Predator-prey equilibrium. Coursework: Qualitative study of the predator-prey model Classical Volterra model

Predator-prey situation model

Let's consider a mathematical model of the dynamics of coexistence of two biological species (populations) interacting with each other according to the “predator-prey” type (wolves and rabbits, pikes and crucian carp, etc.), called the Volter-Lotka model. It was first obtained by A. Lotka (1925), and a little later and independently of Lotka, similar and more complex models were developed by the Italian mathematician V. Volterra (1926), whose work actually laid the foundations of the so-called mathematical ecology.

Let there be two biological species that live together in an isolated environment. This assumes:

• 1. The victim can find enough food to survive;
• 2. Every time a prey meets a predator, the latter kills the victim.

For definiteness, we will call them crucian carp and pike. Let

the state of the system is determined by the quantities x(t) And y(t)- the number of crucian carp and pike at a time G. To obtain mathematical equations that approximately describe the dynamics (change over time) of a population, we proceed as follows.

As in the previous population growth model (see section 1.1), for victims we have the equation

Where A> 0 (birth rate exceeds death rate)

Coefficient A the increase in prey depends on the number of predators (decreases with their increase). In the simplest case a- a - fjy (a>0, p>0). Then for the size of the prey population we have the differential equation

For a population of predators we have the equation

Where b>0 (mortality exceeds birth rate).

Coefficient b The decline of predators is reduced if there are prey to feed on. In the simplest case we can take b - y -Sx (y > 0, S> 0). Then for the size of the predator population we obtain the differential equation

Thus, equations (1.5) and (1.6) represent a mathematical model of the problem of population interaction under consideration. In this model the variables x,y is the state of the system, and the coefficients characterize its structure. The nonlinear system (1.5), (1.6) is the Voltaire-Lotka model.

Equations (1.5) and (1.6) should be supplemented with initial conditions - given values ​​of the initial populations.

Let us now analyze the constructed mathematical model.

Let us construct the phase portrait of system (1.5), (1.6) (in the sense of the problem X> 0, v >0). Dividing equation (1.5) by equation (1.6), we obtain an equation with separable variables

Using this equation, we have

Relation (1.7) gives the equation of phase trajectories in implicit form. System (1.5), (1.6) has a stationary state determined from

From equations (1.8) we obtain (since l* F 0, y* F 0)

Equalities (1.9) determine the equilibrium position (point ABOUT)(Fig. 1.6).

The direction of movement along the phase trajectory can be determined from such considerations. Let there be few crucians. g.e. x ~ 0, then from equation (1.6) y

All phase trajectories (except for the point 0) closed curves covering the equilibrium position. The state of equilibrium corresponds to a constant number of x" and y" of crucian carp and pike. Crucian carp multiply, pike eat them, die out, but the number of them and others does not change. "Closed phase trajectories correspond to a periodic change in the number of crucian carp and pike. Moreover, the trajectory along which the phase point moves depends on the initial conditions. Let us consider how the state changes along the phase trajectory. Let the point be in the position A(Fig. 1.6). There are few crucian carp here, many pike; pikes have nothing to eat, and they gradually die out and almost

completely disappear. But the number of crucian carp also decreases almost to zero and

only later, when there were fewer pikes than at, the number of crucian carp begins to increase; their growth rate increases and their number increases - this happens until about the point IN. But an increase in the number of crucian carp leads to a slowdown in the process of extinction of the shuk and their number begins to grow (there is more food) - plot Sun. Next there are a lot of pikes, they eat crucian carp and eat almost all of them (section CD). After this, the pike begin to die out again and the process repeats with a period of approximately 5-7 years. In Fig. 1.7 qualitatively constructed curves of changes in the number of crucian carp and pike depending on time. The maximums of the curves alternate, and the maximum numbers of pikes lag behind the maximums of crucian carp.

This behavior is typical for various predator-prey systems. Let us now interpret the results obtained.

