Quasinormal Forms for Chains of Coupled Logistic Equations with Delay

Abstract

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state.

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

$$\dot{u}=r[1-u(t-T)]u$$

(1)

is considered. Here, $u(t)>0$ is the normalized population or the population density, $r>0$ is the Malthusian parameter, $T>0$ 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 $u_0\equiv 1$ is asymptotically stable as $rT\ll \pi /2$, but this equilibrium state is unstable and there is a stable cycle $u_0(t)$ as $rT>\pi /2$. Under the condition

$$0<rT-\frac{\pi }{2}\ll 1$$

its asymptotic behavior is:

$$\begin{array}{cc}\hfill u_0(t)=1+& \sqrt{rT-\frac{\pi }{2}}[{\xi }_0(\tau (1+O(\epsilon )))exp\left(i\frac{2\pi }{T_0}t\right)+\hfill \\ \hfill & \overline{{\xi }_0}(\tau (1+O(\epsilon )))exp\left(-i\frac{2\pi }{T_0}t\right)]+\dots .\hfill \end{array}$$

This cycle is relaxational as $rT\gg 1$. 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 $2N+1$ coupled equations

$${\dot{v}}_j(t)=r[1-v_j(t-T)]v_j(t)+r\sum _{m=-N}^N{a}_{mj}\left(v_m(t-h)-v_j(t-h)\right).$$

(2)

Here, $j,m=0,\pm 1,\pm 2,\dots ,\pm N$, $a_{mm}=0$, and $h\ge 0$. Certain constraints may be imposed on the coupling coefficients $a_{mj}$. For example, the biologically derived conditions $u_j(t)\ge 0$ must be satisfied for $h=0$. Therefore, the solutions must keep the quadrant $u_j\ge 0(j=0,\pm 1,\pm 2,\dots ,\pm N)$ invariant for nonnegative initial conditions $u_j(s)\in C_{[-T,0]}(j=0,\pm 1,\pm 2,\dots ,\pm N)$. Hence,

$$a_{mj}\ge 0.$$

(3)

System (2) has the equilibrium state $u_{j0}=u_0=1(j=0,\pm 1,\pm 2,\dots ,\pm N)$. The substitution

$$v_j=1+u_j$$

(4)

leads to the system of equations

$${\dot{u}}_j(t)=-r{u}_j(t-T)+r\sum _{m=-N}^N{a}_{mj}\left(u_m(t-h)-u_j(t-h)\right)-r{u}_j(t-T)u_j.$$

(5)

Type 2: Here, the site of couplings is the only difference between the chains of coupled equations and (2):

$${\dot{v}}_j(t)=r\left[1-v_j(t-T)+\sum _{m=-N}^N{a}_{mj}\left(v_m(t-h)-v_j(t-h)\right)\right]v_j(t).$$

(6)

Substitution (4) converts this system to the system of equations

$$\begin{array}{cc}\hfill {\dot{u}}_j(t)=-& r{u}_j(t-T)+r\sum _{m=-N}^N{a}_{mj}\left(u_m(t-h)-u_j(t-h)\right)-\hfill \\ \hfill & r{u}_j(t-h)u_j+r{u}_j(t)\sum _{m=-N}^N\left(u_m(t-h)-u_j(t-h)\right).\hfill \end{array}$$

(7)

The last term of System (7) distinguishes it from System (5).

It is convenient to associate the element $u_j(t)$ with the value of the two-variable function $u(t,x_j)$. Here, $x_j$ is the point of some circle with the angular coordinate $x_j=2\pi {(2N+1)}^{-1}·j$. Further, the periodicity condition $u_{j+2N+1}(t)=u(t,x_j+2\pi )=u_j(t)=u(t,x_j)$ and $a_{m+2N+1,j}=a_{m,j+2N+1}=a_{mj}$ ($m,j=0,\pm 1,\pm 2,\dots$) holds.

We assume the coupling between elements to be homogeneous, i.e.,

$$a_{mj}=a_{m-j}.$$

(8)

The basic assumption is that the quantity $(2N+1)$ of elements is large enough:

$$0<\epsilon =2\pi {(2N+1)}^{-1}\ll 1.$$

(9)

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

$$a_{mj}=\gamma {(2N+1)}^{-1}{(2\pi )}^{-1}$$

(10)

hold for some coefficient $\gamma$. The expression ${\displaystyle \gamma {\left(2\pi (2N+1)\right)}^{-1}\sum _{j=-N}^{N}f(x_j)}$ is the partial Darboux sum for the expression ${\displaystyle \gamma {(2\pi )}^{-1}{\int }_0^{2\pi }f(x)dx}$. Thus, under condition (10), the boundary value problem

$$\begin{array}{cc}\hfill \frac{\partial u}{\partial t}=-& {\displaystyle ru(t-T,x)+r\gamma \left[{(2\pi )}^{-1}{\int }_0^{2\pi }u(t-h,s)ds-u(t-h,x)\right]-}\hfill \\ \hfill & {\displaystyle ru(t-T,x)u+r\delta \gamma u·\left[{(2\pi )}^{-1}{\int }_0^{2\pi }u(t-h,s)ds-u(t-h,x)\right]}\hfill \end{array}$$

(11)

is the asymptotic approximation for Systems (5) and (7) as $\epsilon \to 0$. Here, $\delta =0$ in the case of (5), and $\delta =1$ in the case of (7).

For Equation (11), the periodic boundary conditions

$$u(t,x+2\pi )\equiv u(t,x)$$

(12)

hold. Note that in the case when coefficients $a_{i-j}$ are not identical, the integral term becomes more complicated. We obtain the expression $\frac{1}{2\pi }{\int }_0^{2\pi }a(s)u(t-h,x+s)ds$ instead of the expression $\frac{1}{2\pi }{\int }_0^{2\pi }u(t-h,s)ds$, and $\frac{1}{2\pi }{\int }_0^{2\pi }a(s)ds=1$.

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

$$a_1=a_{-1}=\gamma \mathrm{and} a_k=0 \mathrm{for} k\ne \pm 1.$$

(13)

Now, from systems (5) and (7), we arrive at the boundary value problem

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial t}}=& -ru(t-T,x)+r\gamma \left[u(t-h,x+\epsilon )-2u(t-h,x)+u(t-h,x-\epsilon )\right]-\hfill \\ \hfill & ru(t-T,x)u(t,x)+r\delta \gamma [u(t-h,x+\epsilon )-\hfill \\ \hfill & 2u(t-h,x)+u(t-h,x-\epsilon )]u(t,x),u(t,x+2\pi )\equiv u(t,x).\hfill \end{array}$$

(14)

If the coefficients $a_k(k\ne \pm 1)$ 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

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial t}}=-& ru(t-T,x)-ru(t-T,x)u(t,x)+\hfill \\ \hfill & {\displaystyle r\gamma \left({\int }_{-\infty }^{+\infty }\left(F_{\epsilon }(s)-2F_0(s)+F_{-\epsilon }(s)\right)u(t-h,x+s)ds\right)\times }\hfill \\ \hfill & \left[1+\delta u(t,x)\right],u(t,x+2\pi )\equiv u(t,x).\hfill \end{array}$$

(15)

Here,

$$\begin{array}{ccc}\hfill F_{\pm \epsilon }(s)& =& {\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}exp\left[-{(2{\sigma }^2)}^{-1}{(s\pm \epsilon )}^2\right],\hfill \\ \hfill F_0(s)& =& {\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}exp\left[-{(2{\sigma }^2)}^{-1}{s}^2\right].\hfill \end{array}$$

Undoubtedly, the integral expression in (15) can be presented in the form of integrals from 0 to $2\pi$ of some function $\tilde{F}(s)$. However, the form of (15) is preferable. Firstly, the asymptotic equality

$${\displaystyle {\int }_{-\infty }^{+\infty }\left(F_{\epsilon }(s)-2F_0(s)+F_{-\epsilon }(s)\right)w(x+s)ds=w(x+\epsilon )-2w(x)+w(x-\epsilon )+o(1)}$$

holds for the fixed function $w(x)$ as $\delta \to 0$. Secondly, it is technically convenient to calculate the importance of further usage of integrals explicitly. For example,

$${\displaystyle {\int }_{-\infty }^{+\infty }{F}_{\epsilon }(s)exp(iks)ds=exp\left(ik\epsilon -\frac{1}{2}{\delta }^2{(\epsilon k)}^2\right).}$$

