At https://www.pbs.org/video/the-serengeti-rules-41dfru/ https://www.pbs.org/video/the-serengeti-rules-41dfru/ is Nature The Serengeti Rules Season 38 Episod

At https://www.pbs.org/video/the-serengeti-rules-41dfru/ https://www.pbs.org/video/the-serengeti-rules-41dfru/ is Nature The Serengeti Rules Season 38 Episode 2 | 53m 14s I has some NICE biology science. The first parts are all about starfish, mussels, sea urchins, kelp, and sea otters and how they do or do not balance. The researchers formulate a theory of keystone species and illustrate that starfish and sea otters can be examples. Then they continue and, in fresh water lakes, count large mouth bass as a keystone species and, then, go to Africa and the Serengeti and count Wildebeest. They propose that keystone species are a general phenomenon. The beginning of the program, the first segment, on starfish in tide pools, shows a clever biology experiment that starts to establish the keystone idea. So, yes, without starfish, some mussels take over. In the kelp forest of the ocean, without sea otters, the urchins take over, eat all the kelp, and the system ends up with only starving urchins. Looking at the video, the urchins do look purple. Sure, for positive integer n and n species, have an n x n matrix A = [a_ij] where a_ij is the propensity or some such of species i to eat species j. Then get rates of predation, start with a distribution of the n species, and then with the matrix A push the distribution forward in time. Then, as just a math problem, might identify a keystone species in terms of the matrix A. Of course, also see if the distribution converges. So, may be looking for eigenvectors with eigenvalue 1?