Despite the fact that the model considered is the simplest and in reality everything happens much more complicated, it made it possible to explain some of the mysterious things that exist in nature. The stories of fishermen about periods when “pike themselves jump into their hands” are understandable; the frequency of chronic diseases, etc., has been explained.

Let us note another interesting conclusion that can be drawn from Fig. 1.6. If at the point R there is a rapid catching of pikes (in other terminology - shooting of wolves), then the system “jumps” to the point Q, and further movement occurs along a closed trajectory of a smaller size, which is intuitively expected. If we reduce the number of pikes at a point R, then the system will go to the point S, and further movement will occur along a larger trajectory. The amplitudes of oscillations will increase. This is counterintuitive, but it explains precisely this phenomenon: as a result of shooting wolves, their numbers increase over time. Thus, the choice of the moment of shooting is important in this case.

Suppose two populations of insects (for example, an aphid and a ladybug, which eats aphids) were in natural equilibrium x-x*,y = y*(dot ABOUT in Fig. 1.6). Consider the effect of a single application of an insecticide that kills x> 0 of the victims and y > 0 of the predators without destroying them completely. A decrease in the number of both populations leads to the fact that the representing point from the position ABOUT will “jump” closer to the origin of coordinates, where x > 0, y 0 (Fig. 1.6) It follows that as a result of the action of an insecticide designed to destroy victims (aphids), the number of victims (aphids) increases, and the number of predators (ladybugs) decreases. It turns out that the number of predators may become so small that they will face complete extinction for other reasons (drought, disease, etc.). Thus, the use of insecticides (unless they almost completely destroy harmful insects) ultimately leads to an increase in the population of those insects whose numbers were controlled by other insect predators. Such cases are described in books on biology.

In general, the growth rate of the number of victims A depends on both "L" and y: A= a(x, y) (due to the presence of predators and due to food restrictions).

With a small change in model (1.5), (1.6), small terms are added to the right-hand sides of the equations (taking into account, for example, the competition of crucian carp for food and pike for crucian carp)

here 0 f.i « 1.

In this case, the conclusion about the periodicity of the process (the return of the system to its original state), which is valid for model (1.5), (1.6), loses its validity. Depending on the type of small amendments/ and g The situations shown in Fig. are possible. 1.8.

In case (1) the equilibrium state ABOUT sustainable. For any other initial conditions, after a sufficiently long time, this is exactly what is established.

In case (2), the system “goes into disarray.” The stationary state is unstable. Such a system ultimately falls into such a range of values X and y that the model ceases to be applicable.

In case (3) in a system with an unstable stationary state ABOUT A periodic regime is established over time. Unlike the original model (1.5), (1.6), in this model the steady-state periodic regime does not depend on the initial conditions. Initially small deviation from steady state ABOUT does not lead to small fluctuations around ABOUT, as in the Volterra-Lotka model, but to oscillations of a well-defined (and independent of the smallness of the deviation) amplitude.

IN AND. Arnold calls the Volterra-Lotka model rigid because its small change can lead to conclusions different from those given above. To judge which of the situations shown in Fig. 1.8, is implemented in this system, additional information about the system (about the type of small corrections/ and g).

Kolmogorov's model makes one significant assumption: since it is assumed that this means that there are mechanisms in the prey population that regulate their numbers even in the absence of predators.

Unfortunately, this formulation of the model does not allow us to answer a question that has been the subject of much debate recently and which we already mentioned at the beginning of the chapter: how can a population of predators exert a regulatory influence on the population of prey so that the entire system is sustainable? Therefore, we will return to model (2.1), in which self-regulation mechanisms (for example, regulation through intraspecific competition) are absent in the prey population (as well as in the predator population); therefore, the only mechanism for regulating the numbers of species included in a community is the trophic relationship between predators and prey.

Here (so, unlike the previous model, Naturally, solutions (2.1) depend on the specific type of trophic function which, in turn, is determined by the nature of predation, i.e., the trophic strategy of the predator and the defensive strategy of the prey. Common to all of these functions (see Fig. I) are the following properties:

System (2.1) has one nontrivial stationary point, the coordinates of which are determined from the equations

under natural limitation.

