Dependences of population growth rate on their density. Proliferative - growth that proceeds by cell multiplication. It is known in two forms: multiplicative and accretionary. "population growth rate" in books

Natural populations are not a set of individuals frozen once and for all, but a dynamic unity of interacting organisms. A change in the size, structure, and distribution of populations in response to environmental conditions is called population dynamics.

The dynamics of populations in a simplified version can be described by such indicators as fertility and mortality. These are the most important population characteristics, based on the analysis of which one can judge the stability and prospective development of the population.

fertility is defined as the number of individuals born in a population for a certain period of time (hour, day, month, year). The term "fertility" refers to individuals of any species, regardless of the way they were born: whether it is the germination of seeds of psyllium or oats, the emergence of young from eggs in a chicken or turtle, the birth of offspring in an elephant, a whale or a person.

Ecologists distinguish the maximum birth rate in the absence of limiting environmental factors (it is very difficult, even impossible, to achieve this). Under maximum birth rate is understood as the theoretically possible maximum rate of formation of new individuals under ideal conditions. The reproduction of organisms is restrained only by their physiological characteristics. Theoretical reproduction rate various kinds can be quite high. If we take as a basis such an indicator as the time for the species to capture the entire surface of the Earth, then for the cholera bacterium Vibrio cholerae it will be 1.25 days, for the diatom Nitschia putrida - 16.8, for the house fly Musca domestica - 366, for the chicken - about 6000, for an elephant - 376,000 days. It should be emphasized that the maximum birth rate is a theoretical concept. Not a single species in nature can reproduce uncontrollably and unlimitedly, otherwise an ecological catastrophe cannot be avoided.

Unlike the maximum ecological, or implemented, fertility (or simply fertility) characterizes the growth or increase in the size of a population under actual or specific environmental conditions.

Mortality is the number of individuals that died in a population per unit of time. Like fertility, mortality can be expressed as the number of individuals who died in a given period (the number of deaths per unit of time), or as specific mortality for the entire population (or part of it). When determining the mortality of a population, all dead individuals are taken into account, regardless of the cause of death (whether they died of old age or died in the claws of a predator, poisoned by pesticides or froze, etc.).

Population growth curves

Any population is theoretically capable of unlimited growth in numbers, if it is not limited by environmental factors. In such a hypothetical case, the population growth rate will depend only on the magnitude biotic potential, characteristic of the species.

General changes in population size are formed due to four phenomena: fertility, mortality, introduction and eviction of individuals (immigration and emigration).

Fertility - the number of new individuals that appear in a population per unit of time per a certain number of its members.

Distinguish absolute and specific fertility. The first is characterized by the total number of born individuals. The specific birth rate is calculated as the average change in the number of individuals over a certain period of time, divided by their original number.

The most common occurrence in nature is increased mortality of individuals in early period life.

The resettlement (eviction) of individuals from a population or its replenishment by newcomers is a natural phenomenon based on one of the most important biological features of a species, its dispersal ability. In each population, some individuals regularly leave it (population dispersion), replenishing neighboring or populating new territories not yet occupied by the species.

Modern theory considers the rate of population growth as an autoregulated process. Any population of organisms under specific conditions is characterized by a certain average level of abundance, around which fluctuations occur.

In one case, the growth rate from the very beginning is high and constant, independent of increasing density, which corresponds to an avalanche-like, exponential increase in the population size (Fig. 6.2a). It is graphically described by the simplest curve that characterizes the change in the size of a population moving towards equilibrium, provided there is an abundance of food. When a certain density is reached, the growth of the population stops. If the limiting factor of the habitat acts very quickly, then the growth of the population stops suddenly (curve "B" in Fig. 626).

However, it has a limiting effect not sharply, but gradually, which leads to a 5-shaped growth curve (Fig. 6.2#). This shape of the growth curve is observed when the population is introduced into a new territory. In this case, accelerated growth occurs first (according to the logarithmic law). Then, under the influence of environmental resistance, growth slows down, and the equilibrium phase begins in the population.

If the population experiences external influences (for example, an attack by predators), then at a constant specific rate of removal of individuals in nature, interacting predator-prey populations steadily exist (curve "D" in Fig. 6.2i), but at a lower population level: N of less N max .

Rice. 6.2. Population growth curves: A - exponential; 5 - exponential with the cessation of growth; B - logistics; D - logistic with the removal of individuals without exceeding the quota; D - in excess of the quota. N is the population size (Nmax is the maximum); (U out is the real rate of withdrawal, U max is the rate of withdrawal of products critical for the population; t is the time

The specific rate of withdrawal is the number of individuals withdrawn per unit of time, related to the size of the population. If a person withdraws bioproducts from the population at a constant (integral, but not specific) rate, then the concept arises. quotas.

Therefore, the quota represents the rate of capture. When the quota does not exceed the established critical value, the equilibrium of the population is maintained. In this case, capture can be carried out for an arbitrarily long time without detrimental consequences for the population. The S-curve is called the logistic growth curve because it is obtained by integrating an equation based on logically sound assumptions. If the quota exceeds the critical value of capture, then the population is completely captured within end time: the population does not have time to self-repair and dies (Fig. 6.2d).

Of great interest to environmental scientists are cyclic populations subject to regular population fluctuations. However, there is still no unified theory that satisfactorily explains the patterns in cyclic populations.

