Stability and steady state of complex cooperative systems: a diakoptic approach

Cooperative dynamics are common in ecology, population dynamics and in generalised compartment models. However, their commonly high degree of complexity with a large number of coupled degrees of freedom makes them difficult to analyse. Here we present a graphical criterion, via a diakoptic approach (`divide-and-conquer') to determine a cooperative system's stability by decomposing the system's interaction graph into its strongly connected components (SCCs). In particular, we show that a linear cooperative system is Lyapunov stable if the SCCs of the associated dependence graph all have non-positive dominant eigenvalues, and if any two SCCs which have dominant eigenvalue zero are not connected by any path.