  We study the probability of stability of a large complex system of size $N$
within the framework of a generalized May model, which assumes a linear
dynamics of each population size $n_i$ (with respect to its equilibrium value):
$ \frac{\mathrm{d}\, n_i}{\mathrm{d}t} = - a_i n_i - \sqrt{T} \sum_{j} J_{ij}
n_j $. The $a_i>0$'s are the intrinsic decay rates, $J_{ij}$ is a real
symmetric $(N\times N)$ Gaussian random matrix and $\sqrt{T}$ measures the
strength of pairwise interaction between different species. Unlike in May's
original homogeneous model, each species has now an intrinsic damping $a_i$
that may differ from one another. As the interaction strength $T$ increases,
the system undergoes a phase transition from a stable phase to an unstable
phase at a critical value $T=T_c$. We reinterpret the probability of stability
in terms of the hitting time of the level $b=0$ of an associated Dyson Brownian
Motion (DBM), starting at the initial position $a_i$ and evolving in `time'
$T$. In the large $N \to \infty$ limit, using this DBM picture, we are able to
completely characterize $T_c$ for arbitrary density $\mu(a)$ of the $a_i$'s.
For a specific flat configuration $a_i = 1 + \sigma \frac{i-1}{N}$, we obtain
an explicit parametric solution for the limiting (as $N\to \infty$) spectral
density for arbitrary $T$ and $\sigma$. For finite but large $N$, we also
compute the large deviation properties of the probability of stability on the
stable side $T < T_c$ using a Coulomb gas representation.
