1. Introduction
Currently, special attention is paid to important objects such as chains of interacting oscillators. Such chains arise when modeling many applied problems in radiophysics (see [
1,
2,
3]), laser optics (see [
4,
5,
6]), mechanics (see [
7,
8]), neural network theory (see [
9,
10,
11,
12,
13]), biophysics (see [
14]), mathematical ecology (see [
15,
16,
17,
18]), etc. This paper investigates the chains of coupled logistic equations with delay that are relevant for biophysics and mathematical ecology.
As a basic example describing certain population size changes, the well-known logistic equation with delay
is considered. Here,
is the normalized population or the population density,
is the Malthusian parameter,
is the delay associated with the reproductive age of individuals.
Let us recall some well-known facts (see, for example, [
19]). In (
1), the equilibrium state
is asymptotically stable as
, but this equilibrium state is unstable and there is a stable cycle
as
. Under the condition
its asymptotic behavior is:
This cycle is relaxational as
. Its asymptotic behavior is presented in [
20].
We examine two types of chains of the coupled logistic equations with delay.
Type 1: The chains of the
coupled equations
Here,
,
, and
. Certain constraints may be imposed on the coupling coefficients
. For example, the biologically derived conditions
must be satisfied for
. Therefore, the solutions must keep the quadrant
invariant for nonnegative initial conditions
. Hence,
System (
2) has the equilibrium state
. The substitution
leads to the system of equations
Type 2: Here, the site of couplings is the only difference between the chains of coupled equations and (
2):
Substitution (
4) converts this system to the system of equations
The last term of System (
7) distinguishes it from System (
5).
It is convenient to associate the element with the value of the two-variable function . Here, is the point of some circle with the angular coordinate . Further, the periodicity condition and () holds.
We assume the coupling between elements to be homogeneous, i.e.,
The basic assumption is that the quantity
of elements is large enough:
The above condition gives ground for moving from the discrete spatial variable x to the continuous variable x.
In this paper, the three most important cases in terms of applications will be considered. In the first case, it is assumed that the system is fully connected and all coupling coefficients are the same: the equalities
hold for some coefficient
. The expression
is the partial Darboux sum for the expression
. Thus, under condition (
10), the boundary value problem
is the asymptotic approximation for Systems (
5) and (
7) as
. Here,
in the case of (
5), and
in the case of (
7).
For Equation (
11), the periodic boundary conditions
hold. Note that in the case when coefficients
are not identical, the integral term becomes more complicated. We obtain the expression
instead of the expression
, and
.
The local (i.e., in the zero equilibrium state neighborhood) dynamics of the boundary value problem (
11) and (
12) are studied in
Section 2.
Then, we consider the case of so-called diffusional coupling, where
Now, from systems (
5) and (
7), we arrive at the boundary value problem
If the coefficients
do not differ much from zero, the dynamic behavior of (
14) could change significantly. Thus, we examine a more general (compared to (
14)) boundary value problem
Undoubtedly, the integral expression in (
15) can be presented in the form of integrals from 0 to
of some function
. However, the form of (
15) is preferable. Firstly, the asymptotic equality
holds for the fixed function
as
. Secondly, it is technically convenient to calculate the importance of further usage of integrals explicitly. For example,
In
Section 3, the boundary value problem (
15) is examined.
According to the equalities in (
13), interactions with one ‘neighbor’ on the right and one ‘neighbor’ on the left make a dominating contribution to the couplings between elements as
. In the next, fourth, section, the case of coupling coefficient interactions for unidirectional couplings is studied. For
as
, the resulting boundary value problem assumes the form
The paper is devoted to the investigation of the behavior of the solutions to the boundary value problems (
11), (
12); Equations (
15) and (
17) with initial conditions from some sufficiently small metric
-independent zero equilibrium state neighborhood for small
. In each of the problems, critical cases are identified in the study of the stability of the zero solution. In critical cases, the characteristic equations of the linearized-at-zero problems have infinitely many roots, with real parts tending to zero as
. This is a distinctive feature of the problems under consideration. Thereby, we may speak of the infinite-dimensional critical cases implementation. The standard methods of integral manifolds (see, for example, [
21,
22,
23]) and normal forms (see, for example, [
24]) are not directly applicable for their study. Thus, we employ the methods developed by the author in [
25,
26,
27,
28]. Their essence lies in the construction of special first-approximation equations called quasinormal forms (QNFs), whose nonlocal dynamics determine the local structure of the initial boundary value problems’ solutions. These QNFs are nonlinear boundary value problems of neutral or parabolic types with either one or two spatial variables. The solutions of these QNFs make it possible to determine the principal terms of the asymptotic representations of the considered boundary value problems’ solutions.
