The model by Levins (1969)

Suppose now that there is no mainland to provide a continuous flow of migrating individuals into the patches and that dispersal events are due to migration between fragments (See Fig. 4B). This case is obviously much more general than the former and this situation is the one taken into account by Levins (1969). Assume that dispersing individuals from each fragment reach a sort of common reservoir from which they are randomly redistributed to any habitat fragment (the propagule rain paradigm previously described).

The probability of colonization per unit time $C(p)$ can be thought of as growing proportionally to the fraction of occupied fragments $p$, because the greater $p$, the greater the amount of dispersing individuals, namely of potential settlers. Note that this assumption is partly arbitrary, because there can be metapopulations in which, although the fraction of occupied fragments is very high, the average number of organisms per fragment is low or vice versa (many empty fragments, but large number of organisms per occupied patch). On the other hand, in boolean models we a priori give up information about the size of the local population and use $p$ as a proxy variable for density, i.e. as an indirect measure of the total abundance of dispersers. Also note that $C(p)=c\cdot p$ vanishes for $p = 0$, as it must indeed be because there is no immigration from outside the metapopulation. The fact that Levins' model does not account for local dynamics does not imply that this cannot be taken into account indirectly. For example, metapopulations with local populations characterized by higher reproductive rates can be described by a larger value of the colonization parameter $c$ .

As for the extinction rate, the simplest assumption is again to set $E(p) = e$ constant. Of course, this is a crude assumption. For instance, one might reasonably wonder whether this parameter might correlate with the dispersal rate (which is actually a component of the colonization parameter c). In fact, dispersal from one patch increases the probability of local extinction, thus implying that in Levins' model the parameters of extinction and colonization might be somehow linked. For the sake of simplicity, we neglect this problem.

We can thus formulate the metapopulation model by Levins as

$$\dot{p}(t)=cp(t)\cdot\left(1-p(t)\right)-e p(t)$$ (6.3)

As remarked by Levins (1969) himself in his original paper, the relationship (33) is mathematically equivalent to the logistic equation (illustrated in chapter ), which describes the dynamics of a single population subject to density dependence. Equation (33) can indeed be rewritten as

$$\dot{p}=(c-e)p\cdot\left[1-\dfrac{p}{1-\frac{e}{c}}\right] \,.$$

Therefore, there exist two equilibria of the system. One is mathematically trivial $\bar{p}_0=0$ and corresponds to metapopulation extinction. The other one is instead $\bar{p}_+=1-\frac {e}{c}$ and is positive only if $e < c$, i.e. only if the probability of colonization per unit time of an empty patch is greater than the probability of local extinction in the same unit time. Note that the necessary constraint $\bar{p}_+\leq 1$ is always respected. Indeed equality would apply only if the extinction probability were zero, which never occurs in a real metapopulation. Because of the similarity with the logistic equation, we can conclude that the equilibrium to which the fraction of occupied patches tends whatever the initial condition $0<p(0)\leq 1$ is given by

$$\bar{p}_+=\left\{ \begin{array}{ccll} 1-\dfrac{e}{c} & \qquad\q... ... & c>e \\ &&&\\ 0 & \qquad\quad & \text{if } & c\leq e \end{array}\right.$$

A good indicator of the rapidity with which the metapopulation tends towards the equilibrium is the absolute value of the eigenvalue $\left\vert c-e\right\vert$ of the linearized system, namely the absolute value of the difference between the colonization and the extinction coefficient. Thus the condition for the metapopulation persistence that is obtained from the model (33) is very simple: a Levins-like metapopulation persists if and only if the colonization rate exceeds the extinction rate.