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Next: Winking patches Up: A Deterministic Size-Structured Metapopulation Previous: Specific growth dynamics

Model comparison

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 tex2html_wrap_inline2210 or fraction of occupied patches tex2html_wrap_inline2212 , we will first present our model in terms of these quantities.

Ordinary differential equations for the proportion of occupied patches tex2html_wrap_inline2212 , disperser density D, and mean population size tex2html_wrap_inline2210 are obtained using linear chain trickery (LCT) [11]. 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 tex2html_wrap_inline2220 , 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 tex2html_wrap_inline2212 , D, and tex2html_wrap_inline2210 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 tex2html_wrap_inline2232 , represented by tex2html_wrap_inline2234 . Note that tex2html_wrap_inline2236 . In general we consider the mapping

displaymath2208

An ode for the change in F(t) can be obtained following Val & Metz [11],

eqnarray337

where tex2html_wrap_inline2240 is defined in Eq. (16). Inserting the proper functions for f(x) we get

     eqnarray352

In the following we will compare equations (29-32) first to the winking patch models and thereafter to models for the total population.




next up previous
Next: Winking patches Up: A Deterministic Size-Structured Metapopulation Previous: Specific growth dynamics

John Val
Wed Feb 26 07:30:07 EST 1997