Section 5 is a natural extension of
Section 4. It studies the model of a chain with a unidirectional coupling under the condition that the coupling coefficient between elements takes sufficiently large values. It is about the boundary value problem
where
. We note that (
18) is interpreted as a logistic delay equation with spatially distributed control. The function
describing spatial interactions is given by
We demonstrate that the dynamic properties of the problem under consideration change significantly depending on the various relations between , and h.
We present without proof two analogs of the classical Lyapunov stability theorems in the first approximation statements.
For the linearized-at-zero Equations (
11), (
15), (
17) and (
18) with periodic boundary conditions (
12), the characteristic equation has the form
Statement 1. Let the roots of Equation (20) have a negative real part and be separated from the imaginary axis as. Then, ancan be found such that, for, the solutions to the problem under consideration from the sufficiently small-independent zero equilibrium state neighborhood tend to zero as. Statement 2. Let Equation (20) have a root with a positive real part separated from zero as. Then, the zero equilibrium state of the boundary value problem is unstable for small, and there is no attractor of this boundary value problem in some sufficiently small-independent zero neighborhood. Thus, in the posed problem about local conditions in the zero equilibrium state neighborhood, the dynamics are trivial under the condition of Statement
Section 1, and it cannot be studied by local analysis methods under the condition of Statement
Section 1.
5. Dynamics of Logistic Delay Equation with Large Coefficient of Spatially Distributed Control
In this section, we study the logistic delay equation with large coefficient of spatially distributed control of the form
The dependence of the functions
on the spatial variable
x is assumed to be periodic:
Thus, we fix the space
as a phase space of the boundary value problem (
79), (
80).
Function
describing spatial interactions is defined by
We note that
. Apparently, study of the problem (
79) and (
80) is of great interest, provided that parameter
appearing in (
81) is sufficiently small:
This condition arises naturally in many applied problems (see, for example, [
31]). The assumption that the coefficient
is large enough, i.e.,
allows us to apply special asymptotic methods. In the next section, we study the local dynamics of solutions to the boundary value problem (
79) and (
80) under the conditions (
82) and (
83). The corresponding constructions are based on the results from [
28].
Section 5.2 considers the local dynamics (i.e., in a small equilibrium state neighborhood) of the problem (
79), (
80). Critical cases are distinguished in the stability problem. The distinctive feature of these critical cases is that they have infinite dimension. As a basic result, nonlinear boundary value problems of parabolic type are constructed that do not contain small and large parameters. Their nonlocal dynamics determine the behavior of the solutions of problem (
79), (
80) from small equilibrium state neighborhood
. The corresponding investigation is based on the papers [
26,
28]. We immediately pay attention to one of the conclusions obtained in
Section 5.1 and
Section 5.2. As it turns out, complicated bifurcation phenomena in the problem (
79), (
80) can appear even for sufficiently small values of the delay time
T. Here, we use the results obtained in the paper [
32].
5.1. Boundary Value Problem Reduction to Parabolic-Type Equation
This section is divided into three parts. The first two sections assume that the deviation
h is asymptotically small, i.e., function
is close to symmetric. In Part 1, the parameters
and
are assumed to be of the same order. It is shown that the nonlocal dynamics of specially constructed boundary value problems of parabolic type determine, in general, the dynamic properties of problem (
79), (
80). The corresponding structures are called slowly oscillating since they are formed mainly on low modes. The diffusive properties of the initial equation are assumed to be small in Part 2. In terms of problem coefficients, this means the qualified smallness of parameters
and
compared to
. The appropriate range of their change is indicated below. In this case, we demonstrate that the rapidly oscillating (i.e., formed on asymptotically large modes) regimes are distinctive for the boundary value problem (
79), (
80). In order to find them, special families of nonlinear parabolic boundary value problems are constructed. In Part 3, we study the dynamics of problem (
79), (
80) under the condition of
essential dissymmetry when parameter
h is not small.
Thus, we demonstrate that the boundary value problem (
79), (
80) dynamics essentially depend on the relations between parameters
, and
h.
