Beginning with one of the most remarkable ecological collapses of recent time, that of the passenger pigeon, Hadlock goes on to survey collapse processes across the entire spectrum of the natural and man-made world. He takes us through extreme weather events, technological disasters, evolutionary processes, crashing markets and companies, the chaotic nature of Earth's orbit, revolutionary political change, the spread and elimination of disease, and many other fascinating cases.

His key thesis is that one or more of six fundamental dynamics consistently show up across this wide range. These "six sources of collapse" can all be best described and investigated using fundamental mathematical concepts. They include low probability events, group dynamics, evolutionary games, instability, nonlinearity, and network effects, all of which are explained in readily understandable terms. Almost the entirety of the book can be understood by readers with a minimal mathematical background, but even professional mathematicians are likely to get rich insights from the range of examples. The author tells his story with a warmly personal tone and weaves in many of his own experiences, whether from his consulting career of racing around the world trying to head off industrial disasters to his story of watching collapse after collapse in the evolution of an ecosystem on his New Hampshire farm.

Creative teachers could use this book for anything from a liberal arts math course to a senior capstone seminar, and one reviewer suggested that it should be required reading for any mathematics graduate student heading off into a teaching career. This book will also be of interest to readers in the fields under discussion, such as business, engineering, ecology, political science, and others.

conditions. Turning next to the subject of diseases, at least those caused by germs (viruses, bacteria, and similar agents), try to think of these in an evolutionary sense as part of the battle for survival involving one species (us) and another (the germ). This is another way to look at the species collapse or extinction issue, which we discussed earlier. If one species is too strong, it can wipe out the other entirely. But look at what might happen then. It might discover that it has wiped out

chapters. But some products clearly demonstrate an improvement in evolutionary fitness. They push out the competition and change the marketplace by their more successful performance. VCRs and VHS tapes collapsed because DVDs showed technological advantages and reduced costs. Direct online access to high-definition video now threatens much or all of the DVD market. With every such advance there are collapses, both of individual products as well as of the whole industrial ecosystem producing,

allowing random mutations. There are alternative choices for each of these steps so let me just sketch a typical approach to give the idea. First, keep in mind that for any individual game, we will assume as earlier that the individual players can remember the history of their previous interactions. For simplicity, let’s only consider strategies that take into account the three most recent games between the two players. Since there are four play combinations possible for each individual play ✐

previously discussed. In particular, these various strategies defect on the first and sometimes second move, in a way that can be interpreted as testing the extent to which the other player can be exploited, before moving towards a more cooperative strategy in most situations for subsequent rounds of play [13]. If we were to shift to a different fitness function, such as average score against the contemporary strategies in the evolving strategy population, then these kinds of less nice strategies

principle of population growth asserts that the rate of increase of a population is proportional to the size of the population at any moment. That’s really very believable (within certain limits of course). If you have twice as many rabbits in the world, then the rate at which new rabbits are being born is probably about twice as large. If you have half as many fish left in a bay after a heavy fishing harvest, then the rate at which new fish are being born is probably going to be about half as