There is one more stationary point (0, 0), corresponding to the trivial equilibrium. It is easy to show that this point is a saddle, and the separatrices are the coordinate axes.

The characteristic equation for a point has the form

Obviously, for the classical Volterra model .

Therefore, the value of f can be considered as a measure of the deviation of the model under consideration from the Volterra model.

a stationary point is the focus, and oscillations appear in the system; when the opposite inequality is satisfied, there is a node, and there are no oscillations in the system. The stability of this equilibrium state is determined by the condition

i.e., it significantly depends on the type of trophic function of the predator.

Condition (5.5) can be interpreted as follows: for the stability of the nontrivial equilibrium of the predator-prey system (and thus for the existence of this system), it is sufficient that in the vicinity of this state the relative proportion of prey consumed by the predator increases with the increase in the number of prey. Indeed, the proportion of prey (out of their total number) consumed by a predator is described by a differentiable function, the condition for which to increase (positive derivative) looks like

The last condition taken at the point is nothing more than condition (5.5) for the stability of equilibrium. With continuity, it must also be fulfilled in a certain neighborhood of the point. Thus, if the number of victims in this neighborhood, then

Let now the trophic function V have the form shown in Fig. 11, a (characteristic of invertebrates). It can be shown that for all finite values ​​(since it is convex upward)

that is, for any value of the stationary number of victims, inequality (5.5) is not satisfied.

This means that in a system with this type of trophic function there is no stable non-trivial equilibrium. Several outcomes are possible: either the numbers of both prey and predator increase indefinitely, or (when the trajectory passes near one of the coordinate axes) due to random reasons, the number of prey or the number of predator will become equal to zero. If the prey dies, after some time the predator will also die, but if the predator dies first, then the number of the prey will begin to increase exponentially. The third option - the emergence of a stable limit cycle - is impossible, which is easily proven.

In fact, the expression

in the positive quadrant is always positive, unless it has the form shown in Fig. 11, a. Then, according to the Dulac criterion, there are no closed trajectories in this region and a stable limit cycle cannot exist.

So, we can conclude: if the trophic function has the form shown in Fig. 11, and then the predator cannot be a regulator that ensures the stability of the prey population and thereby the stability of the entire system as a whole. The system can be stable only if the prey population has its own internal regulatory mechanisms, for example, intraspecific competition or epizootics. This regulation option has already been discussed in §§ 3, 4.

It was previously noted that this type of trophic function is characteristic of insect predators, whose “victims” are also usually insects. On the other hand, observations of the dynamics of many natural communities of the “predator-prey” type, including insect species, show that they are characterized by fluctuations of very large amplitude and of a very specific type.

Usually, after a more or less gradual increase in numbers (which can occur either monotonically or in the form of oscillations with increasing amplitude), a sharp drop occurs (Fig. 14), and then the picture repeats. Apparently, this nature of the dynamics of the numbers of insect species can be explained by the instability of this system at low and medium numbers and the action of powerful intrapopulation regulators of numbers at large numbers.

Rice. 14. Population dynamics of the Australian psyllid Cardiaspina albitextura feeding on eucalyptus trees. (From the article: Clark L. R. The population dynamics of Cardiaspina albitextura.-Austr. J. Zool., 1964, 12, No. 3, p. 362-380.)

If the “predator-prey” system includes species capable of quite complex behavior (for example, predators are capable of learning or prey are able to find shelter), then a stable non-trivial equilibrium may exist in such a system. This statement is proven quite simply.

In fact, the trophic function should then have the form shown in Fig. 11, c. The point on this graph is the tangency point of the straight line drawn from the origin of the trophic function graph. Obviously, at this point the function has a maximum. It is also easy to show that condition (5.5) is satisfied for all. Consequently, a nontrivial equilibrium in which the number of victims is smaller will be asymptotically stable

However, we cannot say anything about how large the region of stability of this equilibrium is. For example, if there is an unstable limit cycle, then this region must lie inside the cycle. Or another option: the nontrivial equilibrium (5.2) is unstable, but there is a stable limit cycle; in this case we can also talk about the stability of the predator-prey system. Since expression (5.7) when choosing a trophic function like Fig. 11, in can change sign when changing at , then the Dulac criterion does not work here and the question of the existence of limit cycles remains open.

