Stability of mathematical models of the predator-prey type. Equilibrium predator-prey. Analogies with chemical kinetics
COMPUTER MODEL "PREDATOR-Prey"
Kazachkov Igor Alekseevich 1 , Guseva Elena Nikolaevna 2
1 Magnitogorsk State Technical University named after V.I. G.I. Nosova, Institute of Construction, Architecture and Art, 5th year student
b Magnitogorsk State Technical University G.I. Nosov, Institute of Energy and automated systems, Candidate of Pedagogical Sciences, Associate Professor of the Department of Business Informatics and Information Technologies
annotation
This article is devoted to the review of the computer model "predator-prey". The study allows us to state that ecological modeling plays a huge role in the study of the environment. This issue is multifaceted.
COMPUTER MODEL "PREDATOR-VICTIM"
Kazatchkov Igor Alekseevich 1 , Guseva Elena Nikolaevna 2
1 Nosov Magnitogorsk State Technical University, Civil Engineering, Architecture and Arts Institute, student of the 5th course
2 Nosov Magnitogorsk State Technical University, Power Engineering and Automated Systems Institute, PhD in Pedagogical Science, Associate Professor of the Business Computer Science and Information Technologies Department
Abstract
This article provides an overview of the computer model "predator-victim". The study suggests that environmental simulation plays a huge role in the study of the environment. This problem is multifaceted.
Ecological modeling is used to study the environment around us. Mathematical models are used in cases where there is no natural environment and there are no natural objects, it helps to predict the influence of various factors on the object under study. This method takes on the functions of checking, constructing and interpreting the results. On the basis of such forms, ecological modeling deals with the assessment of changes in the environment around us.
IN currently similar forms are used to study the environment around us, and when it is required to study any of its areas, mathematical modeling is used. This model makes it possible to predict the influence of certain factors on the object of study. At one time, the “predator-prey” type was proposed by such scientists as: T. Malthus (Malthus 1798, Malthus 1905), Verhulst (Verhulst 1838), Pearl (Pearl 1927, 1930), as well as A. Lotka (Lotka 1925, 1927 ) and V. Volterra (Volterra 1926). These models reproduce the periodic oscillatory regime that occurs as a result of interspecies interactions in nature.
One of the main methods of cognition is modeling. In addition to the fact that it can predict changes in the environment, it also helps to find best way problem solving. For a long time, mathematical models have been used in ecology in order to establish patterns, trends in the development of populations, and help to highlight the essence of observations. The layout can serve as a sample behavior, object.
When recreating objects in mathematical biology, predictions are used various systems, special individualities of biosystems are provided for: the internal structure of an individual, life support conditions, constancy ecological systems, thanks to which the vital activity of the systems is saved.
The advent of computer simulation has greatly pushed the frontier of research capability. There is a possibility of multilateral implementation difficult forms, which do not allow analytical study, appeared latest destinations as well as simulation modeling.
Let's consider what is the object of modeling. “The object is a closed habitat where the interaction of two biological populations takes place: predators and prey. The process of growth, extinction and reproduction takes place directly on the surface of the environment. Prey feed on the resources that are present in the environment, while predators feed on prey. At the same time, nutritional resources can be both renewable and non-renewable.
In 1931, Vito Volterra derived the following laws of the predator-prey relationship.
The law of the periodic cycle - the process of destruction of the prey by a predator often leads to periodic fluctuations in the number of populations of both species, depending only on the growth rate of carnivores and herbivores, and on the initial ratio of their numbers.
Law of conservation of averages - the average abundance of each species is constant, regardless of the initial level, provided that the specific rates of population increase, as well as the efficiency of predation, are constant.
The law of violation of averages - with a reduction in both species in proportion to their number, the average population of prey increases, and predators - decreases.
The predator-prey model is a special relationship between the predator and the prey, as a result of which both benefit. The most healthy and adapted individuals to the environmental conditions survive, i.e. All this is due to natural selection. In an environment where there is no opportunity for reproduction, the predator will sooner or later destroy the prey population, after which it will die out itself.
