In order to arrive at a Stieltjes renewal equation we will follow the lines set out in Diekmann & Metz (). On the patch level the ingredients are again local population growth, disasters and survival up to the next disaster. We consider the following quantities:
Local population size at time t in a patch, which had a local population size at , and experienced a disperser density on the interval . Note that this quantity is a solution of Eq. (2) provided with the proper initial conditions.
The probability that a patch, which had a local population size at , has not been hit by a disaster at time t.
= The probability that in the time interval a patch is hit by a disaster, and after the disaster the patch is suitable for habitation and has survivors.
= The probability that in the time interval a patch is hit by a disaster, and that after the disaster the patch has become unsuitable for habitation.
On the metapopulation level we gather the dynamics of the local populations in the cumulative operator for patch restart (i.e. the birth operator in ), which is defined by
Mass conservation (1) can be rewritten in terms of the birth operator as
Likewise the change in disperser density (11) can be rewritten as
And so our problem is fully posed in terms of the birth operator B and the disperser density D.