Stability of large complex systems with heterogeneous relaxation dynamics

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