There are many living organisms on earth, which, under favorable conditions, increase the number of relatives to enormous proportions. This ability is called: the biotic potential of the species, i.e. an increase in the population of a species over a given period of time. Each species has its own biotic potential, for example, large species of organisms can grow only 1.1 times in a year, while organisms of smaller species, such as crustaceans, etc. can increase their appearance up to 1030 times, but bacteria are still in more. In any of these cases, the population will grow exponentially.
Exponential population growth is a geometric progression of population growth. This ability can be observed in the laboratory in bacteria, yeast. In non-laboratory conditions exponential growth it is possible to see in the example of locusts or in other types of insects. Such an increase in the number of the species can be observed in those places where it has practically no enemies, and there is more than enough food. Eventually the growth of the species, after the population increased for a short time, the population growth began to decline.
Consider a computer model of mammalian reproduction on the example of the Lotka-Volterra model. Let two species of animals live in a certain area: deer and wolves. Mathematical model of population change in the model Trays-Volterra:
The initial number of victims is xn, the number of predators is yn.
Model parameters:
P1 is the probability of meeting with a predator,
P2 is the growth rate of predators at the expense of prey,
d is the predator mortality rate,
a is the increase in the number of victims.
IN learning task the following values were set: the number of deer was 500, the number of wolves was 10, the growth rate of deer was 0.02, the growth rate of wolves was 0.1, the probability of meeting a predator was 0.0026, the growth rate of predators due to prey was 0.000056. Data are calculated for 203 years.
Exploring influence the growth rate of victims for the development of two populations, the remaining parameters will be left unchanged. In Scheme 1, an increase in the number of prey is observed, and then, with some delay, an increase in predators is observed. Then the predators knock out the prey, the number of prey drops sharply, followed by the decrease in the number of predators (Fig. 1).
Figure 1. Population size with low birth rates among victims
Let us analyze the change in the model by increasing the birth rate of the victim a=0.06. In Scheme 2, we see a cyclic oscillatory process leading to an increase in the number of both populations over time (Fig. 2).
Figure 2. Population size at the average birth rate of the victims
Let's consider how the dynamics of populations will change with a high value of the birth rate of the victim a = 1.13. On fig. 3, there is a sharp increase in the number of both populations, followed by extinction of both prey and predator. This is due to the fact that the population of victims has increased to such an extent that resources have begun to run out, as a result of which the victim is dying out. The extinction of predators is due to the fact that the number of victims has decreased and the predators have run out of resources for existence.
Figure 3. Populations with high birth rates in prey
Based on the analysis of computer experiment data, we can conclude that computer modeling allows us to predict the size of populations, to study the influence of various factors on population dynamics. In the above example, we investigated the predator-prey model, the effect of the birth rate of prey on the number of deer and wolves. A small increase in the population of prey leads to a small increase in prey, which after a certain period is destroyed by predators. A moderate increase in the prey population leads to an increase in the size of both populations. A high increase in the population of prey first leads to a rapid increase in the population of prey, this affects the increase in the growth of predators, but then the breeding predators quickly destroy the deer population. As a result, both species become extinct.
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.
aim term paper is the study of the interaction of individuals in the "predator-prey" system. To achieve the goal, the following tasks were set:
predation and its role in the formation of trophic relationships;
the main models of the relationship "predator - prey";
the influence of the social way of life in the stability of the "predator-prey" system;
laboratory modeling of the "predator - prey" system.
The influence of predators on the number of prey and vice versa is quite obvious, but it is rather difficult to determine the mechanism and essence of this interaction. These questions I intend to address in the course work.
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Duke University scientists, in collaboration with colleagues from Stanford University, the Howard Hughes Medical Institute, and the California Institute of Technology, working under the direction of Dr. Lingchong You, have developed a living system of genetically modified bacteria that will allow more detailed study of predator-prey interactions in a population level.
The new experimental model is an example of an artificial ecosystem in which researchers program bacteria to perform new functions to create. Such reprogrammed bacteria could be widely used in medicine, environmental cleanup and biocomputer development. As part of this work, scientists rewrote the "software" of E. coli (Escherichia coli) in such a way that two different bacterial populations formed in the laboratory a typical system of predator-prey interactions, a feature of which was that the bacteria did not devour each other, but controlled the number the opponent population by changing the frequency of "suicides".
