Modeling - Math 056 - Homework 1
1. In class, discussed the Fibonacci problem where Rn represented the number of pairs of rabbits in generation n. Let Nn 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:
an = [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.