In plants, due to the characteristics of their growth, the population density is usually regulated not only by changing the number of individuals per unit area, but also by changing their vegetative capabilities.

In animals, strict forms of regulation of population density usually manifest themselves only in those cases when the supply of food, water, or other resources is sharply limited, and the animals are either not capable of searching for resources in another territory at a given period, or these searches are ineffective.

Among the mechanisms that retard the growth of populations, in many species, an important role is played by chemical interactions of individuals.

Another mechanism for limiting the number of populations is such changes in physiology and behavior with an increase in density, which ultimately lead to the manifestation mass migration instincts.

The most effective mechanism for restraining population growth in a given area is a certain system of instincts - marking and protecting areas that do not allow the reproduction of “foreign” individuals on them.

Genetic processes in populations. It is now known that all natural populations are heterogeneous and saturated with mutations. The genetic heterogeneity of any population in the absence of pressure from external factors should be unchanged, be in a certain balance.

The provision on the genetic unity of a population is one of the most important conclusions population genetics:any population is a complex genetic system in dynamic equilibrium.

The rate of population growth in natural habitats will depend on climate change, food supply and whether reproduction is limited to certain times of the year, etc., which should be taken into account when compiling models or improving them.[ ...]

The rate of population growth is the change in population size per unit of time. The population growth rate can be positive, zero or negative. It depends on the indicators of fertility, mortality and migration (settlement - immigration and eviction - emigration). An increase (profit) in numbers occurs as a result of the birth rate and immigration of individuals, and a decrease (decrease) in numbers - as a result of mortality and emigration of individuals.[ ...]

The general population growth rate in the absence of the limiting influence of the environment (r) depends on the age composition and the contribution of various age groups to reproduction. Thus, a species can be characterized by several values ​​of r depending on the structure of the population. When a stationary and stable distribution of ages is established, the specific growth rate is called the potential growth rate of the population or Gmax. Often this maximum value of r is called otherwise - biotic or reproductive potential. The difference between rmax, or biotic potential, and the actual growth rate under given laboratory or field conditions is used as a measure of environmental resistance, which characterizes the sum of all limiting environmental factors that prevent the realization of biotic potential.[ ...]

At the population level, abiotic factors affect such parameters as fertility, mortality, average life expectancy of an individual, population growth rate and size, often being the most important factors that determine the nature of population dynamics and the spatial distribution of individuals in it. A population can adapt to changes in abiotic factors, firstly, by changing the nature of its spatial distribution and, secondly, through adaptive evolution.[ ...]

Since the population is variable, we are interested not only in its size and composition in each this moment, but also how it changes. Knowing the rate of population change, one can judge many of its important features. The rate can be determined by dividing the amount of change by the time period over which it occurred; speed, expressed in this way, will characterize the speed with which something changes in time. For" means "divided by". For example, the growth rate of a population is the number of organisms by which it increases over time; it is obtained by dividing the population growth rate by the elapsed period of time. The average rate of population change is usually expressed as DL /M where N is the population size (or other significant indicator), and t is time. Instantaneous velocities are denoted as yY1[ ...]

Prey Growth Parameters: r - Specific instantaneous speed prey population growth; values ​​of r must be positive and lie between 0 and 5. By default, r= 1.[ ...]

The instantaneous speed dN/cI, as well as the speed c?M/(Mc?/) cannot be calculated directly, only on the basis of population calculations; it is also necessary to know the nature of the population growth curve. Calculate the instantaneous speed using the equations that will be discussed in Sec. 7. We cannot connect a “speedometer” to the population to determine its instantaneous speed, as in a car. Of course, an approximate calculation could be made by conducting population censuses at very short intervals. At the same time, we would probably establish that it is absolutely impossible to judge from the average value how the growth rate changes at successive points in time. In the above example, the specific growth rate is calculated in relation to the population size at the initial moment (50 protozoa). In other words, for each of the original 50 individuals, 2 new individuals were formed. At the same time, some individuals could split twice, while others would not split at all, and some individuals would disappear from the population altogether. The census at the beginning and end of a certain period of time will not tell us anything about how all this actually happened. The specific growth rate can also be expressed in relation to the average population size during a given time period. When evaluating the annual growth rate of human populations (in percent), the density in the middle of the year is usually taken as the basis. Thus, a rate of 1% per head means that for every 100 people in the population that existed in the middle of the year, one new individual was added. The expression of the rate in relation to the number of individuals available makes it possible to compare the growth rate of populations that differ greatly in size, for example, the population of a small and large state.[ ...]

Thus, the birth rate, the death rate, and the population growth rate can be average and instantaneous. Further in the text, instantaneous speed is implied, but the word "instantaneous" is deliberately omitted so as not to overload the text.[ ...]

In this equation, the population growth rate depends on its size at the moment (t - T), where T is the time lag (time lag) feedback, measured by the number of generations. Even such a simple model makes it possible to identify the main effects of delay on the dynamics of population growth.[ ...]

The described population growth models and differential equations assume that all organisms are similar to each other, have an equal probability of dying and an equal ability to reproduce, so that the population growth rate in the exponential phase depends only on its size and is not limited by environmental conditions, which remain constant. The ideality” of all environmental factors in the initial conditions predetermined what the considered models are called ideal.[ ...]