The field of research known as synthetic biology emerged around 2000, and most of the systems created since then have been based on reprogramming a single bacterium. The model developed by the authors is unique in that it consists of two bacterial populations living in the same ecosystem, the survival of which depends on each other.
The key to the successful functioning of such a system is the ability of two populations to interact with each other. The authors created two strains of bacteria - "predators" and "herbivores", depending on the situation, releasing toxic or protective compounds into the general ecosystem.
The principle of operation of the system is based on maintaining the ratio of the number of predators and prey in a regulated environment. Changes in the number of cells in one of the populations activate reprogrammed genes, which triggers the synthesis of certain chemical compounds.
Thus, a small number of victims in the environment causes the activation of the self-destruction gene in predator cells and their death. However, as the number of victims increases, the compound released by them into the environment reaches a critical concentration and activates the predator gene, which ensures the synthesis of an "antidote" to the suicidal gene. This leads to an increase in the population of predators, which, in turn, leads to the accumulation of a compound synthesized by predators in the environment, pushing victims to commit suicide.
Using fluorescence microscopy, scientists documented interactions between predators and prey.
Predator cells, stained green, cause suicide of prey cells, stained red. Elongation and rupture of the victim cell indicates its death.
This system is not an accurate representation of predator-prey interactions in nature, as predator bacteria do not feed on prey bacteria and both populations compete for the same food resources. However, the authors believe that the system they have developed is a useful tool for biological research.
The new system demonstrates a clear relationship between genetics and population dynamics, which in the future will help in the study of the influence of molecular interactions on population change, which is a central topic of ecology. The system provides virtually unlimited possibilities for modifying variables to study in detail the interactions between environment, gene regulation, and population dynamics.
Thus, by controlling the genetic apparatus of bacteria, it is possible to simulate the processes of development and interaction of more complex organisms.
CHAPTER 3
CHAPTER 3
Ecologists from the United States and Canada have shown that the group lifestyle of predators and their prey radically changes the behavior of the predator-prey system and makes it more resilient. This effect, confirmed by observations of the dynamics of the number of lions and wildebeests in the Serengeti Park, is based on the simple fact that with a group lifestyle, the frequency of random encounters between predators and potential victims decreases.
Ecologists have developed a number of mathematical models that describe the behavior of the predator-prey system. These models, in particular, explain well the observed sometimes consistent periodic fluctuations in the abundance of predators and prey.
Such models are usually characterized by a high level of instability. In other words, with a wide range of input parameters (such as mortality of predators, efficiency of conversion of prey biomass into predator biomass, etc.) in these models, sooner or later all predators either die out or first eat all the prey, and then still die from hunger.
In natural ecosystems, of course, everything is more complicated than in a mathematical model. Apparently, there are many factors that can increase the stability of the predator-prey system, and in reality it rarely comes to such sharp jumps in numbers as in Canada lynxes and hares.
Environmentalists from Canada and the United States published in latest issue magazine " 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 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 ecological models used to describe predator-prey systems, key role It is precisely the nature of the dependence of the intensity of predation (the number of victims eaten by one predator per unit time) on the density of the prey population that plays. 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, in 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.
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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 loot. Obviously, this disadvantage should be compensated by some very significant advantages. Traditionally, it was believed that the social lifestyle of lions is associated with hunting large animals, which are difficult to cope even with a lion alone. However, in Lately many experts (including the authors of the article under discussion) began to doubt the correctness of this explanation. In their opinion, collective action is necessary for lions only when hunting buffaloes, and lions prefer to deal with other types of prey alone.
More plausible is the assumption that prides are needed to regulate purely internal problems, which are many in a lion's life. For example, infanticide is common among them - the killing of other people's cubs by males. It is easier for females kept in a group to protect their children from aggressors. In addition, it is much easier for a pride than for a lone lion to defend its hunting area from neighboring prides.
Source: John M. Fryxell, Anna Mosser, Anthony R. E. Sinclair, Craig Packer. Group formation stabilizes predator–prey dynamics // Nature. 2007. V. 449. P. 1041–1043.
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...
Predator-Victim
Abstract >> EcologyApplications 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...
Synopsis ecology
Abstract >> Ecologyfactor 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. ...
Predation- a form of trophic relationships between organisms different types, for which one of them ( predator) attacks another ( sacrifice) and feeds on his flesh, that is, there is usually an act of killing the victim.
