Homework 2
Due Date: Friday, September 9
Instructions: Feel free to work together with other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with. You can turn in your homework assignment by e-mailing me your solutions.
- Cirsium vulgare (or bull thistle) is a biennial weed that has invaded habitats throughout North America. Each plant produces approximately 300 seeds, with 90% of these germinating the following year, and the remaining 10% germinating after two years.
- Develop a difference equation model for the yearly population of flowering bull thistle (not including rosettes), assuming favorable conditions and no constraint on the spread of the species. Explain your reasoning.
- Regular mowing has been shown to kill approximately 99.9% of thistle plants before they can spread their seeds. Develop a difference eqution model for the yearly population of flowering bull thistle plants in an area of regular mowing. Explain your reasoning.
- An isolated acre of land has a bull thistle population of 500,000 flowering plants, with an equal number of rosettes. Assuming the area is mowed regularly, estimate how long it will take for the bull thistle population to be eradicated completely. Explain your reasoning.
- The following photograph shows the shell of a nautilus. The shell has been cut in half, revealing the chambers and the spiral structure of the shell. (Click on the image for a larger version.)
Estimate the value of \(k\) for the logarithmic spiral (which has the form \(r=Ce^{k\theta}\) for some constants \(C\) and \(k\)) as accurately as you can. Write a short description of the procedure that you used to produce this estimate, and briefly comment on how accurate you expect your estimate to be.