Animals must survive predation in addition to maximizing their rate of food intake. One theory assumes that they try to maximize the ratio of food collected to predation risk. Suppose that different flowers with nectar of quality n (the rate of food collection) attract P(n) predators. For example, flowers with higher-quality nectar (large values of n) might attract more predators (large value of P(n)). Bees must decide which flowers to select. For each of the following forms of P(n), find the function the bees are trying to maximize, and find the optimal n.
1. Suppose that P(n) = 1 + n2. Find the optimal n for the bees.
2. Suppose that P(n) = 1 + n. Find the optimal n for the bees and draw a graph like that for the Marginal Value Theorem. Does this make sense? Why is the result so different?
3. Find the condition for the maximum for a general function P(n) by solving for P'(n). Use this condition to find the optimal n for the cases P(n) = 1 + n2 and P(n) = 1 + n.
4. Find a graphical interpretation of the condition in the previous problem and test it on P(n) = 1 + n2 and P(n) = 1 + n.