"predator-prey" system- a complex ecosystem for which long-term relationships between predator and prey species are realized, a typical example of coevolution.
Co-evolution is the joint evolution of biological species interacting in an ecosystem.
Relations between predators and their prey develop cyclically, being an illustration of a neutral equilibrium.
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.
2. The reproduction of predators is limited by the amount of food they get (the number of prey).
At its core, the Lotka–Volterra model is mathematical description Darwinian principle of the struggle for existence.
The Volterra-Lotka system, often called the predator-prey system, describes the interaction of two populations - predators (for example, foxes) and prey (for example, hares), which live according to somewhat different "laws". Prey maintain their population by eating natural resource, for example, grasses, which leads to exponential population growth if there are no predators. Predators maintain their population only by "eating" their prey. Therefore, if the prey population disappears, then the predator population then exponentially decreases. Eating prey by predators damages the population of prey, but at the same time provides an additional resource for the reproduction of predators.
Question
THE PRINCIPLE OF MINIMUM POPULATION SIZE
a phenomenon that naturally exists in nature, characterized as a kind of natural principle, meaning that each animal species has a specific minimum population size, the violation of which threatens the existence of the population, and sometimes the species as a whole.
population maximum rule, it lies in the fact that the population cannot increase indefinitely, due to the depletion of food resources and breeding conditions (Andrevarta-Birch theory) and limiting the impact of a complex of abiotic and biotic environmental factors (Frederiks theory).
Question
So, as Fibonacci already made clear, population growth is proportional to its size, and therefore, if population growth is not limited by any external factors, it is continuously accelerating. Let's describe this growth mathematically.
Population growth is proportional to the number of individuals in it, that is, Δ N~N, Where N- population size, and Δ N- its change over a certain period of time. If this period is infinitely small, we can write that dN/dt=r × N , Where dN/dt- change in population size (growth), and r - reproductive potential, a variable that characterizes the ability of a population to increase its size. The above equation is called exponential model population growth (Figure 4.4.1).
Fig.4.4.1. Exponential Growth.
It is easy to understand that with increasing time, the population grows faster and faster, and rather soon tends to infinity. Naturally, no habitat can sustain the existence of an infinite population. However, there are a number of population growth processes that can be described using an exponential model in a certain time period. We are talking about cases of unlimited growth, when some population populates an environment with an excess of free resources: cows and horses populate a pampa, flour beetles populate a grain elevator, yeast populate a bottle of grape juice, etc.
Naturally, exponential population growth cannot be eternal. Sooner or later, the resource will be exhausted, and population growth will slow down. What will this slowdown be like? Practical ecology knows the most different variants: both a sharp rise in numbers, followed by the extinction of a population that has exhausted its resources, and a gradual deceleration of growth as it approaches certain level. The easiest way to describe slow braking. The simplest model describing such dynamics is called logistic and proposed (to describe the growth of the human population) by the French mathematician Verhulst back in 1845. In 1925, a similar pattern was rediscovered by the American ecologist R. Perl, who suggested that it was universal.
In the logistic model, a variable is introduced K- medium capacity, the equilibrium population size at which it consumes all available resources. The increase in the logistic model is described by the equation dN/dt=r × N × (K-N)/K (Fig. 4.4.2).
Rice. 4.4.2. Logistic growth
Bye N is small, the population growth is mainly influenced by the factor r× N and population growth is accelerating. When it becomes high enough, the factor begins to have the main influence on the population size (K-N)/K and population growth starts to slow down. When N=K, (K-N)/K=0 and population growth stops.
For all its simplicity, the logistic equation satisfactorily describes many cases observed in nature and is still successfully used in mathematical ecology.
#16 Ecological Survival Strategy- an evolutionarily developed set of properties of a population, aimed at increasing the likelihood of survival and leaving offspring.
So A.G. Ramensky (1938) distinguished three main types of survival strategies among plants: violents, patients, and explerents.
Violents (enforcers) - suppress all competitors, for example, trees that form indigenous forests.
Patients are species that can survive in adverse conditions(“shade-loving”, “salt-loving”, etc.).
Explorents (filling) - species that can quickly appear where indigenous communities are disturbed - on clearings and burnt areas (aspens), on shallows, etc.