Let's designate the population size of the first species as Nu and the population size of the second - N2. We denote the limiting saturation density and the maximum innate population growth rate by Ki Kg, r and r2, respectively.[ ...]

It should be emphasized that the population growth rate and other indicators are affected by strong influence breeding times. Selection pressure can lead to various adaptive changes in the life cycle; this may shift the start time of breeding but not change the total number of offspring, or it may change the production or "clutch size" but not affect the breeding time. These and many other aspects of population dynamics can be identified using tables of population structure by lifespan of individuals.[ ...]

In general, the 'Ollie effect', i.e. an increase in the population growth rate when individual individuals are combined into interacting groups (the most simple example such a union is the emergence of reproductive pairs) can lead to the emergence of several non-trivial equilibrium positions. The transition of a population from one state to another can occur both as a result of the natural evolution of the system, and under the influence of random perturbations. Sometimes the notion of 'elasticity' of the community is associated with such transitions. More precisely, a system is considered to be 'elastic' if random influences do not destroy it, but bring it to another stationary state. Among the equilibrium points of the system, there can be both stable ones, in the vicinity of which the system will spend most of the time, and unstable ones, which are associated with the boundaries of the regions of attraction of stable states.[ ...]

It is clear that the instantaneous specific growth rate of the 6NIN6t population is maximum (r = rmax) when N = 0 and (K - N)tK = 1, and is zero at N = K and (K - N)IK = 0. This means that the population stops growing when the number K is reached, when the habitat is completely occupied.[ ...]

Density-dependent growth models, such as the logistic equation, describe the process of intraspecific competition in which, as the number of individuals increases, resources become more of a limiting factor and the unit growth rate of the population decreases. The Lotka-Volterra model of interspecific competition (Lotka, 1925; Volterra, 1926) is built on the basis of the logistic equation and essentially bears all its shortcomings. However, despite this, this model is the simplest and, from a historical point of view, a very important way to analyze inter-water competition. It can help to identify the main factors that determine the outcome of the competitive interaction of the two kinds.[ ...]

There are absolute and specific population growth rates.[ ...]

If r is positive, the population size increases exponentially. If r is negative, the population decreases exponentially. Hence the rapid increase and decrease in the population. The growth rate of each organism does not depend on the density of the population. Population sizes do not stabilize If N > K, the growth rate is negative. If K > N, the growth rate is positive, then the size of the population tends to K = LH, i.e. the correspondence with the supporting capacity of the medium is given. When K = LH, the population growth rate is zero. Population sizes remain constant.[ ...]

Although the same designations may be used to denote birth rate and population growth rate, the two quantities are by no means the same thing, because AI denotes different quantities in each case. In the case of fertility, ANn is the number of new individuals added to the population. Fertility can be zero or positive, but never negative. As for the population growth rate, here DM is a net increase or decrease in the population, which is a consequence of not only fertility, but also mortality, eviction, settlement, etc. The population growth rate can be negative, zero or positive, since the population can decrease , remain unchanged or increase.[ ...]

Under such conditions of unlimited growth, the change in population size over time is expressed by an exponential curve (Fig. 12.5, A), described by the equation Nt \u003d N0 ■ e ”, where Na-initial number Nt is the number at time t, e is the base of natural logarithms1. If the number is plotted on a logarithmic scale, its changes will be expressed by a straight line, the slope of which in the coordinate system is determined by the value of r (Fig. 12.5, B). The described exponential model of population growth reflects its potential for reproduction. The indicator of the instantaneous specific population growth rate r is often defined as the reproductive potential of a population or its biotic potential (R. Chapman, 1928, 1931). The exponential growth of the population is possible only under the condition of a constant, independent of the number, value of the coefficient r.[ ...]

In addition, the program calculates the net reproduction rate (according to Equation 9), the average cohort generation time (according to Equation 10), approximate (according to Equation 11) and exact (according to Equation 12) values ​​of the specific population growth rate under given initial conditions.[ .. .]

The impact of environmental factors on the rate of population growth can bring the population to a stable (r=0) or reduce it, i.e. exponential growth slows down or stops completely and the 1-shaped curve seems to stop and flatten out, turning into the so-called B-shaped curve (Fig. 3.3). This is exactly what happens in nature - the further development of the population follows a logistic model, which is described by an 8-shaped, or logistic, population growth curve.[ ...]

Three types of dependence of the population size on its density are known (Fig. 7.10). In the first type (curve 1), the population growth rate decreases as density increases. This widespread phenomenon makes it possible to understand why populations of some animals are relatively stable. First, with an increase in population density, a decrease in the birth rate is observed. Thus, in a population of the great tit at a density of less than one pair per 1 ha, there are 14 chicks per nest; when the density reaches 18 pairs per 1 ha, the brood is less than 8 chicks. Second, as population density increases, the age at which puberty occurs changes. For example, the African elephant, depending on population density, can reach sexual maturity between 12 and 18 years of age. In addition, this species, at low density, gives birth to 1 baby elephant per 4 years, while at high density, the birth rate is 1 baby elephant per 7 years.[ ...]

Quantitative indicators (characteristics) of the population can be divided into static and dynamic. Static indicators characterize the state of the population at a given point in time. The main ones are: abundance, density, as well as structure indicators. The dynamic indicators of a population reflect the processes occurring in the population over a certain period of time. The main ones are: birth rate, death rate, population growth rate.[ ...]