Here, in contrast to (3.2.1), the signs of (-012) and (+a2i) are different. As in the case of competition (system of equations (2.2.1)), the origin of coordinates (1) for this system is a special point of the “unstable node” type. Three other possible steady states:

Biological meaning requires positive quantities X y x 2. For expression (3.3.4) this means that

If the coefficient of intraspecific competition of predators A,22 = 0, condition (3.3.5) leads to the condition ai2

Possible types of phase portraits for the system of equations (3.3.1) are presented in Fig. 3.2 a-c. Isoclins of horizontal tangents are straight lines

and isoclines of vertical tangents are straight

From Fig. 3.2 shows the following. The predator-prey system (3.3.1) may have a stable equilibrium position in which the prey population is completely extinct (x = 0) and only predators remained (point 2 in Fig. 3.26). Obviously, such a situation can only be realized if, in addition to the type of victims in question, X predator X2 has additional power sources. This fact is reflected in the model by a positive term on the right side of the equation for xs. Singular points (1) and (3) (Fig. 3.26) are unstable. The second possibility is a stable stationary state, in which the population of predators has completely died out and only prey remains - stable point (3) (Fig. 3.2a). Here the singular point (1) is also an unstable node.

Finally, the third possibility is the stable coexistence of populations of predator and prey (Fig. 3.2 c), the stationary numbers of which are expressed by formulas (3.3.4). Let's consider this case in more detail.

Let us assume that the coefficients of intraspecific competition are equal to zero (ai= 0, i = 1, 2). Let us also assume that predators feed only on prey of the species X and in their absence they die out at a rate of C2 (in (3.3.5) C2

Let us conduct a detailed study of this model, using the notation most widely accepted in the literature. Refurbishment

Rice. 3.2. The location of the main isoclines on the phase portrait of the Volterra system predator-prey for different ratios of parameters: A- about -

WITH I C2 C2

1, 3 - unstable, 2 - stable singular point; V -

1, 2, 3 - unstable, 4 - stable singular point significant

The predator-prey system in these notations has the form:

We will study the properties of solutions of system (3.3.6) on the phase plane N1 ON2 The system has two stationary solutions. They can be easily determined by equating the right-hand sides of the system to zero. We get:

Hence the stationary solutions:

Let's take a closer look at the second solution. Let us find the first integral of system (3.3.6), not containing t. Let's multiply the first equation by -72, the second by -71 and add the results. We get:

Now let's divide the first equation by N and multiply by € 2, and divide the second by JV 2 and multiply by e. Let's add the results again:

Comparing (3.3.7) and (3.3.8), we will have:

Integrating, we get:

This is the desired first integral. Thus, system (3.3.6) is conservative because it has a first integral of motion, a quantity that is a function of the system variables N And N2 and independent of time. This property allows us to construct for Volterra systems a system of concepts similar to statistical mechanics (see Chapter 5), where the energy value of the system, which is constant in time, plays a significant role.

For every fixed from > 0 (which corresponds to certain initial data), the integral corresponds to a certain trajectory on the plane N1 ON2 , serving as the trajectory of the system (3.3.6).

Let's consider the graphical method of constructing a trajectory, proposed by Volterra himself. Note that the right-hand side of formula (3.3.9) depends only on D g 2, and the left-hand side only on N. Let's denote

From (3.3.9) it follows that between X And Y there is a proportional relationship

In Fig. 3.3 shows the first quadrants of four coordinate systems XOY,NOY, N2 OX and D G 1 0N2 so that they all have a common origin.

In the upper left corner (quadrant NOY) a graph of function (3.3.8) is plotted, in the lower right (quadrant N2 OX)- function graph Y. The first function has min at Ni = and the second - max at N2 = ?-

Finally, in the quadrant XOY let's construct straight line (3.3.12) for some fixed WITH.

