Mathematical modeling

1. What is mathematical modeling?

Since the middle of the XX century. in various fields of human activity, mathematical methods and computers began to be widely used. New disciplines such as "mathematical economics", "mathematical chemistry", "mathematical linguistics", etc., have emerged that study mathematical models of relevant objects and phenomena, as well as methods for studying these models.

Mathematical model- this is an approximate description of any class of phenomena or objects of the real world in the language of mathematics. The main purpose of modeling is to explore these objects and predict the results of future observations. However, modeling is also a method of cognition of the surrounding world, which makes it possible to control it.

Mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult for one reason or another. For example, it is impossible to set up a full-scale experiment in history to check “what would happen if...” It is impossible to check the correctness of this or that cosmological theory. In principle, it is possible, but hardly reasonable, to set up an experiment on the spread of some disease, such as the plague, or to carry out nuclear explosion to study its implications. However, all this can be done on a computer, having previously built mathematical models of the phenomena under study.

2. Main stages of mathematical modeling

1) Model building. At this stage, some "non-mathematical" object is specified - a natural phenomenon, construction, economic plan, production process, etc. In this case, as a rule, a clear description of the situation is difficult. First, the main features of the phenomenon and the relationship between them at a qualitative level are identified. Then the found qualitative dependencies are formulated in the language of mathematics, that is, a mathematical model is built. This is the most difficult part of the modeling.

2) Solution mathematical problem, to which the model leads. At this stage, much attention is paid to the development of algorithms and numerical methods for solving the problem on a computer, with the help of which the result can be found with the required accuracy and within the allowable time.

3) Interpretation of the obtained consequences from the mathematical model. The consequences derived from the model in the language of mathematics are interpreted in the language accepted in this field.

4) Checking the adequacy of the model. At this stage, it is found out whether the results of the experiment agree with the theoretical consequences from the model within a certain accuracy.

5) Model modification. At this stage, either the model becomes more complex so that it is more adequate to reality, or it is simplified in order to achieve a practically acceptable solution.

3. Classification of models

Models can be classified according to different criteria. For example, according to the nature of the problems being solved, models can be divided into functional and structural ones. In the first case, all quantities characterizing a phenomenon or object are expressed quantitatively. At the same time, some of them are considered as independent variables, while others are considered as functions of these quantities. A mathematical model is usually a system of equations of various types (differential, algebraic, etc.) that establish quantitative relationships between the quantities under consideration. In the second case, the model characterizes the structure of a complex object, consisting of separate parts, between which there are certain connections. Typically, these relationships are not quantifiable. To build such models, it is convenient to use graph theory. A graph is a mathematical object, which is a set of points (vertices) on a plane or in space, some of which are connected by lines (edges).

By the nature of the initial data and prediction results, models can be divided into deterministic and probabilistic-statistical. Models of the first type give definite, unambiguous predictions. Models of the second type are based on statistical information, and the predictions obtained with their help are of a probabilistic nature.

4. Examples of mathematical models

1) Problems about the movement of the projectile.

Consider the following problem in mechanics.

The projectile is launched from the Earth with an initial velocity v 0 = 30 m/s at an angle a = 45° to its surface; it is required to find the trajectory of its movement and the distance S between the start and end points of this trajectory.

Then, as it is known from the school physics course, the motion of the projectile is described by the formulas:

where t - time, g = 10 m / s 2 - free fall acceleration. These formulas give the mathematical model of the task. Expressing t in terms of x from the first equation and substituting it into the second, we get the equation for the trajectory of the projectile:

This curve (parabola) intersects the x-axis at two points: x 1 \u003d 0 (the beginning of the trajectory) and (the place where the projectile fell). Substituting the given values ​​v0 and a into the obtained formulas, we obtain

answer: y \u003d x - 90x 2, S \u003d 90 m.

Note that a number of assumptions were used in the construction of this model: for example, it is assumed that the Earth is flat, and the air and rotation of the Earth do not affect the movement of the projectile.

2) The problem of a tank with the smallest surface area.

It is required to find the height h 0 and radius r 0 of a tin tank with a volume V = 30 m 3, having the shape of a closed circular cylinder, at which its surface area S is minimal (in this case, the smallest amount of tin will be used to manufacture it).

Let's write down the following formulas for the volume and surface area of ​​a cylinder of height h and radius r:

V = p r 2 h, S = 2p r(r + h).

Expressing h in terms of r and V from the first formula and substituting the resulting expression into the second, we get:

Thus, from a mathematical point of view, the problem is reduced to determining the value of r at which the function S(r) reaches its minimum. Let us find those values ​​of r 0 for which the derivative

goes to zero: You can check that the second derivative of the function S(r) changes sign from minus to plus when the argument r passes through the point r 0 . Therefore, the function S(r) has a minimum at the point r0. The corresponding value h 0 = 2r 0 . Substituting the given value V into the expression for r 0 and h 0, we obtain the desired radius and height

3) Transport task.

There are two flour warehouses and two bakeries in the city. Every day, 50 tons of flour are exported from the first warehouse, and 70 tons from the second to the factories, with 40 tons to the first and 80 tons to the second.

Denote by a ij cost of transporting 1 ton of flour from the i-th warehouse to j-th plant(i, j = 1.2). Let be

a 11 \u003d 1.2 p., a 12 \u003d 1.6 p., a 21 \u003d 0.8 p., a 22 = 1 p.

How should transportation be planned so that their cost is minimal?

Let's give the task mathematical form alignment. Let us denote by x 1 and x 2 the amount of flour to be transported from the first warehouse to the first and second factories, and by x 3 and x 4 - from the second warehouse to the first and second factories, respectively. Then:

x 1 + x 2 = 50, x 3 + x 4 = 70, x 1 + x 3 = 40, x 2 + x 4 = 80. (1)

The total cost of all transportation is determined by the formula

f = 1.2x1 + 1.6x2 + 0.8x3 + x4.

From a mathematical point of view, the task is to find four numbers x 1 , x 2 , x 3 and x 4 that satisfy all given conditions and give the minimum of the function f. Let us solve the system of equations (1) with respect to xi (i = 1, 2, 3, 4) by the method of elimination of unknowns. We get that

x 1 \u003d x 4 - 30, x 2 \u003d 80 - x 4, x 3 \u003d 70 - x 4, (2)

and x 4 cannot be uniquely determined. Since x i i 0 (i = 1, 2, 3, 4), it follows from equations (2) that 30J x 4 J 70. Substituting the expression for x 1 , x 2 , x 3 into the formula for f, we obtain

f \u003d 148 - 0.2x 4.

It is easy to see that the minimum of this function is achieved at the maximum possible value of x 4, that is, at x 4 = 70. The corresponding values ​​of other unknowns are determined by formulas (2): x 1 = 40, x 2 = 10, x 3 = 0.

