Federal Agency for Education

State educational institution

higher professional education

Izhevsk State Technical University

Faculty of Applied Mathematics

Department "Mathematical Modeling of Processes and Technologies"

Course work

in the discipline "Differential Equations"

Theme: " Qualitative research predator-prey models "

Izhevsk 2010

INTRODUCTION

1. PARAMETERS AND BASIC EQUATION OF THE PREDATOR-VICTIM MODEL

2.2 Generalized models of Voltaire type "predator-prey".

3. PRACTICAL APPLICATION OF THE PREDATOR-VICTIM MODEL

CONCLUSION

LIST OF REFERENCES

INTRODUCTION

At present, environmental problems are of paramount importance. An important step in solving these problems is the development of mathematical models of ecological systems.

One of the main tasks of ecology PA the present stage is the study of the structure and functioning of natural systems, the search for general patterns. Big influence ecology was influenced by mathematics, contributing to the formation of mathematical ecology, especially its sections such as the theory of differential equations, the theory of stability and the theory of optimal control.

One of the first works in the field of mathematical ecology was the work of A.D. Lotki (1880 - 1949), who was the first to describe the interaction of different populations, linked by predator-prey relationships. A great contribution to the study of the predator-prey model was made by V. Volterra (1860 - 1940), V.A. Kostycin (1883-1963) At present, the equations describing the interaction of populations are called the Lotka - Volterra equations.

The Lotka - Volterra equations describe the dynamics of the mean values \u200b\u200b- the population size. At present, on their basis, more general models of interaction between populations are constructed, described by integro-differential equations, and controlled models of a predator - prey are being investigated.

One of the important problems of mathematical ecology is the problem of ecosystem sustainability and management of these systems. Management can be carried out with the aim of transferring the system from one stable state to another, with the aim of using or restoring it.

1. PARAMETERS AND BASIC EQUATION OF THE PREDATOR-VICTIM MODEL

Attempts at mathematical modeling of the dynamics of both individual biological populations and communities, including interacting populations of various species, have been undertaken for a long time. One of the first models for the growth of an isolated population (2.1) was proposed back in 1798 by Thomas Malthus:

This model is set by the following parameters:

N is the size of the population;

Difference between fertility and mortality rates.

Integrating this equation, we get:

, (1.2)

where N (0) is the population size at the moment t \u003d 0. Obviously, the Malthus model for\u003e 0 gives an infinite growth in population, which is never observed in natural populations, where the resources providing this growth are always limited. Changes in the number of populations of flora and fauna cannot be described by the simple law of Malthus; many interrelated reasons affect the dynamics of growth - in particular, the reproduction of each species is self-regulating and mutates so that this species is preserved in the process of evolution.

Mathematical ecology deals with the mathematical description of these laws - the science of the relationship between plant and animal organisms and the communities they form with each other and with the environment.

The most serious study of models of biological communities, including several populations of different species, was carried out by the Italian mathematician Vito Volterra:

,

where is the size of the population;

The rates of natural increase (or mortality) of the population; - coefficients of interspecies interaction. Depending on the choice of coefficients, the model describes either the struggle of species for a common resource, or the interaction of the predator - prey type, when one species is food for another. If in the works of other authors the main attention was paid to the construction of various models, then V. Volterra carried out a deep study of the constructed models of biological communities. It was with the book of V. Volterra, according to many scientists, that modern mathematical ecology began.

2. QUALITATIVE STUDY OF THE "PREDATOR-VICTIM" ELEMENTARY MODEL

2.1 Model of trophic interaction of the "predator-prey" type

Let us consider a model of trophic interaction of the "predator-prey" type, constructed by V. Volterre. Let there be a system consisting of two types, one of which eats the other.

Let us consider the case when one of the species is a predator, and the other is a prey, and we will assume that the predator feeds only on prey. Let's accept the following simple hypothesis:

Victim growth rate;

Predator growth rate;

The size of the victim's population;

Predator population size;

The rate of natural growth of the victim;

The rate of prey consumption by the predator;

Predator mortality rate in the absence of prey;

Coefficient of "processing" of prey biomass by a predator into its own biomass.

Then the dynamics of the population size in the predator - prey system will be described by the system of differential equations (2.1):

(2.1)

where all coefficients are positive and constant.

