Kiryanov D.V., Kiryanova E.N., Kozlov N.I., Kuznetsov V.I.
(D.V.Kiriyanov, E.N.Kiriyanova, N.I.Kozlov, V.I.Kuznetsov)

IPM im. M.V.Keldysh RAS

Moscow, 2005

annotation

The work examines several mathematical models of the influence of the age composition of an ecological population on its development. Modeling is carried out by numerically solving a dynamic system differential equations(ordinary and partial derivatives), belonging to the class of Volterra systems and Leslie matrices.

Abstract

The review of models of age structure influence on ecological population dynamics is presented. We consider a number of dynamical systems of ordinary and PDE differential equations based on classic Volterra model and Leslie matrixes approach.

§ 1. Basic model

Lately for a solution practical problems Modeling of the dynamics of ecosystem development based on differential and integro-differential equations is increasingly being used. This approach is widely used to model diverse biological communities and, in particular, forests. The greatest difficulty is presented by two points:

· correct choice of equations, especially the parameters included in them that describe the magnitude of the influence of certain parameters on the state of a given area of ​​the ecosystem;

· adequate modeling of age effects, as well as the spatial distribution of heterogeneous ecosystems.

In this work, we consider various age effects in forest biocenoses based on numerical modeling of ordinary differential and differential-difference equations, as well as partial differential equations. First of all, let us present a simplified model of the development of a two-species forest, which describes the evolution of the population as a whole, not taking into account either spatial distribution or age effects. At this stage, which essentially expresses global environmental requirements, it is necessary to adequately determine the nature of the main interactions.

We will characterize the population by the biomass density vector, i=l (deciduous species), x (coniferous). Let us limit ourselves to a two-level system of trophic interactions of the “resource-consumer” type: soil - two-species forest competing with each other. The condition of the soil is characterized by the third variable - the generalized indicator of fertility P(t). The dynamical system we used to describe this lumped model is as follows:

i = (x, l)(1)

· P – generalized indicator of fertility – resource density (kg/m 2 );

· u l – biomass density of deciduous species (kg/m 2 );

· u x – density of coniferous biomass (kg/m 2 );

· A i is the coefficient of soil restoration due to the fall of the i-th species (1/year);

· B – coefficient of soil self-healing (1/year);

· P 0 – asymptotic value of fertility in the absence of forest (kg/m 2 );

· Vi – rate of resource consumption (trophic function) (1/year);

· с i – correction factor describing competition;

· k i – growth coefficient of the i-th breed;

· D i – natural mortality rate of trees (1/year);

· W – influence of external factors, often detrimental, therefore with a negative sign, (kg/(year× m 2 ))

· t 0 – average time of maturation of a young forest (year)

This system, as is easy to see, is a generalization of the classical Volterra model. The coefficients are a combination of an experimentally established constant (determining the development of the population under normal conditions) and some correction function. We discussed the explicit form of the coefficients in the work, and typical graphs of the solution of system (1) were also given there.

One of the solutions corresponding to forest development under normal conditions (with sufficient moisture) is presented in Fig. 1. It describes the well-known phenomenon of the replacement of deciduous forests with coniferous ones during the development of a population.

Fig.1. Typical solution of system (1).

It should be immediately noted that the concentrated model (1) allows only the roughest consideration of age effects, because the equations include total biomass densities (without division into age groups). For example, in the calculations shown in Fig. 1, we (through the appropriate correction function) took into account that the natural mortality rate D i significantly depends on the average age of the population.

Having made these preliminary remarks, we can move on to the main topic of this work - various models of forest populations with heterogeneous age composition.

§ 2. Leslie matrix model

Matrix calculus to describe the development of complex multispecies populations (applied to lumped models) was proposed by Leslie in the middle of the century. Until now, the mentioned ecological models have relied on differential calculus methods. This in itself is some approximation. When moving to practical calculations, for example, according to demographic tables, one has to deal with discrete quantities. For example, in human demography, five-year time intervals are typically used. In addition, the development of many populations (including forests) has a pronounced seasonal nature. Therefore, for a correct description of the population and for practical calculations, it is more likely that not the methods of differential and integral calculus are applicable, but the methods of discrete mathematics (matrix, etc.).

Leslie proposed to use the so-called transition matrix to describe a complex multi-age population

,(2)

which, when multiplied by a column vector of the numbers of individuals of different age classes (from zero - newborn individuals, to k - the oldest individuals) gives the number of individuals in age groups after a certain unit of time (most often, a year). Thus, in transition matrix Leslie p i is the survival rate (i.e. the probability that an individual of the i-th class will move to the (i+1)-th next year),a i – average fertility of individuals i-th age groups.

Thus, the transition matrix is ​​a square matrix of size (k+1)´ (k+1), and the column vector of the numbers of age groups is the matrix (k+1)´ 1. If the matrix elements are constant, i.e. do not change with time, then from their non-negativity it follows that the maximum absolute value of the matrix eigenvalue is real and positive. If the maximum eigenvalue is less than one, then the population is doomed to extinction; if it is greater, there is unlimited population growth. Primitive Leslie matrices have a maximum eigenvalue of one. This means that the population will tend over time to some stable age distribution, given by the eigenvector corresponding to the maximum eigenvalue, and the population growth rate will be determined by this eigenvalue.

In a population dynamic description, it is necessary first of all to take into account the difference in the ability of individuals to reproduce. For this purpose, three groups are usually distinguished: pregenerative (young, not yet capable of reproduction), generative (capable of reproduction, but not necessarily reproducing at the moment) and postgenerative (senile, having already lost the ability to reproduce). Depending on the characteristics of the life cycle of a particular species and in the presence of reliable diagnostic characters, each of these large groups is divided into smaller age categories.

It should be noted that dividing a population into age groups is justified from a practical point of view in cases where organisms of a given species have characteristics that make it possible to accurately determine the age of an individual. In our models, in the case of a forest population, the age of an individual tree can be accurately determined using tree rings.

