Camels, butterflies and fractal chaos of theoretical ecology

Greetings to the venerable geeks! Happiness, as you know, is "when they understand you." And with this, there are often problems. Especially if you work in the laboratory on the topic “methods of analyzing the states of deterministic chaos in the dynamics of vegetation cover,” and you are asked in a few words to tell you what you do at work. Under the cut, my attempt to present in a simple and understandable form one of the most interesting aspects of theoretical ecology.



Also D.I. Mendeleev said that: "Science begins where they begin to measure." However, the measurement process itself is not as simple as it may seem at first glance. In order to measure, say, the length, the first thing that is needed is to determine the minimum binary units of measurement. Binary means not divisible further. Each of us calculated such units of measure at a school physics lesson when measuring the price of instrument division.

Hardly at that moment we realized how important the calculated values ​​are. At first glance, only the accuracy of the data depends on which tool we choose to measure. The more accurate the device, the lower its division price, the more accurate the result we can get.
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The same rule was guided by the English mathematician Richardson. Testing the hypothesis of the dependence of the frequency of wars on the length of the borders Richardson compared the length of the Spanish-Portuguese and Portuguese-Spanish borders. Paradoxically, the border from Portugal turned out to be 240 kilometers longer. The same effect is observed when comparing any other borders of states of different sizes.

The fact is that, contrary to our beliefs, the concept of “absolutely accurate measurement” is devoid of any meaning. Linear, areal and volume values, and therefore all other units of measurement (remember, the same weight is measured in the length of the deviation of the balance spring, that is, in the same centimeters, which are then converted into kilograms by a predetermined ratio) do not have absolutely exact values. The more accurate the device, the larger it shows. The length of the border, measured by a centimeter ruler will be greater than the length of the border, measured by a meter ruler. The borders of Portugal are smaller than those of Spain; the Portuguese measured them in more detail, thereby increasing their land border by more than two hundred kilometers.

Moreover, in nature there are no geometrically correct objects. All that surrounds us is not a set of points, curves, planes and volumetric bodies, but something in between. It sounds unusual, but strictly speaking, the three-dimensional world of which we know from childhood is nothing more than an invention, which has nothing in common with reality. The world is not just multidimensional, it is infinitely multidimensional. And each dimension in the absolute majority of cases is a fractional number and, in addition, depends on what range of scales we perform the calculations.

Considering such objects in the 70s of the twentieth century, Benoit Mandelbrot proposed the concept of "fractal" and the principles of describing and working with such objects. A fractal is a set consisting of elements, each of which is similar to a whole set. The main property of fractal objects is a power increase in their size with a linear change in the minimum units of measurement.

Fractal geometry, in addition to a tool for describing the complexity of structures, made it possible to numerically express the degree of randomness of the system.

In ecology, apologists of determinism and indeterminism are still found. The first believe that knowing the initial position of all components of the system, we can predict the further dynamics for an indefinitely long time. The latter convince them of the randomness of phenomena and processes occurring in the world. But physicists have long known a compromise. Yes, that there physicists. He even appears in the literature: Bradbury's Butterfly Effect is a story vividly illustrating the theory of deterministic chaos.

Let me remind you that the protagonist of the story, having gone into the past, strikes a butterfly. The latter does not become food for the predator. The predator does not do what could and the fate of the whole world begins to change like an avalanche. As a result, the main character returns to a completely different world, with different values ​​and laws.

The dynamics of natural systems is extremely dependent on the initial conditions. Just as in the “Butterfly Effect”, a slight change in the initial conditions leads to a fundamental discrepancy between the calculated and real values. You can round off the hundredth decimal place and, thereby, instead of increasing the number of species, get its predicted disappearance. We cannot make long-term weather forecasts, since the appearance of any cyclone can depend on how many sighs you make while reading this text.

You can load a lot of straw on a camel. At the same time, there is no method to estimate which straw will break his back. We can also impact the ecosystem for a long time, but there is no information on which broken branch an infection penetrates, which will later spread and destroy all the weakened forest.

In our laboratory, we are developing a method for replacing temporal and spatial axes. If at one point in the area we have a swamp, and at another point a pine forest with lichens, then when moving from one point to another we will observe the same changes that would occur with the swamp in time when replacing the bog growing conditions with the conditions of the lichen pine. Fractal analysis allows you to determine the units of measurement of conditions in which you can plan the maximum load without the risk of destroying communities.

It is impossible to estimate the number of straws that a camel can withstand, but it is possible to break all the hay into bundles in such a way as to exclude getting an armful with critical straw on the ridge of a poor animal. It is impossible to develop maximum permissible standards, but for each subsequent batch of emissions there is a way (albeit, while questionable) to assess the degree of risk of destruction of the natural community. Drawing an analogy with R. Bradbury, we can say that there is a connection between the danger of negative changes in the future and the size of the boots of the main character, and to describe it is our task.

Over and working.