The model has an equilibrium solution (2.2):

According to model (2.1), the proportion of predators in the total mass of animals is expressed by the formula (2.3):

(2.3)

An analysis of the stability of the equilibrium state with respect to small perturbations showed that the singular point (2.2) is "neutral" stable (of the "center" type), that is, any deviations from equilibrium do not damp, but transfer the system to an oscillatory mode with an amplitude that depends on the magnitude of the disturbance. The trajectories of the system on the phase plane have the form of closed curves located at different distances from the equilibrium point (Fig. 1).

Figure: 1 - Phase "portrait" of the classical Volterra system "predator-prey"

Dividing the first equation of system (2.1) by the second, we obtain the differential equation (2.4) for a curve on the phase plane.

(2.4)

By integrating this equation we get:

(2.5)

where is the constant of integration, where

It is easy to show that the movement of a point along the phase plane will occur only in one direction. For this, it is convenient to change the functions and by transferring the origin of coordinates on the plane to the stationary point (2.2) and then introducing polar coordinates:

(2.6)

In this case, substituting the values \u200b\u200bof system (2.6) into system (2.1), we will have:

(2.7)

Multiplying the first equation by, and the second by and adding them, we get:

After similar algebraic transformations, we obtain the equation for:

The quantity, as seen from (4.9), is always greater than zero. Thus, it does not change sign, and the rotation always goes in one direction.

Integrating (2.9), we find the period:

When it is small, then equations (2.8) and (2.9) become the equations of an ellipse. The circulation period in this case is equal to:

(2.11)

Proceeding from the periodicity of solutions of equations (2.1), some consequences can be obtained. For this we represent (2.1) in the form:

(2.12)

and integrate over the period:

(2.13)

Since the substitutions from and vanish due to periodicity, the averages over the period turn out to be equal to the stationary states (2.14):

(2.14)

The simplest equations of the “predator-prey” model (2.1) have a number of significant disadvantages. So, they assume unlimited food resources for the prey and unlimited growth of the predator, which contradicts experimental data. In addition, as can be seen from Fig. 1, none of the phase curves are highlighted from the point of view of stability. In the presence of even small disturbing influences, the trajectory of the system will move further and further from the equilibrium position, the amplitude of oscillations will increase, and the system will collapse rather quickly.

Despite the shortcomings of model (2.1), the concept of the fundamentally oscillatory nature of the dynamics of the "predator-prey" system has become widespread in ecology. Predator-prey interactions were used to explain such phenomena as fluctuations in the numbers of predatory and peaceful animals in fishing zones, fluctuations in populations of fish, insects, etc. In fact, fluctuations in numbers may be due to other reasons as well.

Let us assume that in the predator-prey system there is an artificial extermination of individuals of both species, and consider the question of how the extermination of individuals affects the average values \u200b\u200bof their numbers, if carried out in proportion to this number with proportionality coefficients and, accordingly, for the prey and the predator. Taking into account the assumptions made, we rewrite the system of equations (2.1) in the form:

(2.15)

Suppose that, i.e., the coefficient of extermination of the victim is less than the coefficient of its natural growth. In this case, periodic fluctuations in numbers will also be observed. Let's calculate the average numbers:

(2.16)

Thus, if, then the average number of prey populations increases, and that of the predator decreases.

Consider the case when the coefficient of extermination of the victim is greater than the coefficient of its natural growth, i.e., E. In this case for any, and, therefore, the solution of the first equation (2.15) is bounded from above by the exponentially decreasing function , i.e., at.

Starting from a certain moment of time t, at which the solution of the second equation (2.15) also begins to decrease and tends to zero at. Thus, in case both species disappear.

2.1 Generalized Voltaire Predator-Prey Models

The first models of V. Volterra, naturally, could not reflect all aspects of interaction in the predator-prey system, since they were greatly simplified relative to real conditions. For example, if the number of a predator is zero, then it follows from equations (1.4) that the number of prey increases indefinitely, which is not true. However, the value of these models lies precisely in the fact that they were the basis on which mathematical ecology began to develop rapidly.

A large number of studies of various modifications of the predator-prey system have appeared, where more general models have been built, taking into account, to one degree or another, the real situation in nature.

In 1936 A.N. Kolmogorov suggested using the following system of equations to describe the dynamics of the predator - prey system:

, (2.17)

where it decreases with an increase in the number of predators, and increases with an increase in the number of prey.

