Why the predator-prey model is not accurate. Coursework: Qualitative study of the predator-prey model

federal agency of 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"

Topic: "Qualitative study of the predator-prey model"

Izhevsk 2010

INTRODUCTION

1. PARAMETERS AND MAIN EQUATION OF THE PREDATOR- PREY MODEL

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

3. PRACTICAL APPLICATIONS OF THE PREDATOR- PREY MODEL

CONCLUSION

BIBLIOGRAPHY

INTRODUCTION

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

One of the main tasks of the ecology of pa present stage is the study of the structure and functioning of natural systems, the search for common patterns. Big influence ecology was influenced by mathematics, which contributed to the development of mathematical ecology, especially such sections of it 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 various populations connected by predator-prey relationships. A great contribution to the study of the predator-prey model was made by V. Volterra (1860 - 1940), V.A. Kostitsyn (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 average values ​​- population size. At present, on their basis, more general models of interaction between populations, described by integro-differential equations, are constructed, controlled predator-prey models are being studied.

One of important issues mathematical ecology is the problem of sustainability of ecosystems, 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 it or restoring it.

1. PARAMETERS AND MAIN EQUATION OF THE PREDATOR- PREY MODEL

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

This model is set by the following parameters:

N - population size;

The difference between birth and death rates.

Integrating this equation we get:

, (1.2)

where N(0) is the population size at the moment t = 0. Obviously, the Malthus model for > 0 gives an infinite population growth, which is never observed in natural populations, where the resources that ensure this growth are always limited. Changes in the populations of flora and fauna cannot be described simple law Malthus, the dynamics of growth is influenced by many interrelated reasons - in particular, the reproduction of each species is self-regulated and modified so that this species is preserved in the process of evolution.

The mathematical description of these regularities is carried out by mathematical ecology - the science of the relationship of 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, which include several populations of different species, was carried out by the Italian mathematician Vito Volterra:

,

where is the population size;

Coefficients 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 shared resource, or an 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 conducted a deep study of the constructed models of biological communities. It is from the book of V. Volterra, in the opinion of many scientists, that modern mathematical ecology began.

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

2.1 Predator-prey trophic interaction model

Let us consider the model of trophic interaction according to the "predator-prey" type, built by W. Volterra. Let there be a system consisting of two species, of which one eats the other.

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 the prey. We accept the following simple hypothesis:

Prey growth rate;

Predator growth rate;

Prey population;

Predator population size;

Coefficient of natural increase of the victim;

The rate of prey consumption by the predator;

Predator mortality rate in the absence of prey;

Coefficient of “processing” of the prey biomass by the predator into its own biomass.

Then the population dynamics 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 formula (2.3):

(2.3)

An analysis of the stability of the equilibrium state with respect to small perturbations showed that singular point(2.2) is “neutrally” stable (of the “center” type), i.e., any deviations from equilibrium do not decay, but transfer the system to an oscillatory regime with an amplitude that depends on the magnitude of the perturbation. 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).

Rice. 1 - Phase "portrait" of the classical Volterra system "predator-prey"

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

(2.4)

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. To do this, it is convenient to make a change of functions and , moving the origin of coordinates on the plane to a stationary point (2.2) and then introducing polar coordinates:

(2.6)

In this case, substituting the values ​​of system (2.6) into system (2.1), we 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 value , as can be seen from (4.9), is always greater than zero. Thus, it does not change sign, and the rotation goes in the same direction all the time.

Integrating (2.9) we find the period:

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

(2.11)

Based on the periodicity of solutions of equations (2.1), we can obtain some corollaries. For this, we represent (2.1) in the form:

(2.12)

and integrate over the period:

(2.13)

Since the substitutions from and due to periodicity vanish, 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 drawbacks. Thus, they assume unlimited food resources for the prey and unlimited growth of the predator, which contradicts the experimental data. In addition, as can be seen from Fig. 1, none of the phase curves is highlighted in terms of stability. In the presence of even small perturbing influences, the trajectory of the system will go farther and farther from the equilibrium position, the amplitude of oscillations will increase, and the system will quickly collapse.

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

Let us assume that in the predator-prey system artificial destruction of individuals of both species takes place, and we will consider the question of how the destruction of individuals affects the average values ​​of their numbers, if it is carried out in proportion to this number with proportionality coefficients and, respectively, for prey and predator. Taking into account the assumptions made, we rewrite the system of equations (2.1) in the form:

(2.15)

We assume that , i.e., the coefficient of extermination of the victim is less than the coefficient of its natural increase. In this case, there will also be periodic fluctuations numbers. Let us calculate the average values ​​of the numbers:

(2.16)

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

Let us consider the case when the coefficient of prey extermination is greater than the coefficient of its natural increase, i.e. In this case for any , and, therefore, the solution of the first equation (2.15) is bounded from above by an exponentially decreasing function , i.e. at .