4) The problem of radioactive decay.

Let N(0) be the initial number of atoms of the radioactive substance, and N(t) be the number of undecayed atoms at time t. It has been experimentally established that the rate of change in the number of these atoms N "(t) is proportional to N (t), that is, N" (t) \u003d –l N (t), l > 0 is the radioactivity constant of a given substance. In the school course of mathematical analysis, it is shown that the solution to this differential equation has the form N(t) = N(0)e –l t . The time T, during which the number of initial atoms has halved, is called the half-life, and is an important characteristic of the radioactivity of a substance. To determine T, it is necessary to put in the formula Then For example, for radon l = 2.084 10–6, and hence T = 3.15 days.

5) The traveling salesman problem.

A traveling salesman living in city A 1 needs to visit cities A 2 , A 3 and A 4 , each city exactly once, and then return back to A 1 . It is known that all cities are connected in pairs by roads, and the lengths of roads b ij between cities A i and A j (i, j = 1, 2, 3, 4) are as follows:

b 12 = 30, b 14 = 20, b 23 = 50, b 24 = 40, b 13 = 70, b 34 = 60.

It is necessary to determine the order of visiting cities, in which the length of the corresponding path is minimal.

Let's depict each city as a point on the plane and mark it with the corresponding label Ai (i = 1, 2, 3, 4). Let's connect these points with line segments: they will depict roads between cities. For each “road”, we indicate its length in kilometers (Fig. 2). The result is a graph - a mathematical object consisting of a certain set of points on the plane (called vertices) and a certain set of lines connecting these points (called edges). Moreover, this graph is labeled, since some labels are assigned to its vertices and edges - numbers (edges) or symbols (vertices). A cycle on a graph is a sequence of vertices V 1 , V 2 , ..., V k , V 1 such that the vertices V 1 , ..., V k are different, and any pair of vertices V i , V i+1 (i = 1, ..., k – 1) and the pair V 1 , V k are connected by an edge. Thus, the problem under consideration is to find such a cycle on the graph passing through all four vertices for which the sum of all edge weights is minimal. Let's search through all the different cycles passing through four vertices and starting at A 1:

1) A 1, A 4, A 3, A 2, A 1;
2) A 1, A 3, A 2, A 4, A 1;
3) A 1 , A 3 , A 4 , A 2 , A 1 .

Now let's find the lengths of these cycles (in km): L 1 = 160, L 2 = 180, L 3 = 200. So, the route of the smallest length is the first one.

Note that if there are n vertices in a graph and all vertices are connected in pairs by edges (such a graph is called complete), then the number of cycles passing through all vertices is equal. Therefore, in our case there are exactly three cycles.

6) The problem of finding a connection between the structure and properties of substances.

Consider a few chemical compounds called normal alkanes. They consist of n carbon atoms and n + 2 hydrogen atoms (n = 1, 2 ...), interconnected as shown in Figure 3 for n = 3. Let the experimental values ​​of the boiling points of these compounds be known:

y e (3) = - 42°, y e (4) = 0°, y e (5) = 28°, y e (6) = 69°.

It is required to find an approximate relationship between the boiling point and the number n for these compounds. We assume that this dependence has the form

y » a n+b

where a, b - constants to be determined. For finding a and b we substitute into this formula successively n = 3, 4, 5, 6 and the corresponding values ​​of the boiling points. We have:

– 42 » 3 a+ b, 0 » 4 a+ b, 28 » 5 a+ b, 69 » 6 a+b.

To determine the best a and b there are many different methods. Let's use the simplest of them. We express b in terms of a from these equations:

b" - 42 - 3 a, b » – 4 a, b » 28 – 5 a, b » 69 – 6 a.

Let us take as the desired b the arithmetic mean of these values, that is, we put b » 16 - 4.5 a. Let us substitute this value b into the original system of equations and, calculating a, we get for a the following values: a» 37, a» 28, a» 28, a» 36 a the average value of these numbers, that is, we set a» 34. So, the desired equation has the form

y » 34n – 139.

Let's check the accuracy of the model on the initial four compounds, for which we calculate the boiling points using the obtained formula:

y r (3) = – 37°, y r (4) = – 3°, y r (5) = 31°, y r (6) = 65°.

Thus, the calculation error of this property for these compounds does not exceed 5°. We use the resulting equation to calculate the boiling point of a compound with n = 7, which is not included in the initial set, for which we substitute n = 7 into this equation: y р (7) = 99°. The result turned out to be quite accurate: it is known that the experimental value of the boiling point y e (7) = 98°.

7) The problem of determining the reliability of the electrical circuit.

Here we consider an example of a probabilistic model. First, let's give some information from the theory of probability - a mathematical discipline that studies the patterns of random phenomena observed during repeated repetition of an experiment. Let's call a random event A a possible outcome of some experience. Events A 1 , ..., A k form a complete group if one of them necessarily occurs as a result of the experiment. Events are called incompatible if they cannot occur simultaneously in the same experience. Let the event A occur m times during the n-fold repetition of the experiment. The frequency of the event A is the number W = . Obviously, the value of W cannot be predicted exactly until a series of n experiments has been carried out. However, the nature of random events is such that in practice the following effect is sometimes observed: with an increase in the number of experiments, the value practically ceases to be random and stabilizes around some non-random number P(A), called the probability of the event A. For an impossible event (which never occurs in the experiment) P(A)=0, and for a certain event (which always occurs in the experiment) P(A)=1. If events A 1 , ..., A k form a complete group of incompatible events, then P(A 1)+...+P(A k)=1.

Let, for example, the experience consists in throwing a dice and observing the number of dropped points X. Then we can introduce the following random events A i =(X = i), i = 1, ..., 6. They form a complete group of incompatible equally probable events, therefore P(A i) = (i = 1, ..., 6).

The sum of events A and B is the event A + B, which consists in the fact that at least one of them occurs in the experiment. The product of events A and B is the event AB, which consists in the simultaneous occurrence of these events. For independent events A and B, the formulas are true

P(AB) = P(A) P(B), P(A + B) = P(A) + P(B).

8) Consider now the following task. Suppose that three elements are connected in series in an electric circuit, working independently of each other. The failure probabilities of the 1st, 2nd and 3rd elements are respectively P 1 = 0.1, P 2 = 0.15, P 3 = 0.2. We will consider the circuit reliable if the probability that there will be no current in the circuit is not more than 0.4. It is required to determine whether the given chain is reliable.

Since the elements are connected in series, there will be no current in the circuit (event A) if at least one of the elements fails. Let A i be the event that i-th element works (i = 1, 2, 3). Then P(A1) = 0.9, P(A2) = 0.85, P(A3) = 0.8. Obviously, A 1 A 2 A 3 is the event that all three elements work simultaneously, and

P(A 1 A 2 A 3) = P(A 1) P(A 2) P(A 3) = 0.612.