Let us now consider the application of Leslie matrices to the model introduced in§ 1. Let us recall that this basic model described the forest without taking into account age. Plant resistance to drought, waterlogging, shading, pollution, ground fires, diseases and other factors largely depends on age.

In what follows, we omit the view index, since the calculations do not depend on it. The view index will only appear in the final result. It should also be taken into account that if the intervals of age groups are much larger in time, then it is necessary to take into account the age distribution within the group itself. The number of age groups can be small, for example, 4 groups for each typical age interval: young forest, forest of reproductive age and overmature forest ( does not produce seeds). That is, there are only 12 groups. However, this distribution is unknown before the calculation begins. It is quite possible that at each time step the distribution within the group is clarified, for example, by interpolating according to the values ​​of group variables per time step. Then the group constants are refined. We choose a simpler path: an assumption is made about the age distribution a priori, and then group constants are found. In fact, these "constants" may depend on group variables. This ensures that group constants are adjusted in accordance with the values ​​of group variables.

To determine the coefficients of transition from group to group, we will resort to a discrete scheme (Fig. 2). Let the age group include a rlet and we consider a steady state when the same biomass value is received at the input of the group.

123 4

Fig.2. Diagram explaining Leslie's model

Then for each year by age we will have the following biomass densities:

1 year=,2 year=,3 year=,...r year=

Here C is a coefficient showing how much the initial biomass density increases per year. This value is usually 0.1-0.18 and depends on the group, biomass density, fertility, etc. However, within the group it changes little. If we take groups in order of 10 years, then the linear law of growth in the group that we have adopted is completely justified.

The proportion of biomass from the mass of a group that comes to another group each year can be determined by the ratio:

(3)

This follows from the fact that the total volume of biomass in a group is the sum of all volumes by year: . Taking these remarks into account, we can obtain that

(4)

As we see, the evolution of functions u i [j] is described by equations similar to system (1), but instead of the coefficients C i0 and D i0, the growth and death arrays C i0 [j] and D i0 [j] are introduced, respectively.

Age should obviously be taken into account taking into account the different attitudes of young and old trees to drought, swampiness, etc. For this, appropriate correction functions are applied, by analogy with the basic model (1). In this case, of course, they, just like unknown functions, become vector ones, because represent age-related characteristics in relation to drought, waterlogging, shading, pollution and other factors. For example, they take into account that young trees suffer most from a lack of water, and mature trees are more resistant to waterlogging, since they have a deeper root system. To write the system of equations of model (4), we use the following reduction technique: in the product of functions of the same structure, we will indicate the indices of only the right factor: the notation is equivalent to .

Total number of differential equations: . Here is the maximum number of groups of the th type, two more equations for soil productivity and the thickness of the moistened layer). Age group size p years.

Let us now present some modeling results based on the system of equations (4) taking into account the age composition of the forest population. The width of each age group was 10 years. In Fig. 3 forest development in a generalized model: under normal moisture (sustainable change from mixed forest to coniferous forest, as in the basic model in Fig. 1). However, there are fluctuations in biomass density associated with fluctuations in average age (Fig. 4.). In general, it can be recognized that the age-based model based on Leslie matrices is more realistic and retains the same basic properties as the basic model

Fig.3. .Sustainable tree population

Fig.4. Variations in the average age of trees

§ 3. Single-age planting model

Let us now consider another model that describes the evolution of an array of seedlings of the same age, taking into account annual propagation by seeds, the age threshold for the start of reproduction, as well as intraspecific competition and “ecological” cuttings (determined only by the state of the array), preserving autonomy.

We use the following discrete coordinate system. We will plot the generation number along the ordinate axis. We will plot time along the abscissa axis. If, for ease of recording, we assume the age of the seedlings to be zero, the tones coordinate plane There will be only one point at the initial moment. New generations will appear taking into account the threshold for reproduction, and, having appeared, they begin to move along the time axis, “aging synchronously.”

Then we can write down our own evolutionary balance equation for each generation:

,(5)

where the generation number, the age of the generation at a given time t, the biomass density of the k-th generation, is the difference between the coefficients of natural growth and natural mortality, a certain function that depends on the entire vector of generation density (the length of the vector depends on the time at which we will consider our system, i.e. not constant).

This function can express intraspecific competition:

, (6)

where m is the reproductive age, t-m+1 gives the total number of generations at time t, the coefficients of competitive interaction of generations k and j, i.e. these are quadratic terms of competition.

We will solve the equation for generation over a time interval of one year, considering k() and some well-known functions on the generation number. Then we can write:

(7)

In this entry, we took into account that the quantity is small compared to and therefore can be expanded in a series, limiting ourselves to the first term of the expansion. If we now introduce notation for the second factor in the last product:

,(8)

then you can get a counting scheme for the transition from time layer t-1 to layer t if you add the missing value of the newly created generation on layer t:

(9)

We see that the values ​​of the generation just born on a given time layer t are obtained from the values ​​of generations also in layer t. In the case of taking into account competition, the integral must be calculated using data from the previous time layer.

If you approach the system of equations for generations as a dynamic system, then you can justify the stability and approximation of the scheme and thereby justify the convergence of the scheme.

Since the proposed system of equations (without competition and sampling) is linear, it will behave like a logistic model without the Volterra competition term. This means that it is either unstable or has only one stable equilibrium state, zero. Meanwhile, calculations (Fig. 5) for a certain variant of the forest show that the divergence is very slow. Only by the year 320 there is a noticeable increase in the overall planting density. The diagram also correctly reflects the initial period, when the establishment of an age distribution profile had not yet occurred.

Fig.5.

(for 100, 200 and 300 year populations)

A numerical solution of the same problem, but with competition, leads to results when the establishment of a stationary profile occurs for the 120th year of the landing’s existence. As for the absolute values ​​of biomass, they are established later (about 200).

From this we can conclude that mono-aged plantings do not provide a natural distribution of the forest area by age. For the foreseeable time, we are dealing with a transition period, which is observed in practice.

