  The low-energy constants, namely the spin stiffness $\rho_s$, the staggered
magnetization density ${\cal M}_s$ per area, and the spinwave velocity $c$ of
the two-dimensional (2D) spin-1 Heisenberg model on the square and rectangular
lattices are determined using the first principles Monte Carlo method. In
particular, the studied models have antiferromagnetic couplings $J_1$ and $J_2$
in the spatial 1- and 2-directions, respectively. For each considered
$J_2/J_1$, the aspect ratio of the corresponding linear box sizes $L_2/L_1$
used in the simulations is adjusted so that the squares of the two spatial
winding numbers take the same values. In addition, the relevant finite-volume
and -temperature predictions from magnon chiral perturbation theory are
employed in extracting the numerical values of these low-energy constants. Our
results of $\rho_{s1}$ are in quantitative agreement with those obtained by the
series expansion method over a broad range of $J_2/J_1$. This in turn provides
convincing numerical evidence for the quantitative correctness of our approach.
The ${\cal M}_s$ and $c$ presented here for the spatially anisotropic models
are new and can be used as benchmarks for future related studies.