This system of differential equations, due to its sufficient generality, makes it possible to take into account the real behavior of populations and, at the same time, carry out a qualitative analysis of its solutions.

Later in his work, Kolmogorov explored in detail a less general model:

(2.18)

Various special cases of the system of differential equations (2.18) have been studied by many authors. The table lists various special cases of functions,,.

Table 1 - Various models predator-prey communities

			Authors
			Volterra Lotka
			Gause
			Pislow
			Holing
			Ivlev
			Royama
			Shimazu
			May

mathematical modeling predator prey

3. PRACTICAL APPLICATION OF THE PREDATOR-VICTIM MODEL

Consider a mathematical model of the coexistence of two biological species (populations) of the "predator-prey" type, called the Volterra-Lotka model.

Let two species cohabit in an isolated environment. The environment is stationary and provides an unlimited amount of everything necessary for life, one of the species, which we will call a victim. Another species - the predator is also in stationary conditions, but feeds only on individuals of the first species. These can be crucians and pikes, hares and wolves, mice and foxes, microbes and antibodies, etc. For the sake of clarity, we will call them crucians and pikes.

The following initial values \u200b\u200bare set:

Over time, the number of crucians and pikes changes, but since there are a lot of fish in the pond, we will not distinguish 1020 crucians or 1021 and therefore we will also consider continuous functions of time t. Let's call a pair of numbers (,) the state of the model.

It is obvious that the nature of the state change (,) is determined by the values \u200b\u200bof the parameters. By changing the parameters and solving the system of equations of the model, it is possible to study the patterns of changes in the state of the ecological system over time.

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

This system of equations is called the Volterra-Lotka model. Numerical coefficients,, - are called model parameters. It is obvious that the nature of the state change (,) is determined by the values \u200b\u200bof the parameters. By changing these parameters and solving the system of equations of the model, it is possible to study the patterns of changes in the state of the ecological system.

Let us integrate both equations of the system with respect to t, which will vary from - the initial moment of time to, where T is the period for which changes occur in the ecosystem. Let in our case the period is 1 year. Then the system takes the following form:

;

;

Taking \u003d and \u003d we give similar terms, we get a system consisting of two equations:

Substituting the initial data into the resulting system, we get the population of pikes and crucians in the lake a year later:

Model of the situation of the type "predator-prey"

Let us consider a mathematical model of the dynamics of the coexistence of two biological species (populations) interacting with each other according to the “predator-prey” type (wolves and rabbits, pikes and crucians, 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 feed;
• 2. At each meeting of the prey with the predator, the latter kills the prey.

For the sake of certainty, we will call them crucians and pikes. Let be

the state of the system is determined by the quantities x (t) and y (t) - the number of crucians and pikes at a time g. To obtain mathematical equations that approximately describe the dynamics (change in time) of the population, we will proceed as follows.

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

where a \u003e 0 (fertility exceeds mortality)

Coefficient a the growth of prey depends on the number of predators (decreases with their increase). In the simplest case a - a - fjy (a\u003e 0, p\u003e 0). Then for the size of the victim population we have the differential equation

For the population of predators, we have the equation

where b \u003e 0 (death rate exceeds birth rate).

Coefficient b the decline of predators is reduced if there are prey to eat. In the simplest case, you can take b - y -Sx (y > 0, S \u003e 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 considered problem of interaction between populations. In this model, the variables x, y - 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 \u200b\u200bof 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 \u003e 0, v\u003e 0). Dividing equation (1.5) by equation (1.6), we obtain an equation with separable variables

By ignoring this equation, we have

Relation (1.7) gives the equation of phase trajectories in an 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 motion along the phase trajectory can be determined from such considerations. Let the carp be few. 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 x "and y" of crucians and pikes. Crucian carps breed, pike eat them, die out, but the number of those and others does not change. "Closed phase trajectories correspond to a periodic change in the number of crucians and pikes. Moreover, the trajectory along which the phase point moves depends on the initial conditions. Consider how the state changes along the phase trajectory. Let the point be in the position AND (fig. 1.6). There are few crucians, a lot of pikes; pikes have nothing to eat, and they are gradually dying out and almost

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

only later, when the pike became less than at , the increase in the number of crucians begins; the rate of their growth increases and their number increases - this happens approximately to the point IN. But an increase in the number of crucians leads to inhibition of the extinction process of shuk and their number begins to grow (there is more food) - the site Sun. Then there are a lot of pikes, they eat crucian carp and eat almost all of them (site CD). After that, the pikes begin to die out again and the process repeats with a period of about 5-7 years. In fig. 1.7 qualitatively plotted curves of changes in the number of crucians and pikes, depending on time. The maxima of the curves alternate, and the maxima of the abundance of pikes lag behind the maxima of crucian carp.

