A mean-field forest fire model exhibiting self-organized criticality

We consider a modification of the dynamical Erdos-Renyi random graph model. Beside creation of edges of the graph with uniform rate, `lightnings' hit the sites according to an independent Poisson flow. When lightning hits a site its cluster becomes totally disconnected. Depending on the rate of lightning different asymptotic behaviour of the system develops. In the most interesting regime the system is dynamically driven to a permanent critical state (self-organized criticality).