The ecological strategies of populations are very diverse. But at the same time, all their diversity lies between two types of evolutionary selection, which are denoted by the constants of the logistic equation: r-strategy and K-strategy.
| sign | r-strategies | K-strategies |
| Mortality | Does not depend on density | Density dependent |
| Competition | Weak | Acute |
| Lifespan | short | Long |
| Development speed | Rapid | Slow |
| Timing of reproduction | Early | Late |
| reproductive enhancement | Weak | big |
| Type of survival curve | Concave | convex |
| body size | Small | Large |
| The nature of the offspring | many, small | small, large |
| Population size | Strong fluctuations | Constant |
| Preferred environment | changeable | Constant |
| Succession stages | Early | Late |
Similar information.
Population dynamics is one of the sections of mathematical modeling. It is interesting in that it has specific applications in biology, ecology, demography, and economics. There are several basic models in this section, one of which, the Predator-Prey model, is discussed in this article.
The first example of a model in mathematical ecology was the model proposed by V. Volterra. It was he who first considered the model of the relationship between predator and prey.
Consider the problem statement. Suppose there are two types of animals, one of which devours the other (predators and prey). At the same time, the following assumptions are made: the food resources of the prey are not limited, and therefore, in the absence of a predator, the prey population grows exponentially, while the predators, separated from their prey, gradually die of hunger, also according to an exponential law. As soon as predators and prey begin to live in close proximity to each other, changes in their populations become interconnected. In this case, obviously, the relative increase in the number of prey will depend on the size of the predator population, and vice versa.
In this model, it is assumed that all predators (and all prey) are in the same conditions. At the same time, the food resources of prey are unlimited, and predators feed exclusively on prey. Both populations live in a limited area and do not interact with any other populations, and there are no other factors that can affect the size of the populations.
The mathematical model "predator - prey" itself consists of a pair differential equations, which describe the dynamics of populations of predators and prey in its simplest case, when there is one population of predators and one prey. The model is characterized by fluctuations in the sizes of both populations, with the peak of the number of predators slightly behind the peak of the number of prey. This model can be found in many works on population dynamics or mathematical modeling. It is widely covered and analyzed by mathematical methods. However, formulas may not always give an obvious idea of the ongoing process.
It is interesting to find out exactly how the dynamics of populations depends on the initial parameters in this model and how much this corresponds to reality and common sense, and to see it graphically, without resorting to complex calculations. For this purpose, based on the Volterra model, a program was created in the Mathcad14 environment.
First, let's check the model for compliance with real conditions. To do this, we consider degenerate cases, when only one of the populations lives under given conditions. Theoretically, it was shown that in the absence of predators, the prey population increases indefinitely in time, and the predator population dies out in the absence of prey, which generally speaking corresponds to the model and the real situation (with the stated problem statement).
The results obtained reflect the theoretical ones: predators are gradually dying out (Fig. 1), and the number of prey increases indefinitely (Fig. 2).
Fig.1 Dependence of the number of predators on time in the absence of prey
Fig. 2 Dependence of the number of victims on time in the absence of predators
As can be seen, in these cases the system corresponds to the mathematical model.
Consider how the system behaves for various initial parameters. Let there be two populations - lions and antelopes - predators and prey, respectively, and initial indicators are given. Then we get the following results (Fig. 3):
Table 1. Coefficients of the oscillatory mode of the system
Fig.3 System with parameter values from Table 1
Let's analyze the obtained data based on the graphs. With the initial increase in the population of antelopes, an increase in the number of predators is observed. Note that the peak of the increase in the population of predators is observed later, at the decline in the population of prey, which is quite consistent with real ideas and the mathematical model. Indeed, an increase in the number of antelopes means an increase in food resources for lions, which entails an increase in their numbers. Further, the active eating of antelopes by lions leads to a rapid decrease in the number of prey, which is not surprising, given the appetite of the predator, or rather the frequency of predation by predators. A gradual decrease in the number of predators leads to a situation where the prey population is in favorable conditions for growth. Then the situation repeats with a certain period. We conclude that these conditions are not suitable for the harmonious development of individuals, as they entail sharp declines in the prey population and sharp increases in both populations.