Let's mark the point N on the axis ON. This point corresponds to a certain value Y(N 1), which is easy to find by drawing a perpendicular

Rice. 3.3.

through N until it intersects with the curve (3.3.10) (see Fig. 3.3). In turn, the value K(D^) corresponds to a certain point M on the line Y = cX and therefore some value X(N) = Y(N)/c, which can be found by drawing perpendiculars A.M. And M.D. The found value (this point is marked in the figure with the letter D) two points correspond R And G on the curve (3.3.11). Using these points, drawing perpendiculars, we will find two points at once E" And E", lying on the curve (3.3.9). Their coordinates:

Drawing a perpendicular A.M., we crossed the curve (3.3.10) at one more point IN. This point corresponds to the same R And Q on the curve (3.3.11) and the same N And SCH. Coordinate N this point can be found by dropping the perpendicular from IN per axis ON. So we get the points F" and F", also lying on the curve (3.3.9).

Coming from a different point N, in the same way we obtain a new four points lying on the curve (3.3.9). The exception will be the point Ni= ?2/72- Based on it, we get only two points: TO And L. These will be the lower and upper points of the curve (3.3.9).

Can't start from values N, and from the values N2 . Heading from N2 to the curve (3.3.11), then rising to the straight line Y = cX, and from there crossing the curve (3.3.10), we also find four points of the curve (3.3.9). The exception will be the point No=?1/71- Based on it, we get only two points: G And TO. These will be the leftmost and rightmost points of the curve (3.3.9). By asking different N And N2 and having received quite a lot of points, connecting them, we will approximately construct the curve (3.3.9).

From the construction it is clear that this is a closed curve containing within itself the point 12 = (?2/721?1/71)” starting from certain initial data N Yu and N20. Taking another value of C, i.e. other initial data, we obtain another closed curve that does not intersect the first and also contains the point (?2/721 ?1/71)1 inside itself. Thus, the family of trajectories (3.3.9) is a family of closed lines surrounding point 12 (see Fig. 3.3). Let us study the type of stability of this singular point using the Lyapunov method.

Since all parameters e 1, ?2, 71,72 are positive, period (N[ is located in the positive quadrant of the phase plane. Linearizing the system near this point gives:

Here n(t) and 7i2(N1, N2 :

Characteristic equation of the system (3.3.13):

The roots of this equation are purely imaginary:

Thus, a study of the system shows that trajectories near the singular point are represented by concentric ellipses, and the singular point itself is the center (Fig. 3.4). The Volterra model under consideration also has closed trajectories far from the singular point, although the shape of these trajectories is already different from ellipsoidal. Behavior of Variables Ni, N2 over time is shown in Fig. 3.5.

Rice. 3.4.

Rice. 3.5. Dependence of prey numbers N i and predator N2 from time

A center-type singular point is stable, but not asymptotically. Let us show with this example what this is. Let the hesitation Ni(t) and LGgM occur in such a way that the representing point moves along the phase plane along trajectory 1 (see Fig. 3.4). At the moment when the point is in position M, a certain number of individuals are added to the system from the outside N 2, such that the representing point jumps from the point M to point A/". After this, if the system is again left to itself, the oscillations Ni And N2 will already occur with larger amplitudes than before, and the representing point moves along trajectory 2. This means that the oscillations in the system are unstable: they forever change their characteristics under external influence. In the future, we will consider models that describe stable oscillatory regimes and show that on the phase plane such asymptotic stable periodic motions are depicted using limit cycles.

In Fig. Figure 3.6 shows experimental curves - fluctuations in the number of fur-bearing animals in Canada (according to the Hudson's Bay Company). These curves are constructed based on data on the number of harvested skins. The periods of fluctuations in the numbers of hares (prey) and lynxes (predators) are approximately the same and are on the order of 9–10 years. In this case, the maximum number of hares is, as a rule, ahead of the maximum number of lynxes by one year.