5.1.1. Slowly Oscillating Structures
Here, we assume that the parameters
, and
are of the same order: for some fixed, positive
k and
, we obtain
After dividing (
79) by
, we consider the resulting ‘main’ part, which is the linear boundary value problem
The characteristic equation of (
85) is of the form
Hence, we obtain the asymptotic formulas
for the roots
. Thus, infinitely many roots of Equation (
86) tend to the imaginary axis as
. This gives us reason to regard the considered critical case in the stability problem to be infinite-dimensional. The methodology of studying such systems is developed in [
26,
28]. We use the appropriate results here. For this purpose, we introduce the formal series
where
. We insert (
88) into (
79) and equate the coefficients at the same power of
. We obtain the infinite system of ordinary differential equations to determine the function
. As it turns out, this system can be written as one complex parabolic equation of the Ginzburg–Landau type for the function
where
.
Theorem 9. Let, for some fixed , the boundary value problem (
89), (
90)
be bounded together with the time derivative solution as . Then, as determined from the equality , for the sufficiently small , the function satisfies the boundary value problem (
79), (
80)
up to . Theorem 10. On conditions of Theorem 9, let be a periodic solution of the problem (
89), (
90)
, and let only two of its multipliers be equal to Modulo 1. Then, for all sufficiently small , the boundary value problem (
79), (
80)
has a periodic solution of the same stability as , and 5.1.2. Rapidly Oscillating Structures
Here, we demonstrate that the decrease of diffusion coefficients (k and ) can lead to the appearance of rapid oscillation with respect to spatial and time variable families of structures. Conditionally, we can divide them into two types. Proportional decrease of coefficients k and , which play the role of diffusion, leads to the appearance of the first type. The structures are formed in the neighborhood of asymptotic large modes. However, the product of diffusion coefficients and the (asymptotically large) value of the corresponding modes is of an asymptotically small quantity. In other words, the squares of the corresponding modes coincide in order with the reciprocal of deviation of spatial variables. The second type of structures arises only when one diffusion coefficient, k, decreases. Here also, rapidly oscillating structures are formed due to the interaction of a large number of modes far apart from each other. However, the principal difference from structures of the first type is that the values of modes themselves coincide in order with the reciprocal of the deviation of the spatial variable but not the values of the squares of the modes. Let us consider these cases separately.
Case 1. Structures of the ‘first’ type.
Letfor some fixed positive and . We arbitrarily fix z as real, and let be the value from the semi-open interval that complements the expression to an integer. We consider the integer set For these numbers, the characteristic Equation (86) has the set of roots similar to (
87)
Then, the family of the boundary value problems depending on the parameter zplays a role in the boundary value problem (89), (90) in the considered case. Here, . Theorem 11. Let, for some fixed and , the boundary value problem (
92), (
93)
be bounded together with a derivative with respect to t solution as . Then, as determined from the equality , for the sufficiently small , the function satisfies the boundary value problem (
79), (
80)
up to . The ‘first’ type of structure can also be formed by the interaction of a larger number of modes. We use constructions from [
32] to demonstrate this.
We fix an arbitrarily natural number
and real numbers
. We consider the set of integer numbers
For modes with these numbers, the equation
plays the role of the boundary value problem (
92), (
93) with
-periodic boundary conditions with respect to each spatial variable, and
Similarly to Theorem 11, the corresponding statement about coupling with the solutions of the boundary value problem (
79), (
80) is formulated for the solutions of this boundary value problem.
Remark 1. The structures considered here that rapidly oscillate with respect to spatial variables arise when the coefficients k and in (
89)
are also asymptotically small. In this part, the case of
and
is considered. Of course, one can investigate a more general case, when for some positive
and
, the equalities
hold. For such
k and
h, the changes in the corresponding constructions are not fundamental, so we do not dwell on them.
Case 2. Structures of the ‘second’ type.
Here, the coefficient h is assumed to be the same as in (84). Further, We consider the integer set , where is such that the expression in brackets is an integer, . The characteristic Equation (86) has the set of roots for which We apply the above construction and obtain the resulting boundary value problemwhere We introduce some notation in order to formulate an analogue of Theorem 9 in the considered situation. We fix an arbitrary and the sequence of the roots of equation . Let be the arbitrary limit point of the sequence . Let for the sequence . Finally, denotes the arbitrary limit point of the sequence , and let and .
