Interaction of individuals in the "predator-prey" system

5th year student 51 A group

Departments of Bioecology

Nazarova A. A.

Scientific adviser:

Podshivalov A. A.

Orenburg 2011

INTRODUCTION

INTRODUCTION

In our daily reasoning and observations, we, without knowing it ourselves, and often without even realizing it, are guided by laws and ideas discovered many decades ago. Considering the predator-prey problem, we guess that the prey also indirectly affects the predator. What would a lion eat if there were no antelopes; what would managers do if there were no workers; how to develop a business if customers do not have funds ...

The "predator-prey" system is a complex ecosystem for which long-term relationships between predator and prey species are realized, a typical example of coevolution. Relations between predators and their prey develop cyclically, being an illustration of a neutral equilibrium.

The study of this form of interspecies relationships, in addition to obtaining interesting scientific results, allows us to solve many practical problems:

optimization of biotechnical measures both in relation to prey species and in relation to predators;

improving the quality of territorial protection;

regulation of hunting pressure in hunting farms, etc.

The foregoing determines the relevance of the chosen topic.

The purpose of the course work is to study the interaction of individuals in the "predator - prey" system. To achieve the goal, the following tasks were set:

predation and its role in the formation of trophic relationships;

the main models of the relationship "predator - prey";

the influence of the social way of life in the stability of the "predator-prey" system;

laboratory modeling of the "predator - prey" system.

The influence of predators on the number of prey and vice versa is quite obvious, but it is rather difficult to determine the mechanism and essence of this interaction. These questions I intend to address in the course work.

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CHAPTER 4. LABORATORY MODELING OF THE PREDATOR - PREY SYSTEM

Duke University scientists, in collaboration with colleagues from Stanford University, the Howard Hughes Medical Institute, and the California Institute of Technology, working under the direction of Dr. Lingchong You, have developed a living system of genetically modified bacteria that will allow more detailed study of predator-prey interactions in a population level.

The new experimental model is an example of an artificial ecosystem in which researchers program bacteria to perform new functions to create. Such reprogrammed bacteria could be widely used in medicine, environmental cleanup and biocomputer development. As part of this work, scientists rewrote the "software" of E. coli (Escherichia coli) in such a way that two different bacterial populations formed in the laboratory a typical system of predator-prey interactions, a feature of which was that the bacteria did not devour each other, but controlled the number the opponent population by changing the frequency of "suicides".

The field of research known as synthetic biology emerged around 2000, and most of the systems created since then have been based on reprogramming a single bacterium. The model developed by the authors is unique in that it consists of two bacterial populations living in the same ecosystem, the survival of which depends on each other.

The key to the successful functioning of such a system is the ability of two populations to interact with each other. The authors created two strains of bacteria - "predators" and "herbivores", depending on the situation, releasing toxic or protective compounds into the general ecosystem.

The principle of operation of the system is based on maintaining the ratio of the number of predators and prey in a regulated environment. Changes in the number of cells in one of the populations activate reprogrammed genes, which triggers the synthesis of certain chemical compounds.

Thus, a small number of victims in the environment causes the activation of the self-destruction gene in predator cells and their death. However, as the number of victims increases, the compound released by them into the environment reaches a critical concentration and activates the predator gene, which ensures the synthesis of an "antidote" to the suicidal gene. This leads to an increase in the population of predators, which, in turn, leads to the accumulation of a compound synthesized by predators in the environment, pushing victims to commit suicide.

Using fluorescence microscopy, scientists documented interactions between predators and prey.

Predator cells, stained green, cause suicide of prey cells, stained red. Elongation and rupture of the victim cell indicates its death.

This system is not an accurate representation of predator-prey interactions in nature, as predator bacteria do not feed on prey bacteria and both populations compete for the same food resources. However, the authors believe that the system they have developed is a useful tool for biological research.

The new system demonstrates a clear relationship between genetics and population dynamics, which in the future will help in the study of the influence of molecular interactions on population change, which is a central topic of ecology. The system provides virtually unlimited possibilities for modifying variables to study in detail the interactions between environment, gene regulation, and population dynamics.

Thus, by controlling the genetic apparatus of bacteria, it is possible to simulate the processes of development and interaction of more complex organisms.

CHAPTER 3

CHAPTER 3

Ecologists from the United States and Canada have shown that the group lifestyle of predators and their prey radically changes the behavior of the predator-prey system and makes it more resilient. This effect, confirmed by observations of the dynamics of the number of lions and wildebeests in the Serengeti Park, is based on the simple fact that with a group lifestyle, the frequency of random encounters between predators and potential victims decreases.

Ecologists have developed a number of mathematical models that describe the behavior of the predator-prey system. These models, in particular, explain well the observed sometimes consistent periodic fluctuations in the abundance of predators and prey.

Such models are usually characterized by a high level of instability. In other words, with a wide range of input parameters (such as mortality of predators, efficiency of conversion of prey biomass into predator biomass, etc.) in these models, sooner or later all predators either die out or first eat all the prey, and then still die from hunger.

In natural ecosystems, of course, everything is more complicated than in a mathematical model. Apparently, there are many factors that can increase the stability of the predator-prey system, and in reality it rarely comes to such sharp jumps in numbers as in Canada lynxes and hares.

Ecologists from Canada and the United States published in the latest issue of the journal " nature" an article that drew attention to one simple and obvious factor that can dramatically change the behavior of the predator-prey system. It's about group life.

