The predator-prey model can be applied. Coursework: Qualitative study of the predator-prey model

Model of the "predator-prey" type of situation

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 crucian carp, etc.), called the Voltaire-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.

Suppose there are two biological species that live together in an isolated environment. This assumes:

• 1. The victim can find enough food to live on;
• 2. At each meeting of the victim with the predator, the latter kills the victim.

For definiteness, we will call them crucians and pikes. Let

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

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

where a> 0 (birth rate exceeds death rate)

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

For the population of predators, we have the equation

where b>0 (mortality exceeds birth rate).

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

Thus, equations (1.5) and (1.6) represent a mathematical model of the considered problem of population interaction. 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 the initial conditions - given values initial populations.

Let us now analyze the constructed mathematical model.

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

Using this equation, we will have

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

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

Equalities (1.9) determine the position of equilibrium on the phase plane (the point O)(Figure 1.6).

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

All phase trajectories (with the exception of the point 0) closed curves enclosing the equilibrium position. The state of equilibrium corresponds to a constant number of x' and y' crucians and pikes. Carp 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 position BUT(Fig. 1.6). There are few carp here, a lot of pike; pikes have nothing to eat, and they are gradually dying out and almost

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

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

This behavior is typical for various predator-prey systems. 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 things that exist in nature. The stories of anglers about the periods when “pikes themselves jump into their hands” are understandable, the frequency of chronic diseases, etc., has been explained.

We note another interesting conclusion that can be drawn from Fig. 1.6. If at the point R there is a quick catch of pike (in other terminology - shooting of wolves), then the system "jumps" to the point Q, and further movement occurs along a smaller closed trajectory, which is intuitively expected. If we reduce the number of pikes at the point R, then the system will go to the point S, and further movement will occur along the trajectory bigger size. The oscillation amplitude will increase. This is contrary to intuition, but it just explains such a phenomenon: as a result of shooting wolves, their numbers increase with time. Thus, the choice of the moment of shooting is important in this case.

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

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

With a small change in the 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 (return of the system to the initial state), valid for the 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) equilibrium state O steadily. For any other initial conditions, after enough big time it is installed.

In case (2) the system "goes to the floor". The stationary state is unstable. Such a system eventually falls into such a range of values X and y that the model is no longer applicable.

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

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

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

Suppose lynxes and hares live in some closed area. Lynxes eat only hares, and hares eat plant foods that are available in unlimited quantities. It is necessary to find macroscopic characteristics that describe populations. Such characteristics are the number of individuals in populations.

The simplest model relationship between predator and prey populations, based on the logistic growth equation, is named (as well as the model of interspecific competition) after its creators - Lotka and Volterra. This model greatly simplifies 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 equally ideal environment in which there is a predator whose population growth is limited only by the abundance of prey, (3 ) both populations reproduce continuously according to the exponential growth equation, (4) the rate of prey consumption by predators is proportional to the frequency of meetings 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 =r 1 N 1

where N is the number, and r, is the specific instantaneous speed prey population growth. If predators are present, then they destroy prey individuals at a rate that is determined, firstly, by the frequency of meetings between 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 victims met and eaten by one predator N c is proportional to the hunting efficiency, which we will express through the coefficient C 1; the number (density) of the victim N and the time spent searching T:

N C \u003d C 1 NT(1)

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

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

(3)

where R - predator population. Now we can write the prey population growth equation as follows:

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

(5)

where r2- specific instantaneous mortality in the predator population.

If there are victims, then those individuals of the predator that can find and eat them will multiply. The birth rate in the predator population in this model depends only on two circumstances: 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 in terms of 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 births and deaths in accordance with the equation:

(6)

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

We can explore the properties of this model in exactly the same way as in the case of competition, i.e. by constructing a phase diagram, in which the number of prey is plotted along the ordinate axis, and predator - along the abscissa axis, and drawing isoclines-lines on it, corresponding to a constant number of populations. With the help of such isoclines, the behavior of interacting predator and prey populations is determined.

For the prey population: whence

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

For the predator population:

whence

Because the r2 and C 2 - 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 axis and intersecting the abscissa axis at the point N = r 2 /C 2. To the left of it, the number of predators will decrease, and to the right - to increase.

If we consider these two isoclines together, we can easily see that the interaction between predator and prey populations 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 drop 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 oscillations indefinitely until some external influence changes their numbers, after which the populations perform a new cycle of oscillations with different parameters. . In order for cycles to become stable, populations must, after external influences, strive to return to the original cycle. Such cycles, in contrast to neutrally stable oscillations in the Lotka-Volterra model, are called stable limit cycles.