Let us now set the initial number of the predator equal to 200 individuals, while maintaining the remaining parameters (Fig. 4).
Table 2. Coefficients of the oscillatory mode of the system
Fig.4 System with parameter values from Table 2
Now the oscillations of the system occur more naturally. Under these assumptions, the system exists quite harmoniously, there are no sharp increases and decreases in the number of populations in both populations. We conclude that with these parameters, both populations develop fairly evenly to live together in the same area.
Let's set the initial number of the predator equal to 100 individuals, the number of prey to 200, while maintaining the remaining parameters (Fig. 5).
Table 3. Coefficients of the oscillatory mode of the system
Fig.5 System with parameter values from Table 3
In this case, the situation is close to the first considered situation. Note that with mutual increase in populations, the transitions from increase to decrease in the prey population become smoother, and the predator population remains in the absence of prey at a higher numerical value. We conclude that with a close relationship of one population to another, their interaction occurs more harmoniously if the specific initial numbers of populations are large enough.
Consider changing other parameters of the system. Let the initial numbers correspond to the second case. Let's increase the multiplication factor of prey (Fig.6).
Table 4. Coefficients of the oscillatory mode of the system
Fig.6 System with parameter values from Table 4
Let's compare given result with the result obtained in the second case. In this case, there is a faster increase in prey. At the same time, both the predator and the prey behave as in the first case, which was explained by the low population size. With this interaction, both populations reach a peak with values much larger than in the second case.
Now let's increase the coefficient of growth of predators (Fig. 7).
Table 5. Coefficients of the oscillatory mode of the system
Fig.7 System with parameter values from Table 5
Let's compare the results in a similar way. In this case general characteristics system remains the same, except for the period change. As expected, the period became shorter, which is explained by the rapid decrease in the predator population in the absence of prey.
And finally, we will change the coefficient of interspecies interaction. To begin with, let's increase the frequency of predators eating prey:
Table 6. Coefficients of the oscillatory mode of the system
Fig.8 System with parameter values from Table 6
Since the predator eats the prey more often, the maximum of its population has increased compared to the second case, and the difference between the maximum and minimum values of the populations has also decreased. The oscillation period of the system remained the same.
And now let's reduce the frequency of predators eating prey:
Table 7. Coefficients of the oscillatory mode of the system
Fig.9 System with parameter values from Table 7
Now the predator eats the prey less often, the maximum of its population has decreased compared to the second case, and the maximum of the prey's population has increased, and by 10 times. It follows that, under given conditions, the prey population has greater freedom in terms of reproduction, because a smaller mass is enough for the predator to satiate itself. The difference between the maximum and minimum values of the population size also decreased.
When trying to model complex processes in nature or society, one way or another, the question arises about the correctness of the model. Naturally, when modeling, the process is simplified, some minor details are neglected. On the other hand, there is a danger of simplifying the model too much, thus throwing out important features of the phenomenon along with insignificant ones. In order to avoid this situation, before modeling, it is necessary to study the subject area in which this model is used, to explore all its characteristics and parameters, and most importantly, to highlight those features that are most significant. The process should have a natural description, intuitively understandable, coinciding in the main points with the theoretical model.
The model considered in this paper has a number of significant drawbacks. For example, the assumption of unlimited resources for the prey, the absence of third-party factors that affect the mortality of both species, etc. All these assumptions do not reflect the real situation. However, despite all the shortcomings, the model has become widespread in many areas, even far from ecology. This can be explained by the fact that the "predator-prey" system gives a general idea of the interaction of species. Interaction environment and other factors can be described by other models and analyzed in combination.
Relationships of the "predator-prey" type are an essential feature of various types of life activity in which there is a collision of two interacting parties. This model takes place not only in ecology, but also in economics, politics and other fields of activity. For example, one of the areas related to the economy is the analysis of the labor market, taking into account the available potential employees and job vacancies. This topic would be an interesting continuation of work on the predator-prey model.
The "predator-prey" model and Goodwin's macroeconomic model
Consider the biological model "predator - prey", in which one species is food for another. This model, which has long become a classic, was built in the first half of the 20th century. Italian mathematician V. Volterra to explain fluctuations in fish catches in the Adriatic Sea. The model assumes that the number of predators increases as long as they have enough food, and an increase in the number of predators leads to a decrease in the population of prey fish. When the latter becomes scarce, the number of predators decreases. As a result, from a certain moment, an increase in the number of prey fish begins, which after a while causes an increase in the population of predators. The cycle closes.
