1997 MCM A： The Velociraptor Problem
The velociraptor, Velociraptor mongoliensis, was a predator dinosaur that lived during the late Cretaceous period approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs or even larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologist studying the behaviors of lions, tigers, and similar predatory animals.
The average adult velociraptor was 3 meters long with a hip height of 0.5 meters and an approximate mass of 45 kg. It is estimated that the animal could run extremely fast, at speeds of 60 km/hr, for about 15 seconds. After that burst of speed, the animal needed to stop and recover from a buildup of lactate acid in its muscles.
Suppose that velociraptor preyed on Thescelosaurus neglectus, a bipedal herbivore approximately the same size as the velociraptor. A biomechanical analysis of fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr almost indefinitely.
Part 1: Assuming the velociraptor is a solitary hunter, design a mathematical model which describes a hunting strategy for a single velociraptor stalking and chasing a single prey as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect velociraptor when it gets within 15 meters, but may detect this predator at even greater ranges (up to 50 meters) depending upon the nature of the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 meters.
Part 2: Assuming the more realistic situtation that the velociraptor hunted in pairs, design a new model which describes a hunting strategy for two velociraptors stalking and chasing a single prey as well as the evasive strategy of the prey. Use the same other assumptions and limitations in Part 1.