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

2.1 Generalized Voltaire models of the "predator-prey" type

The first models of V. Volterra, of course, could not reflect all aspects of the interaction in the predator-prey system, since they were largely simplified in relation to real conditions. For example, if the number of predators is equal to 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 rapidly mathematical ecology began to develop.

A large number of studies of various modifications of the predator-prey system have appeared, where more general models have been constructed that take 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 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 well real behavior populations and at the same time conduct a qualitative analysis of its solutions.

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

(2.18)

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

Table 1 - Different models of the "predator-prey" community

			The authors
			Volterra Lotka
			Gause
			Pislow
			Holing
			Ivlev
			Royama
			Shimazu
			May

mathematical modeling predator prey

3. PRACTICAL APPLICATIONS OF THE PREDATOR- PREY MODEL

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

Let two biological species live together in an isolated environment. The environment is stationary and provides an unlimited amount of everything necessary for life to one of the species, which we will call the victim. Another species - a 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 definiteness, we will call them crucians and pikes.

The following initial indicators are 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 between 1020 crucians or 1021 and therefore we will consider and continuous functions time t. We will call a pair of numbers (,) the state of the model.

Obviously, the nature of the state change (,) is determined by the values ​​of 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 carp. We will consider this dependence 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. Obviously, the nature of the state change (,) is determined by the values ​​of 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's integrate the system of both equations 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 equal to 1 year. Then the system takes the following form:

;

;

Taking = and = we bring similar terms, we obtain a system consisting of two equations:

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

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 a stationary 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, a 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).

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

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

4. Mathematical model "predators-prey"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

and extinction, "predators" are absent;

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

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

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

So, we have the following system of differential equations:

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

t is the current time;

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

Q - reproduction of "predators";

P - extinction of "predators".

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

5th level: complete system equations.

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

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

solving the equation we get

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

II. WORK OF STUDENTS DURING THE PRACTICAL LESSON

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

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

Completing of the work

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

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

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

4. After the title splash press "ENTER".

5. Modeling start with 1st level.

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

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

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

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

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

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

Task 3. Control of the final level of knowledge:

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

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

Assumptions:

1. The environment is homogeneous.

2. The number of this species is described by one variable, i.e. we neglect age, sex and genetic differences.

3. We neglect random fluctuations.

4. Interaction is instant.

In the biological literature, there is a huge number of works in which such systems were either observed in nature or modeled on "model" populations in the laboratory.

However, their results are often contradict each other:

- in some experiments, at first glance, incomprehensible phenomena of periodic changes in the number of populations in a homogeneous environment were observed;

- in other observations, the systems were destroyed quite quickly: either the predator dies, and the prey remains, or the prey dies, and the predator follows it.

Built in the 1920s by Vito Voltaire, the predator-prey community model explains many of these features.

This is the first success of mathematical ecology.

When considering this system, we consider the issues of stability: the conditions of stability and the mechanisms of stability.

Classic Volterra Model

The number of victims

The number of predators.

Additional assumptions.

1. The only limiting factor limiting the reproduction of prey is the pressure on them from predators. The limited resources of the environment for the victim are not taken into account (as in the Malthus model).

2. The reproduction of predators is limited by the amount of food they get (the number of prey).

− coefficient of natural increase of prey;

− coefficient of natural predator mortality;

− number (biomass) of prey consumed by one predator per unit of time (trophic function);

- part of the energy obtained from biomass, which is consumed by the predator for reproduction. The rest of the energy is spent on maintaining the basic metabolism and hunting activity.

Equations of the "predator-prey" system

The function is determined in experimental works. So far it has been established that these functions belong to one of the following three types.

This type is typical for invertebrates and some species of predatory fish.

A trophic function with a pronounced saturation threshold is characteristic of filter-feeding predators (mollusks).

This type is typical for vertebrates - organisms capable of learning.

At low numbers of prey, almost all prey become the prey of a predator, which is always hungry and never satiates. The trophic function can be considered linear:

Classic Volterra model:

Initial conditions

System (2) is autonomous, since does not have on the right side. The change in the state of the system is depicted on the phase plane and is a solution to the equation

Let us find the rest points of system (2).

The nontrivial rest point of system (4) has the form

Let us determine the nature of the rest point (5).

Let's make a replacement

Let's open the brackets and get the system

Discarding the nonlinear terms, we obtain the system

The characteristic equation has the form

Roots are purely imaginary numbers. The resting point is the center. In the original variables, the phase trajectories have the form

The arrows indicate the direction in which the state of the system changes over time.

According to this movement along the trajectory, the number of predator and prey populations perform undamped periodic oscillations, and the oscillations in the number of the predator lag behind the oscillations in the number of the prey in phase (by a quarter of the period).

The phase portrait of the solution looks like a spiral:

In the "predator-prey" system, there are damped oscillations. The numbers of prey and predators tend to their equilibrium values ​​(8).

Graphs of the dependence of the number of species.

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 of various species of ungulates was also regularly assessed. 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.

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