(16)

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 $\epsilon \to 0$. In the next, fourth, section, the case of coupling coefficient interactions for unidirectional couplings is studied. For $a_1=\gamma ,a_j=0$ as $j\ne 1$, the resulting boundary value problem assumes the form

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial u}{\partial t}}& =& -ru(t-T,x)-ru(t-T,x)u(t,x)+\hfill \\ \hfill & & {\displaystyle r\gamma \left({\int }_{-\infty }^{+\infty }\left(F_{\epsilon }(s)-F_0(s)\right)u(t-h,x+s)ds\right)·\left[1+\delta u(t,x)\right],}\hfill \\ \hfill & & u(t,x+2\pi )\equiv u(t,x).\hfill \end{array}$$

(17)

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 $C_{[-T,0]}\times W_{[0,2\pi ]}(T_0=max(T_0,h))$$\epsilon$-independent zero equilibrium state neighborhood for small $\epsilon$. 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 $\epsilon \to 0$. 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

$${\displaystyle \dot{N}=r[1-N(t-T,x)]N+\gamma \left[{\int }_{-\infty }^{\infty }F(s)N(t,x+s)ds-N\right],}$$

(18)

$$N(t,x+2\pi )\equiv N(t,x),$$

(19)

where $\gamma \gg 1$. We note that (18) is interpreted as a logistic delay equation with spatially distributed control. The function $F(s)$ describing spatial interactions is given by

$$F(s)={\displaystyle \frac{1}{\sqrt{\mu \pi }}}exp[-{\mu }^{-1}{(s+h)}^2],\mu >0.$$

We demonstrate that the dynamic properties of the problem under consideration change significantly depending on the various relations between $\gamma ,\mu$, 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

$$\lambda +rexp(-\lambda )=d\left({\int }_{-\infty }^{\infty }F(s,\epsilon )exp(iks)ds-1\right),k=0,\pm 1,\pm 2,\dots .$$

(20)

Statement 1. 

Let the roots of Equation (20) have a negative real part and be separated from the imaginary axis as$\epsilon \to 0$. Then, an${\epsilon }_0>0$can be found such that, for$\epsilon \in (0,{\epsilon }_0)$, the solutions to the problem under consideration from the sufficiently small$\epsilon$-independent zero equilibrium state neighborhood tend to zero as$t\to \infty$.

Statement 2. 

Let Equation (20) have a root with a positive real part separated from zero as$\epsilon \to 0$. Then, the zero equilibrium state of the boundary value problem is unstable for small$\epsilon$, and there is no attractor of this boundary value problem in some sufficiently small$\epsilon$-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.

2. Dynamics of Fully Connected Spatially Distributed Chain

In this section, the local dynamics of the boundary value problem (11), (12) are studied. The linearized at zero boundary value problem

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial t}}=-rv(t-T,x)& +r\gamma \left[M\left(v(t-h,s)-v(t-h,x)\right)\right],\hfill \\ \hfill & v(\tau ,x+2\pi )\equiv v(\tau ,x)\hfill \end{array}$$

(21)

has the characteristic equation

$$\lambda =-rexp(-\lambda T)+r\gamma exp(-\lambda h)[{\delta }_k-1],k=0,\pm 1,\pm 2,\dots ,$$

(22)

where

$${\delta }_k=\left\{\begin{array}{cc}1,\hfill & k=0,\hfill \\ 0,\hfill & k\ne 0.\hfill \end{array}\right.$$

We obtain this equation from (21) by substituting the elementary Euler solutions $v_k=exp(ikx+\lambda t)$$(k=0,\pm 1,\pm 2,\dots )$.

The case of the small parameter $\gamma$ is studied in Section 2.1. We fix some ${\gamma }_1$ in such a way that

$$\gamma =\mu {\gamma }_1\mathrm{and}0<\mu \ll 1.$$

(23)

The general case is studied in Section 2.2.

2.1. Case of Small $\gamma$ Values

Under the conditions $rT<\pi /2$ and (23), the roots of Equation (22) have separated from zero negative real parts as $\mu \to 0$. For $rT>\pi /2$, Equation (22) has a root with a positive real part separated from zero as $\mu \to 0$. The local dynamics of (11), (12) are not considered in these cases.

We assume that the conditions

$$r_0{T}_0=\frac{\pi }{2},r=r_0+\mu r_1,T=T_0+\mu T_1,\gamma =\mu {\gamma }_1$$

(24)

hold for some positive $r_0$ and $T_0$. In this case, $v_k=exp(i\pi {(2T_0)}^{-1}t+ikx)$ is the solution of the linear boundary value problem (21) as well as

$$\begin{array}{cc}\hfill v(t,x)=& \sum _{k=-\infty }^{+\infty }{\xi }_{k}exp\left(i\pi {(2T_0)}^{-1}t+ikx\right)=\hfill \\ \hfill & exp\left(i\pi {(2T_0)}^{-1}t\right)\sum _{k=-\infty }^{+\infty }{\xi }_{k}exp(ikx)=exp\left(i\pi {(2T_0)}^{-1}t\right)\xi (x).\hfill \end{array}$$

(25)

Then, we seek the solution to the nonlinear boundary value problem (11), (12) in the form

$$\begin{array}{cc}\hfill u(t,x,\mu )=& {\mu }^{1/2}\left(\xi (\tau ,x)exp\left(i\pi {(2T_0)}^{-1}t\right)+\overline{cc}\right)+\hfill \\ \hfill & \mu u_2(t,\tau ,x)+{\mu }^{3/2}{u}_3(t,\tau ,x)+\dots ,\hfill \end{array}$$

(26)

where $\tau =\mu t$ is the ‘slow’ time, $\xi (\tau ,x)$ is the unknown complex amplitude, the functions $u_j(t,\tau ,x)$ are $4T_0$-periodic with respect to t and $2\pi$-periodic with respect to x. Here, $\overline{cc}$ means the complex conjugate to the previous term expression.

We insert the formal expression (26) into (11) and collect the coefficients at the same powers of $\mu$. At the first step, we equate the coefficients of ${\mu }^{1/2}$ and obtain an identity. Then, we collect the coefficients at the first power of $\mu$ and obtain the equation for $u_2$

$${\dot{u}}_2=-r{u}_2(t-T,x)-r\left[\xi exp\left(-i\frac{\pi }{2}+i\frac{\pi }{2T_0}t\right)+\overline{\xi }exp\left(i\frac{\pi }{2}-i\frac{\pi }{2T_0}t\right)\right]·\left(\xi exp\left(i\frac{\pi }{2T_0}t\right)+\xi \right).$$

Hence,

$$u_2=A{\xi }^{2}exp(i\pi {(T_0)}^{-1}t)+\overline{cc},A=-exp(-i{r}_0){\left(2(i+exp(-2i{T}_0))\right)}^{-1}.$$

At the third step, we obtain the equation for determining $u_3$, the solvability condition of which in the indicated class of functions is formulated as

$$\begin{array}{c}{\displaystyle \frac{\partial \xi }{\partial \tau }}=b\xi +{\gamma }_0\left(M(\xi )-\xi \right)+\beta \xi {\left|\xi \right|}^2,\\ \xi (\tau ,x+2\pi )\equiv \xi (\tau ,x).\end{array}$$

(27)

Here, $M(\xi )$ stands for the mean value with respect to $x\in [0,2\pi ]$ of the function $\xi (\tau ,x)$:

$$M(\xi )=\frac{1}{2\pi }{\int }_0^{2\pi }\xi (\tau ,x)dx.$$

The following equalities hold for the coefficients in (27):

$$\begin{array}{ccc}\hfill b& =& {\left(1+{\displaystyle \frac{{\pi }^2}{4}}\right)}^{-1}\left[\left(\frac{\pi }{2}+i\right)r_1+{\lambda }_0^2{T}_1\left(1-i\frac{\pi }{2}\right)\right],\hfill \\ \hfill {\gamma }_0& =& {\gamma }_1{r}_{0}exp\left(-i\pi h{(2T_0)}^{-1}\right)·{\left[i\pi {(2T_0)}^{-1}-r_{0}exp\left(-i\pi {(2T_0)}^{-1}\right)\right]}^{-1},\hfill \\ \hfill \beta & =& -{\lambda }_0[3\pi -2+i(\pi +6)]{\left(10\left(1+{\displaystyle \frac{4}{{\pi }^2}}\right)\right)}^{-1},\Re \beta <0.\hfill \end{array}$$

In the considered case, the next statement indicates the boundary value problem (27) to be the QNF for the boundary value problem (11), (12).

