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Diana Davis
16 March 2006
Berries and Exercise: Simply Complex
Suppose that a bird lives on a mountain with a large berry bush on one side and a small berry bush on the other. The bird must decide how much time it will spend at each of the bushes to maximize the number of berries it can eat in a day. If the big bush produces an average of four ripe berries each hour, and the small bush averages two ripe berries per hour, and the bird makes 30 berry-bush visits per hour, the average number of berries per visit that the bird would get is shown in Figure 1.
It is clear from the situation that if the bird visits the big bush some number n of times per hour, it will visit the small bush 30 – n times per hour. In Figure 1, the x-axis shows the number of times per hour the bird visits the big bush, and the number of visits to the small bush would be the difference between 30 and the numbers on the x-axis. The y-axis shows the number of berries the bird averages per visit, per hour. The blue diamonds represent the average number of berries per visit to the big bush, and the pink squares represent the average number of berries per visit to the small bush. For instance, if the bird visits the big bush only once within the hour, it will be able to eat an average of four berries. If it visits the small bush only once, it can eat two berries. These two events are represented by the highest blue diamond and the highest pink square in the figure, respectively.
It is natural in this situation to wonder what the bird will do: How much of its time will it allot to the big bush, and how much to the small bush? We can answer this question by examining Figure 1 in greater depth. Suppose first that the bird allots 90% of its time to the big bush, thereby visiting the big bush 27 times and the small bush three times within an hour. This situation is marked “1” just below the x-axis. In this case, the small bush is yielding 2 berries / 3 visits = 0.67 berries / visit, and the big bush yields 4 berries / 27 visits = 0.15 berries / visit. The bird will realize that the small bush yields more berries per visit, and will visit the small bush more, thereby moving left along the x-axis.
Suppose next that the bird visits both buses equally, visiting each 15 times. This situation is marked “2” just below the x-axis. Then the small bush will yield 2 berries / 15 visits = 0.13 berries / visit, and the big bush will yield 4 berries / 15 visits = 0.27 berries / visit. In this case, visiting the big bush provides more berries per visit, so the bird will visit it more, moving right along the x-axis. Finally, suppose that the bird spends 90% of its time on the small bush, visiting the small bush 27 times and the big bush three times. This situation is marked “3.” Then the small bush will yield 2 berries / 27 visits = 0.07 berries / visit, and the large bush will yield 4 berries / 3 visits = 1.33 berries / visit, so the big bush is more productive per visit, so the bird will visit it more, moving right along the x-axis.
All of this analysis, while informative, has not answered our main question, which is what the bird's stable equilibrium choice of time distribution between the two bushes will be. The stable point will occur when each bush yields the same number of berries per visit, so that the bird has no reason to increase its visits to either one. From the graph, we can see that this occurs at 20 visits to the big bush (and thus 10 visits to the small bush). Algebraically, this point occurs when
4 (berries from the big bush) = 2 (berries from the small bush)
n (visits to the big bush) 30 – n (visits to the small bush)
Cross-multiplying, we see that this occurs when 4*(30 – n) = 2n, which is easily simplified to n = 20.
This is a stable point because if the bird happens to spend more time at the big bush, moving to the right in Figure 1, the small bush will be more profitable per visit, and the bird will move back towards the left. Similarly, if the bird spends a bit more time at the small bush, the big bush will become more profitable per visit, and the bird will move back towards the right. Thus, the long-term distribution of the bird’s behavior in this situation will approach 20/30 = 2/3 of its time on the big bush, and the remaining 1/3 of its time on the small bush. This is consistent with the simple matching law, which predicts that, all other things being equal, the amount of time the bird spends on the big bush relative to the amount it spends on the small bush is proportional to the amount of reinforcement the big bush provides relative to the reinforcement from the small bush.
This kind of analysis applies not only to bird behavior, but to human behavior as well. Consider someone who enjoys running, and has to decide how much she will run and how much she will cross train in a given week, assuming that she does some exercise every day. The more she runs, the better she gets at running and the more she enjoys it, but only up to a point. At some point, such as running every day for an extended period of time, she will be more likely to get injured or “burned out.” In Figure 2, the red curve represents a plausible function for the amount of running strength gained per workout (on the y-axis), based on the number of days someone has been running per week over the past several weeks (on the x-axis). If the person barely runs at all, this small amount of running is not very beneficial to her training, because she is not running enough to maintain the cardiovascular benefits she gains. If she has been running a lot, six or seven days a week, each workout is not as beneficial, because the high volume of running takes its toll on the body, and she does not recover as fully, so that the next workout does not strengthen her body as much as it could if she were well rested.