Then P(A) + P(A 1 A 2 A 3) = 1, so P(A) = 0.388< 0,4. Следовательно, цепь является надежной.

In conclusion, we note that the above examples of mathematical models (among which there are functional and structural, deterministic and probabilistic) are illustrative and, obviously, do not exhaust the whole variety of mathematical models that arise in the natural and human sciences.

Lecture 1

METHODOLOGICAL BASES OF MODELING

The current state of the problem of system modeling

Concepts of Modeling and Simulation

Modeling can be considered as a replacement of the investigated object (original) by its conditional image, description or another object, called model and providing behavior close to the original within certain assumptions and acceptable errors. Modeling is usually performed with the aim of knowing the properties of the original by examining its model, and not the object itself. Of course, modeling is justified in the case when it is simpler than creating the original itself, or when the latter, for some reason, is better not to create at all.

Under model a physical or abstract object is understood, the properties of which are in a certain sense similar to the properties of the object under study. In this case, the requirements for the model are determined by the problem being solved and the available means. There are a number of general requirements for models:

2) completeness - providing the recipient with all the necessary information

about the object;

3) flexibility - the ability to reproduce different situations in everything

range of changing conditions and parameters;

4) the complexity of development should be acceptable for the existing

time and software.

Modeling is the process of building a model of an object and studying its properties by examining the model.

Thus, modeling involves 2 main stages:

1) model development;

2) study of the model and drawing conclusions.

At the same time, at each stage, different tasks are solved and

essentially different methods and means.

In practice, various modeling methods are used. Depending on the method of implementation, all models can be divided into two large classes: physical and mathematical.

Mathematical modeling It is customary to consider it as a means of studying processes or phenomena with the help of their mathematical models.

Under physical modeling is understood as the study of objects and phenomena on physical models, when the process under study is reproduced with the preservation of its physical nature or another physical phenomenon similar to the one under study is used. Wherein physical models As a rule, they assume the real embodiment of those physical properties of the original that are essential in a particular situation. For example, when designing a new aircraft, its model is created that has the same aerodynamic properties; when planning a building, architects make a layout that reflects the spatial arrangement of its elements. In this regard, physical modeling is also called prototyping.

HIL Modeling is a study of controlled systems on simulation complexes with the inclusion of real equipment in the model. Along with real equipment, the closed model includes impact and interference simulators, mathematical models of the external environment and processes, for which a sufficiently accurate mathematical description is not known. The inclusion of real equipment or real systems in the circuit for modeling complex processes makes it possible to reduce a priori uncertainty and explore processes for which there is no exact mathematical description. With the help of semi-natural simulation, studies are performed taking into account small time constants and non-linearities inherent in real equipment. In the study of models with the inclusion of real equipment, the concept is used dynamic simulation, in the study complex systems and phenomena - evolutionary, imitation And cybernetic simulation.

Obviously, the real benefit of modeling can only be obtained if two conditions are met:

1) the model provides a correct (adequate) display of properties

the original, significant from the point of view of the operation under study;

2) the model makes it possible to eliminate the problems listed above, which are inherent

conducting research on real objects.

2. Basic concepts of mathematical modeling

The solution of practical problems by mathematical methods is consistently carried out by formulating the problem (development of a mathematical model), choosing a method for studying the obtained mathematical model, and analyzing the obtained mathematical result. The mathematical formulation of the problem is usually presented in the form of geometric images, functions, systems of equations, etc. The description of an object (phenomenon) can be represented using continuous or discrete, deterministic or stochastic and other mathematical forms.

Theory of mathematical modeling ensures the identification of regularities in the flow of various phenomena of the surrounding world or the operation of systems and devices by their mathematical description and modeling without field tests. In this case, the provisions and laws of mathematics are used that describe the simulated phenomena, systems or devices at a certain level of their idealization.

Mathematical Model (MM) is a formalized description of a system (or operation) in some abstract language, for example, in the form of a set of mathematical relations or an algorithm scheme, i.e. e. such a mathematical description that provides an imitation of the operation of systems or devices at a level sufficiently close to their real behavior obtained during full-scale tests of systems or devices.

Any MM describes a real object, phenomenon or process with some degree of approximation to reality. The type of MM depends both on nature real object, as well as on the objectives of the study.

Mathematical modeling social, economic, biological and physical phenomena, objects, systems and various devices is one of the most important means of understanding nature and designing a wide variety of systems and devices. There are known examples of the effective use of modeling in the creation of nuclear technologies, aviation and aerospace systems, in the forecast of atmospheric and oceanic phenomena, weather, etc.

However, such serious areas of modeling often require supercomputers and years of work by large teams of scientists to prepare data for modeling and its debugging. Nevertheless, in this case, too, mathematical modeling of complex systems and devices not only saves money on research and testing, but can also eliminate environmental disasters - for example, it makes it possible to abandon the testing of nuclear and thermonuclear weapons in favor of their mathematical modeling or testing aerospace systems before their real flights. Meanwhile, mathematical modeling at the level of solving simpler problems, for example, from the field of mechanics, electrical engineering, electronics, radio engineering and many other areas of science and technology, has now become available to perform on modern PCs. And when using generalized models, it becomes possible to model quite complex systems, for example, telecommunication systems and networks, radar or radio navigation systems.

The purpose of mathematical modeling is the analysis of real processes (in nature or technology) by mathematical methods. In turn, this requires the formalization of the MM process to be investigated. The model can be a mathematical expression containing variables whose behavior is similar to the behavior of a real system. The model can include elements of randomness that take into account the probabilities of possible actions of two or more"players", as, for example, in game theory; or it may represent the real variables of the interconnected parts of the operating system.

Mathematical modeling for studying the characteristics of systems can be divided into analytical, simulation and combined. In turn, MM are divided into simulation and analytical.

Analytical Modeling

For analytical modeling it is characteristic that the processes of functioning of the system are written in the form of some functional relations (algebraic, differential, integral equations). The analytical model can be investigated by the following methods:

1) analytical, when they strive to obtain in general terms explicit dependencies for the characteristics of systems;

2) numerical, when it is not possible to find a solution to equations in general form and they are solved for specific initial data;

3) qualitative, when, in the absence of a solution, some of its properties are found.

Analytical models can be obtained only for relatively simple systems. For complex systems, large mathematical problems often arise. To apply the analytical method, one goes to a significant simplification of the original model. However, a study on a simplified model helps to obtain only indicative results. Analytical models mathematically correctly reflect the relationship between input and output variables and parameters. But their structure does not reflect the internal structure of the object.