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

Despite the fact that the considered model is the simplest and in reality everything happens much more complicated, it made it possible to explain some of the mysterious that exists in nature. The stories of fishermen about the periods when "the pikes themselves jump into their hands" are understandable, the frequency of chronic diseases has been explained, and so on.

Note one more interesting conclusion that can be drawn from Fig. 1.6. If at the point R there is a quick catch of pikes (in another terminology - shooting 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 the point R,then the system will go to point S, and further movement will follow a larger trajectory. The vibration amplitudes will increase. This is contrary to intuition, but it just explains 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, the aphid and the ladybug, which is aphid) were in natural equilibrium x-x *, y \u003d y * (dot ABOUT in Fig. 1.6). Consider the impact of a single application of an insecticide that kills x\u003e 0 of the victims and y\u003e 0 of the predators without destroying them completely. A decrease in the number of both populations leads to the fact that the representative point from the position ABOUTWill "jump" closer to the origin, where x\u003e 0, у 0 (Fig. 1.6) It follows that as a result of the action of the insecticide, designed to destroy the prey (aphids), the number of victims (aphids) increases, and the number of predators ( ladybirds) decreases. It turns out that the number of predators can become so small that they will be fond of complete disappearance, but 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, the number of which was under the control of other predatory insects. 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 \u003d 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 crucians for food and pikes for crucians)

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 corrections / and g the situations shown in Fig. 1.8.

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

In case (2), the system is “running wild”. The stationary state is unstable. Such a system eventually falls into such a range of values x and that the model is no longer applicable.

In case (3) in a system with an unstable stationary state ABOUTa periodic mode is established over time. In contrast to 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 leads not to small fluctuations about ABOUT , as in the Volterra-Lotka model, but to oscillate a completely definite (and independent of the small 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 indicated in Fig. 1.8 is implemented in this system, additional information about the system is absolutely necessary (about the form of small corrections / and g).

Adaptations developed by prey to counter predators facilitate the development of mechanisms by predators to overcome these adaptations. Long-term coexistence of predators and prey leads to the formation of a system of interaction in which both groups are stably preserved in the study area. Violation of such a system often leads to negative environmental consequences.

The negative impact of the violation of coevolutionary connections is observed during the introduction of species. In particular, goats and rabbits introduced in Australia do not have effective mechanisms of population regulation on this continent, which leads to the destruction of natural ecosystems.

Mathematical model

Let's say that two kinds of animals live in a certain area: rabbits (feeding on plants) and foxes (feeding on rabbits). Let the number of rabbits $x$, number of foxes $y$... Using the Malthus Model with the necessary amendments, taking into account the eating of rabbits by foxes, we come to the following system, which bears the name of the Volterra model - Lots:

\\ begin (cases) \\ dot x \u003d (\\ alpha -c y) x; \\\\

\\ dot y \u003d (- \\ beta + d x) y. \\ end (cases)

Model behavior

The group lifestyle of predators and their prey radically changes the behavior of the model, giving it increased stability.

Rationale: with a group lifestyle, the frequency of chance encounters of predators with potential prey decreases, which is confirmed by observations of the dynamics of the number of lions and wildebeests in the Serengeti Park.

History

The model of coexistence of two biological species (populations) of the "predator-prey" type is also called the Volterra-Lotka model.

see also

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Notes

Literature

• V. Volterra, Mathematical theory of the struggle for existence. Per. with French O. N. Bondarenko. Edited and afterword by Yu. M. Svirezhev. Moscow: Nauka, 1976.287 p. ISBN 5-93972-312-8
• A. D. Bazykin, Mathematical biophysics of interacting populations. Moscow: Nauka, 1985.181 p.
• A. D. Bazykin, Yu. A. Kuznetsov, A. I. Khibnik, Portraits of bifurcations (Bifurcation diagrams - dynamical systems on a plane) / Series “New in life, science, technology. Mathematics, cybernetics "- M .: Knowledge, 1989. 48 p.
• P. V. Turchin,