In the absence of limiting environmental factors, the specific growth rate is equal to the value r, which characterizes the properties of the population itself and is called the specific (innate) population growth rate or the biotic potential of the species.[ ...]

The first terms of the right-hand sides of the system (9.10) characterize the population growth rate in the absence of limiting factors. The second terms take into account those changes in speeds that are caused by the limited feed.[ ...]

Reproductive (reproductive) potential - the ability of a population to increase in numbers, the rate constant of population growth.[ ...]

At the intersection points of the grazing and recruitment curves, the net growth rate of the prey population is zero (grazing equals recruitment). Each of these points is characterized by prey and predator densities, and these pairs of densities, located on the prey isocline, thus characterize the combined populations. Based on these points, the prey isocline shown in the figure below (and in Fig. 10.7) was constructed: its shape is characteristic of a self-limiting prey population and an approximately linear functional response. The arrows in the lower figure show the direction of change in the number of prey.[ ...]

Dynamic indicators characterize the processes occurring in a population over a certain period (interval) of time. The main dynamic indicators (characteristics) of populations are the birth rate, mortality and population growth rate.[ ...]

Competition can lead to a stable equilibrium. - Natural populations cannot be characterized by only one parameter, such as marginal saturation density. - The highest rate of population growth occurs at intermediate density values. Population growth at low density follows a B-shaped curve.[ ...]

The maximum birth rate is the theoretical upper limit that a population could achieve under ideal conditions. Despite the difficulties in the practical definition of this indicator, it is of interest for the following reasons. First, the maximum birth rate serves as a criterion for comparison with the real birth rate. For example, a bird population birth rate of 4 chicks per year would make real sense if an upper limit was known to which it could increase under less restrictive conditions. Secondly, since the maximum birth rate is a constant value for a given population, this indicator can be used for mathematical calculations and predicting the population growth rate.[ ...]

The B-shaped (sigmoid, logistic) curve reflects the logistic type of growth, depending on the population density, in which the population growth rate decreases as the number (density) increases. The growth rate decreases down to zero when the limit is reached.[ ...]

In addition, in the upper field of each graph, you can see the model-calculated values ​​of the net reproduction rate (RQ), the average cohort generation time (G), as well as the exact (r) and approximate (InR0/G) values ​​of the specific instantaneous population growth rate for given initial conditions.[ ...]

In the upper part of the screen, the program displays the values ​​of the initial parameters, as well as the calculated values ​​of the population growth rate (R), the average birth rate (L), the proportion of populations extinct by this generation (Fraction Extinct), and the number of past generations (Generation).[ .. .]

The first and third solutions are stable, the second is not. Depending on the initial values ​​of the population size, the population either degenerates (if the initial size is less than n) or, when the initial size is greater than n, it develops and reaches a maximum value equal to k. 17.5 shows the dependence of the population size on time and the dependence of the population growth rate on the population size.[ ...]

If we neglect the indicators of immigration and emigration of individuals (they are almost absent in plants), then it is possible to estimate the instantaneous rate of population growth, i.e., the balance between birth rate and population per unit of time. For stable populations, the instantaneous velocity is zero, for growing populations it is positive number, for endangered - negative.[ ...]

The balance between absorption and renewal of a resource. Point K is the only point on the IHR (this is not the net population growth rate) at which the concentration of the resource does not change (the rates of consumption and renewal are equal and directed in opposite sides). See the text for more detailed explanations.[ ...]

A resource can be defined as a substance, organism, or other material object that consumes or otherwise uses an organism that results in an increase in the specific growth rate of its population as the availability of the resource increases. If a species consumes a single limited resource, its population may eventually reach an equilibrium where the rate of population growth is equal to the rate of its decline, and the rate of replenishment of the resource in the environment is equal to the rate of its consumption. In the state of equilibrium, when 6NiN6t = 0 = 6Rf6t, the increase in the population is equal to its decrease, and the rate of resource inflow is equal to the rate of its consumption. This equilibrium can only be reached at one specific concentration of the scarce resource, denoted as R. Thus, R is the resource concentration needed by a population of a species to provide an increase that balances its decrease. At the same time, this is such a concentration to which the level of the resource can be reduced by the population consuming it when equilibrium is reached.[ ...]

[ ...]

It can be seen from the figure that in a medium that is neutral in relation to size, early maturity is much more profitable than delay, and monocyclicity is much more profitable than polycyclicity. In early maturing organisms, the generation time is shorter, so the population growth rate is higher.[ ...]

The question of how much the laws of ecology can be transferred to the relationship of man with the environment remains open, since man is different from all other species. For example, in most species, the rate of population growth decreases with increasing population density; in humans, on the contrary, population growth in this case accelerates. Therefore, some of the regulatory mechanisms of nature are absent in humans, and this may serve as an additional reason for technological optimism in some, and for environmental pessimists to testify to the danger of such a catastrophe, which is impossible for any other species.[ ...]