Most of the models available are based on the assumption of a uniform distribution of predators and their prey within a given territory. This is the basis for calculating the frequency of their meetings. It is clear that the higher the density of prey, the more often predators stumble upon them. The number of attacks, including successful ones, and, ultimately, the intensity of predation by predators depend on this. For example, with an excess of prey (if you do not have to spend time searching), the speed of eating will be limited only by the time necessary for the predator to catch, kill, eat and digest the next prey. If the prey is rarely caught, the main factor determining the rate of grazing becomes the time required to search for the prey.

In the ecological models used to describe the “predator–prey” systems, the nature of the dependence of the predation intensity (the number of prey eaten by one predator per unit time) on the prey population density plays a key role. The latter is estimated as the number of animals per unit area.

It should be noted that with a group lifestyle of both prey and predators, the initial assumption of a uniform spatial distribution of animals is not satisfied, and therefore all further calculations become incorrect. For example, with a herd lifestyle of prey, the probability of encountering a predator will actually depend not on the number of individual animals per square kilometer, but on the number of herds per unit area. If the prey were distributed evenly, predators would stumble upon them much more often than in the herd way of life, since vast spaces are formed between the herds where there is no prey. A similar result is obtained with the group way of life of predators. A pride of lions wandering across the savannah will notice few more potential victims than a lone lion following the same path would.

For three years (from 2003 to 2007), scientists conducted careful observations of lions and their victims (primarily wildebeest) in the vast territory of the Serengeti Park (Tanzania). Population density was recorded monthly; the intensity of eating by lions was also regularly assessed various kinds ungulates. Both the lions themselves and the seven main species of their prey lead a group lifestyle. The authors introduced the necessary amendments to the standard ecological formulas to take this circumstance into account. The parametrization of the models was carried out on the basis of real quantitative data obtained in the course of observations. Four versions of the model were considered: in the first, the group way of life of predators and prey was ignored; in the second, it was taken into account only for predators; in the third, only for prey; and in the fourth, for both.

As one would expect, the fourth option corresponded best to reality. He also proved to be the most resilient. This means that with a wide range of input parameters in this model, long-term stable coexistence of predators and prey is possible. The data of long-term observations show that in this respect the model also adequately reflects reality. The numbers of lions and their prey in the Serengeti are quite stable, nothing resembling periodic coordinated fluctuations (as is the case with lynxes and hares) is observed.

The results obtained show that if lions and wildebeest lived alone, the increase in the number of prey would lead to a rapid acceleration of their predation by predators. Due to the group way of life, this does not happen, the activity of predators increases relatively slowly, and the overall level of predation remains low. According to the authors, supported by a number of indirect evidence, the number of victims in the Serengeti is limited not by lions at all, but by food resources.

If the benefits of collectivism for the victims are quite obvious, then in relation to the lions the question remains open. This study clearly showed that the group lifestyle for a predator has a serious drawback - in fact, because of it, each individual lion gets less prey. Obviously, this disadvantage should be compensated by some very significant advantages. Traditionally, it was believed that the social lifestyle of lions is associated with hunting large animals, which are difficult to cope even with a lion alone. However, in Lately many experts (including the authors of the article under discussion) began to doubt the correctness of this explanation. In their opinion, collective action is necessary for lions only when hunting buffaloes, and lions prefer to deal with other types of prey alone.

More plausible is the assumption that prides are needed to regulate purely internal problems, which are many in a lion's life. For example, infanticide is common among them - the killing of other people's cubs by males. It is easier for females kept in a group to protect their children from aggressors. In addition, it is much easier for a pride than for a lone lion to defend its hunting area from neighboring prides.

Source: John M. Fryxell, Anna Mosser, Anthony R. E. Sinclair, Craig Packer. Group formation stabilizes predator–prey dynamics // Nature. 2007. V. 449. P. 1041–1043.

1. Simulation systems "Predator-Victim"

   Abstract >> Economic and mathematical modeling

   ... systems « Predator-Victim" Made by Gizyatullin R.R gr.MP-30 Checked by Lisovets Yu.P MOSCOW 2007 Introduction Interaction... model interactions predators And victims on surface. Simplifying assumptions. Let's try to compare victim And predator some...

2. Predator-Victim

   Abstract >> Ecology

   Applications of mathematical ecology is system predator-victim. The cyclic behavior of this systems in a stationary environment was ... by introducing an additional nonlinear interactions between predator And a victim. The resulting model has on its...

3. Synopsis ecology

   Abstract >> Ecology

   factor for victims. That's why interaction « predator–victim" is periodic and is system Lotka's equations... the shift is much smaller than in system « predator–victim". Similar interactions are also observed in Batsian mimicry. ...

PA88 system, which simultaneously predicts the probability of more than 100 pharmacological effects and mechanisms of action of a substance based on its structural formula. The efficiency of applying this approach to screening planning is about 800%, and the accuracy computer forecast 300% better than experts predicted.

So, one of the constructive tools for obtaining new knowledge and solutions in medicine is the method mathematical modeling. The process of mathematization of medicine is a frequent manifestation of interpenetration scientific knowledge, increasing the effectiveness of treatment and preventive work.

4. Mathematical model "predators-prey"

For the first time in biology, a mathematical model of a periodic change in the number of antagonistic animal species was proposed by the Italian mathematician V. Volterra and his co-workers. The model proposed by Volterra was the development of the idea outlined in 1924 by A. Lotka in the book "Elements of Physical Biology". Therefore, this classical mathematical model is known as the "Lotka-Volterra" model.

Although antagonistic species relations are more complex in nature than in a model, they are nevertheless a good educational model on which to learn the basic ideas of mathematical modeling.