The Lotka-Volterra model, however, is useful in that it allows us to demonstrate the main trend in the predator-prey relationship, the emergence of cyclic conjugate fluctuations in the number of their populations.

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.

CHAPTER 4. LABORATORY MODELING OF THE PREDATOR - PREY SYSTEM

Duke University scientists, in collaboration with colleagues from Stanford University, Howard Hughes Medical Institute and the California Institute of Technology, working under the direction of Dr. Lingchong You (Lingchong You), have developed living system from genetically modified bacteria, which will allow a more detailed study of the interactions between predator and prey at the 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 can be widely used in medicine, cleaning environment and creation of biocomputers. 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 high level instability. In other words, when a wide range input parameters (such as the mortality of predators, the 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 of starvation.

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 recent times 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. ...

Back in the 20s. A. Lotka, and somewhat later, independently of him, W. Voltaire, proposed mathematical models that describe conjugate fluctuations in the number of predator and prey.

The model consists of two components:

C is the number of predators; N is the number of victims;

Suppose that in the absence of predators, the prey population will grow exponentially: dN/dt = rN. But prey is destroyed by predators at a rate determined by the frequency of predator-prey encounters, and the frequency of encounters increases as the number of predator (C) and prey (N) increases. The exact number of met and successfully eaten prey will depend on the efficiency with which the predator finds and captures the prey, i.e. from a' - "search efficiency" or "frequency of attacks". Thus, the frequency of “successful” encounters between the predator and the prey and, consequently, the rate of prey consumption will be equal to a’CN and in general: dN/dt = rN – a’CN (1*).

In the absence of food, individual individuals of the predator lose weight, starve and die. Suppose that in the model under consideration, the size of the predator population in the absence of food due to starvation will decrease exponentially: dC/dt = - qC, where q is mortality. The death is compensated by the birth of new individuals at a rate that, as it is believed in this model, depends on two circumstances:

1) food intake rate, a'CN;

2) the efficiency (f) with which this food passes into the predator's offspring.

Thus, the birth rate of a predator is equal to fa’CN and in general: dC/dt = fa’CN – qC (2*). Equations 1* and 2* constitute the Lotka-Voltaire model. The properties of this model can be studied, line isoclines corresponding to a constant population size can be constructed, with the help of such isoclines the behavior of interacting predator-prey populations is determined.

In the case of the prey population: dN/dt = 0, rN = a'CN, or C = r/a'. Because r and a' = const, the isocline for the victim will be the line for which the value of C is constant:

At a low predator density (C), the number of prey (N) increases, on the contrary, it decreases.

Similarly for predators (equation 2*) with dC/dt = 0, fa’CN = qC, or N = q/fa’, i.e. the isocline for the predator will be the line along which N is constant: At high prey density, the predator population increases, and at low density, it decreases.

Their number undergoes unlimited conjugate fluctuations. When the number of prey is high, the number of predators increases, which leads to an increase in the pressure of predators on the prey population and, thereby, to a decrease in its number. This decrease, in turn, leads to the restriction of predators in food and a decrease in their numbers, which causes a weakening of the pressure of predators and an increase in the number of prey, which again leads to an increase in the population of predators, etc.

Populations perform the same cycle of oscillations indefinitely until some external influence changes their numbers, after which the populations perform new cycles of unlimited oscillations. In fact, the environment is constantly changing, and population numbers will constantly shift by new level. In order for the cycles of fluctuations that a population makes to be regular, they must be stable: if an external influence changes the level of populations, then they must tend to the original cycle. Such cycles are called stable, limit cycles.

The Lotka-Voltaire model makes it possible to show the main trend in the predator-prey relationship, which is expressed in the occurrence of fluctuations in the abundance in the prey population, accompanied by fluctuations in the abundance in the predator population. The main mechanism of such fluctuations is the time delay inherent in the sequence of the state from a high number of prey to a high number of predators, then to a low number of prey and a low number of predators, to a high number of prey, and so on.

5) POPULATION STRATEGIES OF PREDATOR AND PREY

Relationships "predator - prey" represent the links in the process of transfer of matter and energy from phytophages to zoophages or from predators of a lower order to predators higher order. By the nature of these relationships distinguish three variants of predators:

a) collectors. Predator collects small rather numerous moving victims. This variant of predation is characteristic of many species of birds (plotters, finches, pipits, etc.), which expend energy only to search for prey;

b) true predators. The predator pursues and kills the prey;

in) pastures. These predators use the prey repeatedly, for example, gadflies or horseflies.

