As mentioned in Chapter 10, Richard Levins derived the following rule from his metapopulation model [dp/dt = cp(1 – p) – ep]: if a metapopulation is to persist, the ratio e/c must be less than 1.
To obtain this result, first note that, by definition, when dp/dt = 0, the occupied proportion of patches (p) does not change. As long as that (unchanging) proportion p is greater than 0, the metapopulation will persist indefinitely. This fact motivates us to solve for the value of p that makes dp/dt equal 0.
To do this, we set dp/dt equal to 0 and rearrange Equation 10.2 to find that
ep = cp(1 – p)
which can be simplified to read
p = 1 – e/c
Thus, when the ratio e/c is less than 1, p > 0, which means that some patches are occupied, and hence the metapopulation persists. In contrast, if e/c is greater than or equal to 1, p = 0, and the metapopulation goes extinct (as do all populations in it).