So, task: in some ecologically closed area two species of animals live (for example, lynxes and hares). Hares (prey) feed on plant foods, which are always available in sufficient quantities (this model does not take into account the limited resources of plant foods). Lynxes (predators) can only eat hares. It is necessary to determine how the number of prey and predators will change over time in such an ecological system. If the prey population increases, the probability of encounters between predators and prey increases, and, accordingly, after some time delay, the predator population grows. This rather simple model quite adequately describes the interaction between real populations of predators and prey in nature.

Now let's get down to compiling differential equations. Ob-

we denote the number of prey through N, and the number of predators through M. The numbers N and M are functions of time t . In our model, we take into account the following factors:

a) natural reproduction of victims; b) natural death of victims;

c) destruction of victims by eating them by predators; d) natural extinction of predators;

e) an increase in the number of predators due to reproduction in the presence of food.

Since we are talking about a mathematical model, the task is to obtain equations that would include all the intended factors and that would describe the dynamics, that is, the change in the number of predators and prey over time.

Let for some time t the number of prey and predators change by ∆N and ∆M. The change in the number of victims ∆N over time ∆t is determined, firstly, by the increase as a result of natural reproduction (which is proportional to the number of victims present):

where B is the coefficient of proportionality characterizing the rate of natural extinction of victims.

At the heart of the derivation of the equation describing the decrease in the number of prey due to being eaten by predators is the idea that the more often they meet, the faster the number of prey decreases. It is also clear that the frequency of encounters between predators and prey is proportional to both the number of prey and the number of predators, then

Dividing the left and right sides of equation (4) by ∆t and passing to the limit at ∆t→0 , we obtain a first-order differential equation:

In order to solve this equation, you need to know how the number of predators (M) changes over time. The change in the number of predators (∆M ) is determined by an increase due to natural reproduction in the presence of sufficient food (M 1 = Q∙N∙M∙∆t ) and a decrease due to the natural extinction of predators (M 2 = - P∙M∙∆ t):

M = Q∙N∙M∙∆t - P∙M∙∆t

From equation (6) one can obtain a differential equation:

Differential equations (5) and (7) represent the mathematical model "predators-prey". It is enough to determine the values ​​of the coefficient

components A, B, C, Q, P and the mathematical model can be used to solve the problem.

Verification and correction of the mathematical model. In this lab-

In this work, it is proposed, in addition to calculating the most complete mathematical model (equations 5 and 7), to study simpler ones, in which something is not taken into account.

Having considered five levels of complexity of the mathematical model, one can "feel" the stage of checking and correcting the model.

1st level - the model takes into account for "victims" only their natural reproduction, "predators" are absent;

2nd level - the model takes into account natural extinction for "victims", "predators" are absent;

3rd level - the model takes into account for the "victims" their natural reproduction

And extinction, "predators" are absent;

4th level - the model takes into account for the "victims" their natural reproduction

And extinction, as well as eating by "predators", but the number of "predators" remains unchanged;

Level 5 - the model takes into account all the discussed factors.

So, we have the following system of differential equations:

where M is the number of "predators"; N is the number of "victims";

t is the current time;

A is the rate of reproduction of "victims"; C is the frequency of "predator-prey" encounters; B is the extinction rate of "victims";

Q - reproduction of "predators";

P - extinction of "predators".

1st level: M = 0, B = 0; 2nd level: M = 0, A = 0; 3rd level: M = 0; 4th level: Q = 0, P = 0;

5th level: complete system of equations.

Substituting the values ​​of the coefficients into each level, we will get different solutions, for example:

For the 3rd level, the value of the coefficient M=0, then

solving the equation we get

Similarly for the 1st and 2nd levels. As for the 4th and 5th levels, here it is necessary to solve the system of equations by the Runge-Kutta method. As a result, we obtain the solution of mathematical models of these levels.

II. WORK OF STUDENTS DURING THE PRACTICAL LESSON

Exercise 1 . Oral-speech control and correction of the assimilation of the theoretical material of the lesson. Giving permission to practice.

Task 2 . Performing laboratory work, discussing the results obtained, compiling a summary.

Completing of the work

1. Call the "Lab. No. 6" program from the desktop of the computer by double-clicking on the corresponding label with the left mouse button.

2. Double-click the left mouse button on the "PREDATOR" label.

3. Select the shortcut "PRED" and repeat the call of the program with the left mouse button (double-clicking).

4. After the title splash press "ENTER".

5. Modeling start with 1st level.

6. Enter the year from which the analysis of the model will be carried out: for example, 2000

7. Select time intervals, for example, within 40 years, after 1 year (then after 4 years).

2nd level: B = 0.05; N0 = 200;

3rd level: A = 0.02; B = 0.05; N=200;

4th level: A = 0.01; B = 0.002; C = 0.01; N0 = 200; M=40; 5th level: A = 1; B = 0.5; C = 0.02; Q = 0.002; P = 0.3; N0 = 200;

9. Prepare a written report on the work, which should contain equations, graphs, the results of calculating the characteristics of the model, conclusions on the work done.

Task 3. Control of the final level of knowledge:

a) oral-speech report for the performed laboratory work; b) solving situational problems; c) computer testing.

Task 4. Task for the next lesson: section and topic of the lesson, coordination of topics for abstract reports (report size 2-3 pages, time limit 5-7 minutes).