The strategy of obtaining food in predators is aimed at ensuring the energy efficiency of nutrition: the energy consumption for obtaining food should be less than the energy obtained during its assimilation.

True Predators are divided into

"reapers" who feed on abundant resources (n, planktonic fish and even baleen whales), and "hunters" who get less abundant food. In its turn

"hunters" are divided into "ambushes" lying in wait for prey (for example, pike, hawk, cat, mantis beetle), "seekers" (insectivorous birds) and "pursuers". For last group searching for food does not require a lot of energy, but it takes a lot of it to master the victim (lions in the savannahs). However, some predators can combine elements of strategy different options hunting.

As with the “phytophage-plant” relationship, the situation in which all the victims will be eaten by predators, which ultimately will lead to their death, is not observed in nature. ecological balance between predators and prey is maintained special mechanisms that reduce the risk of complete extermination of victims. Yes, victims can:

Run away from a predator. In this case, as a result of adaptations, the mobility of both victims and predators increases, which is especially characteristic of steppe animals, which have nowhere to hide from pursuers;

Acquire a protective color (“pretend” to be leaves or knots) or, on the contrary, a bright color, N .: red, warning a predator about a bitter taste. It is well known that the color change of a hare in different times years, which allows him to disguise himself in the grass in summer, and in winter against the background white snow. An adaptive change in coloration can occur at different stages of ontogeny: seal pups are white (the color of snow), while adults are black (the color of a rocky coast);

Spread in groups, which makes their search and fishing for a predator more energy-intensive;

Hide in shelters;

Switch to active defense measures (herbivores with horns, spiny fish), sometimes joint (musk oxen can take up “all-round defense” from wolves, etc.).

In turn, predators develop not only the ability to quickly pursue victims, but also the sense of smell, which allows them to determine the location of the victim by smell. Many species of predators tear the holes of their victims (foxes, wolves).

At the same time, they themselves do everything possible not to reveal their presence. This explains the cleanliness of small cats, which spend a lot of time on the toilet and burying excrement to eliminate the smell. Predators wear "camouflage robes" (the striping of pikes and perches, making them less visible in the thickets of macrophytes, the striping of tigers, etc.).

Complete protection from predators of all individuals in the populations of prey animals also does not occur, since this would lead not only to the death of starving predators, but ultimately to the catastrophe of prey populations. At the same time, in the absence or decrease in the population density of predators, the gene pool of the prey population deteriorates (sick and old animals remain), and due to a sharp increase in their number, the food base is undermined.

For this reason, the effect of the dependence of the number of prey and predator populations - a pulsation in the number of the prey population, followed by a pulsation in the number of the predator population with some delay ("the Lotka-Volterra effect") - is rarely observed.

A fairly stable ratio is established between the biomasses of predators and prey. So, R. Ricklefs cites data that the ratio of predator and prey biomass ranges from 1:150 to 1:300. In different ecosystems temperate zone In the United States, there are 300 small white-tailed deer (weight 60 kg), 100 large elk deer (weight 300 kg) or 30 elk (weight 350) per wolf. The same pattern was found in the savannas.

With intensive exploitation of phytophage populations, people often exclude predators from ecosystems (in the UK, for example, there are roe deer and deer, but no wolves; in artificial reservoirs where carp and other pond fish are bred, there are no pikes). In this case, the role of a predator is performed by the person himself, removing a part of the individuals of the phytophage population.

A special variant of predation is observed in plants and fungi. In the plant kingdom, there are about 500 species that can catch insects and partially digest them with the help of proteolytic enzymes. Predatory mushrooms form trapping devices in the form of small oval or spherical heads located on short sprigs of mycelium. However, the most common type of trap is sticky three-dimensional networks consisting of a large number of rings resulting from the branching of hyphae. Predatory mushrooms can catch quite large animals, for example, roundworms. After the worm gets entangled in the hyphae, they grow inside the animal's body and fill it quickly.

1.Constant and favorable level of temperature and humidity.

2. Abundance of food.

3. Protection from adverse factors.

4. Aggressive chemical composition habitat (digestive juices).

1. The presence of two habitats: the first order environment - the host organism, the second order environment - the external environment.

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 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 -

FROM I C2 C2

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

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 FROM.

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 AT. 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 AT 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 straight line Y = cX, and from there crossing the curve (3.3.10), we also find four points of the curve (3.3.9). The exception 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, arbitrarily 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 minimum number prey available to the 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 commit in time damped oscillations, the system has a nonzero singular point and 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 extinction appropriate type. 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. The stochastic model of the predator-prey system 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 special form 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.