We have already seen a simpler model for metapopulations, in which the probability colonization for each patch is always the same, due to a constant propagule rain coming from a core area. We have also seen a slightly more complex model, in which the colonization probability varies according to the number of occupied patches - and with this we don't need to assume a constant propagule rain. In the second model, the colonization was internal to the studied area, meaning that the migration of individuals happens between the patches.
Now, you should be asking yourselves: does it make any sense that the extinction probability remains constant? The answer to this is 'no'. Whenever the number of occupied patches increases, the migration to empty patches increases, but the migration to already occupied patches is also increased. This means that the income of propagules from other patches in the landscape may prevent the local extinction! Imagine a forest fragment in which individuals from a plant species germinate and grow, but are unable to reproduce due to lack of a pollinator. After some time, this population will become extinct in this patch - but if there is a constant arrival of seeds from other patches, the species will subsist. This is called rescue effect.
Let's get to work! What do we need to change in our basic model to incorporate the effect of rescue? If the arrival of seeds from other patches is reducing the probability of local extinction, then a smaller fraction of occupied patches will lead to a greater chance of extinction:
$$p_e=e(1-f)$$
here, $e$ is a measure of how much the local extinction probability increases as the fraction of occupied patches $f$ decreases.
So, our model now has the following formula:
$$ \frac{df}{dt}=p_i * (1-f) - ef (1-f)$$
And the fraction of occupied patches in equilibrium ($F$) becomes:
$$F=\frac{p_i}{e} $$
Moreover, in equilibrium:
$$p_e=e-p_i$$