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Excerpt from the Predator-Prey System

- Charmant, charmant, [Charming, charming,] - said Prince Vasily.
- C "est la route de Varsovie peut etre, [This is a Warsaw road, perhaps.]" Said Prince Ippolit loudly and unexpectedly. Everyone looked at him, not understanding what he wanted to say by this. Prince Ippolit also looked around with cheerful surprise He, like the others, did not understand what the words he said meant. During his diplomatic career, he more than once noticed that the words spoken in this way suddenly turned out to be very witty, and he just said these words just in case, the first ones who came to his tongue. “Maybe it will work out very well,” he thought, “but if it doesn’t work out, they will be able to arrange it there.” Indeed, while an awkward silence reigned, that insufficiently patriotic face entered, which was waiting for the appeal Anna Pavlovna, and she, smiling and shaking her finger at Ippolita, invited Prince Vasily to the table, and, bringing him two candles and a manuscript, asked him to start.
- Most Merciful Emperor! - Prince Vasily proclaimed sternly and looked around the audience, as if asking if anyone had anything to say against this. But nobody said anything. - “The capital city of Moscow, New Jerusalem, accepts its Christ,” he suddenly struck at his word, “like a mother into the arms of her zealous sons, and through the arising darkness, foreseeing the brilliant glory of your power, sings in delight:“ Hosanna, blessed is the coming ! " - Prince Vasily in a crying voice uttered these last words.
Bilibin carefully examined his nails, and many, apparently, were shy, as if asking, what are they to blame? Anna Pavlovna in a whisper was already repeating ahead, like an old woman, the prayer of communion: "Let the impudent and insolent Goliath ..." - she whispered.
Prince Vasily continued:
- “Let the impudent and impudent Goliath from the borders of France carry deadly horrors on the edges of Russia; meek faith, this sling of the Russian David, will suddenly slay the head of his bloodthirsty pride. Behold the image saint Sergius, an ancient zealot for the welfare of our fatherland, is brought to your Imperial Majesty. Painful, that my weakening forces prevent me from enjoying your most amiable contemplation. I send warm prayers to heaven, so that the all-powerful will exalt the race of the right and fulfill your Majesty's desires in good ”.
- Quelle force! Quel style! [What a power! What a syllable!] - praise was heard to the reader and writer. Inspired by this speech, Anna Pavlovna's guests spoke for a long time about the state of the fatherland and made various assumptions about the outcome of the battle, which was to be given the other day.
- Vous verrez, [you will see.] - Anna Pavlovna said, - that tomorrow, on the birthday of the sovereign, we will receive news. I have a good feeling.

Anna Pavlovna's premonition was indeed justified. The next day, during a prayer service in the palace on the occasion of the sovereign's birthday, Prince Volkonsky was summoned from the church and received an envelope from Prince Kutuzov. This was Kutuzov's report, written on the day of the battle from Tatarinova. Kutuzov wrote that the Russians did not retreat a single step, that the French had lost much more than ours, that he was reporting in a hurry from the battlefield, not having time to collect the latest information. So it was a victory. And immediately, without leaving the temple, gratitude was given to the creator for his help and for the victory.
Anna Pavlovna's premonition was justified, and a joyful festive mood of spirit reigned in the city all morning. Everyone recognized the victory as perfect, and some have already talked about the capture of Napoleon himself, about his overthrow and the election of a new head for France.
Far from business and amid the conditions of court life, it is very difficult for events to be reflected in all their fullness and power. Involuntarily, general events are grouped around one particular case. So now the main joy of the courtiers was as much in the fact that we won, as in the fact that the news of this victory happened on the birthday of the sovereign. It was like a lucky surprise. In the news of Kutuzov, it was also said about the losses of the Russians, and among them Tuchkov, Bagration, Kutaisov were named. Also the sad side of the event involuntarily in the local, St. Petersburg world was grouped around one event - the death of Kutaisov. Everyone knew him, the emperor loved him, he was young and interesting. On this day, everyone met with the words:
- How amazing it happened. In the most prayer service. And what a loss the Kutaisov! Oh, what a pity!
- What did I tell you about Kutuzov? - Prince Vasily spoke now with the pride of a prophet. - I have always said that he alone is able to defeat Napoleon.
But the next day no news came from the army, and the general voice became alarmed. The courtiers suffered for the suffering of the unknown in which the sovereign was.
- What is the position of the sovereign! - said the courtiers and no longer extolled, as the day before yesterday, and now they condemned Kutuzov, who was the cause of the sovereign's concern. Prince Vasily on that day no longer boasted of his protege Kutuzov, but remained silent when it came to the commander-in-chief. In addition, by the evening of that day, it was as if everything had come together in order to plunge into alarm and uneasiness of the Petersburg residents: another terrible news had joined. Countess Elena Bezukhova died suddenly from this terrible disease, which was so pleasant to pronounce. Officially, in large societies, everyone said that Countess Bezukhova died of a terrible attack of angine pectorale [chest sore throat], but intimate circles told details about how le medecin intime de la Reine d "Espagne [the medic of the Queen of Spain] prescribed Helene small doses some kind of medicine for producing a certain action; but how Helene, tormented by the fact that the old count suspected her, and the fact that her husband, to whom she wrote (this unfortunate, depraved Pierre), did not answer her, suddenly took a huge dose of the medicine prescribed for her and died in agony before they could offer help. ”It was said that Prince Vasily and the old count had taken up the Italian, but the Italian showed such notes from the unfortunate deceased that he was immediately released.