The "predator-prey" model and Goodwin's macroeconomic model

Consider the biological model "predator - prey", in which one species is food for another. This model, which has long become a classic, was built in the first half of the 20th century. Italian mathematician V. Volterra to explain fluctuations in fish catches in the Adriatic Sea. The model assumes that the number of predators increases as long as they have enough food, and an increase in the number of predators leads to a decrease in the population of prey fish. When the latter becomes scarce, the number of predators decreases. As a result, from a certain moment, an increase in the number of prey fish begins, which after a while causes an increase in the population of predators. The cycle closes.

Let N x (t) And N 2 (t) - the number of prey and predator fish at a point in time t respectively. Let us assume that the rate of increase in the number of prey in the absence of predators is constant, i.e.

Where A - positive constant.

The appearance of a predator should reduce the rate of prey growth. We will assume that this decrease linearly depends on the number of predators: the more predators, the lower the growth rate of prey. Then

Where t > 0.

Therefore, for the dynamics of the number of prey fish, we obtain:

Let us now compose an equation that determines the dynamics of the population of predators. Let us assume that their number in the absence of prey decreases (due to lack of food) at a constant rate b, i.e.

The presence of prey causes an increase in the rate of growth of predators. Let us assume that this increase is linear, i.e.

Where n> 0.

Then for the growth rate of predatory fish we obtain the equation:

In the "predator - prey" system (6.17) - (6.18), the decrease in the growth rate of the number of prey fish caused by predators eating them is equal to mNxN2, i.e., proportional to the number of their encounters with the predator. The increase in the growth rate of the number of predator fish caused by the presence of prey is equal to nNxN2, i.e., also proportional to the number of encounters between prey and predators.

We introduce dimensionless variables U = mN 2 /a And V = nN x /b. Variable dynamics U corresponds to the dynamics of predators, and the dynamics of the variable V- victim dynamics. By virtue of equations (6.17) and (6.18), the change in new variables is determined by the system of equations:

Let's assume that at t= 0 the number of individuals of both species is known, therefore, are the initial values ​​of the new variables known?/(0) = U 0 , K(0) = K 0 . From the system of equations (6.19) one can find a differential equation for its phase trajectories:

Separating the variables of this equation, we get:

Rice. 6.10. Building a phase trajectory ADCBA systems of differential equations (6.19)

From here, taking into account the initial data, it follows:

where is the integration constant WITH = b(VQ - In V 0)/a - lnU 0 + U 0 .

On fig. 6.10 shows how the line (6.20) is constructed for a given value of C. To do this, in the first, second and third quarters, respectively, we build graphs of functions x = V - In V, y = (b/a)x, at== In U-U+C.

Due to equality dx/dV = (V- 1)/U function X = V- In K determined at V> 0, increases if V> 1, and decreases if V 1. Due to the fact that cPx/dV 1\u003d 1 / F 2\u003e 0, graph of the function l: \u003d x(V) directed downward. The equation V= 0 specifies the vertical asymptote. This function has no oblique asymptotes. So the graph of the function X = x(y) has the shape of the curve shown in the first quarter of Fig. 6.10.

The function y= In U - U + C, whose graph in Fig. 6.10 is depicted in the third quarter.

If we now place in Fig. 6.10 second quarter function graph y= (b/a)x, then in the fourth quarter we get a line that connects the variables U and V. Indeed, taking the point V t on axle OV, calculate using the function X= V - V relevant knowledge x x. After that, using the function at = (b/a)x, according to the received value X ( find y x(second quarter in Figure 6.10). Next, using the graph of the function at= In U - U + C determine the corresponding values ​​of the variable U(in Fig. 6.10 there are two such values ​​\u200b\u200b- the coordinates of the points M And N). The set of all such points (V; U) forms the desired curve. It follows from the construction that the graph of dependence (6.19) is a closed line containing a point inside it E( 1, 1).

Recall that we obtained this curve by setting some initial values U 0 And V0 and calculating the constant C from them. Taking other initial values, we get another closed line that does not intersect the first one and also contains a point inside itself E( eleven). This means that the family of trajectories of system (6.19) on the phase plane ( V, U) is the set of closed non-intersecting lines concentrating around the point E( 1, 1), and the solutions of the original model U = SCH) And V = V(t) are functions that are periodic in time. In this case, the maximum of the function U = U(t) does not reach the maximum of the function V = V(t) and vice versa, i.e. fluctuations in the number of populations around their equilibrium solutions occur in different phases.

On fig. 6.11 shows four trajectories of the system of differential equations (6.19) on the phase plane ouv, different initial conditions. One of the equilibrium trajectories is the point E( 1, 1), which corresponds to the solution U(t) = 1, V(t)= 1. Points (U(t),V(t)) on the other three phase trajectories, as time increases, they shift clockwise.

To explain the mechanism of change in the size of two populations, consider the trajectory ABCDA in fig. 6.11. As you can see, in the field AB both predators and prey are few. Therefore, here the population of predators is reduced due to lack of food, and the population of prey is growing. Location on sun the number of prey reaches high values, which leads to an increase in the number of predators. Location on SA there are many predators, and this entails a reduction in the number of prey. However, after passing the point D the number of victims decreases so much that the population begins to decrease. The cycle closes.

The predator-prey model is an example of a structurally unstable model. Here, a small change in the right side of one of the equations can lead to a fundamental change in its phase portrait.

Rice. 6.11.

Rice. 6.12.

Indeed, if intraspecific competition is taken into account in the equation for the dynamics of victims, then we will obtain a system of differential equations:

Here at t = 0 the victim population develops according to a logical law.

At t F 0 nonzero equilibrium solution of system (6.21) for some positive values parameter of intraspecific competition AND is a stable focus, and the corresponding trajectories "wind" around the equilibrium point (Fig. 6.12). If h = 0, then in this case the singular point E( 1, 1) of system (6.19) is the center, and the trajectories are closed lines (see Fig. 6.11).

