Historical recovery from mass extinction events
I think the concept of adaptive radiation is what you may be thinking about, since adaptive radiations have been observed following historic mass extinctions.  

When organisms are placed in environments with low diversity (either due to mass extinction or the recent creation of the environment) they can undergo adaptive radiation.  This generally occurs when there are lots of unoccupied niches in the environment - essentially there are lots of potential ways to make a living that are not being exploited by an existing organism.  

In this situation, any organism with a heritable trait that was able to utilize a niche better than other organisms would have a selective advantage and would increase in frequency.  Over time this process can result in new species of organisms.  

There is nothing really different about adaptive radiation and the normal process of evolution and speciation except that in an environment with lots of unoccupied niches, the probability of having a trait that gives you a selective advantage is greater so speciation occurs at a faster rate. 

It is the normal processes of evolution and speciation that give the biome the capacity to increase diversity based on the remaining species.