What does thermal integral have to do with agriculture?

Today we are going to a somewhat more technical concept. It may sound complicated because of the fact that it has the name “integral” in front of it, but we will see how it is very easy to apply this method. The thermal integral is a concept used in agriculture to establish a pattern of how certain organisms and plants will behave according to the environmental temperatures to which they are subjected.


For centuries, humanity has observed how the same crops subjected to warmer and more benevolent climates, completed their life cycles earlier than in cold climates. This observation immediately leads us to think that the ambient temperature to which the culture is subjected has a direct influence on the growth rate. That is, we deduce that there is a relationship between these two variables. The interesting thing about this idea is to be able to quantify this relationship in some way, with the great advantage that if it responds to a more or less constant pattern, great advances can be made in the control of crops and their interactions with the environment. For this, the term thermal integral was constructed .


Plants respond to a life cycle year after year defined by very well marked states (phenological states) that we all know, such as: germination, vegetative state, flowering, fruiting, winter rest (in the case of perennials or trees) … All these phenological stages are completed when the plant has accumulated a more or less specific temperature . The sooner it accumulates that temperature, the sooner it will complete each of the states and therefore its life cycle is shortened. This is when the way to quantify this thermal accumulation comes into play.

Phenological states of an orange tree
Source: hortalbagarces.com


Although it seems that its name is going to refer us to an incomprehensible mathematical formula, the truth is that it is the simplest thing in the world. It is a simple and flat sum, a sum of degrees. In fact, it is also known as degree days or units of heat. Degree-days are called in the case that average daily temperatures are taken (which is usually the most common). This is expressed as accumulated degrees necessary to complete a phenological stage or the complete cycle, which we can calculate by adding the effective temperatures of development day after day until reaching the number indicated for the crop. Let’s take an example:

Wheat needs to accumulate approximately 2000ºC, with a certain margin of variation, to achieve maturity from the moment it is sown. To know when it will reach maturity, we should add the heat units day after day until reaching 2000ºC.

Since it was sown, the effective growing temperatures for wheat day by day have been: 10, 12, 8, 14, 16, 25…. They are added up to the so-called thermal integral of 2000ºC. It will be then when the crop will have reached its maturity and we will know how many days it has taken to reach that number. It is evident then, that the hotter it is day after day, the sooner this integral will be reached.


First of all, before knowing how to calculate the thermal integral of a crop, we must have some clear concepts such as the upper and lower thermal thresholds. Every plant develops in a range of temperatures depending on its adaptation to climatic conditions. Each family, genus, species or subspecies up to variety, have differences regarding the thermal range of development.


Lower thermal threshold, base temperature or zero growth temperature is considered to be the temperature below which the plant stops its growth completely. Therefore, when the thermal integral of a crop is made, any temperature below this minimum threshold will not count in the development of the crop.


Just as there is a base temperature for growth, there is a maximum. The upper threshold is considered to be the one above which the plant stops its development or it is very very slow . Temperatures that are above this threshold will not be counted in the calculation of the thermal integral either.

In the following graph you can see how the yellow shaded area corresponds to the effective growth temperature.


Once we have the previous concepts clear, we can establish that between the maximum and minimum thresholds, the plant grows. But at what speed? Does it grow as fast for 8ºC as for 26ºC on average? It would be the ideal to establish our calculations but then they would not be plants, they would be machines. Growth speed is also influenced by other factors such as soil nutrition, humidity, rainfall, radiation … Even so, with many influencing factors, temperature has a very large weight and therefore we can use it considering that grows linearly. In any case, every day there are more complex agroclimatic models, which are getting closer and closer to reality.


Let’s take an example. Lettuce, for example, has approximately a lower threshold of 6ºC and an upper threshold of around 30ºC. To know the effective growth temperature we would have to subtract the minimum threshold of 6ºC from the average daily temperature. We are going to do a simple calculation for one week.

  • Average daily temperatures for a week: 8, 5, 7, 10, 11, 12, 16.
  • We subtract the 6ºC from the lower threshold at each temperature, therefore:
  • Effective growth temperatures ( average temperature – minimum threshold ): 2, -1, 4, 5, 6, 10.
  • We see that on the second day it gives a negative record. When this happens, it is not taken into account in the sum of degrees , therefore the total result would be: 2 + 4 + 5 + 6 + 10 = 27 ºC accumulated degree days. As simple as that.

In this calculation method with the mean daily temperatures only the lower threshold is used. In other more complex ones, the two thresholds are used to achieve greater calculation precision.

Although it seems simple, this method is widely used in many herbaceous, horticultural, fruit crops … to try to predict the ripening date among other things.


Let’s take the example of lettuce. If the lettuce has an approximate thermal integral of 700ºC, we could see the weeks it will take to reach that number and know approximately when it will be ready. Therefore, several variables can be predicted:

  • The date of planting and harvesting of a crop can be roughly predicted .
  • In fruit trees we can know how much the tree is missing to flower, knowing the degree days it needs to go to the phenological stage of flowering. In this way we can observe the accumulation of degrees and calculate the risk of frost that the tree may suffer   after flowering.
  • The thermal integral is also used to calculate the vital rates of pests and diseases. Insects, fungi, and bacteria also respond and develop based on a thermal integral. In this way, according to the weather forecasts, we can anticipate the appearance of a plague or disease and make the appropriate treatment in a preventive way.

The University of California UC-Davis has a degree day calculator to be able to calculate certain agroclimatic models and prediction models for pests and diseases. It only works for the US but at least gives an idea of ​​what can be done with the thermal integral.

As you can see, something as simple as calculating a sum of degrees can have a great impact on decision-making regarding the management of an agricultural crop.

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