On the other hand, development is the gradual differentiation of parts of an organism, allowing it to perform various functions (for example, to reproduce) at different stages of the life cycle. In many cases, growth and development occur simultaneously. However, these are two completely separate processes. The same stage of ontogeny can correspond to a wide range of sizes, and specimens the same size may vary in development. Rapid development can be advantageous because it brings the onset of reproduction closer, shortens the generation time, and thereby increases the rate of population growth (chap. 4). On the other hand, if an organism has to go through a period of extremely unfavorable conditions during its life, a stop in development (i.e., a stage of rest or diapause) can be beneficial (chap. 5).[ ...]

In contrast, two other Australian ecologists, Andrewartha and Birch, view density-dependent processes in general as secondary and play no role in some species in determining population size. Unlike Nicholson, their work was more concerned with the control of pests in natural conditions. The point of view of Andrewartha and Birch (Andrewartha and Birch, 1954) can be summarized as follows: “The number of animals in the natural population is limited in three ways: (a) the lack of material resources, such as food, nesting sites, etc.; (b) the inaccessibility of these material resources due to the limited ability of animals to disperse and search for them; (c) a limited period of time during which the value of r (population growth rate) has positive value. Apparently, it is the latter option that is most common, and the first option is the least common. Fluctuations in the value of r can be caused by weather, predators, or any other environmental component that affects the rate of population growth.”[ ...]

Nevertheless, in fig. 6.8 in all cases, a bell-shaped curve is observed. The shape of this curve reflects the general nature of density-dependent changes in fertility and mortality whenever intraspecific competition occurs.

If the environment does not have a limiting effect, then the specific population growth rate for given microclimatic conditions is constant and maximum. Under such favorable conditions, the growth rate is characteristic of a certain age structure of the population and serves as the only indicator of the hereditarily determined ability of the population to grow. It is an exponent in the differential equation of population growth in an unlimited environment under specific physical conditions:

dN / dt = rN; r = dN / (Ndt);

log n t = rt - ln(N 0 ); n t = N 0 e rt ,

where N 0 - number at zero time, n t - number at time t, e- base of natural logarithms.

From the equation ln n t = t - ln(N 0) we can calculate the population growth rate:

The indicator r is the actual difference between the specific instantaneous rate of births (b) and the specific instantaneous rate of death of organisms (d). It can be expressed as:

r = b - d.

The general population growth rate in the absence of the limiting influence of the environment depends on the age composition and the contribution of various age groups to reproduction. Therefore, a species can be characterized by r values ​​depending on the structure of the population. When determining a stationary and stable distribution of ages, a specific growth rate is called indicator of potential population growth (r max ). This indicator is often referred to as biotic or reproductive potential(the term was introduced by R. Chapman in 1928). The difference between the biotic potential and the actual growth rate is considered a measure of environmental resistance, which characterizes the sum of all limiting factors that prevent the realization of the biotic potential.

In addition to the above, other population growth equations are also used (R. Whittaker, 1980). With unlimited population growth, geometric and exponential formulas can be used.

Geometric formulas:

N 1 / N 0 = R - population growth rate per unit of time;

N 1 \u003d N 0 R t - population density over time t.

Exponential Formulas:

dN / dt = Nn - population growth rate;

Nt = N 0 e rt - population density over time t. The mathematical method can also be used to describe the unlimited growth of a population. In the case of limited population growth, its growth rate and density can be calculated using the following logistic formulas:

growth rate;

where N is the population density; N 0 - initial population density; N 1 - population density in a unit of population growth time ( t= 1); N t - population density over time t at a constant growth rate, K - the capacity of the medium for the maximum population density, e is the base of the natural logarithm. The value of K is called the maximum allowable load on the environment, or the capacity of the environment for a given population.

The choice of a mathematical model is determined by the objectives of the research and the adequacy of the model for each specific case.

Types of population growth. An idea of ​​the capacity of the habitat. Depending on the nature of population growth, various types of their growth are distinguished. According to the shape of the curves built on an arithmetic scale, two main types of growth can be distinguished, described by the J-shaped and S-shaped curves. These two types of curves can be modified in different ways (Fig. 18).

Rice. 18. Curves of population growth (according to Yu. Odum, 1875)

A - J-shaped, B - S-shaped curve of population growth. A 0 - at first, there is an unlimited increase in the population size; A 1 - at first, an unlimited increase in numbers is observed, then it stops and, under favorable conditions, resumes again, reaching the previous value; A 2 - there is an unlimited increase in the number, then it suddenly stops and then fluctuations are observed at a lower level; At 0 - there is an increase in the population along the S-shaped curve, reaching the K-level; In 2 - at first, a slow increase in the number is observed, then the rate increases and reaches the K-level; IN 1 - upon reaching the K-level, slight deviations from it are observed; In 3 - there are significant deviations from the K-level.

With a J-shaped growth curve, density increases rapidly, but then, when a limiting factor comes into play, population growth suddenly stops. This type of growth can be described by an exponential equation:

The J-curve equation is the same as for the growth rate. The difference is that N has a limit. This means that relatively unrestricted growth suddenly stops when the population exhausts its resources (food, living space) or some other factor intervenes. After the upper limit of N is reached, the density may remain at this level for some time or fall sharply. This is typical for natural populations of insects, algae, etc.

With a logistic (S-shaped) population growth curve, at first the population increase is very slow, then faster, but then, under the influence of environmental resistance factors, the population growth gradually slows down. This slowdown, due to the resistance of the medium, becomes more and more pronounced and eventually reaches a certain value. Then you begin to maintain a more or less stable balance. This type of growth can be expressed by the Verhulst-Pearle equation:

where K - a constant indicating the upper limit of population growth, called the upper asymptote for the S-curve.

