Continuous space-time decay surge population models are those semi-stochastic ones for which deterministically declining populations, bound to fade away, are reinvigorated at random times by bursts or surges of random sizes, resulting in a subtle asymptotic balance. Making use of the notion of scale functions, we exhibit conditions under which such processes either explode or are transient at 0 or at infinity, or recurrent. A description of the structures of both the discrete-time embedded chain and extreme record chain of such continuous-time processes are supplied.