“Ecological” felling turns out to be much more effective in terms of time to reach a stationary profile and a stationary value of biomass. In Fig. Figure 6 shows calculation data according to the same scheme, but instead of competition, cutting is introduced. Cutting is carried out until the total biomass has decreased to a certain critical value. The age of the trees being cut down is 40-45 years, 5% are cut down. The natural profile and asymptotic values ​​of biomass are established much faster than with competition.

Fig.6. Evolution of the age spectrum of mono-age planting

(for 150 and 200 year old populations): model with cutting

§ 4 Continuous diffusion model

Below we consider a continuous age model that allows us to describe the effect of the reproductive threshold. The biological variable here will be a function of two variables: time and age T. In this case, u(t, T) dT is the amount of biomass contained in the interval of the age variable (T, T + dT).

Let's calculate the balance for the age group in the interval T, T + dT for time t:

· u(t, T) T is the amount of biomass included in time (where the same value is perceived as an age interval) from the left end,

· u(t,T+dT) Tsuch amount of biomass leaves the group in time from the right edge of the group,

· u(t,T) change in biomass due to the processes of natural mortality, growth and processes of intraspecific struggle:

Left borderRight border

Fig.7.

In the latter cases, the integral over biomass of all ages is included.

Let us take into account that the biomass of the dT group is equal to u(t,T)dT:

Going to the limit and reducing by the product, we find the full form of the equation:

(10)

boundary condition(11)

Where is the initial condition

In the boundary condition at the left end of the age interval, the integral is calculated over the entire reproductive period. The function gives the productivity of adult biomass by seeds.

Here is the reproductive age, the right border of the reproductive age.

To solve problems related to stability, we reduce this equation to delay equations, for which this issue is well developed. To do this, consider replacing the required function (Fig. 8):

(12)

Fig.8. Towards the construction of a time grid

Substituting both substitutions into the original equation, we obtain for v(t,T) a homogeneous equation with solutions that can be written through two as yet unknown functions:

(13)

The validity of these formulas can be verified by direct verification. All that remains is to find the introduced functions.

The initial and boundary conditions give:

(14)

From these equations we can already obtain u(t,T) for t

(15)

To find solutions at T

(16)

These expressions can be obtained after replacing the functions we introduced under the integral u(t,T`). If we now take into account that the values ​​in the first line are nothing more than a known initial condition specified on the interval of the reproduction period, then, after differentiation, we obtain a differential equation with a retarded argument, but only if the dependence q(s) on the argument is absent or has a special form - exp(s):

(17)

These equations must be supplemented with an initial condition specified on the time interval. After this, the equations can be solved sequentially.

We will not solve these equations, since below we carry out a numerical solution for this type of equations in a more general form. The usefulness of the calculations carried out is that they give a hint on how to solve the problem of stability for such equations.

From the theory of equations with a retarded argument, a characteristic equation is constructed. However, unlike the previously discussed characteristic equations, it is a transcendental equation that has infinitely many roots. The theory associated with solving characteristic equations of this type is analyzed in detail. From this theory it follows that for linear cases it is possible to formulate a theorem similar to the theorem for linear equations without lagging arguments.

The corresponding formulation for almost linear equations of the usual type is given. For our case, as follows from , we can repeat it verbatim: if the real roots of the characteristic equation are negative, and all complex ones have negative real parts, then the equations are asymptotically stable according to Lyapunov. Otherwise, either there is no stability (the roots or their real parts are positive), or it is not asymptotic (there is equality of the zero real part of the roots).

In our case, to find the characteristic equation, there is no need to go over to differential equations, especially since this narrows the area of ​​practically interesting cases. The possibility of such a reduction that we have discussed gives us the right to rely on powerful results obtained for equations with a retarded argument. That’s all. This also applies to theorems on the roots of the transcendental characteristic equation and on the possibility expand the solution in an exponential series. As for obtaining the equation itself, we will use a different technique. We will assume that the solution to the original equation can be represented as a superposition of solutions as follows: .

Substituting this expression into the original equation, we find:

(18)

This technique is widely used in mathematical physics for linear problems.

To find the characteristic equation for , we must substitute the solution representation we have adopted into the boundary condition. After reducing both sides of the equalities to a function containing time, we obtain the characteristic equation:

(19)

The last equality is the characteristic equation. It is interesting that it can be obtained in closed form for very arbitrary functions, since the linear equation for G(T) of the first order is solved in quadratures. However, a simple exponential polynomial may not be obtained. However, the property of roots is largely preserved. We will not dwell on this, since we will study complex cases numerically.

In the simplest case, when the functions do not depend on T, the expression for the characteristic equation looks especially simple:

(20)

Such equations are called exponential polynomials. The location of the roots s of such a polynomial has been well studied. If it turns out that the only real root is negative, then all complex roots have negative real parts. In this case, the stability of the system in a discrete version, the addition of competition led to the stabilization of the forest area. If before the inclusion of competition the forest decreased over time, then competition will only intensify this process. But in an unstable state, the inclusion of competition leads to non-trivial stabilization.

Note that even if only competitive relations are taken into account, the study of stability is theoretically impossible (however, the situation is similar for ordinary nonlinear systems). This shows that the value of analytical

with the same time and age steps, choosing the zero point by age from the previous step:

(21)

To describe the integral in the boundary condition for age, the Simpson formula is used. We take the right side of the equations at the midpoint.

We define all dependencies on age and time in the form of piecewise smooth functions (in the calculations below these are straight line segments). The dependencies used do not affect the research programs in any way and are subject to change.

The programs are organized so that it is possible to calculate a certain number of time steps and display the result on a graph. This way you can trace the process of establishing a profile.

Fig. Figure 10 shows the results of calculations of the age spectrum using a linear model. The initial age distribution U0 and three distributions at successive moments of time are shown. We can observe the process of establishment when there is no competition (the forest is young). It is clear that there is an increase in biomass density, accompanied by a shift in the age spectrum profile to the right.

Fig. 10. Evolution of the age spectrum of a monoage population

When calculating with competition, we introduce the competition integral into the overall coefficient. It is calculated in the previous step. It must be said that such an organization of counting is quite natural for the evolution of a forest: first changes occur, and then they give an effect in the form of competition.