The characteristic shape of the S-shaped curve is due to the gradual increase in the effect of adverse factors as the density of the population increases. This type of growth differs from J-shaped, in which the population begins to experience environmental pressure at the end of growth.

In a simple or ideal case, the increase in the effect of unfavorable factors depending on the density of the population is linear and can be written as follows:

where r - population growth rate or potential growth indicator, N - population size, K - maximum possible population size; e is the base of natural logarithms, a is the integration constant that determines the position of the curve relative to the origin, t- time.

This equation differs from the exponential one in that it contains the expressions (K - N) / K, (r / K) N 2 or (1 - N) / K. These expressions represent three indicators characterizing the resistance of the environment created due to the growth of the population, which as it approaches the limit, it reduces the rate of potential reproduction. This equation reflects the law: the rate of population increase is equal to the maximum possible rate of population growth, multiplied by the degree to which the maximum rate is realized.

It should be noted that there are many mathematical equations to describe changes in population size, the solution of which can be represented graphically in the form of S-shaped curves. This is true for almost any equation in which the increase in negative factors is in any way dependent on population density.

To compare the experimental data with the theoretical curve, one should make sure that the indicators included in the equation characterize the effects that regulate the population density. Situations where the resistance of the medium increases linearly with increasing density may occur in populations with a simple life cycle. In more highly organized populations, with complex biological cycles and long periods of individual development, changes are likely to be delayed in time.

Population development strategies. Depending on the type of population growth curve, a development strategy is distinguished, determined by such properties as the rate of reproduction, the nature of energy transfer from one generation to another, fluctuations in abundance relative to the equilibrium value, or K-level, the rate of change in abundance, the adaptability of a species to a particular territory, size individuals, their lifespan, etc. (Table 9).

Table 9 Characteristics r- and K-species.

The symbols r and K are used to characterize the population strategy. Rapidly reproducing species with a high r value are called K-species. These include typical pioneer species of disturbed habitats.

Species with a relatively low value of r are called k-species. The rate of their reproduction largely depends on the density of the population and is close to the equilibrium value determined by the -level. These species are typical for the late stages of succession development.

It should be noted that there are a number of intermediate strategies. These two population strategies represent two ways of solving the same problem - the problem of the survival of the species. Species belonging to the r-strategy, faster than those belonging to the K-strategy, colonize the disturbed habitats characteristic of early successions (outcrop rock, forest clearings, former quarries), since they spread more easily and multiply faster. K-strategy species are more competitive and eventually outcompete r-strategy species that may move to other disturbed habitats. Since species with an r-strategy have a high reproductive potential, this means that if they remained in any habitat, they would quickly use the available resources and exceed the supporting capacity of the environment, which would lead to the death of the population. Species with an r-strategy are characterized by a J-shaped growth curve with a rapid decline in population size.

