Roots at war

Optimizing crops of cereals has been and will be of vital importance in cultural development. In principle it is possible to obtain some 160 seeds from one seed of classical cereals like barley or rye in a year. Such high yielding factors are not met in practice, however. At present, our highly priced modern technology is able to reach a value of some 10 till 20. In the 18-th century, it was 7 à 8. Extrapolating backwards, it is generally assumed that at the birth of european agriculture, it has been some 3 à 4. Great surprise when Reynolds (1979) obtained yield factors of 60 till 80, with a mean well above 30 during some 15 years, in the Butcher-Hill Ancient Farm Research Project. This project aims to mimic the old Babylonian technique as close as possible (including the use of old races, obtained from eastern Turkey). Looking for explanations of this remarkable result, it has been argued that the Babylonians planted the seeds in a honeycomb pattern, while later technologies use random patterns (obtained by throwing handfuls wide from the lab). In the honeycomb pattern, with an appropriate seed to seed distance, competition between plants would be significantly reduced as compared with random patterns. Let us study this explanation more closely, using the notion of Dirichlett cells.

Figure 9.1: Dirichlett cells of a honeycomb and a random pattern.

The Dirichlett cell belonging to a particular plant is defined as the set of sites nearest to that plant with respect to the other plants, Fig.9.1. Dirichlett cells can be considered as the plane analogue of intervals between points on a time axis. In a honeycomb pattern with mean plant density m per unit of surface area, all Dirichlett cells are identically honeycomb shaped with a surface area of . The 6 nearest plants to a typical plant all are at a distance . Initially, i.e. just after sowing, there are no roots. After some time they will start to grow and approximately occupy a circle with radius . Suppose that the roots of all plants grow equally fast. As long as , the roots do not interfere. The fraction of unused land is then . If the roots continue to grow, they will make contact with that of neighbouring plants. Suppose that they will cease growing at the places of contact, but continue to grow, where no neighbours are felt. We can define a dimensionless index for competition as one minus the ratio of the actual surface area the roots of a plant occupy and the potential one, when no neighbours where present. The potentially occupied surface area is taken to be . Initially, the competition index is 0. If the roots travel further than a distance from the plant, all land will be occupied and the competition index is . If , straightforeward geometry learns that the competition index is , with , Fig.9.2. The fraction of unused land becomes . Suppose that the seed density has been chosen such that the roots of a fully grown (lonely) plant occupy an area of just , i.e. the surface area available for one plant. So, . In a honeycomb pattern, the competition index then 0.037, with the same fraction of land still unoccupied.

Figure 9.2: Competition indices and fraction of unused land for honeycomb and random patterns as function of the size of the roots relative to the sowing density.

Now let us consider random patterns. We idealize the sowing process, such that it is reasonable to assume that the number of seeds falling into a plot of unit size follows a Poisson distribution with parameter . Now no two Dirichlett cells have the same shape and size. It is extremely difficult, indeed, to tell more of the surface area of Dirichlett cells in random patterns than their mean, which is obviously , and their variance, which is a cumbersome expression (Matern, 1960). Their shape, which is determined by the position of neighbouring plants also varies considerably. The number of neighbouring plants, i.e. plants having adjacent Dirichlett cells, can vary between 3 and a lot. So we must follow a totally different approach to find expressions for the competition index and the fraction of land not in use by the plants. The latter fraction is easily found, if we realize that it corresponds with the probability that no plants are present within a circle of radius from a randomly selected site. Since such a circle has surface area , this probability is the zero-probability of a Poisson distribution with parameter , i.e. . To obtain the competition index, it is helpful to note that the ratio of the actually and the potentially occupied area of a plant corresponds with the probability , say. being the probability that a randomly selected site within a circle with radius from a randomly selected plant is within its Dirichlett cell. This means that the distance between such a site and our plant, say , is smaller than any distance to other neighbouring plants. Given a particular site, the latter probability equals . We find , by mixing the latter probability with the probability density, say, of a random site at distance from the plant. This probability density is easily derived from its distribution function. The probability that the distance between a random site within a circle of radius and the centre, is less than , is . So . So, is given by . The competition index is thus , thus , Fig.9.2. If the seed density, , has been chosen such that a full-grown plant just occupies an area equal to the mean surface area of the Dirichlett cells, so , the competition index is 0.368, while the same fraction of land is still unoccupied. In conclusion we can state that in random patterns the competition index and the fraction of unused land is 10 times as high, compared with honeycomb patterns, if plants are sowed as dense as possible. It seems safe to assume that growth, and so yield, is increasingly retarded for increasing competition indices. Wether or not the difference between the yield factors can be fully explained by the difference in competition indices, is beyond the scope of the present analysis. Other factors of importance include nutrient supply. The Babylonians seem to allow a significant amount of weeds growing between the grain, which was sufficiently competitive to the weeds to defeat adverse effects on growth. After the season, the weeds were plight through the soil, supplying adequate nutrients as well as improving soil structure. The reason why the romans did not adapt the babylonian techniques is possibly due to their aim of maximizing yield per man-hour. (The Babylonians possibly used their hands in collecting grain for private use mostly.)