Comment. Usually, the "predator-prey" model is understood as the model (6.19) whose phase trajectories are closed. However, model (6.21) is also a predator-prey model, since it describes the mutual influence of predators and prey.

One of the first applications of the predator-prey model in economics for the study of cyclically changing processes is the Goodwin macroeconomic model, which uses continuous approach to the analysis of the mutual influence of the level of employment and the wage rate.

In the work of V.-B. Zanga presented a variant of the Goodwin model, in which labor productivity and labor supply grow at a constant rate of growth, and the retirement rate of funds is zero. This model formally leads to the equations of the "predator-prey" model.

Below we consider a modification of this model for the case of a non-zero fund retirement rate.

The following notation is used in the model: L- the number of workers; w- the average wage rate of workers; TO - fixed production assets (capital); Y- national income; / - investments; C - consumption; p - the coefficient of disposal of funds; N- supply of labor in the labor market; T = Y/K- return on assets; A = Y/L - labor productivity; at = L/N - employment rate; X = C/Y - consumption rate in national income; TO - increase in capital depending on investment.

Let us write the equations of the Goodwin model:

Where a 0 , b, g, n, N 0 , g are positive numbers (parameters).

Equations (6.22) - (6.24) express the following. Equation (6.22) is the usual equation for the dynamics of funds. Equation (6.23) reflects an increase in the wage rate when employment is high (the wage rate rises if the supply of labor is low) and a decrease in the wage rate when unemployment is high.

Thus, equation (6.23) expresses the Phillips law in linear form. Equations (6.24) mean exponential growth in labor productivity and labor supply. We also assume that C = wL, i.e., all wages are spent on consumption. Now we can transform the equations of the model, taking into account the equalities:

Let's transform equations (6.22)-(6.27). We have:
Where

Where

Therefore, the dynamics of variables in the Goodwin model is described by a system of differential equations:

which formally coincides with the equations of the classical predator-prey model. This means that fluctuations in phase variables also arise in the Goodwin model. The mechanism of oscillatory dynamics here is as follows: with low wages w consumption is low, investment is high, and this leads to an increase in production and employment y. Big busy at causes an increase in the average wage w, which leads to an increase in consumption and a decrease in investment, a fall in production and a decrease in employment y.

Below, the hypothesis about the dependence of the interest rate on the level of employment of the considered model is used in modeling the dynamics of a single-product firm. It turns out that in this case, under some additional assumptions, the firm model has the property of cyclicity of the “predator-prey” model considered above.

• See: Volterra V. Decree, op.; Rizniienko G. Yu., Rubin A. B. Decree. op.
• See: Zang W.-B. Synergistic economy. M., 2000.
• See: Pu T. Nonlinear economic dynamics. Izhevsk, 2000; Tikhonov A.N. Mathematical model // Mathematical encyclopedia. T. 3. M., 1982. S. 574, 575.

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

Biological meaning requires positive values 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 shown in fig. 3.2 a-c. The isoclines of the horizontal tangents are straight lines

and the isoclines of the vertical tangents are straight

From fig. 3.2 shows the following. The predator-prey system (3.3.1) may have a stable equilibrium 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 be realized only if, in addition to the type of victims under consideration, X predator X2 has additional power supplies. This fact is reflected in the model by the 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 completely died out and only victims remained - a 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 predator and prey populations (Fig. 3.2 c), whose stationary numbers are expressed by formulas (3.3.4). Let's consider this case in more detail.

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 carry out a detailed study of this model, using the notation most widely accepted in the literature. Refurbished

Rice. 3.2. The location of the main isoclines in 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 to system (3.3.6) on the phase plane N1 ON2 The system has two stationary solutions. They are easy to determine 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) that does not contain t. Multiply the first equation by -72, the second by -71 and add the results. We get:

Now we 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, since it has the first integral of motion, a quantity that is a function of the variables of the system N And N2 and independent of time. This property makes it possible to construct a system of concepts for Volterra systems similar to statistical mechanics (see Chapter 5), where an essential role is played by the magnitude of the energy of the system, which is unchanged in time.

For every fixed c > 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).

Consider a graphical method for constructing a trajectory, proposed by Volterra himself. Note that the right side of the formula (3.3.9) depends only on D r 2, and the left side depends only on N. Denote

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

On 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) the graph of the function (3.3.8) is constructed, 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 construct the line (3.3.12) for some fixed WITH.

Mark a point N on axle 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 curve (3.3.10) (see Fig. 3.3). In turn, the value of K(A^) corresponds to some point M on the line Y = cX and hence some value X(N) = Y(N)/c which can be found by drawing perpendiculars AM And MD. The found value (this point is marked in the figure by the letter D) match two points R And G on the curve (3.3.11). By these points, drawing perpendiculars, we find two points at once E" And E" lying on the curve (3.3.9). Their coordinates are:

Drawing perpendicular AM, we have 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 axle ON. So we get points F" and F" also lying on the curve (3.3.9).

Coming from another point N, in the same way we obtain a new quadruple of points lying on the curve (3.3.9). The exception is the dot 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 come from values N, and from the values N2 . Heading from N2 to the curve (3.3.11), then rising to the 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 is the dot 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 enough points, connecting them, we approximately construct the curve (3.3.9).

It can be seen from the construction that this is a closed curve containing inside itself the point 12 = (?2/721? N yu and N20. Taking another value of C, i.e. other initial data, we get another closed curve that does not intersect the first one and also contains the point (?2/721?1/71)1 inside itself. Thus, the family of trajectories (3.3.9) is the family of closed lines surrounding the point 12 (see Fig. 3.3). We investigate the type of stability of this singular point using the Lyapunov method.

