Modeling - Math 056 - Homework 1

1. In class, discussed the Fibonacci problem where R_n represented the number of pairs of rabbits in generation n. Let N_n be the number of newborn pairs in generation (month) n. These were denoted by open circles on our tree in class.

(a) Find the biological meaning of N₀ if N₀ = 1 and N₁ = 1. (Hint: In the model in class, R₀ refers to the first month the rabbits were placed in the field.)

(b) Find N₂, N₃, N₄, N₅. What do you notice? Write out the model.

(c) Write N₂, N₃, N₄, and N₅ as functions of N₀ and N₁. What do you notice about the coefficients?

2. Run a search for the golden ratio in biology (ex. google). Find three examples where φ shows up in biological phenomena. Choose one of these examples and write a short paragraph explaining the example. Please cite your sources.

3. For the following questions, consider the growth of an aphid population. Use the model developed in class:

a_n = [fr(1 - m)]ⁿa₀.

(a) If the fractional mortality of aphids is 80% and the sex ratio (ratio of females to the total number of aphids) is 50%, what minimum fecundity is required to prevent extinction?

(b) Establish a general condition on the fecundity of aphids to guarantee population growth given a fixed survivorship and a known sex ratio.