1997 Problem A:  The Velociraptor Problem
 
 
The velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists believe that it was a tenacious hunter, and may have hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild as can be done my 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 biologists studying the behavior 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 the initial burst of speed, the animal needed to stop and recover from the buildup of lactic acid in its muscles.

Suppose that velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that it could run at a speed of 50 km/hr for long periods of time.

Part 1
Assuming the velociraptor is a solitary hunter, design a mathematical model that describes the hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15 meters, but may detect the predator at even greater ranges (up to 50 meters) depending on 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 more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the assumptions and limitations given in Part 1.