Since all parameters e 1, ?2, 71.72 are positive, dot (N[ is located in the positive quadrant of the phase plane. Linearization of 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, the study of the system shows that the 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 already differs from ellipsoidal. Variable behavior Ni, N2 in time is shown in Fig. 3.5.

Rice. 3.4.

Rice. 3.5. The dependence of the number of prey N i and predator N2 from time

A singular point of type center is stable, but not asymptotically. Let's use this example to show what it is. Let the vibrations Ni(t) and LGgM occur in such a way that the representative 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 representative point jumps from the point M point A/". After that, if the system is again left to itself, the oscillations Ni And N2 will already occur with larger amplitudes than before, and the representative point moves along trajectory 2. This means that the oscillations in the system are unstable: they permanently change their characteristics under external influence. In what follows, we consider models describing stable oscillatory regimes and show that such asymptotic stable periodic motions are represented on the phase plane by means of limit cycles.

On fig. 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 built on the basis of data on the number of harvested skins. The periods of fluctuations in the number of hares (prey) and lynxes (predators) are approximately the same and are of the order of 9-10 years. At the same time, the maximum number of hares, as a rule, is ahead of the maximum number of lynxes by one year.

The shape of these experimental curves is much less correct 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 shift between fluctuations in the numbers of predators and prey. A much more serious shortcoming 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 subjected to countless such random influences. As can be seen from the experimental curves, the amplitude of fluctuations in the number of species varies little from year to year.

The Volterra model is a reference (basic) model for mathematical ecology to the same extent that the harmonic oscillator model is basic for classical and quantum mechanics. With the help of 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 abundance of fur-bearing animals According to the Hudson's Bay Fur Company (Seton-Thomson, 1987), a conclusion was drawn by calculus about the qualitative nature of the behavior of such a system - about the presence of population fluctuations in such a system. Without the construction of a mathematical model and its use, such a conclusion would be impossible.

In the simplest form we have considered above, the Volterra system has two fundamental and interrelated shortcomings. Their "elimination" is devoted to extensive ecological and mathematical literature. First, the inclusion in the model of any, however small, additional factors 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 drawbacks, various modifications of the Volterra system have been proposed by different authors. The most interesting of them will be considered in Section 3.5. Here we dwell 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 solutions can change when the 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 members reflect the fact that the population of prey cannot grow indefinitely even in the absence of predators due to limited food resources, limited range of existence. The same "self-limitations" are imposed on the population of predators.

To find the stationary numbers of species iVi and N2 equate to zero the right parts of the equations of system (3.3.15). 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 of the positivity of stationary numbers should be put on the parameters of the system: N> 0 and N2 > 0. The roots of the characteristic equation of a system linearized in a neighborhood of a singular point (3.3.16):

It can be seen from the expression for the characteristic numbers that if the condition

then the numbers of predators and prey perform damped oscillations in time, the system has a nonzero 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 an 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 it cannot be solved explicitly. Therefore, we confine ourselves to general considerations. Suppose, for example, that the equilibrium point is at some distance from each of the axes. Then for phase trajectories on which the values ​​of JVj, N2 remain sufficiently large, 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 relation (3.3.17) between the parameters is fulfilled; b- when performing the inverse relationship between the parameters

phase trajectory, any variable is not very large, then random fluctuations can become significant. They lead to the fact that the representative point will move 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 dies out, either by chance or because its prey population is eliminated first. Stochastic system model predator-prey well explains the experiments of Gause (Gause, 1934; 2000), in which ciliates Paramettum candatum served as a prey for another ciliate Didinium nasatum- predator. The equilibrium numbers expected according to deterministic equations (3.3.6) in these experiments were approximately only five individuals of each species, so there is nothing surprising in the fact that in each repeated experiment, either predators or prey died out fairly quickly (and then predators). ).

So, the analysis of the Volterra models of species interaction shows that, despite the great variety of types of behavior of such systems, there can be no undamped population fluctuations 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 of mechanisms in such a system that seek to preserve its state. However, such fluctuations are observed in nature and experiment. The need for their theoretical explanation was one of the reasons for formulating model descriptions in more general view. Section 3.5 is devoted to consideration of such generalized models.

Predators can eat herbivores, and also weak predators. Predators have a wide range of food, easily switch from one prey to another, more accessible. Predators often attack weak prey. An ecological balance is maintained between prey-predator populations.[ ...]

If the equilibrium is unstable (there are no limit cycles) or outer loop is unstable, then the numbers of both species, experiencing strong fluctuations, leave the neighborhood of equilibrium. Moreover, rapid degeneration (in the first situation) occurs with low adaptation of the predator, i.e. with its high mortality (compared to the rate of reproduction of the victim). This means that a predator that is weak in all respects does not contribute to the stabilization of the system and dies out on its own.[ ...]

The pressure of predators is especially strong when, in predator-prey co-evolution, the balance shifts towards the predator and the range of the prey narrows. Competitive struggle is closely related to the lack of food resources, it can also be a direct struggle, for example, of predators for space as a resource, but most often it is simply the displacement of a species that does not have enough food in a given territory by a species that has enough of the same amount of food. This is interspecies competition.[ ...]

Finally, in the “predator-prey” system described by model (2.7), the occurrence of diffusion instability (with local equilibrium stability) is possible only in the case when the natural mortality of the predator increases with the growth of its population faster than linear function, and the trophic function is different from the Volterra one, either when the prey population is an Ollie-type population.[ ...]

