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Lie down on the or volume. Maybe this mathematical law can help you choose the best solution.

2025-01-19 Update From: SLTechnology News&Howtos shulou NAV: SLTechnology News&Howtos > IT Information >

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This article is from the official account of Wechat: back to Park (ID:fanpu2019), by James V. Stone (Honorary Associate Professor, University of Sheffield, UK), translated by Shi Hao

In nature, animals are often faced with the choice of income and physical strength in order to feed or reproduce. For example, when a great tit hunts in the bushes, it is easy to find caterpillars at first, but it becomes more and more difficult to find after a period of time, and even finding food is not enough to make up for the physical exertion of the search process. So, when should we leave this area and find another way? People are surprised to find that animals often instinctively choose the optimal solution, that is, to maximize the benefits. In mathematics, this is what the marginal value theorem shows.

Knowing when to stop is a big problem in our lives. When faced with diminishing returns, such dilemmas abound in choosing to persist or give up the present in order to get a better return in the future. For every gold mine, it reaches a point-the gold obtained from continued mining is not worth the cost, and once the time is up, it is time to give up mining and start looking for new gold deposits.

Similarly, for birds that feed on caterpillars in the bushes, when they don't have enough calories to keep looking for more caterpillars, it's time to give up preying here and switch to another bush. Similarly, for bees, when the weight of collecting pollen does not match the energy required, the collection should be stopped and it is time to fly back to the hive.

Fortunately, there is a mathematical solution-the marginal value Theorem (Marginal value theorem,1976 year was proposed by American ecologist Eric Charnov). The marginal benefit theory clearly points out when to give up in order to get the maximum return. More importantly, from the original optimal foraging theory to how the brain processes information, the marginal value theorem has been widely used. In essence, no matter what you give or receive, marginal value theory provides a general strategy to maximize what you get from each contribution.

Ladybugs in prey

Ladybugs and aphids? photo Source: Greyson Orlando ladybugs feed on aphids, but watching ladybugs eat aphids is like watching humans eat lobster, because the exoskeleton of aphids is to ladybugs what lobster shells are to humans. After eating the most juicy part at first, it often takes more and more effort to get less and less food. Human beings often know how to stop when faced with this problem and turn to enjoy other delicacies. Similarly, when ladybugs are faced with such a problem, they also know when to give up and find another aphid.

If the ladybug gives up the present one prematurely, it will waste some of the food that is easily available. On the contrary, if you give up too late, you will get less and less, and these twos and threes can be easily obtained from another aphid. Obviously, there is a "just right" time to give up, neither too early nor too late. In order to find out when this "just right" time will come, we need to learn more about ladybugs' eating behavior.

The anatomical properties of aphids similar to lobsters mean that if a ladybug is allowed to eat only one aphid, the cumulative weight will increase with the feeding time t, as shown in the following figure. (this data comes from a classic article on foraging theory published in 1978 by R. M. Cook and B. J. Cockrell, which inspired us to give an example of ladybugs. )

The black spot represents the food quality eaten by the ladybug at t minutes, and the black curve represents the fitted benefit curve E (t). These data points can be used to synthesize the return function E (t) = a (1-e-bt). In our ladybug example, we can use this return function to define two quantities:

1. Real-time rate of return, r (t) = Δ E (t) / Δ t, that is, the increased calories Δ E (t) per time period Δ t (e.g. calories per minute)

two。 Moving average rate of return. This is based on the cumulative food quality of an aphid after feeding for t minutes, divided by the time it takes to find the aphid and the time it takes to eat.

These quantities have quite intuitive geometric explanations. The income function fitted by the above ladybug's data points is shown in the following figure, showing the cumulative income E (t) after the food is discovered at T time.

The black curve in the picture is the return function E (t), and the slope of the curve is the immediate rate of return r (t). The red dotted line represents the moving average return R (t). This picture reflects the premature abandonment of aphids by ladybugs. At the point t after the aphid is found, we can define a right triangle with a base length of t and a height of E (t) (a black vertical dotted line). The slant of this right triangle is a red dotted line whose slope is equal to the moving average rate of return R (t). This means that the problem of maximizing the moving average yield can be expressed as maximizing the slope of the red dotted line. Specifically, which red dotted line has the greatest slope?

In the image above, the feeding time is t=10min, and the red dotted line has a smaller slope; if the ladybug delays giving up feeding, the slope will increase, and it will give up too early. If the feeding time is more than 10 minutes, the moving average rate of return R (t) will be larger.