The shape of these experimental curves is much less regular than the theoretical ones. However, in this case, it is sufficient that the model ensures the coincidence of the most significant characteristics of the theoretical and experimental curves, i.e. amplitude values ​​and phase shifts between fluctuations in the numbers of predators and prey. A much more serious drawback of the Volterra model is the instability of solutions to the system of equations. Indeed, as mentioned above, any random change in the abundance of one or another species should lead, following the model, to a change in the amplitude of oscillations of both species. Naturally, in natural conditions animals are exposed to countless such random influences. As can be seen from the experimental curves, the amplitude of fluctuations in species numbers varies little from year to year.

The Volterra model is a reference (basic) for mathematical ecology to the same extent that the harmonic oscillator model is basic for classical and quantum mechanics. Using this model, based on very simplified ideas about the nature of the patterns that describe the behavior of the system, purely mathematical

Chapter 3

Rice. 3.6. Kinetic curves of the number of fur-bearing animals according to the data of the Hudson Bay Fur Company (Seton-Thomson, 1987) By means of a conclusion was drawn about the qualitative nature of the behavior of such a system - about the presence of population fluctuations in such a system. Without constructing a mathematical model and using it, such a conclusion would be impossible.

In the simplest form we considered above, the Volterra system has two fundamental and interrelated disadvantages. An extensive ecological and mathematical literature is devoted to their “elimination.” Firstly, the inclusion of any additional factors, however small, in the model qualitatively changes the behavior of the system. The second “biological” drawback of the model is that it does not include the fundamental properties inherent in any pair of populations interacting according to the predator-prey principle: the effect of saturation of the predator, the limited resources of the predator and prey even with an excess of prey, the possibility of a minimum number of prey available for predator, etc.

In order to eliminate these shortcomings, various modifications of the Volterra system have been proposed by different authors. The most interesting of them will be discussed in section 3.5. Here we will focus only on a model that takes into account self-limitations in the growth of both populations. The example of this model clearly shows how the nature of decisions can change when system parameters change.

So, we consider the system

System (3.3.15) differs from the previously considered system (3.3.6) by the presence of terms of the form -7 on the right-hand sides of the equations uNf,

These terms reflect the fact that the population of prey cannot grow indefinitely even in the absence of predators due to limited food resources and limited habitat. The same “self-restrictions” are imposed on the population of predators.

To find stationary numbers of species iVi and N2 Let us equate the right-hand sides of the equations of system (3.3.15) to zero. Solutions with zero numbers of predators or prey will not interest us now. Therefore, consider a system of algebraic

equations Her decision

gives us the coordinates of the singular point. Here the condition for the positivity of stationary numbers should be imposed on the parameters of the system: N> 0 and N2 > 0. The roots of the characteristic equation of the system linearized in the neighborhood of the singular point (3.3.16):

From the expression for characteristic numbers it is clear that if the condition is met

then the numbers of predators and prey undergo damped oscillations over time, the system has a non-zero singular point and a stable focus. The phase portrait of such a system is shown in Fig. 3.7 a.

Let us assume that the parameters in inequality (3.3.17) change their values ​​in such a way that condition (3.3.17) becomes equality. Then the characteristic numbers of the system (3.3.15) are equal, and its singular point will lie on the boundary between the regions of stable foci and nodes. When the sign of inequality (3.3.17) is reversed, the singular point becomes a stable node. The phase portrait of the system for this case is shown in Fig. 3.76.

As in the case of a single population, a stochastic model can be developed for model (3.3.6), but an explicit solution cannot be obtained for it. Therefore, we will limit ourselves to general considerations. Let us assume, for example, that the equilibrium point is located at a certain distance from each of the axes. Then for phase trajectories on which the values ​​of JVj, N2 remain large enough, a deterministic model will be quite satisfactory. But if at some point