On the other hand, cross training is quite beneficial when used occasionally, because it trains different muscles from those used in running. However, cross training every day would lead to the degradation of her running muscles, so if she did a lot of cross training, each workout would not be very beneficial. The blue line in Figure 2 depicts the benefits of cross-training. If the athlete ran five days out of the past week, she cross trained two days. Thus, the scale on the x-axis for cross training is seven minus the number of days running. As described above, the less cross training the athlete has been doing, the more effective each cross-training session is in building strength, whereas if she has been doing a lot of cross training, each workout is not as beneficial. We will assume that the strength gained per cross training workout is a linear function of the number of days cross training, as shown.
The first point we will examine is labeled “1” in Figure 2, which is where the athlete is only cross training, with no running. If she then tries running, she will find that it is so difficult that it provides less strength per workout than cross-training (i.e., the red curve is below the blue line), so she will go back to only cross-training. In fact, if she runs any amount below an average of about 1.5 times a week, she will find that cross training is better at increasing her strength than is running, so she will run less, and will move left on the x-axis until reaching point 1 again. Thus, 1 is a stable point. A person at this point can be interpreted as someone who never runs and does not enjoy running, the kind of person who claims to “only run when being chased.” This is where everyone starts out in life.
If such a person happens to run more than usual, running two days a week, she will find that running provides more strength per workout than cross training (since the pink curve is above the blue line). Thus, she will run more and cross train less, moving right on the x-axis. Thus, point “2” in Figure 2 is an unstable point, because being on its left will encourage an athlete to move more to the left, and being on its right will encourage an athlete to move more to the right, so that no one stays at point 2.
Once the athlete has passed point 2 and is running more than two days a week, she will continue to find that running is more beneficial than cross training, so she will continue to increase the number of days per week that she runs, until she averages about 6.5 days a week of running (marked as “3” in Figure 2). If she runs more than this (point “4”), such as running seven days a week (moving to the right of point 3 on the x-axis), she will find that cross-training yields more strength per workout than running (since the blue line is above the red curve to the right of point 3). This would mean that she is feeling overtired or not fully recovered from each workout, so she will run less and do more cross training, thus moving back to point 3, where she runs an average of 6.5 days a week. This situation is precisely analogous to the bird finding a stable matching point at 20 visits per hour to the big bush; in both cases, deviating away slightly from the current position yields unfavorable results, so the bird and the athlete both move back to the stable matching point. In this case, running 6.5 days a week is the mark of the dedicated runner, probably a varsity athlete, who disdains days off and believes that the more she runs, the faster and stronger she will get, only taking a day off every other week.
While it is interesting to analyze what birds and humans actually do, it is often more interesting to determine what their optimal course of action would be. To do this, we will examine Figure 3. In this figure the red and blue curves are as before, and we have marked the two stable points found above, as 1 and 2, respectively. It is clear from the figure that an athlete at point 2 is gaining more strength per workout than an athlete at point 1, and will therefore be better off. But is this the best the athlete can do? The answer, perhaps surprisingly, is no.
In this figure we have added a third curve (black), which is the average benefit per workout from both activities, weighted by how much the athlete partakes in each. For instance, when the athlete is not running at all, the global average strength gained per workout is the same as the average strength gained from a cross-training workout, since the athlete is only cross training. Similarly, when the athlete is only running, the average strength per workout is the same as the average strength gained from a running workout, since the athlete is not doing any cross training. In the middle of the graph, when the athlete is engaging in both activities equally, the average is exactly between the two curves.
In order to maximize the benefit from exercising, the athlete should maximize the average strength gained per workout (both running and cross training workouts). Thus, the athlete’s goal is to run and cross train in just the right proportion so as to fall at the maximum of the black curve (labeled as point “3” on Figure 3), running about 5.5 days a week. Importantly, point 3 is not a stable point, so athletes will not naturally train in the right proportions to land there. Instead, some athletes will run less than they should, and some will run more than they should, and neither will derive the greatest possible benefit from each workout, so they will not improve as much as would be possible.
Thus, we have identified three types of runners: Those who don’t run at all (who are not actually runners), those who train too hard for their own good, and those who train just enough to improve, but not enough to get injured or burned out. Many runners will spend their childhood in the first situation, their competitive years in the second, and their adult years in the third. Many runners run more than they should, and become very fast, but are often injured, as our model predicts (they are not getting the maximum strength if they are injured). However, some noteworthy runners have excelled on plans of less running than one would expect for their level of achievement, such as Nicole Blood (an exceptional high school runner from Saratoga Springs, NY), validating our conclusion that the maximum benefit is derived from running less than one would naturally be driven to run.