Ladybugs give up aphids too late similarly, as shown in the picture above, the feeding time is t=55min, and the slope of the red dotted line is relatively small. If the ladybug gives up feeding in advance, the slope will increase, and it will leave too late. If the food intake is less than 55 minutes, the moving average rate of return R (t) will become larger.

Ladybugs give up eating at the right time. Finally, the eating time t=35min produces the largest possible slope. This is the optimal feeding time, and if you give up eating at this time, the moving average rate of return R (t) will be the highest.

Notice that the "just right" eating time occurs when the slope of the income curve r (t) is equal to the slope of the red dotted line, that is, the moving average rate of return R (t). In other words, the right time to eat occurs when the immediate rate of return is equal to the moving average rate of return: r (t) = R (t).

It can be geometrically verified that the immediate rate of return r (t) is represented by a red dotted line in the following figure, and the moving average rate of return R (t) is represented by a black solid line.

The red dotted line indicates the immediate rate of return, and the black solid line indicates the moving average rate of return, as we expected, and the two curves intersect at the "right" time (that is, r (t) = R (t)). From this geometric point of view, the marginal value theorem is effectively proved, that is, the best exit time is when the immediate rate of return is equal to the moving average rate of return, that is, when r (t) = R (t).

The mathematical proof is as follows: under the following three rather weak (mild) conditions, the marginal value theorem is true:

1. Fixed cost T is greater than 0

two。 The income function E (t) increases with the increase of time t

3. The slope of the return function decreases with the increase of time (that is, E (t) is the income increment decreasing function).

We hope to prove that when R (t) is maximum, the immediate rate of return r (t) is equal to the average rate of return R (t). To do this, we need to find the value of r (t) when R (t) is maximum. In order to find the maximum average rate of return, we need to use the fact that it is maximum at its slope of 0.

The average return is defined as:

And its derivative is

According to the definition

Is the real-time rate of return, and

By substituting equations (3) and (4) into equation (2), you can get

And because

It's the average rate of return, so formula (5) becomes

At the maximum, make the above equation equal to 0

Finally, the two sides of the equation are multiplied by Tantt, and the shift term is obtained.

This proves that when the immediate rate of return is equal to the average rate of return, the average rate of return will reach the maximum.

Ladybugs understand the marginal value Theorem the above analysis assumes a constant search time T. However, a proper test of the marginal value theorem should involve variable search time. In essence, the longer search time indicates that food is scarce, so it makes sense to increase the time it takes to eat each aphid accordingly. In other words, the time spent eating on an aphid should increase with the time it takes to find the aphid. However, in contrast to this correct but vague conclusion, marginal value theory accurately predicts how eating time will increase as search time increases, as shown in the following figure. (the return function based on the experimental data of Cook and Cockrell in 1978 is used again).

If it takes 40 minutes to find an aphid, it will take 50 minutes for the ladybug to feed on the aphid (red dotted line); if it takes 10 minutes to find an aphid, it will take 28 minutes for the ladybug to eat the aphid (blue dotted line).

As we can see from the picture above, the marginal value theorem predicts that the time it takes to feed on aphids increases with the time spent looking for aphids. If it takes 40 minutes to find an aphid, it should take about 50 minutes for ladybugs to eat (red dotted line above). If it takes 10 minutes to find an aphid, it should only take about 28 minutes for ladybugs to feed (blue dotted line).

The question is, will ladybugs give up eating aphids at the time predicted by the marginal value theorem? To answer this question, Cook and Cockrell randomly scattered the aphids in a tray and measured the time it took the ladybug larvae to find each aphid and the time it took to eat each aphid. (Cook and Cockrell use ladybug larvae, but this is not very important. )

How the average feeding time of ladybug larvae varies with the time when an aphid is found: the dot represents the observed value in the experiment, and the black line represents the feeding time predicted by the marginal value theory.

The image above shows how the average feeding time of ladybug larvae varies with the search time observed by Cook and Cockrell: the dots represent the results they observed in the experiment, and the black curve represents the feeding time predicted by the marginal value theorem. After many experiments, the number of aphids gradually decreased, making the search for new aphids correspondingly longer. As predicted, as the search time increases (the density of aphids decreases), the average time each aphid will be eaten will also increase. More importantly, the way that eating time increases with search time is to some extent consistent with the prediction of the marginal value theorem, as shown in the figure above.

In this particular case, the fitting between the data and the theory is not so impressive, which may be because the moving average income function R (t) does not have a peak, so the loss of giving up eating a little earlier or later is relatively small. On the contrary, if the return function rises steeply, then the moving average yield curve will have a higher front, then the experimental data may be more consistent with the marginal value theorem predictions.

