December 6, 2014 at 00:49Camels, butterflies and fractal chaos of theoretical ecology
Greetings to the venerable Geeks! Happiness, as you know, is "when they understand you." And there are often problems with this. Especially if you are working 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 nutshell to tell what you do at work. Under the cut, my attempt to state in a simple and understandable way one of the most interesting aspects of theoretical ecology.
More D.I. Mendeleev said that: "Science begins where they begin to measure." However, the measurement process itself is not as simple as it might seem at first glance. In order to measure, say length, the first thing that is needed is to determine the minimum binary units. Binary - henceforth not divisible. Each of us calculated such units in a school physics lesson when we measured the unit's division price.
At that moment, we hardly realized how important the calculated values are. At first glance, the accuracy of the data depends on which instrument we choose for measurement. The more accurate the device, the lower its division price, the more accurate the result we can get.
The English mathematician Richardson was guided by the same rule. Testing the hypothesis of the dependence of the frequency of wars on the extent of borders, Richardson compared the length of the Spanish-Portuguese and Portuguese-Spanish borders. Paradoxically, the border from Portugal turned out to be more than 240 kilometers. 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 volumetric quantities, and therefore all other units of measurement (I recall that the same weight is measured in the deviation length of the balance spring, that is, in the same centimeters, which are then converted into kilograms according to a predetermined ratio) do not have absolutely accurate values. The more accurate the device, the greater the value it shows. The length of the border measured with a centimeter ruler will be greater than the length of the border measured with a meter ruler. The borders of Portugal are smaller than the Spanish ones, 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 that we have known about since childhood is nothing more than a fiction that has nothing to do 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 which scale range we perform calculations.
Considering such objects in the 70s of the twentieth century, Benoit Mandelbrot proposed the concept of "fractal" and the principles of description and work 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-law increase in their size with a linear change in minimum units.
In addition to the tool for describing the complexity of structures, fractal geometry made it possible to numerically express the degree of randomness of the system.
In ecology, apologists of determinism and indeterminism are still encountered. The former believe that, knowing the initial position of all components of the system, we can predict further dynamics for an indefinitely long time. The second convinces them of the randomness of the phenomena and processes taking place in the world. But physicists have long known a compromise option. Yes, what is there for physicists. He even appears in the literature: “The Butterfly Effect” by Bradbury - a story that clearly illustrates the theory of deterministic chaos.
Let me remind you that the protagonist of the story, setting off for the past, steps on a butterfly. The latter does not become food for the predator. The predator does not do what he could and the fate of the whole world begins to change in 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. 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 the hundredth decimal place and, thus, instead of increasing the number of species, get its predicted extinction. We cannot do long-term weather forecasts since the appearance of any cyclone may depend on how many sighs you take while reading this text.
You can load a lot of straw on a camel. At the same time, there is no way to evaluate which straw will break his back. We can also affect the ecosystem for a long time, but there is no information through which broken branch the infection will penetrate, which will subsequently spread and destroy the entire weakened forest.
In our laboratory, we are developing a method of replacing temporal and spatial axes. If at one point in the terrain we have a swamp, and at another pine with lichens, then when we move from one point to another, we will observe the same changes that would occur with a swamp in time when replacing the bog growing conditions with the conditions of 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 destruction of 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 bunches in such a way as to exclude an armful of critical straw from falling onto 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 (although, so far, dubious) to assess the degree of risk of destruction of the natural community. Drawing an analogy with R. Bradbury, we can say that between the danger of negative changes in the future and the size of the protagonist’s boots there is, nevertheless, a connection, and to describe it is our task.
We are working on that.
More D.I. Mendeleev said that: "Science begins where they begin to measure." However, the measurement process itself is not as simple as it might seem at first glance. In order to measure, say length, the first thing that is needed is to determine the minimum binary units. Binary - henceforth not divisible. Each of us calculated such units in a school physics lesson when we measured the unit's division price.
At that moment, we hardly realized how important the calculated values are. At first glance, the accuracy of the data depends on which instrument we choose for measurement. The more accurate the device, the lower its division price, the more accurate the result we can get.
The English mathematician Richardson was guided by the same rule. Testing the hypothesis of the dependence of the frequency of wars on the extent of borders, Richardson compared the length of the Spanish-Portuguese and Portuguese-Spanish borders. Paradoxically, the border from Portugal turned out to be more than 240 kilometers. 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 volumetric quantities, and therefore all other units of measurement (I recall that the same weight is measured in the deviation length of the balance spring, that is, in the same centimeters, which are then converted into kilograms according to a predetermined ratio) do not have absolutely accurate values. The more accurate the device, the greater the value it shows. The length of the border measured with a centimeter ruler will be greater than the length of the border measured with a meter ruler. The borders of Portugal are smaller than the Spanish ones, 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 that we have known about since childhood is nothing more than a fiction that has nothing to do 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 which scale range we perform calculations.
Considering such objects in the 70s of the twentieth century, Benoit Mandelbrot proposed the concept of "fractal" and the principles of description and work 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-law increase in their size with a linear change in minimum units.
In addition to the tool for describing the complexity of structures, fractal geometry made it possible to numerically express the degree of randomness of the system.
In ecology, apologists of determinism and indeterminism are still encountered. The former believe that, knowing the initial position of all components of the system, we can predict further dynamics for an indefinitely long time. The second convinces them of the randomness of the phenomena and processes taking place in the world. But physicists have long known a compromise option. Yes, what is there for physicists. He even appears in the literature: “The Butterfly Effect” by Bradbury - a story that clearly illustrates the theory of deterministic chaos.
Let me remind you that the protagonist of the story, setting off for the past, steps on a butterfly. The latter does not become food for the predator. The predator does not do what he could and the fate of the whole world begins to change in 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. 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 the hundredth decimal place and, thus, instead of increasing the number of species, get its predicted extinction. We cannot do long-term weather forecasts since the appearance of any cyclone may depend on how many sighs you take while reading this text.
You can load a lot of straw on a camel. At the same time, there is no way to evaluate which straw will break his back. We can also affect the ecosystem for a long time, but there is no information through which broken branch the infection will penetrate, which will subsequently spread and destroy the entire weakened forest.
In our laboratory, we are developing a method of replacing temporal and spatial axes. If at one point in the terrain we have a swamp, and at another pine with lichens, then when we move from one point to another, we will observe the same changes that would occur with a swamp in time when replacing the bog growing conditions with the conditions of 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 destruction of 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 bunches in such a way as to exclude an armful of critical straw from falling onto 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 (although, so far, dubious) to assess the degree of risk of destruction of the natural community. Drawing an analogy with R. Bradbury, we can say that between the danger of negative changes in the future and the size of the protagonist’s boots there is, nevertheless, a connection, and to describe it is our task.
We are working on that.