Problem Set 3

Problem Set 3 - MS&E201 Dynamic Systems Spring 2008...

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MS&E201 Dynamic Systems Spring 2008 Professor Edison Tse Problem Set 3 (not to be turned in) 1. (Optimal Control) 2. (Catastrophe Theory)
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3. (Catastrophe Theory) The Size of Bee Societies There are about 20,000 species of bee. Of these, 5% are solitary and they do not live in cooperational colonies. The remainders are social species. The social bees fall into three groups: the bumblebees (Bombinae), tropical stingless bees (Melponinae), and honeybees (Apinae). Consider the environment producing sufficient nectar to maintain a population density B of bees/km 2 . A colony of N bees will be supported by in equilibrium by a territory of area N/B, of radius R km. Roughly (since territories may overlap), we have B R N 2 π = . Notice that trivially, this implies 1 N ) /( 1 min B R R = . The average productivity of a bee is given by the following formula R B A C R P = ) ( where A and C are constants depending on the bees’ collection technique, velocity, load capacity, and so on. It is a basic ecological principle that less efficient species among those
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This note was uploaded on 06/16/2010 for the course MS&E 201 taught by Professor Edisontse during the Spring '08 term at Stanford.

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Problem Set 3 - MS&E201 Dynamic Systems Spring 2008...

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