The fact that animals know when to stop ladybugs preying on aphids is just one of many examples of animals giving up eating at the right time for maximum benefit. The same applies to the high cost of obtaining valuable resources.

The great wisdom of the great tit assumes that a great tit takes some time to find bushes with a large number of caterpillars. After finding the bushes, the tits began to eat a large number of caterpillars per minute, but after a while, the number of caterpillars that could be found decreased, so so did the number of caterpillars that could be eaten per minute. Given the diminishing marginal returns, when should the great tits abandon this bush and choose to look for new bushes?

As in the ladybug example above, the answer is derived from the marginal value theorem. By doing experiments similar to the ladybug experiment, scientist Richard Cowie discovered in 1977 that the theory of marginal value could correctly predict when the great tit would give up the bushes.

For a male dung fly, knowing how long it takes to mate with a female represents a typical tradeoff between the current guaranteed return and an uncertain but likely greater benefit. There is sperm competition (Sperm competition) during fecal fly reproduction. Females mate with multiple males, and each male replaces the sperm of the other males as much as possible. The longer the male copulates with the female, the more sperm will be replaced by other males who have previously mated with the female, and the less likely the female will mate with other males. In other words, every minute of the male mating could have been used to mate with other females.

We can find that the time it takes to find a new female fly is similar to the time it takes to find new food sources (new aphids and new bushes) in the previous example, and the time it takes to mate with a female fly corresponds to an increased but incrementally reduced benefit. Therefore, we should not be surprised that the marginal value theorem can predict how long a male dung fly should mate with each female to ensure as many offspring as possible. In 1976, scientists G. A. Parker and R. A. Stuart observed that the mating time between male and female fecal flies was 36 minutes, which was equal to 41 minutes predicted by the marginal value theorem.

Are bees too busy to be inefficient? As bees collect more and more pollen, it takes more energy to bring it back to the hive. The dilemma that every bee faces is: when should bees stop collecting pollen and return to the hive?

At one extreme, if the bee collects only one grain of pollen and then returns, it almost certainly consumes more energy to and from the hive than it gets from the pollen grain; at the other extreme, if the bee collects a large amount of pollen, the pollen will be collected very quickly, but the energy consumed on the way back to the hive will account for a large proportion of the energy obtained from the pollen. There is an appropriate carrying capacity between the two extremes-it provides the maximum energy for each calorie consumed during the flight. It is worth noting that the continuous collection of flowers by bees leads to more load and diminishing returns.

Scientists Paul Schmid-Hempel, Alejandro Kacelnik and Alasdair Houston observed that the load of honeybee pollen was consistent with that predicted by the marginal value theorem in 1985. In short, bees collect pollen as efficiently as possible in terms of energy consumption per gram of collected pollen. In contrast, in a model based on maximizing the collection rate per minute, it fits very poorly with the amount of loaded pollen. This suggests that bees prefer energy efficiency to the rate of pollen collection.

The main function of brain and marginal value theory neurons is to process information, but the cost of maintaining neurons is very high, and the cost of transmitting information is even higher, so it is necessary for them to operate with the highest efficiency.

According to Shannon's information theory, the common unit of information is the bit, which provides enough information to choose between two possible events, such as tossing a coin. However, Shannon's information defines a general law of diminishing returns, which means that the energy required to process each bit of information increases as the number of bits processed per second increases. Therefore, information can be processed at slow speed and low cost or at high speed and high cost, but not at high speed and low cost.

This law of diminishing returns is similar to the diminishing return on the amount of food ladybugs eat per second. The difference is that ladybugs want to maximize the average amount of food eaten per second, while neurons want to maximize the average amount of information per joule of energy consumption. In view of these similarities, both the ladybug problem and the neuron problem can be solved by the marginal value theorem.

Energy efficiency varies with the speed at which neurons emit. The peak of the curve predicts the average emission frequency of neurons, which is consistent with the idea of neurons maximizing energy efficiency.

The marginal value theorem can be redefined in terms of energy consumption (instead of eating time) and information processing (instead of eating income). The resulting efficiency curve shows that the maximum efficiency predicted by the marginal value theory (in bits / joule) occurs at the emission frequency of about 2 neuronal peaks per second. Importantly, scientists William Levy and Robert Baxter in 1996 found that this prediction was consistent with observations of the average emission frequency of neurons.

So, no matter how much time it takes to find aphids, or the energy consumed by neuron discharges, as measured by eating aphids per hour or processing information bits per joule, the marginal value theorem specifies how much to pay to maximize the benefits.

references

[1] https://www.jstor.org/stable/3799

This article is translated from James V. Stone, Knowing when to quit

Original link: https://plus.maths.org/ content / knowing-when-quit

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