Theorem 12. Let, for some , and , the boundary value problem (
94), (
95)
be bounded together with a derivative with respect to t solution . Then, for , , , and , the function satisfies the boundary value problem (
79), (
80)
up to . 5.1.3. Case of Essentially Asymmetric Function
Here, we assume parameter
h to not be small but to be close to some number rationally commensurate with
. This means, that for some relatively prime integers
and
,
We assume that
, where
,
if
is even, or
if
is odd. Here, we repeat the constructions from
Section 5.1.1 and obtain, similar to (
89), (
90), the boundary value problem
The formula
establishes a relation between its solutions and the solutions of problem (
79), (
80). Accordingly, for (
91), we arrive at the boundary value problem
The solutions of this boundary value problem and of the boundary value problem (
79), (
80) are coupled by the equality
Satisfaction of only one condition
is a more interesting situation. Let
be a value which complements the expression
to an integer multiple of
M. We examine the set of integers
,
. Then, for the roots of Equation (
86) with numbers
m from
K and
, we obtain
Here, the boundary value problem
is the analogue of problem (
97). Definitely in the situations under consideration, there is also a connection between the solutions of the constructed boundary value problems and the asymptotic (with respect to residual) solutions of the boundary value problem (
79), (
80). As in
Section 5.1.1, boundary value problems similar to those given but with a large number of spatial variables appear when taking into account the larger number of modes. We do not dwell on this in more detail.
5.2. Large Coefficient -Induced Bifurcations
Upon condition (
83), we study the behavior of solutions of the boundary value problem (
79), (
80) in a sufficiently small equilibrium state
neighborhood. The characteristic quasipolynomial of a boundary value problem linearized on
has the form
As in
Section 5.1, we restrict ourselves to the most interesting and important situations depending on the relationship between the parameters
, and
. In
Section 5.2.1, we assume that condition (
84) holds:
,
. Below, we demonstrate that Andronov–Hopf bifurcation occurs in (
79), (
80) even for small values of the delay parameter
. The periodic solution bifurcating from the equilibrium state
turns out to be rapidly oscillating in time. In
Section 5.2.2, the relations
,
are assumed to be valid. In this case, the corresponding bifurcation process has an infinite dimension: infinitely many roots of the characteristic Equation (
98) tend to the imaginary axis as
. We construct quasinormal forms, i.e., the families of complex parabolic (and degenerate parabolic) boundary value problems whose nonlocal dynamics determine the behavior of the initial boundary value problem (
79), (
80) solutions in the small neighborhood of
for small
. In
Section 5.2.3, even more complicated families of quasinormal forms are constructed to determine the dynamic properties of problem (
79), (
80) under the constraint
. The conclusions are given in
Section 5.2.4.
It is natural to choose the delay time
T as the main bifurcation parameter. We recall that the bifurcation value for
T is determined from the equality
in Equation (
79). In the cases considered here, we demonstrate that the bifurcation parameter can be asymptotically small.
5.2.1. Bifurcation Analysis upon Condition ,
Then, we obtain
up to
. If it exists, let
,
be the least positive root of Equation (
99). Obviously, there is a finite number of such roots. Let
be the least of them. Let
. We formulate two statements about the roots of Equation (
98) that are simple but cumbersome proofs that are omitted.
Lemma 6. Let . Then, for all sufficiently small ε, the Equation (98) roots separate from zero negative real parts as . Lemma 7. Let . Then, for all sufficiently small ε, Equation (98) has a root separate from the zero positive real part as . Thus, in the context of the above lemmas, the issue of the boundary value problem (
79), (
80) local dynamics is solved trivially.
Let us study the behavior of the solutions of the problem (
79), (
80) in the
neighborhood close to the critical case condition
Then, the characteristic Equation (
98) has the coupling of the roots
of the form
The real parts of the remaining roots of Equation (
98) are negative and zero-separated as
. In this case, for small
, the boundary value problem (
79), (
80) has a two-dimensional stable local invariant integral manifold in the (small)
neighborhood, on which this boundary value problem can be represented in the form of the scalar complex equation up to the summands of the
order:
where
is a ‘slow’ time. The solution
on this manifold is coupled to solutions of Equation (
100) by the relation
Here, the functions are periodic with respect to first and third arguments, with periods and , respectively.
Regarding the dynamics of Equation (
100) (and hence the boundary value problem (
79), (
80) in the
neighborhood), one needs to find the coefficients
and
d. To accomplish this, we insert the formal series (
101) into (
79) and equate the coefficients at the same powers of
. At the second step, we obtain
and
From the solvability condition obtained at the third-step equation with respect to
, we obtain
We note that each of the quantities
and
can be either positive or negative, depending on the values of the parameters
,
,
r,
k, and
. As an example, we make one statement about the dynamics of problem (
79), (
80).