Mathematical modeling of biological processes began with the creation of the first simplest models of the ecological system.

Let's say lynxes and hares live in a certain closed area. Lynxes feed only on hares, and hares - on plant food, available in unlimited quantities. It is necessary to find the macroscopic characteristics that describe the populations. These characteristics are the number of individuals in populations.

The simplest model The relationship between predator and prey populations, based on the logistic equation of growth, is named (like the model of interspecific competition) after its creators, Lotka and Volterra. This model oversimplifies the situation under study, but is still useful as a starting point in the analysis of the predator-prey system.

Suppose that (1) a prey population exists in an ideal (density-independent) environment, where its growth can be limited only by the presence of a predator, (2) an environment in which a predator exists, whose population growth is limited only by the abundance of prey, is just as ideal, (3 ) both populations reproduce continuously according to the exponential growth equation, (4) the rate of prey eating by predators is proportional to the frequency of encounters between them, which, in turn, is a function of population density. These assumptions underlie the Lotka-Volterra model.

Let the prey population grow exponentially in the absence of predators:

dN / dt \u003d r 1 N 1

where N is the number, and r is the specific instantaneous growth rate of the victim's population. If predators are present, then they destroy individuals of the prey at a rate that is determined, firstly, by the frequency of encounters of predators and prey, which increases as their numbers increase, and, secondly, by the efficiency with which the predator detects and catches its prey when meeting. The number of prey met and eaten by one predator N c is proportional to the hunting efficiency, which we express through the coefficient C 1; the number (density) of the victim N and the time spent on searching T:

N C \u003d C 1 NT(1)

From this expression, it is easy to determine the specific rate of prey consumption by a predator (i.e., the number of prey eaten by one predator per unit of time), which is often also called the functional response of a predator to the prey population density:

In the considered model C 1is a constant. This means that the number of prey taken from the population by predators increases linearly with an increase in its density (the so-called type 1 functional response). It is clear that the total rate of prey eating by all individuals of the predator will be:

(3)

where R -the size of the predator population. We can now write the growth equation for the prey population as follows:

In the absence of a prey, individuals of the predator starve and die. Let us also assume that in this case the size of the predator population will decrease exponentially according to the equation:

(5)

where r 2 - specific instant mortality in the predator population.

If victims are present, then those individuals of the predator who can find and eat them will breed. Fertility in the predator population in this model depends only on two factors: the rate of prey consumption by the predator and the efficiency with which the consumed food is processed by the predator into its offspring. If we express this efficiency through the coefficient s, then the birth rate will be:

Since C 1 and s are constants, their product is also a constant, which we will denote as C 2. Then the growth rate of the predator population will be determined by the balance of fertility and mortality in accordance with the equation:

(6)

Equations 4 and 6 together form the Lotka-Volterra model.

We can investigate the properties of this model in the same way as in the case of competition, i.e. by constructing a phase diagram, on which the number of prey is plotted along the ordinate, and that of the predator, along the abscissa, and drawing isocline lines on it corresponding to the constant population size. These isoclines are used to determine the behavior of interacting populations of predator and prey.