It can be seen that in the case of competition we reach a stationary regime: a young forest turns into a mature one (Fig. 11).

Fig. 11. Evolution of the age spectrum

mono-age population (with competition)

Conclusion

In this paper, we presented three models of forest evolution taking into account age, which were built based on different methods. First (§ 2) is a discrete model based on Leslie matrices, taking into account the distribution of the population across finite age groups. The other two models are continuous, and the first of them (§ 3) refers to ordinary differential equations, and the last one (§ 4) - to partial differential equations.

Bibliography

Svirezhev Yu.M., Logofet D.O. Stability of biological communities. M., Nauka, 1978.

Fedorov V.D., Gilmanov T.G. Ecology. M., Ed. Moscow State University, 1980.

Williamson M. Analysis of biological populations. M.: Mir, 1975.

Volterra V. Mathematical theory of the struggle for existence. M.: Nauka, 1976.

V.I. Kuznetsov “Mathematical model of forest evolution”, dissertation for the degree of candidate of physical and mathematical sciences, M, 1998

Kozlov N.I., Kuznetsov V.I., Kiryanov D.V., Kiryanova E.N. Dynamic models of mid-latitude forest development. Preprint IAM RAS M., 2005.

Leslie P.H. On the use of matrices in certain population mathematics. Biometrica, v.33(1945), N3, p.183

Godunov S.K., Ryabenkiy V.S. Difference schemes."Science", M. 1973.

Bellman R., Cook K.L. Differential-difference equations. "Mir", M., 1967.

Godunov S.K. Ordinary differential equations with constant coefficients. Volume 1. Ed. NSU, ​​1994.

Kalitkin N.N. Numerical methods. "Mir", M., 1978.

UDK577.4:517.9

MODIFICATION OF THE HETEROGENEOUS LESLIE MODEL FOR THE CASE OF NEGATIVE FERTILITY RATES

BALAKIREVA A.G.

that at each fixed point in time (for example, t0) the population can be characterized using a column vector

A heterogeneous Leslie model with negative fertility coefficients is analyzed. The age dynamics of the teaching staff within a particular university is studied and predicted based on this model.

1. Introduction

where xi(tj) is the number of the i-th age group at time tj, i = 1,...,n.

Vector X(ti), characterizing the population at the next point in time, for example, in a year, is connected with vector X(to) through the transition matrix L:

Forecasting and calculating population size taking into account its age distribution is an urgent and difficult task. One of its modifications is to predict the age structure of a homogeneous professional group within a specific enterprise or industry as a whole. Let us consider an approach to solving this class of problems using a structural model of age distribution. The formalism of this approach is based on the Leslie model, well known in population dynamics.

The purpose of this work is to show the possibility of using the heterogeneous Leslie model in the case of negative birth rates to predict the development of population dynamics.

2. Construction of a model of population dynamics taking into account age composition (Leslie model)

To build the Leslie model, it is necessary to divide the population into a finite number of age classes (for example, n age classes) of single duration, and the number of all classes is regulated in discrete time with a uniform step (for example, 1 year).

Under the above assumptions and the condition that food resources are not limited, we can conclude that 40

Thus, knowing the structure of the matrix L and the initial state of the population (column vector X(t0)), we can predict the state of the population at any given point in time:

X(t2) = L X(ti) = LL X(t0) = L* 2 X(t0),

X(tn) = LX(tn-i) =... = LnX(t0). (1)

The Leslie matrix L has the following form:

^ai a2 . .. a n-1 a > u-n

0 Р 2 . .. 0 0 , (2)

v 0 0 . .. Р n-1 0 V

where a i are age-specific birth rates, characterizing the number of individuals born from the corresponding groups; Pi - survival rates equal to the probability of transition from age group i to i +1 group by the next point in time (at-

than ^Pi can be greater than 1). i=1

RI, 2011, No. 1

The matrix L defines a linear operator in n-dimensional Euclidean space, which we will also call the Leslie operator. Since the quantities x;(t) have the meaning of numbers, they are non-negative, and we will be interested in the action of the Leslie operator in the positive octant of Pn n -dimensional space. Since all elements of the matrix are non-negative (in this case the matrix itself is called non-negative), it is clear that any positive octant vector is not taken beyond its limits by the Leslie operator, i.e. the trajectory X(t j) (j = 1,2,...) remains in Pn. All further properties of the Leslie model follow from the non-negativity of the matrix L and its special structure.

The asymptotic behavior of solutions to equation (1) is significantly related to the spectral properties of the matrix L, the main of which are established by the well-known Perron-Frobenius theorem.

Definition. A heterogeneous Leslie model is a model of the form

X(tj+i) = L(j)X(to), L(j) = Li L2 ... Lj, j = 1,2,...,

where Lj is the Leslie matrix of the jth step.

The dynamics of the inhomogeneous model have been studied very poorly (while being largely similar to the dynamics of model (1), it also has some differences). At the same time, this model is undoubtedly more realistic.

3. Spectral properties of the Leslie operator

Following the work, we will consider the concept of the imprimitivity index of the Leslie matrix.

An indecomposable matrix L with nonnegative elements is called primitive if it carries exactly one characteristic number with a maximum modulus. If a matrix has h > 1 characteristic numbers with a maximum modulus, then it is called imprimitive. The number h is called the imprimitivity index of the matrix L. It can be shown that the imprimitivity index of the Leslie matrix is ​​equal to the largest common divisor numbers of those age groups in which the birth rate is different from zero. In particular, for the primitivity of the Leslie matrix

it is enough that a 1 > 0, or that the birth rate takes place in any two consecutive groups, i.e. there existed a j such that a j Ф 0 and

Considering the above, we can note some properties of the Leslie matrix.

1. The characteristic polynomial of the matrix L is equal to

An(P) = l1^-L = pn -“gr.n 1

Easy sprt,

which is easily proven by the method of mathematical induction.