In analytical modeling, its results are presented in the form of analytical expressions. For example, by connecting RC- circuit to a constant voltage source E(R, C And E are the components of this model), we can make an analytical expression for the time dependence of the voltage u(t) on the capacitor C:

This is a linear differential equation (DE) and is an analytical model of this simple linear circuit. Its analytical solution, under the initial condition u(0) = 0 , meaning a discharged capacitor C at the beginning of the simulation, allows you to find the required dependence - in the form of a formula:

u(t) = E(1− exp(- t/RC)). (2)

However, even in this simplest example, certain efforts are required to solve differential equation (1) or to apply computer mathematics systems(SCM) with symbolic calculations - computer algebra systems. For this quite trivial case, the solution of the problem of modeling a linear RC-circuit gives an analytical expression (2) of a rather general form - it is suitable for describing the operation of the circuit for any component ratings R, C And E, and describes the exponential charge of the capacitor C through a resistor R from a constant voltage source E.

Undoubtedly, finding analytical solutions in analytical modeling turns out to be extremely valuable for revealing the general theoretical laws of simple linear circuits, systems and devices. However, its complexity increases sharply as the impact on the model becomes more complex and the order and number of equations of state that describe the modeled object increase. You can get more or less visible results when modeling objects of the second or third order, but even with a higher order, analytical expressions become excessively cumbersome, complex and difficult to comprehend. For example, even a simple electronic amplifier often contains dozens of components. However, many modern SCMs, such as systems of symbolic mathematics Maple, Mathematica or Wednesday MATLAB are capable of automating to a large extent the solution of complex problems of analytical modeling.

One type of modeling is numerical simulation, which consists in obtaining the necessary quantitative data about the behavior of systems or devices by any suitable numerical method, such as the Euler or Runge-Kutta methods. In practice, the modeling of nonlinear systems and devices using numerical methods is much more efficient than the analytical modeling of individual private linear circuits, systems or devices. For example, to solve DE (1) or DE systems in more complex cases, the solution in an analytical form is not obtained, but numerical simulation data can provide sufficiently complete data on the behavior of the simulated systems and devices, as well as plot graphs describing this behavior of dependencies.

Simulation

At imitation In modeling, the algorithm that implements the model reproduces the process of the system functioning in time. The elementary phenomena that make up the process are imitated, with the preservation of their logical structure and the sequence of flow in time.

The main advantage of simulation models compared to analytical ones is the ability to solve more complex problems.

Simulation models make it easy to take into account the presence of discrete or continuous elements, non-linear characteristics, random effects, etc. Therefore, this method is widely used at the design stage of complex systems. The main tool for the implementation of simulation modeling is a computer that allows digital modeling of systems and signals.

In this regard, we define the phrase " computer modelling”, which is increasingly used in the literature. We will assume that computer modelling- this is mathematical modeling using computer technology. Accordingly, computer simulation technology involves the following actions:

1) definition of the purpose of modeling;

2) development of a conceptual model;

3) formalization of the model;

4) software implementation of the model;

5) planning of model experiments;

6) implementation of the experiment plan;

7) analysis and interpretation of simulation results.

At simulation modeling the used MM reproduces the algorithm (“logic”) of the functioning of the system under study in time for various combinations of values ​​of the parameters of the system and the environment.

An example of the simplest analytical model is the equation of uniform rectilinear motion. When studying such a process using a simulation model, observation of the change in the path traveled over time should be implemented. Obviously, in some cases, analytical modeling is more preferable, in others - simulation (or a combination of both). To make a good choice, two questions must be answered.

What is the purpose of modeling?

To what class can the simulated phenomenon be assigned?

Answers to both of these questions can be obtained during the execution of the first two stages of modeling.

Simulation models not only in properties, but also in structure correspond to the object being modeled. In this case, there is an unambiguous and explicit correspondence between the processes obtained on the model and the processes occurring on the object. The disadvantage of simulation modeling is that it takes a long time to solve the problem in order to obtain good accuracy.

The results of simulation modeling of the work of a stochastic system are realizations of random variables or processes. Therefore, to find the characteristics of the system, multiple repetition and subsequent data processing are required. Most often, in this case, a type of simulation is used - statistical

modeling(or the Monte Carlo method), i.e. reproduction in models of random factors, events, quantities, processes, fields.

Based on the results of statistical modeling, estimates of probabilistic quality criteria, general and particular, characterizing the functioning and efficiency of the controlled system are determined. Statistical modeling is widely used to solve scientific and applied problems in various fields of science and technology. Methods of statistical modeling are widely used in the study of complex dynamic systems, evaluation of their functioning and efficiency.

The final stage of statistical modeling is based on the mathematical processing of the obtained results. Here, methods of mathematical statistics are used (parametric and non-parametric estimation, hypothesis testing). An example of a parametric assessment is the sample mean of a performance measure. Among the nonparametric methods, the most widely used histogram method.

The considered scheme is based on multiple statistical tests of the system and methods of statistics of independent random variables. This scheme is far from always natural in practice and optimal in terms of costs. Reduction of system testing time can be achieved by using more accurate estimation methods. As is known from mathematical statistics, effective estimates have the highest accuracy for a given sample size. Optimal filtering and the maximum likelihood method give general method obtaining such estimates. In the problems of statistical modeling, the processing of realizations of random processes is necessary not only for the analysis of output processes.

It is also very important to control the characteristics of input random effects. The control consists in checking whether the distributions of the generated processes correspond to the given distributions. This task is often formulated as hypothesis testing task.

The general trend in computer-assisted simulation of complex controlled systems is the desire to reduce the simulation time, as well as to conduct research in real time. Computational algorithms are conveniently represented in a recurrent form that allows their implementation at the pace of current information.

PRINCIPLES OF A SYSTEM APPROACH IN MODELING

Fundamentals of systems theory

The main provisions of the theory of systems arose in the course of the study of dynamic systems and their functional elements. A system is understood as a group of interrelated elements acting together to perform a predetermined task. Systems analysis allows you to determine the most real ways accomplishment of the set task, ensuring the maximum satisfaction of the set requirements.

The elements that form the basis of systems theory are not created with the help of hypotheses, but are discovered experimentally. In order to start building a system, it is necessary to have general characteristics of technological processes. The same is true for the principles of creating mathematically formulated criteria that a process or its theoretical description must satisfy. Modeling is one of the most important methods of scientific research and experimentation.

When building models of objects, a systematic approach is used, which is a methodology for solving complex problems, which is based on the consideration of an object as a system operating in a certain environment. The system approach involves the disclosure of the integrity of the object, the identification and study of its internal structure, as well as connections with the external environment. In this case, the object is presented as a part of the real world, which is identified and studied in connection with the problem of building a model being solved. Besides, systems approach involves a consistent transition from the general to the particular, when the consideration is based on the design goal, and the object is considered in relation to the environment.

A complex object can be divided into subsystems, which are parts of the object that meet the following requirements:

1) the subsystem is a functionally independent part of the object. It is connected with other subsystems, exchanges information and energy with them;

2) for each subsystem, functions or properties that do not coincide with the properties of the entire system can be defined;

3) each of the subsystems can be further subdivided to the level of elements.

