In order to arrive at a Stieltjes renewal equation we will follow the lines set out in Diekmann & Metz ([1]). 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 [1]), 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.
