In this section we will compare the size-structured metapopulation model with two kinds of models describing single species population dynamics. In continuous time winking patch models, local dynamics is assumed to be fast compared to extinction and re-colonization, and patch occupancy is the resulting sole variable. The other models are those in which it is assumed that migration is fast, and that the whole metapopulation can be regarded as a single population at average population size. Since these kinds of models keep track of total population size or fraction of occupied patches , we will first present our model in terms of these quantities.
Ordinary differential equations for the proportion of occupied patches , disperser density D, and mean population size are obtained using linear chain trickery (LCT) . LCT transforms infinite dimensional systems into a set of ordinary differential equations. Unfortunately, in our model this transformation only results in a finite dimensional representation for the case in which , and v(x) is linear in x. In comparing the different models we think that for the other cases, where no finite dimensional representation can be obtained, this approach remains fruitful. We derive equations for , D, and and if these do not form a closed set of ode's, we add additional equations for the moments around the mean of the probability density distribution , represented by . Note that . In general we consider the mapping
An ode for the change in F(t) can be obtained following Val & Metz ,
where is defined in Eq. (16). Inserting the proper functions for f(x) we get
In the following we will compare equations (29-32) first to the winking patch models and thereafter to models for the total population.