In this case, an element is understood as a subsystem of the lower level, the further division of which is inexpedient from the standpoint of the problem being solved.

Thus, a system can be defined as a representation of an object in the form of a set of subsystems, elements, and relationships for the purpose of its creation, research, or improvement. At the same time, an enlarged representation of the system, which includes the main subsystems and connections between them, is called a macrostructure, and a detailed disclosure of the internal structure of the system to the level of elements is called a microstructure.

Along with the system, there is usually a supersystem - a system of a higher level, which includes the object under consideration, and the function of any system can be determined only through the supersystem.

It is necessary to single out the concept of the environment as a set of objects of the external world that significantly affect the efficiency of the system, but are not part of the system and its supersystem.

In connection with the systematic approach to building models, the concept of infrastructure is used, which describes the relationship of the system with its environment (environment). In this case, the selection, description and study of the properties of an object that are essential within a specific task is called the stratification of an object, and any model of an object is its stratified description.

For a systematic approach, it is important to determine the structure of the system, i.e. set of links between the elements of the system, reflecting their interaction. To do this, we first consider the structural and functional approaches to modeling.

With a structural approach, the composition of the selected elements of the system and the links between them are revealed. The totality of elements and relationships makes it possible to judge the structure of the system. The most general description of a structure is a topological description. It allows you to define the components of the system and their relationships using graphs. Less general is the functional description when individual functions are considered, i.e., algorithms for the behavior of the system. At the same time, a functional approach is implemented that determines the functions that the system performs.

On the basis of a systematic approach, a sequence of model development can be proposed, when two main stages of design are distinguished: macro-design and micro-design.

At the macro-design stage, a model of the external environment is built, resources and constraints are identified, a system model and criteria for assessing adequacy are selected.

The stage of microdesign largely depends on the specific type of model chosen. In the general case, it involves the creation of information, mathematical, technical and software support for the modeling system. At this stage, the main technical characteristics of the created model are established, the time of working with it and the cost of resources to obtain the specified quality of the model are estimated.

Regardless of the type of model, when building it, it is necessary to be guided by a number of principles of a systematic approach:

1) consistent progress through the stages of creating a model;

2) coordination of information, resource, reliability and other characteristics;

3) the correct ratio of different levels of model building;

4) the integrity of the individual stages of model design.

Instruction

The method of statistical modeling (statistical tests) is commonly known as the "Monte Carlo" method. This method is a special case of mathematical modeling and is based on the creation of probabilistic models of random phenomena. The basis of any random is a random variable or a random process. In this case, a random process from a probabilistic point of view is described as an n-dimensional random variable. Full probabilistic random variable gives its probability density. The knowledge of this distribution law makes it possible to obtain digital models of random processes on a computer, not full-scale experiments with them. All this is possible only in a discrete form and in discrete time, which must be taken into account when creating static models.

In static modeling, one should move away from considering a specific phenomenon, focusing only on its probabilistic characteristics. This makes it possible to use for modeling the simplest phenomena that have probabilistic indicators with the simulated phenomenon. For example, any event that occurs with a probability of 0.5 can be simulated by simply tossing a symmetrical coin. Each individual stage of statistical modeling is called a raffle. So, to determine the estimate of the mathematical expectation, N draws of a random variable (CV) X will be required.

The main tool for modeling on a computer are the sensors of random numbers uniform on the interval (0, 1). So, in the Pascal environment, such a random number is called using the Random command. On calculators, the RND button is provided for this case. There are also tables of such random numbers (up to 1,000,000 in size). The value of the uniform on (0, 1) SW Z is denoted by z.

Consider a technique for modeling an arbitrary random variable using a nonlinear transformation of the distribution function. This method does not have methodological errors. Let the law of distribution of continuous SW X be given by the probability density W(x). From here, start preparing for the simulation and its implementation.

Find the distribution function X - F(x). F(x)=∫(-∞,x)W(s)ds. Take Z=z and solve the equation z=F(x) with respect to x (this is always possible since both Z and F(x) range from zero to one). Write down the solution x=F^(-1)( z). This is the modeling algorithm. F^(-1) is the inverse of F. It remains only to consistently obtain the values ​​xi of the digital model X* CD X using this algorithm.

Example. SW is given by the probability density W(x)=λexp(-λx), x≥0 (exponential distribution). Find a digital model. Solution.1.. F(x)=∫(0,x)λ∙exp(-λs)ds=1- exp(-λx).2. z=1-exp(-λx), x=(-1/λ)∙ln(1-z). Since both z and 1-z have values ​​in the interval (0, 1) and they are uniform, then (1-z) can be replaced by z. 3. The procedure for modeling exponential SW is carried out according to the formula x=(-1/λ)∙lnz. More precisely, xi=(-1/λ)ln(zi).

What is a mathematical model?

The concept of a mathematical model.

A mathematical model is a very simple concept. And very important. It is mathematical models that connect mathematics and real life.

talking plain language, a mathematical model is a mathematical description of any situation. And that's it. The model can be primitive, it can be super complex. What is the situation, what is the model.)

In any (I repeat - in any!) business, where you need to calculate something and calculate - we are engaged in mathematical modeling. Even if we don't know it.)

P \u003d 2 CB + 3 CB

This record will be the mathematical model of the expenses for our purchases. The model does not take into account the color of the packaging, expiration date, politeness of cashiers, etc. That's why she model, not a real purchase. But the costs, ie. what we need- we'll know for sure. If the model is correct, of course.

It is useful to imagine what a mathematical model is, but this is not enough. The most important thing is to be able to build these models.

Compilation (construction) of a mathematical model of the problem.

To compose a mathematical model means to translate the conditions of the problem into a mathematical form. Those. turn words into an equation, formula, inequality, etc. Moreover, turn it so that this mathematics strictly corresponds to the original text. Otherwise, we will end up with a mathematical model of some other problem unknown to us.)

More specifically, you need

There are an infinite number of tasks in the world. Therefore, to propose a clear step by step instructions on drawing up a mathematical model any tasks are impossible.

But there are three main points that you need to pay attention to.

1. In any task there is a text, oddly enough.) This text, as a rule, has explicit, open information. Numbers, values, etc.

2. In any task there is hidden information. This is a text that assumes the presence of additional knowledge in the head. Without them - nothing. In addition, mathematical information is often hidden behind in simple words and ... slips past attention.

3. In any task there must be given communication between data. This connection can be given in clear text (something equals something), or it can be hidden behind simple words. But simple and clear facts are often overlooked. And the model is not compiled in any way.

I must say right away that in order to apply these three points, the problem has to be read (and carefully!) several times. The usual thing.

And now - examples.