2. The characteristic equation A n(p) = 0 has a unique positive root р1 such that

where p is any other eigenvalue of the matrix L. The number p1 corresponds to a positive eigenvector X1 of the matrix L.

Statement 2 of the property follows directly from the theorem on non-negative matrices and Descartes' theorem.

3. The equal sign in (3) occurs in the exceptional case when only one of the fertility rates is different from zero:

and k > 0, and j = 0 for j = 1,2,...,k - 1,k + 1,...,n.

4. The value p1 determines the asymptotic behavior of the population. The population size increases indefinitely when I1 >1 and asymptotically tends to zero when I1< 1. При И1 =1 имеет место соотношение

X1 = [-I-----,-I------,...,-^,1]"

Р1Р2 -Pn-1 P2---Pn-1 Pn-1

The positive eigenvector of the matrix L, determined up to a factor.

An indicator of property 4 for an indecomposable Leslie matrix of the form (4) is the quantity

R = а1 + £а iP1...Pi-1, i=2

which can be interpreted as the reproductive potential of the population (generalized parameter of the reproduction rate), i.e. if R > 1, then p1 > 1 (the population grows exponentially), if R< 1, то И1 < 1 (экспоненциально убывает), если R = 1, то И1 = 1 (стремится к предельному распределению).

4. Modification of the Leslie model for the case of negative fertility rates

The works considered only the Leslie model with non-negative coefficients. The rationale for this choice, in addition to the obvious mathematical advantages, was that both survival probabilities and fertility rates cannot be inherently negative. However, already in the earliest works on population reproduction models, the relevance of developing models with, generally speaking, non-positive coefficients of the first row of the Leslie matrix was noted. In particular, models of reproduction of biological populations with “anti-reproductive” behavior of non-reproductive individuals have negative coefficients.

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which age groups (destruction of eggs and young individuals, etc.). Competition for resources between newborns and representatives of other age groups can also lead to this. In this regard, the relevant question is whether the property of ergodicity, which is true for Leslie models with non-negative coefficients, is preserved in a wider class of models for the reproduction of demographic potential.

The following theorem answers this question.

Theorem (On the circle of instability of the demographic potential reproduction model).

Let the age structure of demographic potential and the number of people living be given. Then there is a circle l = (p: |p|< рmin }, такой, что режим воспроизводства с указанными выше показателями обладает свойством эргодичности тогда и только тогда, когда истинный коэффициент воспроизводства не принадлежит этому кругу.

We will call this circle the circle of instability, and its radius the radius of instability.

Remark 1. An important conclusion follows from the theorem - whatever the structure of the demographic potential, at certain values ​​of the true reproduction rate the property of ergodicity will be observed. In particular, models with negative elements in the first row of the reproduction matrix and even negative values ​​of demographic potentials can have the ergodicity property.

Remark 2. It follows from the theorem that if for a certain value of the true reproduction coefficient a model has the property of ergodicity, then it also has this property for all reproduction coefficients that are large in magnitude.

5. Study of the age dynamics of the teaching staff of the university. Numerical experiment

Let's consider the forecast of the dynamics of the number and age distribution of the teaching staff according to data from one of the universities in Kharkov. The standard, so-called “compressed” age structure of the teaching staff is formed by statistics in the form of 5 age categories. The table shows the number N of each age category by year and the percentage that this age category constitutes in relation to the total number.

Let us compose transition matrices L j such that

X(tj+i) = LjX(tj) (Lj (5 x 5)). (4)

To do this, it is necessary to determine the birth rate and survival rates in a matrix of the form (2). Survival rates can be obtained by

directly solving equation (4) using data from the table.

Structure of the teaching staff

1 <40 322 38 242 38 236 36 273 40

2 40;49 117 14 88 14 95 15 90 14

3 50;59 234 27 163 26 160 25 156 24

4 60:65 88 10 68 11 79 12 69 11

5 65> 93 11 68 11 79 12 69 11

Total 854 629 649 657

As for fertility rates, additional assumptions need to be made. Let the number of teaching staff increase by ten people every year. Since fertility rates are a; interpreted as the average fertility of individuals of the i-th age group, it can be assumed that a1, a 5 = 0, and a 2 = 7, and 3 = 3. Based on the initial data, we find that a 4 are negative. This condition is interpreted as the departure of some members of the teaching staff from the university. From the above it follows that the matrices L j have the form:

0 0 in 3 0 0 . (5)

We will only consider reproductive classes. To do this, you need to change the form of the reduced matrix (let's get rid of the last zero column). And we calculate post-reproductive classes as shown in paragraph 2.

Thus, taking into account the above and the initial data, we obtain two matrices:

Matrix Li of the form (5) with coefficients а4 = 15, Р1 = 0.27, р2 = 1.39, р3 = 0.29;

Matrix L2 of type (5) with coefficients а 4 = 11, Р1 = 0.381, р2 = 1.64, р3 = 0.43.

Matrices L1 and L2 correspond to the transitions of 2005-2006 and 2007-2008, respectively. For the initial age distribution we take the vector X(t0) = T.

These matrices have reproduction coefficients p1, which do not fall within the circle of instabilization. It follows that a population with a given reproduction regime has the property of ergodicity.

Applying the heterogeneous Leslie model with a given initial distribution, we find that, starting from n=30 for the total number, the condition is satisfied

RI, 2011, No. 1

stabilization of the following form: X(tj+1) = ^1X(tj), j = 20,..., where q = 1.64 is the largest eigenvalue of the matrix L 2.

After stabilization, the percentage ratio of age categories is as follows: first category - 39%, second - 14%, third - 22%, fourth - 12%, fifth -13%.

Since the largest eigenvalue is greater than one, our model is open. In this regard, we will not consider total number teaching staff, and the ratio of this number to the degree of the greatest

eigenvalue of matrix L2:

L(j)X(t0)/cc, where j = 1,2,....

The figure shows the dynamics of the age structure of the teaching staff until 2015.