Theoretically, in "one predator - two prey" models, equivalent predation (lack of preference for one or another type of prey) can affect the competitive coexistence of prey species only in those places where a potentially stable equilibrium already exists. Diversity can only increase under conditions where species with less competitiveness have a higher population growth rate than dominant species. This makes it possible to understand the situation when even grazing leads to an increase in plant species diversity where a larger number of species that have been selected for rapid reproduction coexist with species whose evolution is aimed at increasing competitiveness.[ ...]

In the same way, the choice of prey, depending on its density, can lead to a stable equilibrium in theoretical models of two competing types of prey, where no equilibrium existed before. To do this, the predator would have to be capable of functional and numerical responses to changes in prey density; it is possible, however, that switching (disproportionately frequent attacks on the most abundant victim) will be more important in this case. Indeed, switching has been found to have a stabilizing effect in "one predator - n prey" systems and is the only mechanism capable of stabilizing interactions in cases where the prey niches completely overlap. This role can be played by unspecialized predators. The preference of more specialized predators for a dominant competitor acts in the same way as predator switching, and can stabilize theoretical interactions in models in which there was no equilibrium between prey species before, provided that their niches are to some extent separated.[ ...]

Also, the community is not stabilized and the predator is 'strong in all respects', i.e. well adapted to a given prey and with low relative mortality. In this case, the system has an unstable limit cycle and, despite the stability of the equilibrium position, degenerates in a random environment (the predator eats away the prey and, as a result, dies). This situation corresponds to slow degeneration.[ ...]

Thus, with a good adaptation of a predator in the vicinity of a stable equilibrium, unstable and stable cycles can arise, i.e. depending on the initial conditions, the “predator-prey” system either tends to equilibrium, or, oscillating, leaves it, or stable fluctuations in the numbers of both species are established in the vicinity of the equilibrium.[ ...]

Organisms that are classified as predators feed on other organisms, destroying their prey. Thus, among living organisms, one more classification system should be distinguished, namely “predators” and “victims”. Relationships between such organisms have evolved throughout the evolution of life on our planet. Predatory organisms act as natural regulators of the number of prey organisms. An increase in the number of "predators" leads to a decrease in the number of "prey", which, in turn, reduces the supply of food ("prey") for the "predators", which generally dictates a decrease in the number of "prey", etc. Thus, in In the biocenosis, there are constant fluctuations in the number of predators and prey, in general, a certain balance is established for a certain period of time within fairly stable environmental conditions.[ ...]

This eventually comes to an ecological balance between predator and prey populations.[ ...]

For a trophic function of the third type, the equilibrium state will be stable if where N is the inflection point of the function (see Fig. 2, c). This follows from the fact that the trophic function is concave in the interval and, consequently, the relative share of prey consumption by the predator increases.[ ...]

Let Гг = -Г, i.e. there is a community of the “predator-prey” type. In this case, the first term in expression (7.4) is equal to zero, and in order to fulfill the condition of stability with respect to the probability of the equilibrium state N, it is required that the second term is not positive either.[ ...]

Thus, for the considered community of the predator-prey type, we can conclude that the generally positive equilibrium is asymptotically stable, i.e., for any initial data provided that N >0.[ ...]

So, in a homogeneous environment that does not have shelters for reproduction, a predator sooner or later destroys the prey population and then dies out itself. Waves of life” (changes in the number of predator and prey) follow each other with a constant shift in phase, and on average the number of both predator and prey remains approximately at the same level. The duration of the period depends on the growth rates of both species and on the initial parameters. For the prey population, the influence of the predator is positive, since its excessive reproduction would lead to the collapse of its numbers. In turn, all the mechanisms that prevent the complete extermination of the prey contribute to the preservation of the predator's food base.[ ...]

Other modifications may be due to the behavior of the predator. The number of prey individuals that the predator is able to consume in given time, has its limit. The effect of saturation of the predator when approaching this boundary is shown in Table. 2-4, B. The interactions described by equations 5 and 6 may have stable equilibrium points or exhibit cyclical fluctuations. However, such cycles differ from those reflected in the Lotka-Volterra equations 1 and 2. The cycles conveyed by equations 5 and 6 may have constant amplitude and average densities as long as the medium is constant; after a violation has occurred, they can return to their previous amplitudes and average densities. Such cycles, which are restored after violations, are called stable limit cycles. The interaction of a hare and a lynx can be considered a stable limit cycle, but this is not a Lotka-Volterra cycle.[ ...]

Let us consider the occurrence of diffusion instability in the "predator-prey" system, but first we write out the conditions that ensure the occurrence of diffusion instability in the system (1.1) at n = 2. It is clear that the equilibrium (N , W) is local (i.e. [ .. .]

Let us turn to the interpretation of cases related to the long-term coexistence of predator and prey. It is clear that in the absence of limit cycles, a stable equilibrium will correspond to population fluctuations in a random environment, and their amplitude will be proportional to the dispersion of perturbations. Such a phenomenon will occur if the predator has a high relative mortality and at the same time a high degree of adaptation to a given prey.[ ...]

Let us now consider how the dynamics of the system changes with an increase in the fitness of the predator, i.e. with decreasing b from 1 to 0. If the fitness is low enough, then there are no limit cycles, and the equilibrium is unstable. With the growth of fitness in the vicinity of this equilibrium, the emergence of a stable cycle and then an external unstable one is possible. Depending on the initial conditions (the ratio of predator and prey biomass), the system can either lose stability, i.e. leave the neighborhood of equilibrium, or stable oscillations will be established in it over time. Further growth of fitness makes the oscillatory nature of the system's behavior impossible. However, when b [ ...]