Rice. 3.7. Phase portrait of the system (3.3.15): A - when the relationship (3.3.17) between the parameters is satisfied; b- when performing an inverse relationship between parameters

phase trajectory, any variable is not very large, then random fluctuations can become significant. They lead to the fact that the representing point moves to one of the axes, which means the extinction of the corresponding species. Thus, the stochastic model turns out to be unstable, since the stochastic “drift” sooner or later leads to the extinction of one of the species. In this kind of model, the predator eventually goes extinct, either by chance or because its prey population is eliminated first. The stochastic model of the predator-prey system explains well the experiments of Gause (Gause, 1934; 2000), in which ciliates Paramettum candatum served as a victim for another ciliate Didinium nasatum- predator. The equilibrium numbers expected according to the deterministic equations (3.3.6) in these experiments were approximately only five individuals of each species, so it is not surprising that in each repeated experiment either predators or prey died out quite quickly (and then the predators ).

So, the analysis of Volterra models of species interaction shows that, despite the wide variety of types of behavior of such systems, there cannot be undamped fluctuations in numbers in the model of competing species at all. In the predator-prey model, undamped oscillations appear due to the choice of a special form of the model equations (3.3.6). In this case, the model becomes non-rough, which indicates the absence in such a system of mechanisms seeking to preserve its state. However, such oscillations are observed in nature and experiment. The need for their theoretical explanation was one of the reasons for formulating model descriptions in a more general form. Section 3.5 is devoted to consideration of such generalized models.

to the agreement dated ___.___, 20___ on the provision of paid educational services

Ministry of Education and Science of the Russian Federation

Lysvensky branch

Perm State Technical University

Department of Economics

Course work

in the discipline "System Modeling"

Topic: Predator-prey system

Completed:

Student gr. BIVT-06

------------------

Checked by the teacher:

Shestakov A. P.

Lysva, 2010

Essay

Predation is a trophic relationship between organisms in which one of them (the predator) attacks the other (the prey) and feeds on parts of its body, that is, there is usually an act of killing the victim. Predation is contrasted with eating corpses (necrophagy) and organic products of their decomposition (detritophagy).

Another definition of predation is also quite popular, which proposes that only organisms that eat animals be called predators, in contrast to herbivores that eat plants.

In addition to multicellular animals, protists, fungi and higher plants can act as predators.

The population size of predators affects the population size of their prey and vice versa, population dynamics are described by the Lotka-Volterra mathematical model, however, this model is a high degree of abstraction and does not describe the real relationship between predator and prey, and can only be considered as a first degree of approximation of mathematical abstraction.

In the process of coevolution, predators and prey adapt to each other. Predators appear and develop means of detection and attack, and victims have means of secrecy and defense. Therefore, the greatest harm to victims can be caused by predators that are new to them, with whom they have not yet entered into an “arms race.”

Predators can specialize in one or more species for prey, which makes them more successful at hunting on average, but increases their dependence on those species.

Predator-prey system.

Predator-prey interaction is the main type of vertical relationship between organisms, in which matter and energy are transferred through food chains.

Equilibrium of V. x. - and. is most easily achieved if there are at least three links in the food chain (for example, grass - vole - fox). At the same time, the density of the phytophage population is regulated by relationships with both the lower and upper links of the food chain.

Depending on the nature of the prey and the type of predator (true, grazer), different dependences on the dynamics of their populations are possible. Moreover, the picture is complicated by the fact that predators are very rarely monophagous (i.e., feeding on one type of prey). Most often, when the population of one type of prey is depleted and catching it requires too much effort, predators switch to other types of prey. In addition, one population of prey can be exploited by several species of predators.

For this reason, the effect of pulsating the prey population size, often described in the environmental literature, followed by the pulsating population size of the predator with a certain delay, is extremely rare in nature.

The balance between predators and prey in animals is maintained by special mechanisms that prevent the complete extermination of the victims. Thus, victims can:

• run away from a predator (in this case, as a result of competition, the mobility of both victims and predators increases, which is especially typical for steppe animals that have nowhere to hide from their pursuers);
• acquire a protective color (<притворяться>leaves or twigs) or, on the contrary, a bright (for example, red) color, warning the predator about the bitter taste;
• hide in shelters;
• move to active defense measures (horned herbivores, spiny fish), often joint (prey birds collectively drive away the kite, male deer and saigas occupy<круговую оборону>from wolves, etc.).