Theorem 1.

Let the conditions (23) and (24) be satisfied, and let the boundary value problem (27) have the solution $\xi (\tau ,x)$ for $\tau \ge {\tau }_0$. Then, the function

$$u(t,x,\mu )={\mu }^{1/2}\left(\xi (\tau ,x)exp\left(i\pi {(2T_0)}^{-1}t\right)+\overline{cc}\right)+\mu \left(A{\xi }^2(\tau ,x)exp(i\pi {(T_0)}^{-1}t)+\overline{cc}\right)$$

satisfies the boundary value problem (11), (12) up to $O({\mu }^{3/2})$.

We note that the conditions of zero solution in (27) asymptotic stability consist in the fulfillment of the inequalities

$$\Re b<0,\Re (b-{\gamma }_0)<0.$$

(28)

For $\Re b>0$, the QNF (27) has the homogeneous cycle ${\rho }_{0}exp(i{\omega }_{0}t)$, and

$${\rho }_0={\left(-\Re b·{(\Re \beta )}^{-1}\right)}^{1/2},{\omega }_0=\Im b+{\rho }_0^2\Im \beta .$$

The same cycle under the same condition exists in the logistic Equation (1). The cycle in Equation (1) is orbitally stable, while the orbital stability in (27) requires the following inequalities to be valid:

$$\begin{array}{ccc}\hfill (1)& & {\rho }_0\Re \beta -\Re {\gamma }_0<0,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill (2)& & |{\gamma }_0{|}^2-2{\rho }_0^2\left(\Re \beta ·\Re {\gamma }_0+\Im \beta ·\Im {\gamma }_0\right)>0.\hfill \end{array}$$

(30)

A value of ${\gamma }_0$ can be selected such that either (29) or (30) does not hold. We note that stability (instability) condition fulfillment can be achieved by delay coefficient h variation.

In addition to one cycle, the QNF (27) can have the cycles

$${\rho }_{k}exp(i{\omega }_{k}t+ikx)(k=0,\pm 1,\pm 2,\dots ).$$

These cycles exist under the condition $\Re (b-{\gamma }_0)>0$ and

$${\rho }_k={\rho }^0={\left(-\Re (b-{\gamma }_0)·{(\Re \beta )}^{-1}\right)}^{1/2},{\omega }_k={\omega }^0=\Im (b-{\gamma }_0)+{\rho }_k^2\Im \beta .$$

(31)

The above cycles are orbitally stable if the inequalities

(1)

${({\rho }^0)}^2\Re \beta +\Re {\gamma }_0<0$,

(2)

$|{\gamma }_0{|}^2+2{({\rho }^0)}^2\left(\Re \beta ·\Re {\gamma }_0+\Im \beta ·\Im {\gamma }_0\right)>0$

hold. Strict unfulfillment of at least one of these inequalities implies instability of all cycles.

The problem of the spatially inhomogeneous step-like cycle’s existence is more intriguing.

At the first instance, we note that Equation (27) is periodic with respect to t and $2\pi$-periodic piecewise continuous with respect to x solution

$$\rho (x)exp(i{\omega }^{0}t),\rho (x)=\left\{\begin{array}{cc}{\rho }^0,\hfill & x\in (0,\pi ),\hfill \\ -{\rho }^0,\hfill & x\in (\pi ,2\pi ).\hfill \end{array}\right.$$

One can construct families of $2\pi {\omega }_0^{-1}$-periodic with respect to t and $2\pi$-periodic piecewise continuous with respect to x solutions $\rho (x,k_1,k_2,\alpha )exp(i{\omega }_{0}t)$, where

$$\rho (x,k_1,k_2,\alpha )=\left\{\begin{array}{cc}{\rho }^{0}exp(i2\pi {\alpha }^{-1}{k}_{1}x),\hfill & x\in (0,\alpha ),k_1=\pm 1,\pm 2,\dots ,\hfill \\ {\rho }^{0}exp(i2\pi {(2\pi -\alpha )}^{-1}{k}_{2}x),\hfill & x\in (\alpha ,2\pi -\alpha ),k_2=\pm 1,\pm 2,\dots .\hfill \end{array}\right.$$

Obviously, constructions of this kind extend to solutions with an arbitrary number of ‘steps’.

What is more intriguing, these are the cycles consisting of two different steps with respect to ‘amplitude’ on the interval $[0,2\pi ]$. To construct them, we arbitrarily fix the parameters $\alpha \in (0,2\pi )$ and ${\phi }_{1,2}\in (0,2\pi )$. We assume

$$u_0(t,x)=\rho (x)exp(i\omega t),\rho (x)=\left\{\begin{array}{cc}{\rho }_{1}exp(i{\phi }_1),\hfill & x\in (0,\alpha ),\hfill \\ {\rho }_{2}exp(i{\phi }_2),\hfill & x\in (\alpha ,2\pi ).\hfill \end{array}\right.$$

(32)

Let $\phi ={\phi }_2-{\phi }_1$. We insert (32) into (26). Then, we obtain the system of polynomial equations

$$i\omega {\rho }_j=(b-{\gamma }_0){\rho }_j+{\gamma }_{0}P+\beta {\rho }_j^3,(j=1,2).$$

(33)

Here, $P={(2\pi )}^{-1}{\gamma }_0\left(\alpha {\rho }_1+(1-\alpha ){\rho }_{2}exp(i\phi )\right)$.

The system (33) represents two complex or four real equations of five real variables ${\rho }_1,{\rho }_2,\alpha ,\phi$ and $\omega$. We proceed with the real-valued notation of this system:

$$\begin{array}{ccc}\hfill {\rho }_1& =& \Re (b-{\gamma }_0){\rho }_1+{\rho }_1^3\Re \beta +\Re {\gamma }_{0}P,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill \omega {\rho }_1& =& \Im (b-{\gamma }_0){\rho }_1+{\rho }_1^3\Im \beta +\Im {\gamma }_{0}P,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\rho }_2& =& \Re (b-{\gamma }_0){\rho }_2+{\rho }_2^3\Re \beta +\Re \left({\gamma }_{0}exp(-i\phi )P\right),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill \omega {\rho }_2& =& \Im (b-{\gamma }_0){\rho }_2+{\rho }_2^3\Im \beta +\Im \left({\gamma }_{0}exp(-i\phi )P\right),\hfill \end{array}$$

(37)

First, we provide the expressions for ${\rho }_{1,2}^3$ from (34) and (36):

$$\begin{array}{ccc}\hfill {\rho }_1^3& =& \left[\left(1-\Re (b-{\gamma }_0)\right){\rho }_1-\Re ({\gamma }_{0}P)\right]{(\Re \beta )}^{-1},\hfill \\ \hfill {\rho }_2^3& =& \left[\left(1-\Re (b-{\gamma }_0)\right){\rho }_2-\Re ({\gamma }_{0}exp(-i\phi )P)\right]{(\Re \beta )}^{-1}.\hfill \end{array}$$

Then, we substitute them into (35) and (37) instead of ${\rho }_{1,2}^3$, respectively. As a result, we obtain the linear system with the $2\times 2$ matrix B of the form

$$B\left(\begin{array}{c}{\rho }_1\\ {\rho }_2\end{array}\right)=\omega \left(\begin{array}{c}{\rho }_1\\ {\rho }_2\end{array}\right).$$

If the real positive eigenvalue $\omega =\omega (\alpha ,\phi )$ of this matrix can be determined, we obtain the eigenvector ${\rho }_2=c(\alpha ,\phi ){\rho }_1$. Taking this equality into consideration in (34) and (36), we obtain the expressions for ${\rho }_j={\rho }_j(\alpha ,\phi )(j=1,2)$. Finally, another variable is eliminated via the equality

$$c(\alpha ,\phi )={\rho }_2(\alpha ,\phi ){\left({\rho }_1(\alpha ,\phi )\right)}^{-1}.$$

Numerical investigations are carried out in this way.

