a dynamical system determined from the Cauchy problem of the model equations. 
It is also shown that the dynamical system possesses a bounded absorbing set and every trajectory 
has a nonempty $\omega$-limit set in a suitable weak topology. 
These results are then a modification of those obtained in our previous paper \cite{ChuYa} 
from the Neumann condition case to the Dirichlet condition case.