mathematical biology is the theory of mathematical models biological processes and phenomena. Mathematical biology can be classified as applied mathematics and actively uses its methods. The criterion of truth in it is a mathematical proof. critical role it plays mathematical modeling using computers. Unlike pure mathematical sciences, in mathematical biology, purely biological tasks and problems are studied by the methods of modern mathematics, and the results have a biological interpretation. The tasks of mathematical biology are the description of the laws of nature at the level of biology and the main task is the interpretation of the results obtained in the course of research, an example is the Hardy-Weinberg law, which is provided by means that do not exist for some reason, but it proves that the population system can be and also predicted on the basis of this law. Based on this law, we can say that a population is a group of self-sustaining alleles, in which natural selection provides the basis. Then, in itself, natural selection is, from the point of view of mathematics, as an independent variable, and the population is a dependent variable, and under the population is considered a certain number of variables that affect each other. This is the number of individuals, the number of alleles, the density of alleles, the ratio of the density of dominant alleles to the density of recessive alleles, etc., etc. Natural selection also does not stand aside, and the first thing that stands out here is strength natural selection, which refers to the impact of environmental conditions that affect the characteristics of the individuals of the population that have developed in the process of phylogenesis of the species to which the population belongs.
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This abstract is based on an article from the Russian Wikipedia. Synchronization completed 07/10/11 17:38:26
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The laws of evolution, although based on facts, do not have a strict mathematical justification. This is what allows scientists various directions interpret them differently, or even not recognize them at all. But all this until mathematics got to these laws.
The first application of mathematics in biology is associated with the processing of observational results. This is how most of the experimental regularities were established ... However, this is in the highest degree the useful application of mathematics to biology is not only not the only, but not even the most important.
Experimental laws exist not only in biology. There are many of them in physics, technology, economics and other areas of human knowledge. But no matter what science such a law belongs to, it always has one serious flaw: although it answers the question "how", it does not answer the question "why".
Even alchemists knew how substances dissolve. When measuring the concentration of a solution, it is easy to draw a curve that clearly shows that at first the substance goes into solution in large doses, then these doses gradually decrease until the substance stops dissolving altogether.
Similar curves can be found in forestry books. They are obtained as a result of hundreds and thousands of measurements and show that the tree grows quickly at first, then growth slows down and stops completely.
These laws are experimental. They quite accurately describe the phenomenon - quite enough for practice. But it is difficult to predict, knowing only them: we can only say that given substance will dissolve in this way if the conditions under which we studied it are repeated. It's the same with trees. Without knowing why they grow one way or another, it is impossible to predict what will happen to their growth in other conditions.
"The sciences differ greatly in the degree of predictability of the facts relating to them, and some argue that biology is not a science. Because biological phenomena cannot always be predicted." This sad remark of the scientist K. Willy hits right on target. To gain the rank of a modern science, it is no longer enough for biology to have detailed information about numerous and disparate facts. We need laws that answer the question "why". And this is where the very essence of mathematical biology lies.
Just as in physics, when studying a biological phenomenon, one tries to reveal its mathematical characteristics. For example, if a patient is being examined, then numerical data is required to analyze his condition - body temperature, pressure and blood composition, pulse rate, etc., etc.
But after all, usually only one aspect is studied, something is the main thing, and something can be neglected. In astronomy, for example, the entire globe is represented as a point devoid of dimensions. Rougher, it would seem, nowhere. Nevertheless, these calculations have been regularly used for more than 300 years in determining the timing of eclipses, and in our years - when launching satellites.
Often, however, biologists refuse to make any simplifications at all. At one very representative biological seminar, the model of tree growth was discussed. The speaker, a well-known specialist in his field, was received favorably by the audience. Everything was going well until he uttered the phrase: "Since the energy of photosynthesis is proportional to the area of the leaf, for simplicity we will consider the leaf to be flat, having no thickness." Perplexed questions immediately rained down: "How so? After all, even the thinnest sheet has a thickness!". They also remembered conifers, in which it is generally difficult to distinguish thickness from width. With some difficulty, however, it was possible to explain that in the task that the speaker is engaged in, the thickness of the sheet does not play any role and can be neglected. But instead of a living sheet with all its endless complexities, we can study a simple model.
The mathematical model is being studied mathematical means. Therefore, we can digress for a while from the biological content of the model and focus our attention on its mathematical essence.
Of course, all this hard work, which requires special knowledge, the biologist carries out in close alliance with the mathematician, and some moments are entirely entrusted to the mathematician-specialist. As a result, such joint work a biological law is obtained, written mathematically.
Unlike the experimental one, it answers the question "why", reveals internal mechanism the process being studied. This mechanism is described by mathematical relations included in the model. In the tree growth model, for example, such a mechanism is a differential equation expressing the energy conservation law. Having solved the equation, we obtain the theoretical growth curve - it coincides with the experimental one with amazing accuracy.
Back in 1931, a book by the famous mathematician W. Volterra "Mathematical Theory of the Struggle for Existence" was published in Paris. In it, in particular, the problem of "predator-prey" was also considered. The mathematician reasoned as follows: "The increase in the number of prey will be the greater, the more parents, that is, the greater the number of prey in this moment. But, on the other hand, the greater the number of prey, the more often it will be encountered and destroyed by predators. Thus, the decrease in the prey is proportional to its number. In addition, this decline increases with the increase in the number of predators.
And what changes the number of predators? Its decline occurs only due to natural mortality and is therefore proportional to the number of adults. And its profit can be considered proportional to nutrition, that is, proportional to the amount of prey destroyed by predators.
The last of these problems is very interesting. Its essence is that chemical methods control of harmful species often does not satisfy biologists. Some chemicals are so strong that along with harmful animals, they destroy many useful ones. It also happens vice versa: the suppressed species very quickly adapts to chemical poisons and becomes invulnerable. Experts assure, for example, that DDT powder, the smell of which alone killed the bedbugs of the 30s, is successfully eaten by today's bedbugs.
And here is another small example of how a mathematical approach has clarified a confusing biological situation. In one of the experiments, an amazing thing was observed: as soon as a drop of sugar syrup was placed in a colony of the simplest microorganisms living in water, all the inhabitants of the colony, even the most distant ones, began to move towards the drop. The amazed experimenters were ready to assert that microorganisms have a special organ that senses the bait at a great distance and helps them move towards it. A little more, and they would have rushed to look for this unknown organ.
Fortunately, one of the biologists, familiar with mathematics, offered another explanation for the phenomenon. His version was that, away from the bait, the movement of microorganisms is not much different from the usual diffusion characteristic of inanimate particles. The biological characteristics of living organisms appear only in the immediate vicinity of the bait, when they linger around it. Due to this delay, the next layer from the drop becomes less saturated with inhabitants than usual, and microorganisms from the neighboring layer rush there according to the laws of diffusion. According to the same laws, the inhabitants of the next, even more remote layer rush into this layer, etc., etc. As a result, the flow of microorganisms to the drop is obtained, which the experimenters observed.
This hypothesis was easy to verify mathematically, and there was no need to look for a mysterious organ.
Mathematical methods made it possible to give answers to many specific questions of biology. And these answers are sometimes striking in their depth and elegance. However, it is too early to speak of mathematical biology as an established science.
Fundamentals of mathematical modeling
In this section of the course of lectures "Mathematical models in biology" are considered basic concepts mathematical modeling. On the example of the simplest systems, the main regularities of their behavior are analyzed. The focus is not on the biological system itself, but on the approaches used to create its model.
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