Often members of one species (population) feed on members of another species.

The Lotka-Volterra model is a model of the mutual existence of two populations of the “predator-prey” type.

The predator-prey model was first developed by A. Lotka in 1925, who used it to describe the dynamics of interacting biological populations. In 1926, independently of Lotka, similar (and more complex) models were developed by the Italian mathematician V. Volterra, whose deep research in the field of environmental problems laid the foundation for the mathematical theory of biological communities or the so-called. mathematical ecology.

In mathematical form, the proposed system of equations has the form:

where x is the number of prey, y is the number of predators, t is time, α, β, γ, δ are coefficients that reflect interactions between populations.

Formulation of the problem

Consider an enclosed space in which two populations exist—herbivores (“prey”) and predators. It is believed that no animals are imported or exported and that there is enough food for herbivores. Then the equation for the change in the number of victims (only victims) will take the form:

where $α$ is the birth rate of victims,

$x$ is the size of the prey population,

$\frac(dx)(dt)$ is the growth rate of the prey population.

When predators do not hunt, they can become extinct, which means that the equation for the number of predators (predators only) becomes:

Where $γ$ is the predator loss rate,

$y$ is the size of the predator population,

$\frac(dy)(dt)$ is the growth rate of the predator population.

When predators and prey meet (the frequency of meetings is directly proportional to the product), predators destroy victims with a coefficient; well-fed predators can reproduce offspring with a coefficient. Thus, the system of equations of the model will take the form:

The solution of the problem

Let's build a mathematical model of the coexistence of two biological populations of the “predator-prey” type.

Let two biological populations live together in an isolated environment. The environment is stationary and provides unlimited quantities of everything necessary for life for one of the species - victims. Another species - a predator - also lives in stationary conditions, but feeds only on prey. Cats, wolves, pikes, foxes can act as predators, and chickens, hares, crucian carp, and mice can act as victims, respectively.

To be specific, let’s consider cats as predators, and chickens as victims.

So, chickens and cats live in some isolated space - a farm yard. The environment provides chickens with unlimited food, and cats eat only chickens. Let us denote by

$x$ – number of chickens,

$у$ – number of cats.

Over time, the number of chickens and cats changes, but we will consider $x$ and $y$ to be continuous functions of time t. Let's call a pair of numbers $x, y)$ the state of the model.

Let's find how the state of the model $(x, y).$ changes

Let's consider $\frac(dx)(dt)$ – the rate of change in the number of chickens.

If there are no cats, then the number of chickens increases, and the faster the more chickens there are. We will assume the dependence is linear:

$\frac(dx)(dt) a_1 x$,

$a_1$ is a coefficient that depends only on the living conditions of chickens, their natural mortality and birth rates.

$\frac(dy)(dt)$ – the rate of change in the number of cats (if there are no chickens), depends on the number of cats y.

If there are no chickens, then the number of cats decreases (they have no food) and they die out. We will assume the dependence is linear:

$\frac(dy)(dt) - a_2 y$.

In an ecosystem, the rate of change in the number of each species will also be considered proportional to its quantity, but only with a coefficient depending on the number of individuals of another species. So, for chickens this coefficient decreases with an increase in the number of cats, and for cats it increases with an increase in the number of chickens. We will also assume that the dependence is linear. Then we obtain a system of differential equations:

This system of equations is called the Volterra-Lotka model.

a1, a2, b1, b2 – numerical coefficients, which are called model parameters.

As you can see, the nature of the change in the state of the model (x, y) is determined by the values ​​of the parameters. By changing these parameters and solving the system of model equations, it is possible to study the patterns of changes in the state of the ecological system.

Using the MATLAB program, the Lotka-Volterra equation system is solved as follows:

In Fig. 1 shows the solution of the system. Depending on the initial conditions, the solutions are different, which corresponds to different colors of the trajectories.

In Fig. 2 presents the same solutions, but taking into account the time axis t (i.e., there is a dependence on time).