Let's start with a simple problem:

Petrovich returned from fishing and proudly presented his catch to his family. Upon closer examination, it turned out that 8 fish come from northern seas, 20% of all fish are from the south, and there is not a single one from the local river where Petrovich fished. How many fish did Petrovich buy in the Seafood store?

All these words need to be turned into some kind of equation. To do this, I repeat, establish a mathematical relationship between all the data of the problem.

Where to start? First, we will extract all the data from the task. Let's start in order:

Let's focus on the first point.

What is here explicit mathematical information? 8 fish and 20%. Not a lot, but we don't need a lot.)

Let's pay attention to the second point.

Are looking for covert information. She is here. These are the words: "20% of all fish". Here you need to understand what percentages are and how they are calculated. Otherwise, the task is not solved. This is exactly the additional information that should be in the head.

There is also here mathematical information that is completely invisible. This task question: "How many fish did you buy... It's also a number. And without it, no model will be compiled. Therefore, let us denote this number by the letter "X". We do not yet know what x is equal to, but such a designation will be very useful to us. For more information on what to take for x and how to handle it, see the lesson How to solve math problems? Let's write it right away:

x pieces - the total number of fish.

In our problem, southern fish are given as a percentage. We need to translate them into pieces. What for? Then what's in any the task of the model should be in the same quantities. Pieces - so everything is in pieces. If we are given, let's say hours and minutes, we translate everything into one thing - either only hours, or only minutes. It doesn't matter what. It is important to all values ​​were the same.

Back to disclosure. Whoever does not know what a percentage is will never reveal, yes ... And who knows, he will immediately say that interest here is from total number fish are given. We don't know this number. Nothing will come of it!

The total number of fish (in pieces!) is not in vain with the letter "X" designated. It will not work to count the southern fish in pieces, but can we write it down? Like this:

0.2 x pieces - the number of fish from the southern seas.

Now we have downloaded all the information from the task. Both explicit and hidden.

Let's pay attention to the third point.

Are looking for mathematical connection between task data. This connection is so simple that many do not notice it... This often happens. Here it is useful to simply write down the collected data in a bunch, and see what's what.

What do we have? There is 8 pieces northern fish, 0.2 x pieces- southern fish and x fish- total amount. Is it possible to link this data somehow together? Yes Easy! total number of fish equals sum of southern and northern! Well, who would have thought ...) So we write down:

x = 8 + 0.2x

This will be the equation mathematical model of our problem.

Please note that in this problem we are not asked to fold anything! It was we ourselves, out of our heads, who realized that the sum of the southern and northern fish would give us the total number. The thing is so obvious that it slips past attention. But without this evidence, a mathematical model cannot be compiled. Like this.

Now you can apply all the power of mathematics to solve this equation). This is what the mathematical model was designed for. We solve this linear equation and get the answer.

Answer: x=10

Let's make a mathematical model of another problem:

Petrovich was asked: "How much money do you have?" Petrovich wept and answered: “Yes, just a little bit. If I spend half of all the money, and half of the rest, then I will have only one bag of money left ...” How much money does Petrovich have?

Again, we work point by point.

1. We are looking for explicit information. You won't find it right away! Explicit information is one money bag. There are some other halves... Well, we'll sort it out in the second paragraph.

2. We are looking for hidden information. These are halves. What? Not very clear. Looking for more. There is another issue: "How much money does Petrovich have?" Let's denote the amount of money by the letter "X":

X- all the money

And read the problem again. Already knowing that Petrovich X money. This is where the halves work! We write down:

0.5 x- half of all money.

The remainder will also be half, i.e. 0.5 x. And half of the half can be written like this:

0.5 0.5 x = 0.25x- half of the remainder.

Now all the hidden information is revealed and recorded.

3. We are looking for a connection between the recorded data. Here you can simply read the sufferings of Petrovich and write them down mathematically):

If I spend half of all the money...

Let's write down this process. All money - X. Half - 0.5 x. To spend is to take away. The phrase becomes:

x - 0.5 x

and half of the rest...

Subtract another half of the remainder:

x - 0.5 x - 0.25 x

then only one bag of money will remain with me ...

And there is equality! After all the subtractions, one bag of money remains:

x - 0.5 x - 0.25x \u003d 1

Here it is, the mathematical model! This is again a linear equation, we solve, we get:

Question for consideration. Four is what? Ruble, dollar, yuan? And in what units do we have money in the mathematical model? In bags! So four bag Petrovich's money. It's not bad too.)

The tasks are, of course, elementary. This is specifically to capture the essence of drawing up a mathematical model. In some tasks, there may be much more data in which it is easy to get confused. This often happens in the so-called. competency tasks. How to pull mathematical content out of a pile of words and numbers is shown with examples

One more note. In classical school problems (pipes fill the pool, boats are sailing somewhere, etc.), all the data, as a rule, is chosen very carefully. There are two rules:
- there is enough information in the problem to solve it,
- there is no extra information in the task.

This is a hint. If there is some unused value in the mathematical model, think about whether there is an error. If there is not enough data in any way, most likely, not all hidden information has been revealed and recorded.

In competence and other life tasks, these rules are not strictly observed. I don't have a hint. But such problems can also be solved. Unless, of course, practice on the classic.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

According to the textbook of Sovetov and Yakovlev: "a model (Latin modulus - measure) is an object-substitute of the original object, providing the study of some properties of the original." (p. 6) "Replacing one object with another in order to obtain information about the most important properties of the original object with the help of a model object is called modeling." (p. 6) “Under mathematical modeling we will understand the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model, and the study of this model, which allows obtaining the characteristics of the real object under consideration. The type of mathematical model depends on both the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem.

Finally, the most concise definition of a mathematical model: "An equation expressing the idea».

Model classification

Formal classification of models

The formal classification of models is based on the classification of the mathematical tools used. Often built in the form of dichotomies. For example, one of the popular sets of dichotomies is:

etc. Each constructed model is linear or non-linear, deterministic or stochastic, ... Naturally, mixed types are also possible: concentrated in one respect (in terms of parameters), distributed models in another, etc.

Classification by the way the object is represented

Along with the formal classification, the models differ in the way they represent the object:

• Structural or functional models

Structural Models represent an object as a system with its own device and mechanism of functioning. functional models do not use such representations and reflect only the externally perceived behavior (functioning) of the object. In their extreme expression, they are also called "black box" models. Combined types of models are also possible, which are sometimes referred to as "models" gray box».

Content and formal models

Almost all authors describing the process of mathematical modeling indicate that first a special ideal construction is built, content model. There is no established terminology here, and other authors call this ideal object conceptual model , speculative model or premodel. In this case, the final mathematical construction is called formal model or just a mathematical model obtained as a result of the formalization of this content model (pre-model). The construction of a meaningful model can be carried out using a set of ready-made idealizations, as in mechanics, where ideal springs, solid bodies, ideal pendulums, elastic media, etc. give ready-made structural elements for meaningful modeling. However, in areas of knowledge where there are no fully completed formalized theories (the cutting edge of physics, biology, economics, sociology, psychology, and most other fields), the creation of meaningful models is dramatically more complicated.