2.2. Case of Parameter $\gamma$ ‘Middle’ Values

We restrict ourselves to considering the boundary value problem (11), (12) in the case when the parameters h and T coincide, namely $h=T$, which is interesting for mathematical ecology problems. Then, the characteristic Equation (22) splits into two:

(1)

$\lambda =-rexp(-\lambda T),$

(2)

$\lambda =-r(1+\gamma )exp(-\lambda T)$.

Moreover, each root of the last equation is repeated infinitely many times. Let the inequality $rT<\pi /2$ hold. Thus, the size of each isolated population does not oscillate in the positive equilibrium state neighborhood. We assume that the problem of the stationery stability in (11), (12) has a critical case: the relations

$$r_0(1+{\gamma }_0)T_0=\frac{\pi }{2}$$

(38)

hold for some $r=r_0,\gamma ={\gamma }_0$ and $T=T_0$. From here, the linear boundary value problem (21) has infinitely many periodical solutions $u_k(t,x)=exp(ikx+i{\omega }_{0}t)$, $k=1,2,\dots$, ${\omega }_0=\pi {(2T_0)}^{-1}=r_0(1+{\gamma }_0)$. This implies that under the condition $M(\xi (x))=0$, the function $u_0(t,x)=\xi (x)exp(i{\omega }_{0}t)$ is also a solution of (21). We introduce a small parameter $\mu :0<\mu \ll 1$. Let

$$r=r_0+\mu r,\gamma ={\gamma }_0+\mu \gamma ,T=T_0+\mu T_1$$

in (11), (12). We seek solutions to this boundary value problem in the form of formal asymptotic series

$$u(t,x,\mu )={\mu }^{1/2}\left(\xi (\tau ,x)exp\left(i{\omega }_{0}t\right)+\overline{cc}\right)+\mu u_2(t,\tau ,x)+{\mu }^{3/2}{u}_2(t,\tau ,x)+\dots ,$$

(39)

where $\tau =\mu t$, $\xi (\tau ,x)$ is the unknown amplitude, and the functions $u_j(t,\tau ,x)$ are $2\pi /{\omega }_0$-periodic with respect to t and $2\pi$-periodic with respect to x. The key condition is that the function $\xi (\tau ,x)$ has a zero mean with respect to the spatial variable:

$$M(\xi (\tau ,s))=0.$$

(40)

We insert (39) into (11). First, we perform standard actions to obtain the equation for $u_2$:

$$\begin{array}{cc}\hfill \frac{\partial u_2}{\partial t}=-& r_0{u}_2(t-T,x)+r_0{\gamma }_0\left[M\left(u_2(t-T,s)\right)-u_2(t-T,x)\right]-\hfill \\ \hfill & r_0(1+\delta {\gamma }_0){\xi }^2(\tau ,x)exp(-i{\omega }_0{T}_0+2i{\omega }_{0}t)+\overline{cc}.\hfill \end{array}$$

(41)

We seek the solution of (41) in the form

$$u_2(t,\tau ,x)=u_{20}(t,\tau )exp(2i{\omega }_{0}t)+\overline{cc}+u_{21}(t,\tau ,x)exp(2i{\omega }_{0}t)+\overline{cc}$$

and $M(u_{21}(t,\tau ,s))=0$. Then,

$$\begin{array}{ccc}\hfill u_{20}(t,\tau )& =& C_{1}M({\xi }^2(\tau ,s)),\hfill \\ \hfill u_{21}(t,\tau ,x)& =& C_2\left({\xi }^2(\tau ,x)-M({\xi }^2(\tau ,s))\right),\hfill \\ \hfill C_1& =& -{\left(2i{\omega }_0+r_{0}exp(-i{\omega }_0{T}_0)\right)}^{-1}{r}_0(1+\delta {\gamma }_0)exp(-i{\omega }_0{T}_0),\hfill \\ \hfill C_2& =& -{\left(2i{\omega }_0+r_0(1+{\gamma }_0)exp(-i{\omega }_0{T}_0)\right)}^{-1}{r}_0(1+\delta {\gamma }_0)exp(-i{\omega }_0{T}_0).\hfill \end{array}$$

At the next step, we collect the coefficients at ${\mu }^{3/2}$ in the formal identity and obtain the equation for $u_3$ in the form

$$\begin{array}{cc}\hfill \frac{\partial u_3}{\partial t}=-& r_0{u}_3(t-T,x)+r_0{\gamma }_0\left[M\left(u_3(t-T,s)\right)-u_3(t-T,x)\right]+\hfill \\ \hfill & D_1(\tau ,x)exp(i{\omega }_{0}t)+\overline{cc}+D_3(\tau ,x)exp(3i{\omega }_{0}t)+\overline{cc}.\hfill \end{array}$$

The $D_3(\tau ,x)$-independent in the indicated class of functions solvability condition of this equation is the validity of the equality

$$D_1(\tau ,x)-M(D_1(\tau ,x))=0.$$

We take into account the explicit form of $D_1(\tau ,x)$ and obtain from this equality the boundary value problem as the QNF to find $\xi (\tau ,x)$:

$${\displaystyle \frac{\partial \xi }{\partial \tau }}=b_1\xi +{\beta }_1\xi \left|{\xi }^2\right|+{\beta }_2\overline{\xi }M({\xi }^2),$$

(42)

$$\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x),M(\xi (\tau ,s))=0.$$

(43)

The formulas

$$\begin{array}{ccc}\hfill b_1& =& {\left(1+{\displaystyle \frac{{\pi }^2}{4}}\right)}^{-1}\left[\left(\frac{\pi }{2}+i\right)(r_1(1+{\gamma }_0)+{\gamma }_1{r}_0)+r_0^2(1+{\gamma }_0^2)T_1\left(1-i\frac{\pi }{2}\right)\right],\hfill \\ \hfill {\beta }_1& =& -r(1+\delta {\gamma }_0)\left(exp(-2i{\omega }_0{T}_0)+exp(i{\omega }_0{T}_0)\right)C_2{\left(1+i{\displaystyle \frac{\pi }{2}}\right)}^{-1},\hfill \\ \hfill {\beta }_2& =& -r[(1+\delta {\gamma }_0)\left(exp(-2i{\omega }_0{T}_0)+exp(i{\omega }_0{T}_0)\right)(C_1-C_2)+\hfill \\ & & \delta {\gamma }_{0}exp(-2i{\omega }_0{T}_0)C_1]{\left(1+i{\displaystyle \frac{\pi }{2}}\right)}^{-1}\hfill \end{array}$$

hold for the coefficients in (42).

Let us sum things up.

Theorem 2.

Let condition (38) be satisfied, and the boundary value problem (42), (43) has the bounded solution $\xi (\tau ,x)$ as $\tau \to \infty$. Then, the function

$$u(t,x,\mu )={\mu }^{1/2}\left(\xi (\tau ,x)exp\left(i{\omega }_{0}t\right)+\overline{cc}\right)+\mu u_2(t,\tau ,x)$$

satisfies the boundary value problem (11), (12) up to $O({\mu }^{3/2})$.

Periodic with respect to t and $2\pi$-periodic piecewise continuous with respect to x solutions to the boundary value problem (11), (12) could also be determined in explicit form. However, it is not considered in this paper. We only refer to the paper [29]. In a similar situation, families of step-like solutions to the QNF are constructed, its stability is studied, and a comparison with experimental data is performed.

