Even More Bunnies

The original problem that Fibonacci was investigating (in the year 1202) was about how quickly bunnies could breed. In his thought experiment, each pair of child bunnies matures into a pair of adult bunnies in one timestep (producing no offspring), and each pair of adult bunnies produces one pair of child bunnies on each timestep. We'll analyze the original system, and then some variations. Let A be a signal representing the number of pairs of adult bunnies, C be a signal representing the number of pairs of child bunnies, and N be a signal representing the number of pairs of new bunnies added to the bunny colony.

1. The number of pairs of adults in a generation is the sum of the number of pairs of adults from the previous generation, the number of pairs of children from the previous generation, and the current number of new pairs of bunnies:
   A = {\cal R} A + {\cal R} C + N

2. The number of pairs of child bunnies in a generation is the same as the number of adults in the previous generation:
   C = {\cal R} A

3. Combining these, we get Fibonacci's equation in operator form:
   A = {\cal R} A + {\cal R}^2 A + N

Answer the questions below, which involve similar systems.

1) Problem-Solving Pythons

Around the same time as Fibonacci's thought experiment, a lesser-known cousin of Fibonacci, called Fabonucci, started to think about the benefits of using Pythons to solve problems (this was part of the inspiration for 6.01). Fabonucci, concerned about the exploding bunny populations in his cousin's model, introduced several Pythons into the bunnies' home, in an experiment in population control.

Fabonucci tried his experiment with several species of Python:

1. Snakes of the P. minusculus species can only eat child bunnies. They eat a proportion 1-\alpha of each generation of young bunnies before they can mature into adults.

2. Snakes of the P. voluminus species only bother eating adult bunnies. They eat a proportion 1-\alpha of each generation of adult bunnies, but only after they've had a chance to procreate.

3. Snakes of the P. monstrous species only bother eating adult bunnies. They eat a proportion 1-\alpha of each generation of adult bunnies, before they have a chance to procreate.

4. Snakes of the P. eatsalottus species eat a proportion 1-\alpha of both child and adult bunnies, and get to the adults before they can procreate.

Assume that none of these snakes prey on the newly-added bunnies (represented by the signal N) during the timestep in which those new bunnies are introduced.

Fabonucci also made several models of population growth, in which 0 \lt \alpha \lt 1:

• Model A: ~\displaystyle{\frac{A}{N} = \frac{1}{1 - {\cal R} - {\cal R}^2}}

• Model B: ~\displaystyle{\frac{A}{N} = \frac{1}{1 - \alpha {\cal R} - \alpha^2 {\cal R}^2}}

• Model C: ~\displaystyle{\frac{A}{N} = \frac{1}{1 - \alpha {\cal R} - {\cal R}^2}}

• Model D: ~\displaystyle{\frac{A}{N} = \frac{1}{1 - \alpha {\cal R} - \alpha {\cal R}^2}}

• Model E: ~\displaystyle{\frac{A}{N} = \frac{1}{1 - {\cal R} - \alpha {\cal R}^2}}

Which population model corresponds to which species of Python? For each species of Python below, choose the letter corresponding to the appropriate model for that species, or None if none of the models is appropriate.

Which population model corresponds to Python species 1?  -- A B C D E None

Which population model corresponds to Python species 2?  -- A B C D E None

Which population model corresponds to Python species 3?  -- A B C D E None

Which population model corresponds to Python species 4?  -- A B C D E None