Meaningful classification of models

No hypothesis in science can be proven once and for all. Richard Feynman put it very clearly:

“We always have the ability to disprove a theory, but note that we can never prove that it is correct. Let's suppose that you put forward a successful hypothesis, calculate where it leads, and find that all its consequences are confirmed experimentally. Does this mean that your theory is correct? No, it simply means that you failed to refute it.

If a model of the first type is built, then this means that it is temporarily recognized as true and one can concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of the model of the first type can only be temporary.

Type 2: Phenomenological model (behave as if…)

The phenomenological model contains a mechanism for describing the phenomenon. However, this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or does not agree well with the available theories and accumulated knowledge about the object. Therefore, phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown and it is necessary to continue the search for "true mechanisms". Peierls refers, for example, the caloric model and the quark model of elementary particles to the second type.

The role of the model in research may change over time, it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Likewise, new knowledge may gradually come into conflict with models-hypotheses of the first type, and they may be transferred to the second. Thus, the quark model is gradually moving into the category of hypotheses; atomism in physics arose as a temporary solution, but with the course of history it passed into the first type. But the ether models have gone from type 1 to type 2, and now they are outside of science.

The idea of ​​simplification is very popular when building models. But simplification is different. Peierls distinguishes three types of simplifications in modeling.

Type 3: Approximation (something is considered very large or very small)

If it is possible to construct equations describing the system under study, this does not mean that they can be solved even with the help of a computer. A common technique in this case is the use of approximations (models of type 3). Among them linear response models. The equations are replaced by linear ones. The standard example is Ohm's law.

And here is type 8, which is widely used in mathematical models of biological systems.

Type 8: Possibility demonstration (the main thing is to show the internal consistency of the possibility)

These are also thought experiments. with imaginary entities demonstrating that supposed phenomenon consistent with basic principles and internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions.

One of the most famous of these experiments is Lobachevsky's geometry (Lobachevsky called it "imaginary geometry"). Another example is the mass production of formally kinetic models of chemical and biological oscillations, autowaves, etc. The Einstein-Podolsky-Rosen paradox was conceived as a type 7 model to demonstrate inconsistency quantum mechanics. In a completely unplanned way, it eventually turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information.

Example

Let us consider a mechanical system consisting of a spring fixed at one end and a load of mass , attached to the free end of the spring. We will assume that the load can move only in the direction of the spring axis (for example, the movement occurs along the rod). Let us construct a mathematical model of this system. We will describe the state of the system by the distance from the center of the load to its equilibrium position. Let us describe the interaction of a spring and a load using Hooke's law() after which we use Newton's second lawto express it in the form of a differential equation:

where means the second derivative of with respect to time: .

The resulting equation describes the mathematical model of the considered physical system. This pattern is called the "harmonic oscillator".

According to the formal classification, this model is linear, deterministic, dynamic, concentrated, continuous. In the process of its construction, we made many assumptions (about the absence of external forces, the absence of friction, the smallness of deviations, etc.), which in reality may not be fulfilled.

In relation to reality, this is most often a type 4 model. simplification(“we omit some details for clarity”), since some essential universal features (for example, dissipation) are omitted. In some approximation (say, as long as the deviation of the load from equilibrium is small, with little friction, for a not too long time and subject to certain other conditions), such a model describes a real mechanical system quite well, since the discarded factors have a negligible effect on its behavior . However, the model can be refined by taking into account some of these factors. This will lead to a new model, with a wider (though again limited) scope.

However, when the model is refined, the complexity of its mathematical study can increase significantly and make the model virtually useless. Often, a simpler model allows you to better and deeper explore the real system than a more complex (and, formally, “more correct”) one.

If we apply the harmonic oscillator model to objects that are far from physics, its meaningful status may be different. For example, when applying this model to biological populations, it should most likely be attributed to type 6 analogy(“Let’s take into account only some features”).

Hard and soft models

The harmonic oscillator is an example of a so-called "hard" model. It is obtained as a result of a strong idealization of a real physical system. To resolve the issue of its applicability, it is necessary to understand how significant are the factors that we have neglected. In other words, it is necessary to investigate the "soft" model, which is obtained by a small perturbation of the "hard" one. It can be given, for example, by the following equation:

Here - some function, which can take into account the friction force or the dependence of the coefficient of stiffness of the spring on the degree of its stretching - some small parameter. The explicit form of the function does not interest us at the moment. If we prove that the behavior of a soft model does not fundamentally differ from the behavior of a hard one (regardless of the explicit form of the perturbing factors, if they are small enough), the problem will be reduced to studying the hard model. Otherwise, the application of the results obtained in the study of the rigid model will require additional research. For example, the solution to the equation of a harmonic oscillator are functions of the form , that is, oscillations with a constant amplitude. Does it follow from this that a real oscillator will oscillate indefinitely with a constant amplitude? No, because considering a system with an arbitrarily small friction (always present in a real system), we get damped oscillations. The behavior of the system has changed qualitatively.

If a system retains its qualitative behavior under a small perturbation, it is said to be structurally stable. The harmonic oscillator is an example of a structurally unstable (non-rough) system. However, this model can be used to study processes over limited time intervals.

Universality of models

The most important mathematical models usually have important property universality: fundamentally different real phenomena can be described by the same mathematical model. Let's say that the harmonic oscillator describes not only the behavior of the load on the spring, but also other oscillatory processes, often having a completely different nature: small oscillations of a pendulum, fluctuations in the level of a liquid in a -shaped vessel, or a change in the current strength in an oscillatory circuit. Thus, studying one mathematical model, we study at once a whole class of phenomena described by it. It is this isomorphism of laws expressed by mathematical models in various segments scientific knowledge, Ludwig von Bertalanffy's feat of creating "General Systems Theory".

Direct and inverse problems of mathematical modeling

There are many problems associated with mathematical modeling. First, it is necessary to come up with the basic scheme of the object being modeled, to reproduce it within the framework of the idealizations of this science. So, a train car turns into a system of plates and more complex bodies made of different materials, each material is given as its standard mechanical idealization (density, elastic moduli, standard strength characteristics), after which equations are drawn up, along the way some details are discarded as insignificant , calculations are made, compared with measurements, the model is refined, and so on. However, for the development of mathematical modeling technologies, it is useful to disassemble this process into its main constituent elements.

Traditionally, there are two main classes of problems associated with mathematical models: direct and inverse.

