The mainland-island model

Assume that the habitat of the metapopulation is a real archipelago of small islands facing the mainland (as in the diagram of Fig. 4A). A landscape of this kind is apt to characterize not only real archipelagos in the geographical sense, but is useful to describe all those situations in which one of the habitat patches has a much larger size than the others, and a resource availability which is sufficient to make the probability of extinction within the mainland virtually vanishing. Consider, for example, the case of the Edith's checkerspot (Euphydryas editha) in Santa Clara County described in Fig. 1, where the large vegetation patch of Morgan Hill effectively acts as a permanent source of individuals for other fragments. As the size of the mainland is incomparably larger than those of the islands, it is reasonable to think that it can host many more individuals of the considered species than each of the small islands. One can imagine, therefore, as a first approximation, that it is possible to neglect, because of that numerical imbalance, any immigration between islands. If the mainland acts as the source of permanent immigration of individuals, the probability of an empty patch colonization $C(p)$ can be considered as independent of $p$, namely a constant equal to $c$. As for the extinction probability, let us suppose that it is also independent of $p$ namely $E(p) = e$, a constant. This second hypothesis is in some sense equivalent to that of density-independence in birth-death processes. Thus the model (31) becomes

$$\dot{p}(t)=c\cdot\left(1-p(t)\right)-e p(t)$$ (6.2)

Figure 4: Outlines of a mainland-island metapopulation on the left and a metapopulation à la Levins (1969) on the right.

The dynamics of patch occupation as predicted by equation 32 is easy to study. Equating the right-hand-side of equation 32 to zero provides only one equilibrium $\bar{p}=\frac {c}{(c+e)}$ which is always feasible, because $0<\bar{p}<1$ for any positive value of parameters $c$ and $e$. This equilibrium is also monotonically increasing with $c$ and is always asymptotically stable, as one can easily prove, for instance graphically. From the conservation viewpoint, we can therefore conclude that the persistence of a mainland-island metapopulation is guaranteed - at least on the theoretical ground - independently of the colonization ability of the considered species and its probability of local extinction.