For the victim population: whence

Thus, since r, and С 1, are constants, the isocline for the prey will be the line on which the number of the predator (R)is constant, i.e. parallel to the abscissa and intersecting the ordinate at the point P \u003d r 1 /From 1. Above this line, the number of victims will decrease, and below this line, it will increase.

For the predator population:

whence

Insofar as r 2and С 2 are constants, the isocline for the predator will be the line on which the number of prey (N) is constant, i.e. perpendicular to the ordinate and intersecting the abscissa at the point N \u003d r 2 / C 2. To the left of it, the number of the predator will decrease, and to the right, it will increase.

If we consider these two isoclines together, then we can easily notice that the interaction of populations of predator and prey is cyclical, since their numbers undergo unlimited conjugate fluctuations. When the number of prey is high, the number of predators increases, which leads to an increase in the pressure of predation on the prey population and thereby to a decrease in its number. This decrease, in turn, leads to a shortage of food for predators and a decrease in their numbers, which causes a weakening of the pressure of predation and an increase in the number of prey, which again leads to an increase in the prey population, etc.

This model is characterized by the so-called "neutral stability", which means that populations perform the same cycle of fluctuations for an unlimited time until some external influence changes their numbers, after which the populations make a new cycle of fluctuations with different parameters ... In order for the cycles to become stable, populations must, after external influence strive to return to the original cycle.Such cycles, in contrast to neutral stable oscillations in the Lotka-Volterra model, are usually called stable limit cycles.

The Lotka-Volterra model, however, is useful in that it allows us to demonstrate the main tendency in the predator-prey relationship, the occurrence of cyclical conjugate fluctuations in the size of their populations.

Predators can eat herbivores as well as 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 between prey-predator populations is maintained. [...]

If the equilibrium is unstable (there are no limit cycles) or the outer cycle is unstable, then the numbers of both species, experiencing strong fluctuations, leave the vicinity of equilibrium. Moreover, rapid degeneration (in the first situation) occurs at 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 itself is dying out. [...]

The pressure of predators is especially strong when in coevolution the predator-prey balance shifts towards the predator and the prey's area is narrowed. Competitive struggle is closely related to a shortage 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 displacement of a species that lacks food in a given territory by a species that is quite enough of the same amount of food. This is already interspecies competition. [...]

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

Theoretically, in the “one predator - two prey” models, equivalent grazing (lack of preference for one or another prey type) 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 when the population growth rate is higher in species with less competitiveness than in dominants. This allows us to understand the situation when uniform grazing leads to an increase in the species diversity of plants where a greater number of species selected for rapid reproduction coexist with species whose evolution is aimed at increasing competitiveness. [...]

Likewise, the choice of a victim, depending on its density, can lead to a stable equilibrium in theoretical models of two competing types of victim, where no equilibrium existed before. For 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, it has been established that switching has a stabilizing effect in systems "one predator - n prey" and is the only mechanism capable of stabilizing interactions in cases where niches of prey completely overlap. This role can be played by non-specialized predators. The preference for more specialized predators of a dominant competitor acts in the same way as predator switching and can stabilize theoretical interactions in models in which there was previously no equilibrium between prey species, provided that their niches are somewhat separated. [...]

Also, the predator “strong in all respects” does not stabilize the community, i.e. well adapted to a given victim and with a 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 a slow degeneration. [...]

Thus, with good adaptation of the predator in the vicinity of stable equilibrium, unstable and stable cycles can arise, i.e. depending on the initial conditions, the predator-prey system either tends to equilibrium, or oscillates away from it, or stable fluctuations in the numbers of both species are established in the vicinity of 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 "prey". The relationship between such organisms developed 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 food ("prey") supplies for "predators", which generally dictates a decrease in the number of "victims", etc. Thus, in biocenosis constantly fluctuates in the number of predators and prey, in general, a certain equilibrium is established for a certain period of time within fairly stable environmental conditions. [...]

This ultimately comes to an ecological balance between the populations of predator and prey. [...]

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

Let Γr \u003d -Γ, i.e. there are communities of the "predator-prey" type. In this case, the first term in expression (7.4) is equal to zero, and to satisfy the stability condition with respect to the probability of an 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 about the asymptotic stability of an overall positive equilibrium position, i.e., for any initial data 1H (0)\u003e 0, evolution occurs in such a way that N (7) - ■ K at provided that N\u003e 0. [...]

