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A partial differential equation at the metapopulation level.

In this paragraph we will concentrate on measures for which


Note that because of Eq. (1) p(t,x) can be regarded as the (defective) probability distribution of patches over local population density x.

Following Metz & Diekmann [9] we can represent the metapopulation dynamics by the forward Kolmogorov functional partial differential equation


for which we must define a boundary condition at x=0.

When total disasters may occur, i.e. tex2html_wrap_inline2004 , empty patches are formed at a rate


When a total disaster does not result in a loss of habitat quality, the patch will be re-colonized by dispersers instantly. The resulting boundary condition is


which simply states that the flux into habitable patches of zero population size is the same as the rate of total disasters. A result of habitat quality being unaffected by total disasters is that there can be no empty patches. This is in contrast to what was assumed by Gyllenberg & Hanski [4]. In their model, patches hit by a total disaster are set aside in a pool of empty patches ( tex2html_wrap_inline1962 ), instead of contributing to the pool of suitable patches. Colonization of empty patches, at the arbitrarily level x=1, happens due to the law of mass action involving dispersers and empty patches. The rate of change in the fraction of empty patches is then


resulting in the boundary condition


which is dependent on the density of dispersers.

Suppose now that there is a probability u that after a total disaster a patch becomes unsuitable for habitation, and that the time required to regain habitat quality is taken from an exponential distribution with parameter tex2html_wrap_inline2012 , we then get




independent of D.

This is just a simple model describing habitat recovery. More complex models might require the incorporation of an age distribution of empty patches. The resulting model will certainly be different from the model of Gyllenberg & Hanski, except for the case where the size of the instream of dispersers affects the recovery of habitat, i.e. tex2html_wrap_inline2012 is a function of D.

The problem becomes well posed if we specify the initial conditions. For ease of reference we repeat the full model


next up previous
Next: A Stieltjes renewal equation Up: The model Previous: The model

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