Percent

2004 2005 2007 2008 2013 2015

Changes in shares of age categories over time

In this figure, a scale of 10 to 40 was chosen because the percentage of age categories is in this range.

The forecast model data generally maintains a general trend towards an increase in the proportion of employees over 50 years of age, which indicates that the trend towards “aging” of the age composition of the university continues. It was determined that it was necessary to increase the first two age categories by at least 23% with a corresponding decrease in the remaining age categories to reverse this trend.

The scientific novelty lies in the fact that for the first time the heterogeneous Leslie model was considered in the case of negative fertility rates. This allows the model to take into account not only the birth rate, but also the death rate of individuals in the pregenerative period, which makes the model more realistic. The presence of negative coefficients fundamentally changes the methodology for studying the dynamics of the Leslie model by considering the corresponding region of localization of the main eigenvalue (the circle of instability).

Practical significance: this model makes it possible to predict changes in population size and its age structure, taking into account both fertility and mortality in each age group. In particular, using real statistical data covering several universities in the city of Kharkov, a forecast was made of the dynamics of age-related changes in the teaching staff. Forecast data correlates quite well with real data.

Literature: 1. Leslie P.H. On the use of matrices in certain population mathematics // Biometrica. 1945.V.33, N3. P.183212. 2. Zuber I.E., Kolker Yu.I., Poluektov R.A. Controlling the size and age composition of populations // Problems of cybernetics. Issue 25. P.129-138. 3. Riznichenko G.Yu., Rubin A.B. Mathematical models biological production processes. M.: Publishing house. Moscow State University, 1993. 301 p. 4. Svirezhev Yu.M., Logofet D.O. Stability of biological communities. M.: Nauka, 1978.352 p. 5. Gantmakher F. P. Theory of matrices. M.: Nauka, 1967.548 p. 6. Logofet D.O, Belova I.N. Non-negative matrices as a tool for modeling population dynamics: classical models and modern generalizations // Fundamental and Applied Mathematics. 2007.T. 13. Vol. 4. P.145-164. 7. Kurosh A. G. Course of higher algebra. M.: Nauka, 1965. 433 p.

Style Icon: Leslie Winer

TEXT: Alla Anatsko

Model, poet and singer, Leslie Winer became disillusioned with fashion because she was judged by her appearance. But fashion is once again fascinated by Winer. And that's why.

The first androgynous model in the world, a friend of Basquiat and Burroughs, the face of Valentino and Miss Dior, a brawler of remarkable intelligence, a poet and musician, without whom Massive Attack and Portishead would not have existed - all this is Leslie Winer, an intellectual and an outsider of her own free will, who, perhaps , invented trip-hop. Why, after several decades, does the fashion industry not forget about Leslie?

First androgynous model

New York, 1979. The phrase OK, Leslie, time to work your magic performed by Vincent Gallo, with whom model and cult musician Leslie Winer will record the track I Sat Back, is more than thirty years old. Young Winer moves to the main metropolis of the world from Massachusetts - to enter the School fine arts to a course by concept art pioneer Joseph Kosuth. To pay for housing and study materials, Leslie helps her neighbor write porn novels, and later becomes William Burroughs' assistant and protégé. Very quickly she concludes a contract with Elite Model Management - her first composition contains five photographs. They are a completely conventional girl: so far there is no hint of the trademark prickly look and androgyny.

Already in 1980, Leslie cut off her hair - shots taken by Paolo Roversi and Peter Lindbergh appeared in her portfolio. Thus begins the career of “the world’s first androgynous model,” as Jean-Paul Gaultier dubbed her. Leslie behaves badly and has fun at parties, has a short affair with Jean-Michel Basquiat, but works well - she is photographed by Helmut Newton and Irving Penn, she is put on the covers of Italian and French Vogue, the great The Face and Mademoiselle magazine, popular in those years. She acquires a special, well-practiced angle by which she is recognized, a sideways glance and a predatory male squint, which will later become almost a cliché popular culture- Hilary Swank will repeat them in the film “Boys Don’t Cry” and vulgarize Ruby Rose.

Vogue US, October 1981

Vogue US, November 1982

Vogue US, July 1982

Now Leslie is called a supermodel of the 80s, although Winer herself venomously jokes: “What kind of crap is this? Back then, even such a concept did not exist. I did a lot of things, and I was an alcoholic, I used tampons - much longer than I worked as a model, and with much more enthusiasm.”

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Fashion disappointment and the Witch album

WITCH album cover

Vogue Italia, September 1989

Leslie actively filmed and traveled around the world, but she also successfully scandalized in clubs - access to the most fashionable establishments from Paris to Tokyo was forever closed to her. In the mid-1980s, she found herself in London, where she shared accommodation with representatives of the local underground, and began hanging out at the Leigh Bowery club Taboo. At some point, Winer became accustomed to her new glossy image - a man's shirt, disheveled hair, a cigarette in her teeth and a middle finger at the lenses; but the understanding that she was wasting her life and not using her literary talent to the fullest did not allow her to come to terms with a career as a model or muse. To stay in London, Leslie quickly marries former bassist Adam and the Ants - for the sake of documents; The wedding witnesses are her neighbors and Bowery friends: director John Maybury and artist Trojan, who dies of an overdose a few months after the wedding. This death indirectly makes Winer a singer: Max, her husband's new band, decides to record a tribute to the artist, and Leslie, who previously only wrote lyrics, tries herself as a vocalist. Her debut track was called 337.5537's Little Ghost, where the dialing code actually turned out to be a tag invented by Basquiat, and represented Winer's name spelled out in numbers - LESSLEE.

Later, Winer and her husband would come up with a track for Sinead O'Connor, but Leslie herself would remain dissatisfied - she did not like the way the Max group recorded music, she did not feel any energy in her colleagues. Fortunately, a role model appeared in her life: the legendary producer Trevor Horn - his style of work forced Leslie to gain strength and release the first track Kind of Easy, pirated copies of which suddenly became popular in narrow circles. The next step was the full-length album Witch, which Leslie recorded under the graphic pseudonym, a copyright symbol, three years before the public was faced with the phenomenon called “the singer formerly known as Prince.” But ironically, the recording was released only three years later - in 1993.