So, in a homogeneous environment that does not have shelter for reproduction, the predator sooner or later destroys the prey population and then dies out on its own. Waves of life ”(changes in the number of predator and prey) follow each other with a constant phase shift, 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 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 threshold is shown in table. 2-4, B. The interactions described by equations 5 and 6 can 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 represented 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 recover from violations, are called stable limit cycles. The interaction between the hare and the lynx can be considered a stable limit cycle, but this is not the Lotka-Volterra cycle. [...]

Let us consider the onset of diffusion instability in the "predator-prey" system, but first we write down the conditions that ensure the onset of diffusion instability in system (1.1) for n \u003d 2. It is clear that the equilibrium (N, U) is local (ie [.. .]

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

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

An example of a negative (stabilizing) feedback is the relationship between a predator and a prey or the functioning of the ocean carbonate system (CO2 solution in water: CO2 + H2O -\u003e H2CO3). Usually the amount of carbon dioxide dissolved in ocean water is in partial equilibrium with the concentration 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. When 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 noted by R. Ricklefs (1979), there are factors that contribute to the stabilization of relationships in the “predator-prey” system: the inefficiency of the predator, the availability of alternative food resources for the predator, a decrease in the delay in the predator's response, and environmental restrictions imposed external environment for a particular population. The interactions between predator and prey populations are very diverse and complex. So, if predators are efficient enough, they can regulate the density of the prey population, keeping it below the capacity of the environment. Through the influence they exert on prey populations, predators influence the evolution of various prey traits, which ultimately leads to an ecological balance between the predator and prey populations. [...]

If one of the conditions is met: 0 1/2. If 6\u003e 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 upset the ecological balance of the community, since under natural conditions they cannot increase their numbers with a constant number of prey. Predators not only have to be locomotive themselves, but can only feed on locomotive animals. [...]

No other fish are as widespread as pike. In a few places in stagnant or flowing water bodies, there is no pressure from the pike to maintain a balance between the prey and the predator. Only modern artificial reservoirs, in which pikes are undesirable fish due to the breeding of other fish there, are not purposefully populated by them. Pike are exceptionally well represented in the world. They are caught throughout the northern) hemisphere from the United States and Canada to North America, through Europe to northern Asia. [...]

Another possibility of sustainable coexistence arises here, in a narrow range of relatively high adaptation. In the 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 inflow into the system (high prey productivity). Then a curious situation arises when the most probable are two characteristic values \u200b\u200bof the amplitude of random oscillations. Some occur near equilibrium, others - near the limit cycle, and more or less frequent transitions between these regimes are possible. [...]

Hypothetical populations that behave according to the vectors in Fig. 10.11 A, shown in Fig. 10.11, -B using 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 predators and prey in time. In the prey population, as it moves from equilibrium at low density to equilibrium at high density and returns back, a "flash" of numbers occurs. And this outbreak in numbers 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 small 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 difficult cases population dynamics in natural populations. [...]

The most important property of an ecosystem is its stability, balance of exchange and 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 is not required external management... An example of homeostasis is the “predator-prey” subsystem, in which the population density of the predator and prey is regulated. [...]

The natural ecosystem (biogeocenosis) functions steadily with the constant interaction of its elements, the circulation of substances, the transfer of chemical, energetic, genetic and other energy and information along the chains-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 sustainability of ecosystems is provided automatically through a feedback mechanism. Feedback is about using the received data from the managed components of the ecosystem to make adjustments to the control components in the process. The relationships "predator" - "prey" considered above can be described in a little more detail in this context; so, in the aquatic ecosystem predatory fish (pike in a pond) eat other types of prey fish (crucian carp); if the number of crucian carp will increase, this is an example of a positive feedback; pike, feeding on crucian carp, reduces its number - this is an example of negative feedback; with an increase in the number of predators, the number of prey 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. Equilibrium is constantly maintained, which 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 no additional terms are introduced into the equation that characterize the action of factors of self-limitation of numbers, then oscillations occur continuously and do not damp (see Levontin, 1969). Pimentel (1968; see also Pimentel and Stone, 1968) has experimentally shown that such additional terms can 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 possible their coexistence with a more stable equilibrium.