"A decreases with B"

Without context, theres no way of knowing whats meant. It could be a petri dish and its referring to bacteria colonization that was isolated by a specific difference within it. In which case its not really saying, "as isolation increases, colonization decreases" rather there was a decline in colonization in our isolated group.

This answer is derived from the contributions of John Lawler, in raising the need for context for the question, and Mostafa Taha, in providing it. So I take no credit for this answer.

The context allows us to see that this is ornithological research, setting out to establish the effect on bird populations of area (how much room there is on an island) and isolation (how easy it is to get to and leave).

Here we address both deficiencies to reveal, for island birds, the empirical shape of the general relationships that determine how colonization, extinction and speciation rates co-vary with the area and isolation of islands.

The word co-vary reflect the question whether, for example, the more isolated an island is, (a) the more species become extinct, or (b) the fewer species become extinct.

In case (c), the conclusion with be that they do not co-vary. The degree of isolation does not effect the rate of species extinction at all.

In cases (a) and (b), they do co-vary, in (a) positively, in (b) inversely

The paper concludes:

We found a decline in colonization with isolation, a decline in extinction with area and an increase in speciation with area and isolation

I have to say that the this sentence is not clearly written. A knowledge of language alone will not permit an answer. Only scientific common sense, albeit of an elementary nature allows a definite interpretation. It could mean that the two decline together, or it could involve ellipse and mean that colonisation declines with isolation. In fact, common sense dictates that it must be so. The further away an island is from other islands or the mainland, the harder it is going to be for would-be avian colonists to reach it. So they vary inversely. Similarly, common sense rather than the English language dictates that "with area" must again indicate an inverse co-variance: the greater the area of the island, the more room there is for all the existing species to survive. I suspect, though the English doesn't show it, that the speciation increased in direct (not inverse) proportion with area (more room = less competition/more diversity of living conditions), and with islolation (less threat of immigrant predators).

So I am afraid the editor of Nature missed an important fault of clarity. The lesson is that when writing about co-variance or correlation, it is essential to qualify the items co-vary positively (both go up - or down - together) or inversely (one goes up while the other goes down).