Vogue UK, May 1990

Leslie Winer and illustrator Tony Viramontes

The album became precisely the embodiment of Leslie Winer’s special magic: she detachedly, as if completely without reflection, pronounces her lyrics, in which sharp political and social problems they sound so ordinary and creepy that it’s impossible to tear yourself away - and all this with deep bass. At that time, Winer turned out to be almost the most politicized performer, but she remained in the underground - she didn’t particularly strive for the charts, but, without wanting to, she came up with trip-hop. Winer's work and techniques increasingly appear in tracks by Massive Attack, Tricky and Portishead, although some critics consider MNE magazine's opinion that Winer is the “grandmother of trip-hop” to be somewhat controversial: by the time the album was released, the same Massive Attack were already active, and the thick bass became the basis for almost every second musical experiment of the early 1990s. On the other hand, when the famous Bristol sound was just taking shape, something common was in the air, not only the manner of performance, but also the mood and, most importantly, the characteristic dystopian lyrics - and Leslie caught it before anyone else.

Twiggy- real name Lesley Hornby. The 60s - the era of youth revolts - when many young people did not want to adapt, obey, or abandon themselves, they wanted to live in pleasure. They rebelled against the authority of their parents, church and state, and began to search for new values. Such conflicts between generations have always occurred. What was unusual was that young people not only protested, but also created new values, a new culture.

Of course, in this era a new one had to arise. At that time, types and Brigitte Bardot remained popular. But the embodiment of the new ideal was the model Twiggy - a sixteen-year-old Englishwoman weighing only 45 kilograms and height 169 cm. She was born in the suburbs of London, at the age of 16 Twiggy met the hairdresser Leonardo and became the face of his beauty salon. Twiggy's first photo shoot as a short-haired model was done by Barry Lategan. It was he who came up with a memorable pseudonym for Leslie Hornby - Twiggy.

One of the journalists from a London newspaper saw a photograph of Twiggy in a salon window and published her portrait in the newspaper under the title “Face of 1966.” In the same year, Twiggy became the most popular model in the world.

After working as a model for only three years, she became so rich that at the age of 19 she was able to retire. Twiggy - translated as a thin twig - was the first model to become an idol of millions. When she went out in public, crowds gathered around her.

Twiggy model For many years in a row she remained the undisputed queen of fashion models. She was the first fashion model who started the process that made models an integral part of pop culture along with musicians and actors.

Twiggy best reflected the image, breathing youth and purity.

Let x i(k) , where is the number of individuals in the population in i th age group at discrete points in time k. The processes of reproduction, death and transition of individuals from one age group to another can be formalized as follows (Rosenberg, 1984). First, let us establish how the state of the population at the moment of time k+ 1 depends on the state at the moment of time k. The number of the first group ( k= 1) represents the number of newborn descendants of all other groups during a single time interval; It is believed that individuals of a certain age group produce offspring in direct proportion to the number of individuals in this group:

Where f i– birth rate i th age group. If we denote by dj<1 коэффициент выживаемости при переходе от возрастной группы j to the group j+ 1, then we can write n– 1 type ratio:

Then, combining and , we can write the system n difference equations representing a discrete model of the age composition of the population. In matrix form we have:

x(k + 1) = Lx(k),

Where x(k) = {x i(k)) is the vector of numbers of individual age groups, and

– matrix of fertility and survival rates

If we describe it in more detail, we get:

The leftmost column vector reflects the number of individuals of different age groups at a point in time k+1, and the rightmost column vector is the number of individuals of different age groups at a point in time k. The matrix of fertility and survival rates is a matrix of transition from one state to another.

To calculate the age composition of the population at any time, we use simple relationships:

x(k + 1) = Lx(k)

x(k + 2) = Lx(k+1) =LLx(k) = L 2 x(k)

x(k+m) = L m x(k)

This model is known as Leslie's model (Leslie, 1945).

Square matrix L is non-negative (all its elements are non-negative). In order for the Leslie matrix to be indecomposable (i.e., it could not be reduced to the form by any permutation of rows and corresponding columns):

Where A And B are square submatrices), it is necessary and sufficient that . Biologically, this condition means that as n It is not the maximum possible, but the greatest reproductive age of individuals.

The characteristic equation of the system has the following form:

Where E– a matrix with ones on the main diagonal, and all its other terms are equal to zero.

Since the Leslie matrix is ​​non-negative and indecomposable, then, in accordance with the Perron-Frobenius theorem, the characteristic equation has a real positive characteristic number (the maximum among all other characteristic numbers), which is the simple root of this equation. In addition, since , the equation has no zero roots. From these conditions it follows that the asymptotic solution of the system for sufficiently large k will be determined by the eigenvalue λ 1 (the maximum of all) and the corresponding eigenvector b 1 Leslie matrix:

Where With 1 – some constant depending on the coordinates of the initial distribution of the vector x(0).

If λ 1 >1, then the population grows ( x(k) increases with growth k). If λ 1<1, то популяция гибнет. Наконец, если λ 1 =1, то общая численность популяции асимптотически стремиться к постоянной величине. P(1)<0 эквивалентно выражению λ 1 >1, i.e. condition of population growth (see formula 5), ​​similar to P(1)>0 corresponds to death, and P(1) = 0 – stationary population size. Thus, from the form of the matrix without determining the eigenvalue λ 1, one can draw qualitative conclusions about the nature of the simulated population over time.

The disadvantage of the Leslie model is similar to the disadvantage of the Malthus model - it is unlimited population growth with λ 1 >1, which corresponds only to the initial phases of growth of some populations (Rosenberg, 1984).