3. Chain Dynamics in Case of Diffusional Couplings

The boundary value problem (15) is examined. Let $\delta =0$ for definiteness. Then, the linearized on the zero equilibrium state boundary value problem has the form

$${\displaystyle \frac{\partial u}{\partial t}}=-ru(t-T,x)+r\gamma {\int }_{-\infty }^{+\infty }F(s,\epsilon )u(t-h,x+s)ds,$$

(44)

$$u(t,x+2\pi )\equiv u(t,x).$$

(45)

Here, we assume that function $F(s,\epsilon )$ is given by

$$F(s,\epsilon )=F_{\epsilon }(s)-2F_0(s)+F_{-\epsilon }(s).$$

(46)

Depending on the parameter $\sigma$, three fundamentally different events can be distinguished. The first and the simplest of them assumes the parameter $\sigma >0$ to be somehow fixed and, naturally, independent of the small parameter $\epsilon$. This case is studied in Section 3.1. In Section 3.2, we assume that there is a value of ${\sigma }_0>0$ such that

$$\sigma =\epsilon {\sigma }_0.$$

(47)

The critical case of infinite dimension mentioned above is realized under this condition. Finally, in Section 3.3, we assume the parameter $\sigma$ to be even smaller: $\sigma =o(\epsilon )$. More precisely, for some fixed ${\sigma }_0>0$, we consider the relation

$$\sigma ={\epsilon }^2{\sigma }_0.$$

(48)

This case is the most complicated and intriguing. It naturally generalizes the case of ‘purely diffusional’ couplings for which $\sigma$∼0.

3.1. Chain Dynamics for Fixed $\sigma$ Value

In Formula (46), we arbitrarily fix the value ${\sigma }_0>0$. The inequality

$$0<r<\frac{\pi }{2}$$

(49)

is the necessary and sufficient condition for the real parts of all eigenvalues of the characteristic Equation (20) to be negative.

For $r=\pi /2$, Equation (20) has exactly two pure imaginary roots ${\lambda }_{\pm }=\pm i\pi /2$, and the real parts of the remaining roots are negative and separated from zero as $\epsilon \to 0$. Thus, the conditions of the well-studied Andronov–Hopf bifurcation are satisfied. Let

$$r=\frac{\pi }{2}+{\epsilon }^2{r}_1$$

(50)

for the arbitrarily fixed value $r_1$.

Then, for $\epsilon \ll 1$, the roots ${\lambda }_{\pm }(\epsilon )$ of Equation (20) close to ${\lambda }_{\pm }$ are as follows:

$$\begin{array}{cc}\hfill {\lambda }_{+}(\epsilon )=& {\overline{\lambda }}_{-}(\epsilon ),\hfill \\ \hfill {\lambda }_{+}(\epsilon )=& i\frac{\pi }{2}+{\epsilon }^2{\lambda }_{10}+O({\epsilon }^4),\hfill \\ \hfill \mathrm{where}{\lambda }_{10}=& {(1+\frac{{\pi }^2}{4})}^{-1}(\frac{\pi }{2}+i)r_1.\hfill \end{array}$$

(51)

Under these conditions and for sufficiently small $\epsilon$, the boundary value problem (15) has a two-dimensional stable local integral invariant manifold $\mathcal{M}(\epsilon )$ in the zero equilibrium state neighborhood, on which this boundary value problem can be written as a special scalar complex ordinary differential equation

$$\frac{d\xi }{d\tau }={\lambda }_{10}\xi +g\xi {\left|\xi \right|}^2,$$

(52)

where $\tau ={\epsilon }^{2}t$ is a slow time, and $\xi (\tau )$ is a slowly varying amplitude in the asymptotic presentations of solutions on the manifold $\mathcal{M}(\epsilon )$

$$u=\epsilon \left[\xi (\tau )exp\left(i\frac{\pi }{2}t\right)+\overline{\xi }(\tau )exp\left(-i\frac{\pi }{2}t\right)\right]+{\epsilon }^2{u}_2(t,\tau )+{\epsilon }^3{u}_3(t,\tau )+\dots .$$

(53)

Here, the functions $u_j(t,\tau )$ are 4-periodic with respect to t. We insert the formal expression (53) into (15) and collect the coefficients at the same powers of $\epsilon$. First, we equate the coefficients at ${\epsilon }^2$ to obtain

$$u_2=\frac{2-i}{5}{\xi }^{2}exp(i\pi t)+\frac{2+i}{5}{\xi }^{2}exp(-i\pi t).$$

(54)

At the next step, from the solvability condition of the resulting equation with respect to $u_3$, we obtain the necessity of satisfying the relation (52), where

$$g=-\frac{\pi }{2}\left[3\pi -2+i(\pi +6)\right]·{\left(10\left(1+\frac{4}{{\pi }^2}\right)\right)}^{-1}.$$

(55)

Let us formulate the resulting statements. Their proofs are well-known (see, for example, [19]).

Theorem 3.

Let $r_1<0$. Then, for all sufficiently small ε, the solution of the boundary value problem (15) from some sufficiently small ε-independent equilibrium state $u_0=0$ neighborhood tends to zero as $t\to \infty$.

Theorem 4.

Let $r_1>0$. Then, all the solutions of Equation (52) except the zero solution tend to an orbitally stable cycle

$$\begin{array}{cc}\hfill {\xi }_0(\tau )=& {\left[10\frac{\pi }{2}{r}_1{(3\pi -2)}^{-1}\right]}^{\frac{1}{2}}{\xi }_{0}exp(i{\varphi }_0\tau ),\hfill \\ \hfill {\varphi }_0=& \Im {\lambda }_{10}+{\xi }_0^2\Im g,\hfill \end{array}$$

(56)

and the solutions ($\neg \not\equiv 1$) from $\mathcal{M}(\epsilon )$ tend to cycle

$$u_0(t,\epsilon )=\epsilon \left[{\xi }_0({\epsilon }^{2}t)exp\left(i\frac{\pi }{2}t\right)+{\overline{\xi }}_0({\epsilon }^{2}t)exp\left(-i\frac{\pi }{2}t\right)\right]+O({\epsilon }^2)$$

(57)

as $t\to \infty$.

Thus, in the considered case, the boundary value problem (15) can have only a homogeneous cycle $1+u_0(t,\epsilon )$ in a zero neighborhood, which is a logistic Equation (1) under the condition (50). Apparently, the case considered here is of no interest.

3.2. Chain Dynamics for $\sigma$ Values of $\epsilon$ Order

Here, we assume that the condition (47) holds. Then, the characteristic Equation (20) has the set of roots ${\lambda }_m(\epsilon )$ and ${\overline{\lambda }}_m(\epsilon )$ ($m=0,\pm 1,\pm 2,\dots$), the real parts of which tend to zero as $\epsilon \to \infty$. The representation

$$\begin{array}{cc}\hfill {\lambda }_m(\epsilon )+\left(\frac{\pi }{2}+{\epsilon }^2{r}_1\right)exp(-{\lambda }_m(\epsilon ))=& d\left({\int }_{-\infty }^{\infty }F(s,\epsilon )exp(ims)ds-1\right)=\hfill \\ \hfill & d(cos(\epsilon m)exp(-{\epsilon }^2{m}^2{\sigma }_0^2)-1).\hfill \end{array}$$

(58)

holds for these roots.

From here, we obtain that the asymptotic equality

$${\lambda }_m(\epsilon )=i\frac{\pi }{2}+{\epsilon }^2{\lambda }_1+\dots ,{\lambda }_1={\lambda }_{10}-{\left(1+i\frac{\pi }{2}\right)}^{-1}d\left(\frac{1}{2}+{\sigma }_0^2\right)m^2$$

holds for each integer m.

Each of the above roots corresponds to the solution $v_m(t,x)$ of the boundary value problem (44), (45) for which

$$v_m(t,x)=exp(i\frac{\pi }{2}t+imx){\nu }_m(\tau ),$$

where ${\nu }_m(\tau )={\nu }_{m}exp((-{\epsilon }^2{\lambda }_1+O({\epsilon }^4))t)$.

Let us introduce the formal series

$$\begin{array}{cc}\hfill u(t,x,\epsilon )=& \epsilon [exp\left(i\frac{\pi }{2}t\right)\sum _{m=-\infty }^{\infty }{\xi }_m(\tau )exp(imx)+\hfill \\ \hfill & exp\left(-i\frac{\pi }{2}t\right)\sum _{m=-\infty }^{\infty }{\overline{\xi }}_m(\tau )exp(-imx)]+\hfill \\ \hfill & {\epsilon }^2{u}_2(t,\tau ,x)+{\epsilon }^3{u}_3(t,\tau ,x)+\dots .\hfill \end{array}$$

(59)

Here, $\tau ={\epsilon }^{2}t$ is a slow time, ${\xi }_m(\tau )$ are the unknown slowly varying amplitudes, and the functions $u_j(t,\tau ,x)$ are periodic with respect to t and x. We note that Formula (59) defines a solution set of the linear boundary value problem (44), (45) in the linear approximation, i.e., for $u_j\equiv 0$.