Let N x (t) And N 2 (t) - the number of prey and predator fish at a point in time t respectively. Let us assume that the rate of increase in the number of prey in the absence of predators is constant, i.e.
Where A - positive constant.
The appearance of a predator should reduce the rate of prey growth. We will assume that this decrease linearly depends on the number of predators: the more predators, the lower the growth rate of prey. Then
Where t > 0.
Therefore, for the dynamics of the number of prey fish, we obtain:
Let us now compose an equation that determines the dynamics of the population of predators. Let us assume that their number in the absence of prey decreases (due to lack of food) at a constant rate b, i.e.
The presence of prey causes an increase in the rate of growth of predators. Let us assume that this increase is linear, i.e.
Where n> 0.
Then for the growth rate of predatory fish we obtain the equation:
In the "predator - prey" system (6.17) - (6.18), the decrease in the growth rate of the number of prey fish caused by predators eating them is equal to mNxN2, i.e., proportional to the number of their encounters with the predator. The increase in the growth rate of the number of predator fish caused by the presence of prey is equal to nNxN2, i.e., also proportional to the number of encounters between prey and predators.
We introduce dimensionless variables U = mN 2 /a And V = nN x /b. Variable dynamics U corresponds to the dynamics of predators, and the dynamics of the variable V- victim dynamics. By virtue of equations (6.17) and (6.18), the change in new variables is determined by the system of equations:
Let's assume that at t= 0 the number of individuals of both species is known, therefore, are known initial values new variables?/(0) = U 0 , K(0) = K 0 . From the system of equations (6.19) one can find a differential equation for its phase trajectories:
Separating the variables of this equation, we get: 
Rice. 6.10. Building a phase trajectory ADCBA systems of differential equations (6.19)
From here, taking into account the initial data, it follows:
where is the integration constant WITH = b(VQ - In V 0)/a - lnU 0 + U 0 .
On fig. 6.10 shows how the line (6.20) is constructed for a given value of C. To do this, in the first, second and third quarters, respectively, we build graphs of functions x = V - In V, y = (b/a)x, at== In U-U+C.
Due to equality dx/dV = (V- 1)/U function X = V- In K determined at V> 0, increases if V> 1, and decreases if V 1. Due to the fact that cPx/dV 1\u003d 1 / F 2\u003e 0, graph of the function l: \u003d x(V) directed downward. The equation V= 0 specifies the vertical asymptote. This function has no oblique asymptotes. So the graph of the function X = x(y) has the shape of the curve shown in the first quarter of Fig. 6.10.
The function y= In U - U + C, whose graph in Fig. 6.10 is depicted in the third quarter.
If we now place in Fig. 6.10 second quarter function graph y= (b/a)x, then in the fourth quarter we get a line that connects the variables U and V. Indeed, taking the point V t on axle OV, calculate using the function X= V - V relevant knowledge x x. After that, using the function at = (b/a)x, according to the received value X ( find y x(second quarter in Figure 6.10). Next, using the graph of the function at= In U - U + C determine the corresponding values of the variable U(in Fig. 6.10 there are two such values \u200b\u200b- the coordinates of the points M And N). The set of all such points (V; U) forms the desired curve. It follows from the construction that the graph of dependence (6.19) is a closed line containing a point inside it E( 1, 1).
Recall that we obtained this curve by setting some initial values U 0 And V0 and calculating the constant C from them. Taking other initial values, we get another closed line that does not intersect the first one and also contains a point inside itself E( eleven). This means that the family of trajectories of system (6.19) on the phase plane ( V, U) is the set of closed non-intersecting lines concentrating around the point E( 1, 1), and the solutions of the original model U = SCH) And V = V(t) are functions that are periodic in time. In this case, the maximum of the function U = U(t) does not reach the maximum of the function V = V(t) and vice versa, i.e. fluctuations in the number of populations around their equilibrium solutions occur in different phases.
