  We consider continuous space-time decay-surge population models which are
semi- stochastic processes for which deterministically declining populations,
bound to fade away, are rein- vigorated 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 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.