Expression (59) can be significantly simplified. For this purpose, we assume

$$\xi (\tau ,x)=\sum _{m=-\infty }^{\infty }{\xi }_m(\tau )exp(imx).$$

Then, it follows from (59) that

$$\begin{array}{cc}\hfill u(t,x,\epsilon )=& \epsilon \left[\xi (\tau ,x)exp\left(i\frac{\pi }{2}t\right)+\overline{\xi }(\tau ,x)exp\left(-i\frac{\pi }{2}t\right)\right]+\hfill \\ \hfill & {\epsilon }^2{u}_2(t,\tau ,x)+{\epsilon }^3{u}_3(t,\tau ,x)+\dots .\hfill \end{array}$$

(60)

We insert (60) into (15) and equate the coefficients of the same powers of $\epsilon$ in the resulting formal identity. At the first step, the identity holds for ${\epsilon }^1$. At the second step, we collect the coefficients at ${\epsilon }^2$ and obtain the equality (54), where $\xi =\xi (\tau ,x)$. At the next step, we obtain the boundary value problem for determining $\xi (\tau ,x)$ from the solvability condition of the resulting equation with respect to $u_3$:

$$\begin{array}{c}\hfill \frac{\partial \xi }{\partial \tau }=d_0\frac{{\partial }^2\xi }{\partial x^2}+{\lambda }_{10}\xi +g\xi {\left|\xi \right|}^2,\\ \hfill \xi (t,x+2\pi )\equiv \xi (\tau ,x).\end{array}$$

(61)

Here, $d_0={(1+i\pi /2)}^{-1}(1/2+{\sigma }_0^2)$ and the coefficients ${\lambda }_{10}$ and g are the same as in (51) and (55), respectively.

We formulate the basic results.

Theorem 5.

Let the boundary value problem (61) have the bounded solution ${\xi }_0(\tau ,x)$ as $\tau \to \infty$. Then, the function

$$\begin{array}{cc}\hfill u_0(t,x,\epsilon )=& \epsilon \left[{\xi }_0(\tau ,x)exp\left(i\frac{\pi }{2}t\right)+{\overline{\xi }}_0(\tau ,x)exp\left(-i\frac{\pi }{2}t\right)\right]+\hfill \\ \hfill & {\epsilon }^2\frac{2-i}{5}{\xi }^{2}exp(i\pi t)+\frac{2+i}{5}{\xi }^{2}exp(-i\pi t)\hfill \end{array}$$

satisfies the boundary value problem (15) up to $O({\epsilon }^4)$.

The problem of the existence and stability of an exact solution to (15), which is close to the corresponding solution of the boundary value problem (61) as $\epsilon \to 0$, arises. It can be solved, for example, if ${\xi }_0(\tau ,x)$ is a periodic solution with the property of coarseness. By coarseness, we mean the following: If ${\xi }_0(\tau ,x)\equiv const·exp(i\omega \tau +imx)$, then only one multiplier of the linearized on ${\xi }_0(\tau ,x)$ boundary value problem is equal to Modulo 1. In other cases, the coarseness condition is that only two multipliers of the linearized on ${\xi }_0(\tau ,x)$ boundary value problem are equal to Modulo 1.

Theorem 6.

Let ${\xi }_0(\tau ,x)$ be the coarse periodic solution of the boundary value problem (61) with period ${\omega }_0$. Then, for all sufficiently small ε, the boundary value problem (15) has the periodic with respect to t solution $u_0(t,x,\epsilon )$ with period ${\omega }_0+O(\epsilon )$ with the same stability as ${\xi }_0(\tau ,x)$. The asymptotic equality

$$\begin{array}{cc}\hfill u_0(t,x,\epsilon )=& \epsilon ({\xi }_0((1+O(\epsilon )){\epsilon }^{2}t,x)exp\left(i\frac{\pi }{2}t\right)+\hfill \\ \hfill & {\overline{\xi }}_0((1+O(\epsilon )){\epsilon }^{2}t,x)exp\left(-i\frac{\pi }{2}t\right)+\dots +O({\epsilon }^4).\hfill \end{array}$$

(62)

holds for this solution.

The proof of Theorem 5 follows directly from the above construction of the boundary value problem (15) solution asymptotics. The justification of Theorem 6 is standard but cumbersome, so we omit it.

3.3. Chain Dynamics for $\sigma =O({\epsilon }^2)$

Here, we assume that the condition (48) holds. We distinguish the roots of the characteristic Equation (20) with real parts tending to zero as $\epsilon \to 0$. This equation’s roots $\lambda =\lambda (k)$ are calculated from the formula

$$\begin{array}{cc}\hfill \lambda +(r_0+{\epsilon }^2{r}_1)exp(-\lambda )=& d\left({\int }_{-\infty }^{\infty }F(s,\epsilon )exp(iks)ds-1\right)=\hfill \\ \hfill & d(cos(z)exp(-{\epsilon }^4{\sigma }_0^2{z}^2)-1),\hfill \end{array}$$

(63)

where $z=\epsilon k$. The vanishing of the right-hand side in (63) up to $O(1)$ (as $\epsilon \to 0$) is responsible for the real parts of the roots tending to zero. Those numbers $k=k(\epsilon )$ satisfy this condition for which $cos(z)\sim 1$. We introduce the notation to describe such numbers. We fix an arbitrary integer n and let ${\theta }_n={\theta }_n(\epsilon )\in [0,1)$ be the expression that complements the value $2\pi n{\epsilon }^{-1}$ to an integer. It appears that the function ${\theta }_n(\epsilon )$ can be considered identically zero. The point is that the parameter $\epsilon$ introduced above is determined as $\epsilon =2\pi {(1+2N)}^{-1}$. Therefore, $2\pi n{\epsilon }^{-1}=n(1+2N)$, which is an integer.

Then, the set of numbers $k(\epsilon )$ of the roots $\lambda (k(\epsilon ))$ consists of the values

$$k(\epsilon )=2\pi n{\epsilon }^{-1}+m,m,n=0,\pm 1,\pm 2,\dots$$

(64)

in the considered case.

It is convenient to denote these roots by ${\lambda }_{m,n}(\epsilon )$. We obtain the asymptotic expression

$${\lambda }_{m,n}(\epsilon )=i\frac{\pi }{2}-{\epsilon }^2{\left(1+i\frac{\pi }{2}\right)}^{-1}(m^2+4{\pi }^2{\sigma }_0^2{n}^2)+O({\epsilon }^4)$$

for them.

We follow the algorithm investigated above and introduce the formal series

$$\begin{array}{cc}\hfill u=& \epsilon [exp\left(i\frac{\pi }{2}t\right)\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{\xi }_{m,n}(\tau )exp(i(2\pi n{\epsilon }^{-1}+m)x)+\hfill \\ \hfill & \epsilon \left(-i\frac{\pi }{2}t\right)\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{\overline{\xi }}_{m,n}(\tau )exp(-i(2\pi n{\epsilon }^{-1}+m)x)]+\hfill \\ \hfill & {\epsilon }^2{u}_2(t,\tau ,x)+{\epsilon }^3{u}_3(t,\tau ,x)+\dots ,\hfill \end{array}$$

(65)

where $\tau ={\epsilon }^{2}t$, and the functions $u_j(t,\tau ,x)$ are periodic with respect to t and x.

Let $y=2\pi {\epsilon }^{-1}x$ and

$$\xi (\tau ,x,y)=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{\xi }_{m,n}(\tau )exp(iny+imx).$$

Then, we can simplify expression (65)

$$u=\epsilon \left[exp\left(i\frac{\pi }{2}t\right)\xi (\tau ,x,y)+exp\left(-i\frac{\pi }{2}t\right)\overline{\xi }(\tau ,x,y)\right]+{\epsilon }^2{u}_2+{\epsilon }^3{u}_3+\dots .$$

(66)

We insert (66) into (15) and equate the coefficients at the same powers of $\epsilon$. First, we determine $u_2(\tau ,t,x)$. Then, from the solvability condition of the equation with respect to $u_3$, we obtain the expression for $\xi (\tau ,x,y)$, determining:

$$\frac{\partial \xi }{\partial \tau }={\left(1+i\frac{\pi }{2}\right)}^{-1}·\left(\frac{{\partial }^2\xi }{\partial x^2}+4{\pi }^2{\sigma }_0^2\frac{{\partial }^2\xi }{\partial y^2}\right)+{\lambda }_{10}\xi +g\xi {\left|\xi \right|}^2,$$

(67)

$$\xi (\tau ,x+2\pi ,y)\equiv \xi (\tau ,x,y+2\pi )\equiv \xi (\tau ,x,y),$$

(68)

where the coefficients ${\lambda }_{10}$ and g are the same as in (52).

