The class lecture notes examined the idea of optimal foraging by Northwestern crows (Corvus caurinus) on whelks (Thais lamellosa). A similar behavior of dropping butter clams (Saxidomus gigantea) for food is observed with the Glaucous-winged gull (Larus glaucescens) on the San Juan Islands [1]. As with the crows, the seagulls select clams from the intertidal zone, then fly back to an area with rocks, cement, or wood before dropping the clams to access the soft body parts. (Some gulls break the clams on the rocks while standing, but this foraging behavior is not explored in this problem.) The clams break open more readily than whelks from the data reported in the article [1]. In this question, you will reproduce some of the results in the notes to determine the optimal foraging strategy for Glaucous gulls.

a. Below is a table that simulates an experiment of dropping clams from different heights to determine how many drops are required to break open the clam. (The table was created on an interpretation of numbers cited in the article and does not reflect an actual experiment.) The average number of drops, N(H), to open a clam as a function of the drop height, H, in meters is shown below
 

H(m)
0.5
1
1.5
2
3
4
5
6
10
15
N(H)
15.2
3.7
2.5
2.1
1.6
1.5
1.4
1.3
1.2
1.1

As in the lecture notes, the proposed function to fit these data has the form

Use Excel's solver to find the best fit to this data. (See the Lab Help for more details on this procedure.) That is, you must find the values of a and b in the formula above that finds the least squares best fit to the data. Write this function in your lab report and show the graph of both the data and this function. Give the value ofthe sum of squares error.

b. The energy expended by a gull to open a clam is dependent on how high the bird flies times the number of drops that it takes to open the clam. Thus, the energy function has the form

Use your values of a and b, and k = 1 to graph the energy function between 0.5 and 5 m. Give the location of any vertical or horizontal asymptotes. Find the derivative of this function and write the formula in your report. Determine the height which produces the minimum expenditure of energy.

c. The authors of the gull study [1] noted that the gulls usually flew about 4-7 m before dropping the clams, which you will notice is higher than the value you obtained in Part b. Thus, for some reason, the gulls choose to spend more energy flying higher to guarantee that the clams are more likely to break open on the first drop. Write a brief explanation for why the birds might choose this strategy. Note that gulls are notorious for mobbing behavior around sources of food, as you have probably seen at our local beaches. This biological idea is called kleptoparasitism, which means that the gulls often steal food from other gulls rather than searching for the food themselves. You should comment on how kleptoparasitism might affect the energy function that should be used in determining calories gained from a broken clam. (The gull only benefits from the energy spent dropping the clam if it eats the clam.) You should also reason why the gulls might not go even higher to better guarantee that the clam dropped opens.

[1] D. P. Barash, P. Donovan, and R. Myrick, Clam dropping of the Glaucous-winged gull (Larus glaucescens) (1975), Wilson Bull. (87) p.60-64.