On decay–surge population models

Abstract

We consider continuous space–time decay–surge population models, which are semi-stochastic processes for which deterministically declining populations, bound to fade away, are reinvigorated at random times by bursts or surges of random sizes. In a particular separable framework (in a sense made precise below) we provide explicit formulae for the scale (or harmonic) function and the speed measure of the process. The behavior of the scale function at infinity allows us to formulate conditions under which such processes either explode or are transient at infinity, or Harris recurrent. A description of the structures of both the discrete-time embedded chain and extreme record chain of such continuous-time processes is supplied.

An Interacting Neuronal Network with Inhibition: theoretical analysis and perfect simulation

“…where M i is a random Poisson measure with intensity dtdrF i (dy) and for all i, the M i are all independent. This model extends that of Goncalves et al [12] in the multidimensional case.…”