Population growth is associated with characteristics such as fertility and mortality, and represents an increase in the number of individuals (or biomass) over time. This value can be either positive (excess of birth rate over death rate) or negative (predominance of death rate over birth rate) and is characterized by growth. Growth characterizes the growth rate of populations, which can be constant or fading.
Population Growth- the difference between its value at the beginning and end of a certain period of time.
A positive population growth depends on the breeding potential, which plays a large role in the survival of the species. Populations reduced to low level, with a favorable change in conditions, can recover as a result of a high biotic potential.
Biotic Potential — individual characteristics species, hypothetically determining the rate of population growth, showing the theoretical maximum of descendants for a certain period of time.
Due to mass reproduction, some species are able to withstand grazing by various consumers. Also, a high birth rate ensures the rapid development of new spaces by the species. In addition to the positive aspects of mass reproduction, there are also dangers associated with exceeding the capacity of the medium. The undermining of resources (lack of food, shelters, territory) can lead to a weakening of the population. Thus, overcrowding is unfavorable for any species. The population, which is an open self-regulating system and is in constant interaction with the environment, is able to control its growth.
So, for example, with an increase in the supply of fish with food, the following ways of self-regulation of abundance are possible:
1) an increase in the growth rate, earlier maturity, a reduction in the age range of individuals maturing for the first time, an increase in fecundity;
2) increase in fat content;
3) decrease in consumption of own juveniles;
4) increase in viability and decrease in mortality of larvae at the first stages of active feeding;
5) increase in the number of fertilized eggs of females approaching spawning grounds;
6) a decrease in the amplitude of the variability of the sizes of eggs.
The reverse process is noted with a reduction in food resources. At the same time, it is observed:
1) a decrease in the growth rate, later maturity, an increase in the length of the row of first-maturing fish, a decrease in the fecundity of one-sized individuals;
2) reduction of fat content;
3) increase in consumption of own juveniles;
4) a decrease in hardiness and a decrease in the survival of larvae at the first stages of active feeding;
5) decrease in the number of fertilized eggs of females approaching spawning grounds;
6) an increase in the amplitude of the variability of the sizes of eggs.
Black musculus (Musculus niger) lives on sandy-silty and gravel-silty ground, surrounding a shallow hole with a byssal nest. Juveniles hatch in filamentous clutches attached to the byssal threads of the nest.
According to the laws of nature, each parental pair in each generation must leave a pair of offspring. If, on average, the size of the reproductive group of the next generations exceeds the size of the previous generation by at least a small percentage, then it will grow. Otherwise, she will die.
Thus, the dynamics of the number and biomass of a population is the result of the interaction between living conditions and the specific adaptive properties of the population. It is determined by the ratio of birth and death rates of individuals, as well as their migration.
This fact should be taken into account when planning the exploitation of hydrobiont populations. This is especially true for fishing. It is established that the size of catches undergoes periodicity. The dynamics of the number of commercial fish can be associated with temperature indicators, which is manifested in a change in the food supply, spawning conditions, wintering, etc. This dependence has been determined for sturgeon, salmon, herring, and cod. Taking into account these features, catch quotas are set.
On store shelves, we are used to seeing mackerel the size of a herring. Real mackerel, for example, the most common Japanese, often reaches 60 cm. The same family includes tuna
Mortality and survival
As a result of natural extinction, destruction by other organisms or death due to adverse environmental factors, the number of individuals of each generation decreases. The rate of this decline characterizes the mortality of the population.
Mortality is the rate of population decline.
Distinguish between mortality for a given period, per unit of time and the level of mortality. To a certain extent, the fecundity of the population serves as an indicator of mortality. If the average number of individuals remains similar over significant periods of time, this means that as many organisms die off as there are formed. Fertility can be seen as an adaptation to a particular mortality. In some hydrobionts, it is expressed in astronomical numbers: the moon-fish, for example, spawns up to 300 million eggs at a time. In populations of large animals, the death of old individuals may play a leading role.
The mortality rate does not characterize the specifics of the death of organisms of different age groups in the population. It is reflected in the mortality curve, which is typical for each population.
Population mortality curve- a value showing the degree of decrease in the number of individuals of this generation from the moment of birth to the end of the life cycle.
This or that mortality at different stages of development is associated not only with age-related changes in the constitutional protection of organisms (an increase in size, speed of movement, the formation of protective structures, etc.). A change in the way of life of one or another stage of development, for example, the transition of larvae from the water column to the ground or penetration into it, can be essential. Increased mortality is often due not to a decrease in the protective properties of the organism, but to an increase in the number of its consumers or a sharp deterioration in the abiotic environment. For populations of hydrobionts, the most common type of mortality is characterized by a decrease in the death of organisms as they grow and an increase in the effectiveness of various protective agents.
The reciprocal of mortality characterizes the number of individuals surviving to a particular age. This is survival.
population survival- a value characterizing the number of individuals who survived to a certain age.
Survival depends on environmental conditions. The characteristics of embryonic survival and survival in the postembryonic period differ. Reducing the mortality of embryos is provided by environmental conditions, as well as protection from destruction by other organisms. The safety of offspring is largely determined by their protection from enemies. The highest survival rate of embryos is observed in the case of live birth. It is more often found in organisms living in extreme conditions of the abiotic environment, in particular in water bodies with low temperatures. Survival in the postembryonic period is achieved using different methods of development. In favorable conditions, species with indirect development are more common, and in more severe conditions, with direct development. Development with metamorphosis, that is, the existence of a species in the form of individuals that differ sharply from each other in their structure and needs, has great importance to increase populations.
Sea snakes (Fig. 25) live only in the Pacific and Indian Oceans. Rounding the Cape of Good Hope on the southern coast of Africa prevents them cold water. For snakes, the optimum temperature is +24 °C. Already at 20 °C they lose their mobility.
The survival of individuals depends on the nature and rate of their growth. In accordance with environmental conditions and the state of the population, the growth rate and its nature in individuals can change significantly, ensuring the flexibility in the use of vital resources and the greatest survival of organisms. Thus, under unfavorable environmental conditions, many hydrobionts cease their active existence, at this time acquire greater resistance to external influences and, by reducing the rate of development, increase their survival. Under unfavorable conditions, the growth of individuals slows down, and the population can be completely switched off from the process of its reproduction.

The opposite picture is observed when environmental conditions improve: the growth rate increases, and the population, using the current situation, rapidly increases its biomass. Thus, there is a periodic growth in individuals of populations that live not only in high, but also in low latitudes.

If, with little emigration and immigration, the birth rate exceeds the death rate, then the population will grow. Population growth is a continuous process if all age groups exist in it. The rate of population growth in the absence of any environmental constraints is described by the differential equation:

dN/df = rN, (1)

where N is the number of individuals in the population; f - time; r is the rate constant of natural growth.

J-shaped model of population growth. If r > 0, then over time the population becomes larger. Growth occurs slowly at first, and then rapidly increases according to an exponential law, i.e., the population growth curve takes a J-shaped form (Fig. 2, a). This model is based on the assumption that population growth does not depend on population density. It is believed that almost any species is theoretically capable of increasing its population to populate the entire Earth with sufficient food, water, space, constancy of environmental conditions and the absence of predators. This idea was put forward at the turn of the 18th and 19th centuries. English economist Thomas R. Malthus, the founder of the theory of Malthusianism.

Rice. 2. Types of population growth curves (population growth models): a - J-shaped; b - S-shaped; K is the holding capacity of the medium.