An example of negative (stabilizing) feedback is the relationship between predator and prey or the functioning of the ocean carbonate system (solution of CO2 in water: CO2 + H2O -> H2CO3). Normally, the amount of carbon dioxide dissolved in ocean water is in partial equilibrium with the concentration of carbon dioxide in the atmosphere. Local increases in carbon dioxide in the atmosphere after volcanic eruptions lead to the intensification of photosynthesis and its absorption by the carbonate system of the ocean. As the level of carbon dioxide in the atmosphere decreases, the carbonate system of the ocean releases CO2 into the atmosphere. Therefore, the concentration of carbon dioxide in the atmosphere is quite stable.[ ...]

[ ...]

As R. Ricklefs (1979) notes, there are factors that contribute to the stabilization of relationships in the “predator-prey” system: the inefficiency of the predator, the presence of alternative food resources in the predator, a decrease in the delay in the reaction of the predator, as well as environmental restrictions imposed by external environment for one population or another. Interactions between predator and prey populations are very diverse and complex. Thus, if predators are efficient enough, they can regulate the density of the prey population, keeping it at a level below the capacity of the environment. Through the influence they have on prey populations, predators affect the evolution of various prey traits, which ultimately leads to an ecological balance between predator and prey populations.[ ...]

If one of the conditions is met: 0 1/2. If 6 > 1 (kA [ ...]

The stability of the biota and the environment depends only on the interaction of plants - autotrophs and herbivorous heterotrophic organisms. Predators of any size are not able to disturb the ecological balance of the community, since under natural conditions they cannot increase their numbers with a constant number of prey. Predators not only must be themselves moving, but can only feed on moving animals.[ ...]

No other fish are as widely distributed as pikes. In the few fishing areas in stagnant or flowing waters, there is no pressure from pikes to maintain a balance between prey and predator. Pike are exceptionally well represented in the world. They are caught throughout the northern) hemisphere from the United States and Canada in North America, through Europe to northern Asia.[ ...]

Another possibility of stable coexistence arises here, in a narrow range of relatively high adaptation. Upon transition to an unstable regime with a very “good” predator, a stable external limiting cycle may arise, in which the dissipation of biomass is balanced by its influx into the system (high productivity of the prey). Then a curious situation arises when the most probable are two characteristic values ​​of the amplitude of random oscillations. Some occur near equilibrium, others near the limit cycle, and more or less frequent transitions between these modes are possible.[ ...]

Hypothetical populations that behave according to the vectors in Fig. 10.11 A, shown in fig. 10.11,-B with the help of a graph showing the dynamics of the ratio of the numbers of predator and prey and in fig. 10.11.5 in the form of a graph of the dynamics of the number of predator and prey over time. In the prey population, as it moves from a low-density equilibrium to a high-density equilibrium and returns back, a "flash" of numbers occurs. And this outbreak is not the result of an equally pronounced change in environment. On the contrary, this change in numbers is generated by the impact itself (with a low level of "noise" in the environment) and, in particular, it reflects the existence of several equilibrium states. Similar reasoning can be used to explain more complex cases of population dynamics in natural populations.[ ...]

The most important property of an ecosystem is its stability, the balance of exchange and the processes occurring in it. The ability of populations or ecosystems to maintain a stable dynamic balance in changing environmental conditions is called homeostasis (homoios - the same, similar; stasis - state). Homeostasis is based on the principle of feedback. To maintain balance in nature, no external control is required. An example of homeostasis is the "predator-prey" subsystem, in which the density of predator and prey populations is regulated.[ ...]

A natural ecosystem (biogeocenosis) functions stably with the constant interaction of its elements, the circulation of substances, the transfer of chemical, energy, genetic and other energy and information through chain-channels. According to the principle of equilibrium, any natural system with a flow of energy and information passing through it tends to develop a stable state. At the same time, the stability of ecosystems is provided automatically due to the feedback mechanism. Feedback consists in using the data received from the managed components of the ecosystem to make adjustments to the management components in the process. The relationship "predator" - "prey" discussed above in this context can be described in somewhat more detail; so, in the aquatic ecosystem predatory fish(pike in the pond) eat other types of prey fish (crucian carp); if the number of crucian carp will increase, this is an example of positive feedback; pike, feeding on crucian carp, reduces its numbers - this is an example of negative feedback; with an increase in the number of predators, the number of victims decreases, and the predator, lacking food, also reduces the growth of its population; in the end, in the pond under consideration, a dynamic balance is established in the abundance of both pike and crucian carp. A balance is constantly maintained that would exclude the disappearance of any link in the trophic chain (Fig. 64).[ ...]

Let's move on to the most important generalization, namely that negative interactions become less noticeable over time if the ecosystem is sufficiently stable and its spatial structure provides the possibility of mutual adaptation of populations. In model systems of the predator-prey type, described by the Lotka-Volterra equation, if additional terms are not introduced into the equation that characterize the effect of factors of population self-limitation, then fluctuations occur continuously and do not die out (see Levontin, 1969). Pimentel (1968; see also Pimentel and Stone, 1968) showed experimentally that such additional terms may reflect mutual adaptations or genetic feedback. When new cultures were created from individuals that had previously coexisted in a culture for two years, where their numbers were subject to significant fluctuations, it turned out that they developed an ecological homeostasis, in which each of the populations was “suppressed” by the other to such an extent that it turned out their coexistence at a more stable equilibrium.