  We study an isotropic Heisenberg spin-1/2 model on a trellis ladder which is
composed of two $J_1-J_2$ zigzag ladders interacting through anti-ferromagnetic
rung couplings $J_3$. The $J_1$ and $J_2$ are ferromagnetic zigzag spin
interaction between two legs and anti-ferromagnetic interaction along each leg
of a zigzag ladder. A quantum phase diagram of this model is constructed using
the density matrix renormalization group (DMRG) method and linearized spin wave
analysis. In small $J_2$ limit a short range stripe collinear phase is found in
the presence of $J_3$, whereas, in the large $J_2/J_3$ limit non-collinear
quasi-long range phase is found. The system shows a short range non-collinear
state in large $J_3$ limit. The short range order phase is the dominant feature
of this phase diagram. We also show that the results obtained by DMRG and
linearized spin wave analysis show similar phase boundary between stripe
collinear and non-collinear short range phases, and the collinear phase region
shrinks with increasing $J_3$. We apply this model to understand the magnetic
properties of CaV$_2$O$_5$ and also fit the experimental data of susceptibility
and magnetization. The variation of magnetic specific heat capacity as function
of external magnetic field is also predicted. We note that $J_3$ is a dominant
interaction in this system, whereas $J_1$ and $J_2$ are approximately half of
$J_3$.
