## 4. Marginal Value Theorem

In the last section, we determined that the amount of time a bee should spend at any given flower is given by maximizing the average rate per visit , where is the total amount of food collected at time and is the flying time between flowers.

We found that when and , the bee should remain at the flower for minutes. In this section, we will consider a general and an unknown .

In general, we have .

Use the quotient rule to differentiate with respect to . Your result will be in terms of and .

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Show that the critical points of occur at times that satisfy .

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We set . Thus,

Since , we know that the critical points for occur exactly when .

This suggests that has a local maximum or local minimum (we do not know which) at a time that satisfies .

In fact, whenever is concave (concave down), there will be a local maximum. (As a challenge, you could try and prove this. Hint: If is concave, then . Consider evaluating at the critical point from above).

In other words, the maximum rate of food collection occurs precisely when the instantaneous rate of food collection is equal to the average rate of food collection .

This means that the bee should leave the flower exactly when the instantaneous rate of food collection is equal to the average rate of food collection. The bee should leave one flower and head to the next when it becomes advantageous.

This result is known as the Marginal Value Theorem. In general, it states that the rate of benefit of a given food source is maximized by exploiting each food source until the rate of benefit falls to the maximum average rate that can be sustained over a long period.

We can interpret the Marginal Value Theorem graphically as follows:

The slope of the red line is . But since the red line is also tangent to the curve at the point , we know that it has slope . Thus, the instantaneous rate of food collection is equal to . Once again, to maximize the average rate of food collection, the bee should leave the flower when the instantaneous rate of food collection is equal to the average rate of food collection.

To help understand the Marginal Value Theorem, you can interact with the simulator below. In particular, you can use the slider to adjust the value of .

What occurs if the travel time becomes longer?

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If the travel time becomes longer, then the optimal time to remain at the flower becomes longer. This is because the point moves to the left as increases, and the point of tangency moves to the right.