Theorem 13. Let and . Then, for all sufficiently small ε, the boundary value problem (
79), (
80)
has a stable periodic solution in the neighborhood, for which where Remark 2. The results presented here could be obtained by studying the bifurcations from the equilibrium state in the boundary value problem (
89), (
90).
5.2.2. High-Mode Bifurcations upon Condition
It is assumed here that the conditions
,
are satisfied. Under these conditions, we study the behavior of the solutions of the problem (
79), (
80) from a small
neighborhood. First, we consider the characteristic Equation (
98). For
, its roots have negative real parts (separated from zero as
). Let us demonstrate that there are roots with a close to zero real part as
, and the numbers (modes)
correspond to them. In (
98), we assume
where
is the value that complements the previous summand
to an integer. Then, we obtain the equality
(up to
). Let us determine the least value of
for which this equation is solvable with respect to
c, i.e., for some
, the relation
holds. We denote by
the least positive root of the equation
Then, the following simple statements hold.
Lemma 8. For and , the equalitieshold. Lemma 9. Let . Then, for all sufficiently small ε, the roots of Equation (
98)
have negative real parts (separated from zero as ). Lemma 10. Let . Then, for all sufficiently small ε, Equation (
98)
has a root with a positive real part (separated from zero as ). In the context of Lemmas 9 and 10, the behavior of the solutions of the problem (
79), (
80) in small (
-independent) neighborhood of
is determined in the standard way.
Below, we assume that a case close to critical in the stability problem of
is realized. Let, for some arbitrary constant
, the equality
hold. In this case, infinitely many roots of Equation (
98) tend to the imaginary axis as
, and there are no roots with a positive zero-separated as
real part. Thus, the critical case of an infinite dimension is realized in the
stability problem. Let us apply the technique of [
26,
28] to study the local dynamics of problem (
79), (
80) for small
.
First, we note that all modes that correspond to roots of Equation (
98) close to the imaginary axis are asymptotically large and have the leading asymptotic term
. In this regard, we consider all modes with numbers
for which
where
,
b is arbitrarily fixed, and
complements the value of
to an integer. The roots
of Equation (
98) with numbers
satisfy the asymptotic equalities
Here, the following designations are accepted:
where
Another notation is used below. Let be the set of all such values of b for which the values of are integers from to ∞.
Let us introduce the formal series
where
The dependence on the first, third, and fourth arguments of function
is periodic, with periods
,
, and
, respectively. We insert (
105) into (
79) and perform standard operations. At the second step, we obtain
and
At the third step, from the solvability condition of the resulting equation with respect to
, we obtain the equation for determining the unknown amplitude
where
We note that there are no boundary conditions for Equation (
106). The point is that the function with respect to argument
y contains an arbitrary set of harmonics. We present one of the variants of strict statements about relations between the solutions of (
106) and the solutions of the boundary value problem (
79), (
80).
Theorem 14. Let Equation (
106)
have an R-periodic with respect to argument y solution . Then, for , the function satisfies the boundary value problem (
79), (
80)
up to , where the expression complements the summand to an integer multiple of . 5.2.3. Reduction to Spatially Two-Dimensional Parabolic Equations
As noted above, the local dynamics of the problem (
79), (
80) essentially depend on the relations between the parameters
, and
. Two of the most important cases are analyzed in
Section 5.2.1 and
Section 5.2.2 with some model ‘relations’ between small parameters. Here, we dwell on another important scenario that is fundamentally different from the previous ones. We construct nonlinear parabolic equations with two spatial variables called the quasinormal forms to study local dynamics of the problem (
79), (
80) in
neighborhood.
We consider a model situation where
Let
be such asymptotically large modes for which
where
, the parameter
c is determined below, the values of
b are arbitrary, the value
complements the sum of two previous terms to an integer, and
complements the expression
to an integer. For the roots of the characteristic Equation (
98) with numbers (
108), the asymptotic formulas
hold. In order to determine the stability boundary of
for the problem (
79), (
80) in the range of parameter
, we assume
At the first step of the asymptotic analysis, from (
109), we arrive at the equality
The least value of
for which Equation (
110) is solvable for some
is determined from the relations (
103) as
, i.e.,
Here, the statements of Lemmas 9 and 10 also hold. Thus, we assume below that a case close to critical in the stability problem of
is realized. Let, for some constant
, the equality
hold. Similar to the previous part, infinitely many roots of Equation (
98) tend to the imaginary axis as
, and there are no roots with a positive zero-separated as
real part. However, there are ‘essentially’ more such roots in this case. Let us explain the above. For this purpose, we obtain expressions for
and
:
The expression (
111) differs from the similar formula for
in the previous part by the presence of parameter
n, which takes all integer values
. Hence, we obtain that the quantity
is also an
-periodic function with respect to
x in the basic formula of the form (
105) in the considered case:
. Thus, we repeat the Equation (
106) construction technique and obtain the more complicated equation
with
-periodic boundary conditions with respect to
x, and
differs from the value
d appearing in (
106) only by the presence of
in the appropriate formula.