Ideologically, the basic results of this subsection repeat Theorems 5 and 6. We cite an analogue of Theorem 5 as an example.

Theorem 7.

Let ${\xi }_0(\tau ,x,y)$ be the bounded solution of the boundary value problem (67), (68) as $\tau \to \infty$. Then, the function

$$\begin{array}{cc}\hfill u_0(t,x,\epsilon )=& 1+\epsilon [exp\left(i\frac{\pi }{2}t\right){\xi }_0({\epsilon }^{2}t,x,2{\pi }^{-1}x)+\hfill \\ \hfill & exp\left(-i\frac{\pi }{2}t\right){\overline{\xi }}_0({\epsilon }^{2}t,x,2\pi {\epsilon }^{-1}x)]+{\epsilon }^2{u}_2\hfill \end{array}$$

(69)

satisfies the boundary value problem (15) up to $O({\epsilon }^3)$.

The boundary value problems (61) and (67), (68) have been numerically investigated by many authors (see, for example, [30]). It has been shown that complicated and irregular oscillations are typical for such boundary value problems, especially for (67), (68). The formulas (60) and (66), which couple these boundary value problem solutions with the boundary value problem (15) solutions, allow us to formulate the same conclusion about the solutions of (15).

4. Cooperative Dynamics in Chains with Unidirectional Coupling

We consider the boundary value problem (17) in which the equality

$$F(s,\epsilon )=F_{\epsilon }(s)-F_0(s)$$

holds for the function $F(s,\epsilon )$. Further, relation (47) holds for the parameter $\sigma$.

We assume that the equilibrium state $u_0=1$ of logistic Equation (1) is asymptotically stable in the absence of couplings. Thus, the values of parameter r satisfy the inequality

$$0<r<\frac{\pi }{2}.$$

(70)

Section 4.1 analyses the linearized at zero boundary value problem, and the nonlinear boundary value problem, which is the QNF, is constructed in Section 4.2.

4.1. Linear Analysis

In the considered case, the characteristic equation for the linearized on $u_0$ boundary value problem takes the form

$$\lambda +rexp(-\lambda )=d(exp(iz-{\sigma }_0^2{z}^2)-1),$$

(71)

where $d>0$, $z=\epsilon k$, $k=0,\pm 1,\pm 2,\dots$. We study the location of the roots of Equation (71) in order to make a conclusion about the stability of the equilibrium state in the boundary value problem (17).

We present several simple statements about the roots of (71) without proofs.

Lemma 1.

For $z\in [\pi (2n+1),\pi (2n+2)]$ ($n=0,\pm 1,\pm 2,\dots$) Equation (71) has no roots with a zero real part as $d>0$.

Lemma 2.

For every $z\in (2\pi n,\pi (2n+1))$, there exists $d>0$ such that Equation (71) has a root with a zero real part as $d=d_z$.

We introduce the notation: $d(r)=\underset{-\infty <z<\infty }{min}{d}_z=d_{z(r)}$.

Then, the equilibrium state of the boundary value problem (17) is asymptotically stable as $d\in (0,d(r))$. For $d=d(r)$ and $z=z(r)$, Equation (71) has roots ${\lambda }_{\pm }(r)$ with zero real part: ${\lambda }_{\pm }(r)=\pm i\omega (r)$ ($\omega (r)>0$).

Lemma 3.

The inequalities

$$0<z(r)<\pi ,\frac{\pi }{2}<\omega (r)<\frac{3\pi }{2}$$

hold for the values of $\omega (r)$ and $z(r)$.

We consider the questions about the asymptotic behavior of the expressions $d(r)$, $\omega (r)$ and $z(r)$ for $r\to 0$ and for $r\to \pi /2$ separately.

First, let $r\to 0$. We denote by ${\omega }_0$ the root from the interval $(\pi /2,\pi )$ of the equation $tan\omega =-{\omega }^{-1}$. Let

$$c_0=(1+{\sigma }_0^2){\omega }_0^2({\omega }_0^2+4),z_0={\omega }_0{c}_0^{-1}.$$

Lemma 4.

The asymptotic equalities

$$d(r)=c_0{r}^{-1}(1+o(1)),\omega (r)={\omega }_0+o(1),z(r)=z_{0}r(1+o(1))$$

hold as $r\to 0$.

Then, let $r=\pi /2-\mu$ and $0<\mu \ll 1$. Now, by $z_{00}\in (0,\pi /2)$, we denote the least root of equation $cosz={(\pi /4+1)}^{-1/2}$. Then, $min{z}_{00}=\pi /2{({\pi }^2/4+1)}^{-1/2}$. We assume

$$c_{00}=\frac{\pi }{2}\left[{\left(\frac{{\pi }^2}{4}+1\right)}^{\frac{1}{2}}exp(-{\sigma }_0^2{z}_{00}^2)-1\right],(c_0<0),$$

$${\omega }_{00}=c_{00}\frac{\pi }{2}{\left(\frac{{\pi }^2}{4}+1\right)}^{-\frac{1}{2}exp(-{\sigma }_0^2{z}_{00}^2)}-1.$$

Lemma 5.

For all sufficiently small μ, the asymptotic equalities

$$\begin{array}{cc}\hfill d(r)& =c_{00}\mu (1+o(1)),\hfill \\ \hfill \omega (r)& =\frac{\pi }{2}+{\omega }_{00}\mu (1+o(1)),\hfill \\ \hfill z(r)& =z_{00}+o(1)\hfill \end{array}$$

hold.

The justifications of Lemmas (4) and (5) are quite simple but cumbersome. Therefore, we omit them.

Further, we fix the value $r_0\in (0,\pi /2)$ and arbitrary values $r_1$ and $d_1$. Let

$$r=r_0+{\epsilon }^2{r}_1,d=d(r_0)+{\epsilon }^2{d}_1$$

(72)

in (17).

Below, let $\theta =\theta (\epsilon )\in [0,1)$ be the expression that complements the quantity $z(r_0){\epsilon }^{-1}$ to an integer. We study the asymptotic behavior of the close to imaginary axis roots of Equation (71). We denote them by ${\lambda }_m(\epsilon )$ and ${\overline{\lambda }}_m(\epsilon )$ ($m=0,\pm 1,\pm 2,\dots$). The equalities

$${\lambda }_m(\epsilon )=i\omega (r)+\epsilon i{R}_1(\theta +m)+{\epsilon }^2(R_{20}+{(\theta +m)}^2{R}_2)+\dots$$

(73)

hold, where

$$\begin{array}{ccc}\hfill R_1& =& {(1-r_{0}exp(-i\omega (r_0)))}^{-1}d(r_0)z(r_0)\times \hfill \\ & & (1+2i{\sigma }_0^2)exp(-{\sigma }_0^2{z}^2(r_0)+i{z}_0(r_0)),\hfill \\ \hfill R_{20}& =& {(1-r_{0}exp(-i\omega (r_0)))}^{-1}\times \hfill \\ & & [d_1(1-exp(-{\sigma }_0^2{z}^2(r_0)+i{z}_0(r_0))-1)-r_{1}exp(-i\omega (r_0))],\hfill \\ \hfill R_2& =& {(1-r_{0}exp(-i\omega (r_0)))}^{-1}[\frac{1}{2}{r}_{0}exp(-i\omega (r_0))R_1^2+\hfill \\ & & d(r_0)\left(2{\sigma }_0^2{z}^2(r_0)-\left({\sigma }_0^2+\frac{1}{2}\right)\right)exp(-{\sigma }_0^2{z}^2(r_0)+iz(r_0))].\hfill \end{array}$$