Direct problem: the structure of the model and all its parameters are considered known, the main task is to study the model to extract useful knowledge about the object. What static load can the bridge withstand? How it will react to a dynamic load (for example, to the march of a company of soldiers, or to the passage of a train at different speeds), how the plane will overcome the sound barrier, whether it will fall apart from flutter - these are typical examples of a direct task. Setting the correct direct problem (asking the correct question) requires special skill. If the right questions are not asked, the bridge can collapse, even if a good model has been built for its behavior. So, in 1879, a metal bridge across the River Tey collapsed in Great Britain, the designers of which built a bridge model, calculated it for a 20-fold margin of safety for the payload, but forgot about the winds constantly blowing in those places. And after a year and a half it collapsed.

In the simplest case (one oscillator equation, for example), the direct problem is very simple and reduces to an explicit solution of this equation.

Inverse problem: many possible models are known, it is necessary to choose a specific model based on additional data about the object. Most often, the structure of the model is known and some unknown parameters need to be determined. Additional information may consist in additional empirical data, or in the requirements for the object ( design task). Additional data can come regardless of the process of solving the inverse problem ( passive observation) or be the result of an experiment specially planned during the solution ( active surveillance).

One of the first examples of a virtuoso solution of an inverse problem with the fullest possible use of available data was the method constructed by I. Newton for reconstructing friction forces from observed damped oscillations.

Another example is mathematical statistics. The task of this science is the development of methods for recording, describing and analyzing observational and experimental data in order to build probabilistic models of mass random phenomena. Those. the set of possible models is limited by probabilistic models. In specific problems, the set of models is more limited.

Computer simulation systems

To support mathematical modeling, computer mathematics systems have been developed, for example, Maple, Mathematica, Mathcad, MATLAB, VisSim, etc. They allow you to create formal and block models of both simple and complex processes and devices and easily change model parameters during simulation. Block Models are represented by blocks (most often graphical), the set and connection of which are specified by the model diagram.

Additional examples

Malthus model

The growth rate is proportional to the current population size. It is described by the differential equation

where is a certain parameter determined by the difference between the birth rate and the death rate. The solution to this equation is exponential function. If the birth rate exceeds the death rate (), the population size increases indefinitely and very quickly. It is clear that in reality this cannot happen due to limited resources. When a certain critical population size is reached, the model ceases to be adequate, since it does not take into account the limited resources. A refinement of the Malthus model can be the logistic model , which is described by the Verhulst differential equation

where is the "equilibrium" population size, at which the birth rate is exactly compensated by the death rate. The population size in such a model tends to the equilibrium value , and this behavior is structurally stable.

predator-prey system

Let's say that two types of animals live in a certain area: rabbits (eating plants) and foxes (eating rabbits). Let the number of rabbits, the number of foxes. Using the Malthus model with the necessary corrections, taking into account the eating of rabbits by foxes, we arrive at the following system, which bears the name tray models - Volterra:

This system has an equilibrium state where the number of rabbits and foxes is constant. Deviation from this state leads to fluctuations in the number of rabbits and foxes, similar to fluctuations in the harmonic oscillator. As in the case of the harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, taking into account the limited resources needed by rabbits) can lead to a qualitative change in behavior. For example, the equilibrium state can become stable, and population fluctuations will fade. The opposite situation is also possible, when any small deviation from the equilibrium position will lead to catastrophic consequences, up to the complete extinction of one of the species. To the question of which of these scenarios is realized, the Volterra-Lotka model does not give an answer: additional research is required here.

Notes

1. "A mathematical representation of reality" (Encyclopaedia Britanica)
2. Novik I. B., ABOUT philosophical questions cybernetic simulation. M., Knowledge, 1964.
3. Sovetov B. Ya., Yakovlev S. A., Systems Modeling: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
4. Samarsky A. A., Mikhailov A. P. Mathematical modeling. Ideas. Methods. Examples. - 2nd ed., corrected. - M .: Fizmatlit, 2001. - ISBN 5-9221-0120-X
5. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., Rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
6. Sevostyanov, A.G. Modeling of technological processes: textbook / A.G. Sevostyanov, P.A. Sevostyanov. - M.: Light and food industry, 1984. - 344 p.
7. Wiktionary: mathematical models
8. CliffsNotes.com. Earth Science Glossary. 20 Sep 2010
9. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII+562 pp. ISBN 3-540-35885-4
10. “A theory is considered to be linear or non-linear, depending on what - linear or non-linear - mathematical apparatus, what - linear or non-linear - mathematical models it uses. ... without denying the latter. A modern physicist, if he happened to redefine such an important entity as non-linearity, would most likely act differently, and, preferring non-linearity as the more important and common of the two opposites, would define linearity as “non-non-linearity”. Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Synergetics: from the past to the future series. Ed.2. - M.: URSS, 2006. - 208 p. ISBN 5-484-00183-8
11. "Dynamical systems modeled finite number ordinary differential equations are called lumped or point systems. They are described using a finite-dimensional phase space and are characterized by a finite number of degrees of freedom. The same system in various conditions can be considered either concentrated or distributed. Mathematical models of distributed systems are differential equations in partial derivatives, integral equations or ordinary equations with retarded argument. The number of degrees of freedom of a distributed system is infinite, and an infinite number of data are required to determine its state. Anishchenko V.S., Dynamic Systems, Soros Educational Journal, 1997, No. 11, p. 77-84.
12. “Depending on the nature of the studied processes in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling displays deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. … Static modeling is used to describe the behavior of an object at any point in time, while dynamic modeling reflects the behavior of an object over time. Discrete modeling serves to describe processes that are assumed to be discrete, respectively, continuous modeling allows you to reflect continuous processes in systems, and discrete-continuous modeling is used for cases where you want to highlight the presence of both discrete and continuous processes. Sovetov B. Ya., Yakovlev S. A. ISBN 5-06-003860-2
13. Usually, the mathematical model reflects the structure (device) of the object being modeled, the properties and interconnections of the components of this object that are essential for the purposes of the study; such a model is called structural. If the model reflects only how the object functions - for example, how it reacts to external influences - then it is called a functional or, figuratively, a black box. Combined models are also possible. Myshkis A. D. ISBN 978-5-484-00953-4
14. “Obvious, but the most important initial stage of constructing or choosing a mathematical model is to obtain the clearest possible idea of ​​the object being modeled and to refine its content model based on informal discussions. Time and efforts should not be spared at this stage; the success of the entire study largely depends on it. More than once it happened that considerable work spent on solving a mathematical problem turned out to be ineffective or even wasted due to insufficient attention to this side of the matter. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., Rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4, p. 35.
15. « Description of the conceptual model of the system. At this sub-stage of building a system model: a) the conceptual model M is described in abstract terms and concepts; b) a description of the model is given using typical mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of a procedure for approximating real processes when building a model is substantiated. Sovetov B. Ya., Yakovlev S. A., Systems Modeling: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.
16. Blekhman I. I., Myshkis A. D.,