S-shaped model of population growth. The situation develops differently with limited food resources or with the accumulation of toxic products (waste) of metabolism. The initial exponential growth under initially favorable conditions cannot continue over time and gradually slows down. The density of the population regulates the depletion of food resources, the accumulation of toxicants, and therefore affects the growth of the population. With increasing density, the population growth rate gradually decreases to zero, and the curve reaches a certain stable level (the graph forms a plateau). The curve of such growth (Fig. 2, b) has an S-shape, and therefore the corresponding model of the development of events is called S-shaped. It is typical, for example, for yeast, the factor limiting their growth is the accumulation of alcohol, as well as for algae, self-shadowing each other. In both cases, the population size does not reach the level at which the lack of nutrients (nutrients) begins to affect.

Overpopulation also influences population growth, in which space plays a significant (perhaps even the main) role. Laboratory experiments with rats showed that after reaching a certain population density, the fertility of animals decreases sharply even with an excess of food. There are hormonal changes that affect sexual behavior; infertility, eating of cubs by parents, etc., is more common. Parental care for offspring is sharply weakened, cubs leave the nest earlier, as a result of which the probability of their survival decreases. Animal aggression is on the rise. Similar phenomena are also encountered in populations of a number of mammals, not only under laboratory conditions, but also under natural conditions.

The population growth rate in the S-shaped model determines the differential equation:

dN/df = rN(l - N/K), (2)

where K is the supporting capacity of the environment, i.e., the maximum size of the population that can exist under given conditions, satisfying its needs indefinitely.

If N > K, the growth rate is negative. If N< К, скорость роста положительна и величина популяции N стремится к К, т. е. приводится в соответствие с поддерживающей емкостью среды. Если N = К, скорость роста популяции равна нулю. При нулевом росте популяция стабильна, т. е. ее размеры не меняются, хотя отдельные организмы по-прежнему растут, размножаются и отмирают. Происходящее размножение уравновешивается смертностью.

In specialized literature, J- and S-shaped population growth models are often called exponential and logistic, respectively.

The supporting capacity plays a decisive role not only with the growth of the population according to the S-shaped, but also according to the J-shaped model, because at some point in time, the exhaustion of any resource of the environment still occurs, i.e. he (or even several at the same time) becomes limiting. After a boom with a sudden exit of the J-shaped curve beyond the level K, the population collapses, that is, a catastrophe leading to a sharp decrease in numbers. The cause of the collapse is often a sudden sharp change in environmental conditions (environmental factors), which reduces the carrying capacity of the environment. Then a huge number of individuals unable to emigrate die.

Under the most favorable set of circumstances for the population, the new level of abundance corresponds to the supporting capacity of the environment, or, in other words, the growth curve turns from J-shaped to S-shaped. However, the depletion of food resources can also lead to other difficulties for the population, such as the development of diseases. Then the abundance decreases to a level much lower than the supporting capacity of the environment, and in the limit, the population may even be doomed to extinction.

For the S-shaped model, in cases of delay in the action of regulatory mechanisms for any reason, for example, due to the time spent on reproduction or for other reasons, the time delay takes into account the differential equation:

N/df = rNK - rN2(f - T)/K, (3)

where T is time, required by the system to respond to external influences.

The subtrahend on the right side of the equation containing N2 makes it possible to predict the moment when the system leaves the equilibrium state in cases where the delay time is relatively large compared to the relaxation time (1/r) of the system. As a result, with an increase in the delay time in the system, instead of an asymptotic approximation to the equilibrium state, the number of organisms fluctuates relative to the theoretical S-shaped curve. In cases where food resources are limited, the population does not reach a stable equilibrium, because the number of one generation depends on the number of another, which affects the rate of reproduction and leads to predation and cannibalism. Fluctuations in population size, which is characterized by large values ​​of r, short reproduction time x and a simple regulatory mechanism, can be quite significant.

The described population growth models and differential equations assume that all organisms are similar to each other, have an equal probability of dying and an equal ability to reproduce, so that the population growth rate in the exponential phase depends only on its size and is not limited by environmental conditions, which remain constant. They accurately describe the processes of growth and interaction of individuals in most artificial and some natural populations. The "ideality" of all environmental factors in the initial conditions predetermined what the considered models are called ideal.

For natural populations, the accepted assumptions are most often incorrect. Under natural conditions, J- and S-shaped patterns of population growth can mainly be observed in cases when certain animals are introduced or they themselves spread to new areas for them. However, theoretical growth models allow a better understanding of the processes occurring in natural conditions. Most of the principles used to model animal populations apply to plant population modeling as well.

It should be noted that for any model (both J- and S-shaped), the phase of exponential population growth is initially characteristic. Therefore, with a combination of favorable (optimal) values ​​of all environmental factors, a "population explosion" occurs, i.e., an especially rapid increase in the population of a particular species.

Migration or dispersal, as well as a sudden decrease in the rate of reproduction, can contribute to a decrease in population size. Dispersal may be associated with a particular stage of the life cycle, such as the formation of seeds.

With regard to the conditions of real natural environment it is customary to use the concepts of biotic potential - the totality of all environmental factors that contribute to an increase in the population, or the species' ability to reproduce in the absence of environmental restrictions, as well as environmental resistance - a combination of factors limiting growth (limiting factors).

Any changes in a population are the result of an imbalance between its biotic potential and environmental resistance.