Leslie's model was used to describe the age structure of the Schell's sheep coenopopulation ( Helictotrichon schellinum). This is a loose-bush small-turf grass of the northern meadow steppes. A.N. Cheburaeva (1977) studied the distribution of the number of individuals of this cereal by age groups in the Poperechenskaya steppe of the Penza region on a watershed plateau on a total area of ​​50 m2 in different years (1970-1974). Every year, counts of sheep individuals were carried out on 200 plots of 0.5×0.5 m. Such a large repetition of observations allows us to consider the obtained estimates of the number of individuals in each age group to be quite stable. The researcher identified nine age groups:

· sprouts And shoots

· pregenerative individuals ( juvenile, immature And young vegetative)

· generative individuals ( young, mature And old)

· postgenerative individuals ( subsenile And senile)

To take into account the influence of weather conditions on the dynamics of the Shell sheep cenopopulation (1972 was the year of drought), it is necessary to move from absolute numbers to relative ones. At equal intervals for each age group the following ratio must be satisfied: x i + 1 (k + 1) < x i( k), i.e. at a subsequent point in time there should not be more individuals in the older age group than there were at the current point in time in the younger group. In this regard, the first seven age classes of A.N. Cheburaeva were united. The initial data for building the model are given in table. 1.

Table 1

Absolute and relative numbers of the Shell sheep cenopopulation for different age groups (according to A.N. Cheburaeva, 1977)

Despite the modification, the 1972 data is still different, so Leslie's model should not be expected to accurately predict abundance. To obtain a more accurate forecast, the Leslie matrix coefficients must be made dependent on weather conditions.

To construct the matrix L We use some ideas about the possible values ​​of its coefficients. Thus, birth rates f i during the transition from the first group, which includes all generative states, to older plants, they should decrease. Survival rates d i taken approximately equal (half of the individuals move from the first group to the second, slightly less from the second to the next). Finally, the Leslie matrix looks like this:

Characteristic equation for Leslie's model in in this case is a third degree polynomial:

It is easy to verify that P(1) = 0.23>0 according to P. Leslie’s theory indicates the aging and withering of a given coenopopulation in the observed time interval.

Let's calculate the roots of the characteristic equation. For this we will use Cardano formula. Consider an algorithm for solving a cubic equation of the form:

Let's make a replacement:

We get the equation:

Suppose that the value of the root is represented as the sum of two quantities y = α + β, then the equation will take the form:

Let us equate the expression (3 αβ + p), then we can move from the equation to the system:

which is equivalent to the system:

We have obtained Vieta's formulas for two roots quadratic equation (α 3 – first root; β 3 – second root). From here:

– discriminant of the equation.

If D>0, then the equation has three different real roots.

If D = 0, then at least two roots coincide: either the equation has a double real root and another, different real root, or all three roots coincide, forming a root of multiple three.

If D<0, то уравнение имеет один вещественный и пару комплексно-сопряженных корней.

Thus, the roots of the cubic equation in canonical form are:

Where i= is an imaginary number.

You need to apply this formula for each value of the cube root (the cube root always gives three values!) and take the value of the root so that the condition is met:

The following relationships can be used to check:

Where d≠ 0

Where d≠ 0

Finally:

In our case: a = 1; b = –0,6; c = –0,15; d = –0,02;

D= – 0,03888, D<0. Уравнение имеет один вещественный и пару комплексно-сопряженных корней.

Next, using the above formulas, we find the eigenvalues ​​of the characteristic equation: λ 1 = 0.814; λ 2 = – 0.107 + 0.112 i; λ 3 = – 0.107 – 0.112 i, Where i= is an imaginary number. Thus, the characteristic equation has one real and two complex roots. λ 1 is the maximum root of this equation, and since λ 1<1, то вывод об увядании данной ценопопуляции остается без изменения.

In addition, according to Yu.M. Svirzhev and D.O. Logofet (1978), a simple and sufficient condition for the existence of periodic fluctuations in the total number are the expressions:

In this regard, one should expect the existence of periodic fluctuations in the size of the Shell sheep cenopopulation, since λ 1 >max (0.5; 0.4).

Within the framework of Leslie's model, the observed A.N. Cheburaeva phenomenon is the aging of the sheep coenopopulation and the presence of fluctuations in the distribution of individuals along the age spectrum over a number of years. In Fig. Figure 1 shows the dynamics of the number of individuals for each of the identified age groups. In order for the model to give a satisfactory forecast, it is necessary that the matrix coefficients L were not constant, but dependent on weather conditions. If we supplement the Leslie model with normalization conditions for the resulting vector x(k+1) so that the sum of the size of the entire population equals the observed total size at the time k+1, then the influence of weather conditions is indirectly taken into account. The model in this case will look like this:

x(k+1) = Lx(k), ,

Where X(k+1) – total population size at a time k+1 (other notations are similar to Leslie’s model). Thus, knowing the total number of individuals of a given cenopopulation in different years, constructing the Leslie matrix from general biological considerations and taking as x(1) the distribution of sheep individuals by age groups in 1970, it is possible to plausibly restore the distribution of individuals by age groups in other years.

Calculation of the absolute size of the cenopopulation Helictotrichon schellinum for different age groups in different years is carried out as follows. We take the original data for 1970 and substitute them into the matrix. We perform matrix multiplication according to the appropriate rules. We obtain a new matrix with the numbers of different age groups for 1971.

We repeat this every time for every year. We enter the results into a table, calculate the total number of individuals using the Leslie model, and compare it with empirical data. Next, we introduce a correction factor and bring the calculations according to the model into line with the total number (Table 2).

table 2

Absolute size of the Shell sheep cenopopulation for different age groups according to Leslie’s model and empirical data

Age group
empirical data	model Leslie	empirical data	model Leslie		empirical data	model Leslie	Leslie model adjusted for total population	empirical data	model Leslie	Leslie model adjusted for total population	empirical data	model Leslie	Leslie model adjusted for total population
Seedlings, pregenerative and generative individuals				280,1	160,9		231,9	31,5		188,9	158,1		153,7	75,1
Subsenile individuals				193,0	110,9		140,1	19,0		116,0	97,1		94,5	46,2
Senile individuals				59,6	34,2		77,2	10,5		56,0	46,9		46,4	22,7
Total number				532,7			449,2			360,9			294,6