For the dynamics of (
79), (
80), Equation (
112) plays the same role as Equation (
106) with the conditions from
Section 5.1. We do not dwell on this in more detail.
5.2.4. Case of ‘Intelligently’ Small Parameters h and
To complete the picture, we briefly consider the simplest situation when both parameters
h and
are ‘intelligently’ small:
The bifurcation value of the delay coefficient
satisfies the equality
. Let
in (
79), (
80). Then, infinitely many roots
of Equation (
98) tend to the imaginary axis as
, and there are no roots with a positive zero-separated as
real part. Therefore, the critical case of an infinite dimension is realized here, too.
Let us introduce the formal series
where
, and
are periodic with respect to first and third arguments with
and
periods, respectively. We insert (
114) into (
79) and perform standard operations. At the third step, we arrive at the boundary value problem for determining the unknown, slowly varying amplitude
:
where
The coupling between solutions of the problem (
115), (
116) and asymptotic with respect to residual solutions of the problem (
79), (
80) is determined by Formula (
114). We note that for a periodic solution of (
115), (
116) of the form
one can formulate a stronger result about the existence (and inheritance of the stability properties) of a periodic solution of the problem (
79), (
80), which is close to
as
.
6. Conclusions
It has been shown that the considered critical cases in the stability problem of the distributed chain of logistic equations with delay have an infinite dimension. This leads to the fact that description of their local dynamics is reduced to the study of the nonlocal behavior of boundary value problems of Ginzburg–Landau type solutions. It is known (see, for example, [
30]) that the dynamics of such objects can be complicated, and they are characterized by irregular oscillations, multistability phenomena, etc. The dynamic effects essentially depend on the choice of couplings. It has been shown that in a number of cases the solutions are rapidly and slowly oscillating with respect to spatial variable components. The basic results define the structure of problems that are asymptotic with respect to residual solutions to the initial boundary value. The problem of existence, stability, and more complicated asymptotic expansions of exact solutions close to those constructed can be solved, for example, for the case of periodic solutions of normalized equations.
We considered separately the role of the above parameter
. We recalled that the dynamic properties of the initial system are determined by the QNF (
49), (
50), which includes the parameter
. The dynamics of (
49), (
50) and hence of the boundary value problem (
9), (
10) may change for different values of this parameter. This is shown in detail in [
31]. This implies that an infinite process of forward and reverse bifurcations can occur as
.
Below, we formulate one more conclusion of the general plan. It was shown above that the quasinormal forms that determine the dynamics of the initial boundary value problem are equations of Ginzburg–Landau type. We note that parabolic boundary value problems with one and two spatial variables can act as quasinormal forms depending on the coefficient
of the function
((
12), (
13)). The stability of the simplest solutions of these equations is studied in [
33]. In particular, it has been established that their stability properties are determined to a large extent by the imaginary components of the diffusion coefficients and of the Lyapunov quantity (coefficients
g and
q in (
49) and (
50)). Numerical analysis of the corresponding criterion makes it possible to formulate the conclusion about the instability of all the simplest solutions of the form
. Thus, solution synchronization is a rather rare phenomenon in the considered chains.
It has been demonstrated that the study of the dynamics of logistic equations with delay is reduced to nonlinear dynamics analysis of special families of the parabolic and degenerately parabolic boundary value problems for large values of the coefficient of spatially distributed control. In particular, the phenomenon of hypermultistability is described.
In the study of local dynamics, bifurcation phenomena can be realized in the equilibrium state neighborhood even for asymptotically small delays. Here, the critical case has an infinite dimension in the stability problem. Analogues of the normal form, so-called quasinormal forms, are constructed in this situation, which are universal nonlinear boundary value problems of the parabolic type. Their nonlocal dynamics determine the local behavior of the solutions of the initial boundary value problem.