On fig. 6.11 shows four trajectories of the system of differential equations (6.19) on the phase plane ouv, different initial conditions. One of the equilibrium trajectories is the point E( 1, 1), which corresponds to the solution U(t) = 1, V(t)= 1. Points (U(t),V(t)) on the other three phase trajectories, as time increases, they shift clockwise.
To explain the mechanism of change in the size of two populations, consider the trajectory ABCDA in fig. 6.11. As you can see, in the field AB both predators and prey are few. Therefore, here the population of predators is reduced due to lack of food, and the population of prey is growing. Location on Sun the number of prey reaches high values, which leads to an increase in the number of predators. Location on SA there are many predators, and this entails a reduction in the number of prey. However, after passing the point D the number of victims decreases so much that the population begins to decrease. The cycle closes.
The predator-prey model is an example of a structurally unstable model. Here, a small change in the right side of one of the equations can lead to a fundamental change in its phase portrait.
Rice. 6.11.
Rice. 6.12.
Indeed, if intraspecific competition is taken into account in the equation for the dynamics of victims, then we will obtain a system of differential equations:
Here at t = 0 the victim population develops according to a logical law.
At t F 0 nonzero equilibrium solution of system (6.21) for some positive values parameter of intraspecific competition AND is a stable focus, and the corresponding trajectories "wind" around the equilibrium point (Fig. 6.12). If h = 0, then in this case the singular point E( 1, 1) of system (6.19) is the center, and the trajectories are closed lines (see Fig. 6.11).
Comment. Usually, the "predator-prey" model is understood as the model (6.19) whose phase trajectories are closed. However, model (6.21) is also a predator-prey model, since it describes the mutual influence of predators and prey.
One of the first applications of the "predator-prey" model in economics to study cyclically changing processes is the Goodwin macroeconomic model, which uses a continuous approach to the analysis of the mutual influence of the level of employment and the rate wages.
In the work of V.-B. Zanga presented a variant of the Goodwin model, in which labor productivity and labor supply grow at a constant rate of growth, and the retirement rate of funds is zero. This model formally leads to the equations of the "predator-prey" model.
Below we consider a modification of this model for the case of a non-zero fund retirement rate.
The following notation is used in the model: L- the number of workers; w- the average wage rate of workers; TO - fixed production assets (capital); Y- national income; / - investments; C - consumption; p - the coefficient of disposal of funds; N- supply of labor in the labor market; T = Y/K- return on assets; A = Y/L - labor productivity; at = L/N - employment rate; X = C/Y - consumption rate in national income; TO - increase in capital depending on investment.
Let us write the equations of the Goodwin model:
Where a 0 , b, g, n, N 0 , g are positive numbers (parameters).
Equations (6.22) - (6.24) express the following. Equation (6.22) is the usual equation for the dynamics of funds. Equation (6.23) reflects an increase in the wage rate when employment is high (the wage rate rises if the supply of labor is low) and a decrease in the wage rate when unemployment is high.
Thus, equation (6.23) expresses the Phillips law in linear form. Equations (6.24) mean exponential growth in labor productivity and labor supply. We also assume that C = wL, i.e., all wages are spent on consumption. Now we can transform the equations of the model, taking into account the equalities:
Let's transform equations (6.22)-(6.27). We have: 
Where
Where 
Therefore, the dynamics of variables in the Goodwin model is described by a system of differential equations:
which formally coincides with the equations of the classical predator-prey model. This means that fluctuations in phase variables also arise in the Goodwin model. The mechanism of oscillatory dynamics here is as follows: with low wages w consumption is low, investment is high, and this leads to an increase in production and employment y. Big busy at causes an increase in the average wage w, which leads to an increase in consumption and a decrease in investment, a fall in production and a decrease in employment y.
Below, the hypothesis about the dependence of the interest rate on the level of employment of the considered model is used in modeling the dynamics of a single-product firm. It turns out that in this case, under some additional assumptions, the firm model has the property of cyclicity of the “predator-prey” model considered above.
- See: Volterra V. Decree, op.; Rizniienko G. Yu., Rubin A. B. Decree. op.
- See: Zang V.-B. Synergistic economy. M., 2000.
- See: Pu T. Nonlinear economic dynamics. Izhevsk, 2000; Tikhonov A.N. Mathematical model // Mathematical encyclopedia. T. 3. M., 1982. S. 574, 575.
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