Spatially implicit boolean models

Assume that the metapopulation habitat consists of an infinite number of identical fragments, as shown in Fig. 3A. Imagine also that the fragments are arranged in such a way as to make it possible for individuals that disperse from each habitat patch to reach and colonize any other fragment with the same probability and with no time delay due to displacement. This is of course such an abstract idealization of the actual process that it is difficult to find examples of real cases. It is not totally unrealistic, however, especially in cases where the average dispersal distance is relatively large compared to the size of the fragments and the movement takes a time which is much shorter than the average lifetime of the organisms (think for example of the dispersal of light seeds carried by wind). Indicate with $p(t)$ the fraction of patches occupied by the species being studied. This proportion - in the case of infinite fragments - can be identified with the probability that any patch is occupied at time $t$. Quoting from Gotelli (2008), we can say that this single variable is as informative of the state of the metapopulation as the proportion of vacant places in a large parking lot: we content ourselves with knowing how many places are still vacant no matter where they are located. The equation that describes the time change of the variable $p(t)$ is a balance between the fluxes of extinction and occupation of empty fragments:

$$\dot{p}(t)=C'(p(t))-E'(p(t))=C(p(t))\cdot\left(1-p(t)\right)-E(p(t))p(t)$$ (6.1)

where the function $C'(p)$ is the colonization rate (i.e. the fraction of fragments successfully colonized in the unit time) while $E'(t)$ is the rate of extinction (i.e. the proportion of fragments in which the local population becomes extinct in the unit time). The careful reader will recognize a modelling approach which is quite of the same kind as the budget proposed by MacArthur and Wilson (1967) to analyse the variation of the number of species in the context of island biogeography, treated in chapter .

The function $C'(p)$ can be described in general terms as the product of the probability of patch colonization per unit time $C(p(t))$ times the fraction of empty fragments $(1-p(t))$, namely those that can be colonized. Similarly, $E'(p)$ is the product of the probability of extinction per unit time $E(p(t))$ of a local population times the fraction of occupied fragments $p(t)$, namely the only fragments where a local population may become extinct. The forms of dependence on the fraction of occupied fragments $p$ of both the colonization probability ($C$) and the local extinction probability ($E$) can differ depending on the type of metapopulation being studied. We now proceed to analysing in more detail two particularly noteworthy cases.