Food Chain and Pyramidal Hierarchy
Lo! Big Fishes are eating Small Fishes. That I kew for Long Time
So says the caption inn the famous painting by Peter Brughel, the flamish master. Planktons are eaten by small fishes, which in tern are eaten by Big fishes, which end up in the stomack of human. Everybody is familiar with this type of hierachical construction of nature. In closer examination, we can find exponential increase in the number of individuals as we go down the steps of hierarchy. It is the same for human society too. Historically, it used to be the norm to have a royal family of fifty supppoted by one thouand magnates, who were supported by fity thousand counts, barons and knights, who in turn was supported by one million peasants.
| Modern-day institutions with their 'democratic" facade, are in fact not much different. Unreasonable deviation from this universal structure tends to result in structural schelosis, as has been amply demonstrated in the bureaucracy of the past century in which there are twenty "heads" of various capacity within an organization of a hundred. Recent management theory puts emphasis on "re-engineering" and "de-layering" to advise fewer layers of hierarchy to match the size of an organization. |
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Still, we ocasionary encounter old style managers who show around intractably tangled diagram for cooperate structre. in an offcial meeting. It is impossible to resist the impulse of lough, but to be polite, try to express it as noiseless giggle, face down.Pyrmidal structure of exponential increase in numbers at the bottom is so ubiquitous, that we suspect some sort of divine scheme of design.
Optimal Sustainable Yield and Elton's Pyramid
In order to explain this fact, we start by considering just two species of fishes, one predator and the other its prey. Assuming you are a big fish, how much small fish you would want to eat? If you re too greedy, you are left with fewer prey later on, so there must be some optimal amount you would want to eat for your own benefit. Claculation utilizing the Lotka-Volterra system reveals that you would be better off, in longer term, by eating up to half of the natural population of small fish (means stable population of small fish when you predators are absent). We learn in Oceanography that there is a concept called "ecologically sustainable maximum yield" which turns out to be half of the available fish. Since different approaches are adopted there, we can guess that this conclusion is more or less independent on the model used, thus something more universal.
If this calculation is applied to the food chain with three trophic levels, The top fishes eat half of the available second-tier fishes, the decrease of the second tier reflected in the decrease in the preying of the second tier on the bottom fishes, and the net result turns out to be 3/4-th of the bottom fish is left uneaten in the system. Repeating this type of calculation to the system with N-trophic levels, we obtain the conclusionn that, as we go down one trophic level, the number of individual increases by factor4Bn/Bn-1. (Here Bnは is a variant of "Fibonacci number" given by B０=1, B１=1, B２=3, B３=5, B４＝11, B５=21, ...). This is the universal pyramid of ecohierarchy. After the british biologist, it is also called Elton's pyramid.
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