It is significant that

$$\Im R_1=0\mathrm{and}\Re R_2<0.$$

(74)

4.2. Construction of Quasinormal Form

We introduce the formal series

$$\begin{array}{cc}\hfill u=& \epsilon (exp(i\omega (r_0)t)\sum _{m=-\infty }^{\infty }{\xi }_m(\tau )exp(i(z(r_0){\epsilon }^{-1}+\theta +m)x+\epsilon i{R}_1(\theta +m)t)+\hfill \\ \hfill & exp(-i\omega (r_0)t)\sum _{m=-\infty }^{\infty }{\overline{\xi }}_m(\tau )exp(-i(z(r_0){\epsilon }^{-1}+\theta +m)x-\hfill \\ \hfill & \epsilon i{R}_1(\theta +m)t))+{\epsilon }^2{u}_2(t,\tau ,x,\epsilon )+{\epsilon }^3{u}_3(t,\tau ,x,\epsilon )+\dots ,\tau ={\epsilon }^{2}t.\hfill \end{array}$$

(75)

The above expression can be simplified significantly. Let

$$\xi (\tau ,y)=\sum _{m=-\infty }^{\infty }{\xi }_m(\tau )exp(imy),y=x+\epsilon R_{1}t.$$

Then, it is possible to proceed from (75) to the presentation

$$\begin{array}{cc}\hfill u=& \epsilon (exp(i(\omega (r_0)+\epsilon R_1\theta )t+i(z(r_0){\epsilon }^{-1}+\theta )x)\xi (\tau ,y)+\hfill \\ \hfill & exp(-i(\omega (r_0)+\epsilon R_1\theta )t-i(z(r_0){\epsilon }^{-1}+\theta )x)\overline{\xi }(\tau ,y))+\hfill \\ \hfill & {\epsilon }^2{u}_2(t,\tau ,x,y)+{\epsilon }^3{u}_3(t,\tau ,x,y)+\dots .\hfill \end{array}$$

(76)

The functions appearing here, $u_j(t,\tau ,x,y)$, are periodic with respect to t, x, and y.

We insert (76) into (17). Then, performing standard techniques, we determine $u_2(t,\tau ,x,y)$:

$$\begin{array}{cc}\hfill u_2(t,\tau ,x,y)=& u_{20}{\left|\xi (\tau ,y)\right|}^2+u_{21}{\xi }^2(\tau ,y)exp((2i(\omega (r_0)+\epsilon R_1\theta )t+2i(z(r_0){\epsilon }^{-1}+\theta )x)+\hfill \\ \hfill & {\overline{u}}_{21}{\overline{\xi }}^2(\tau ,y)exp((-2i\omega (r_0)+\epsilon R_1\theta )t-2i(z(r_0){\epsilon }^{-1}+\theta )x),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill u_{20}=-& 2cos\omega (r_0),\hfill \\ \hfill u_{21}=-& 2rcos(2\omega (r_0))[2i\omega (r_0)+r_{0}exp(-2i\omega (r_0))-\hfill \\ \hfill & d(r_0)(exp(-2i\omega (r_0)-4{\sigma }_0^2{z}^2(r_0))-1){]}^{-1}.\hfill \end{array}$$

At the next step, we obtain the equation for $u_3(t,\tau ,x,y)$. From its solvability condition in the indicated class of functions, we arrive at the boundary value problem for $\xi (\tau ,y)$, determining:

$$\frac{\partial \xi }{\partial \tau }=R_2\frac{{\partial }^2\xi }{\partial y^2}-i\theta R_2\frac{\partial \xi }{\partial y}+(R_{20}+{\theta }^2{R}^2)\xi +q\xi {\left|\xi \right|}^2,$$

(77)

$$\xi (\tau ,y+2\pi )\equiv \xi (\tau ,y).$$

(78)

The equality

$$\begin{array}{cc}\hfill q=& r_0{(1-r_{0}exp(-i\omega (r_0)))}^{-1}[2cos(\omega (r_0))(1+exp(-i\omega (r_0))-\hfill \\ \hfill & u_{21}(exp(i\omega (r_0))+exp(-2i\omega (r_0)))].\hfill \end{array}$$

holds for the coefficient q.

We introduce the notation to formulate the basic result. We arbitrarily fix the value ${\theta }_0\in [0,1)$, and let ${\epsilon }_n({\theta }_0)$ be a sequence such that ${\epsilon }_n({\theta }_0)\to 0$ as $n\to \infty$, and the equality $\theta ({\epsilon }_n({\theta }_0))={\theta }_0$ holds for each n.

From above, we obtain the following statement.

Theorem 8.

Let, for some $\theta ={\theta }_0$, the boundary value problem (77), (78) have the bounded solution ${\xi }_0(\tau ,y)$ as $\tau \to \infty$, $y\in [0,2\pi ]$. Then, under condition (47) and for $\epsilon ={\epsilon }_n({\theta }_0)$, the function

$$\begin{array}{cc}\hfill u_0(t,x,\epsilon )=& \epsilon (exp(i(\omega (r_0)+\epsilon R_1{\theta }_0)t+i(z(r_0){\epsilon }^{-1}+{\theta }_0)x){\xi }_0(\tau ,y)+\hfill \\ \hfill & exp(-i(\omega (r_0)+\epsilon R_1{\theta }_0)t-i(z(r_0){\epsilon }^{-1}+{\theta }_0)x){\overline{\xi }}_0(\tau ,y)),\hfill \end{array}$$

$$\tau ={\epsilon }^{2}t,y=x+\epsilon R_{1}t$$

satisfies the boundary value problem (17) up to $O({\epsilon }^2)$.

Therefore, the boundary value problem (77), (78) is the QNF for (17).

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

$$\dot{N}=r[1-N(t-T,x)]N+\gamma \left[{\int }_{-\infty }^{\infty }F(s)N(t,x+s)ds-N\right].$$

(79)

The dependence of the functions $N(t,x)$ on the spatial variable x is assumed to be periodic:

$$N(t,x+2\pi )\equiv N(t,x).$$

(80)

Thus, we fix the space $C_{[-T,0]\times [0,2\pi ]}$ as a phase space of the boundary value problem (79), (80).

Function $F(s)$ describing spatial interactions is defined by

$$F(s)={\displaystyle \frac{1}{\sqrt{\mu \pi }}}exp[-{\mu }^{-1}{(s+h)}^2],\mu >0.$$

(81)

We note that ${\int }_{-\infty }^{\infty }F(s)ds=1$. Apparently, study of the problem (79) and (80) is of great interest, provided that parameter $\mu$ appearing in (81) is sufficiently small:

$$0<\mu \ll 1.$$

(82)

This condition arises naturally in many applied problems (see, for example, [31]). The assumption that the coefficient $\gamma$ is large enough, i.e.,

$$\gamma \gg 1$$

(83)

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 $N_0=1$. 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 $F(s)$ is close to symmetric. In Part 1, the parameters ${\gamma }^{-1},\mu$ and $h^2$ 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 $\mu$ and $h^2$ compared to ${\gamma }^{-1}$. 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 $F(s)$ 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 ${\gamma }^{-1},\mu$, and h.

5.1.1. Slowly Oscillating Structures

Let

$$\epsilon ={\gamma }^{-1},0<\epsilon \ll 1.$$

Here, we assume that the parameters $\epsilon ,\mu$, and $h^2$ are of the same order: for some fixed, positive k and $h_1$, we obtain

$$\mu =k\epsilon ,h={\epsilon }^{1/2}{h}_1.$$

(84)

After dividing (79) by $\gamma$, we consider the resulting ‘main’ part, which is the linear boundary value problem

$$\epsilon {\displaystyle \frac{\partial u}{\partial t}}={\int }_{-\infty }^{\infty }F(s)u(t,x+s)ds-u,u(t,x+2\pi )\equiv u(t,x).$$

(85)

The characteristic equation of (85) is of the form

$$\epsilon {\lambda }_m=exp[i{\epsilon }^{1/2}{h}_{1}m-\epsilon k{m}^2]-1,m=0,\pm 1,\pm 2,\dots .$$

(86)

Hence, we obtain the asymptotic formulas

$${\lambda }_m=i{\epsilon }^{-1/2}{h}_{1}m-\left(k+{\displaystyle \frac{1}{2}}{h}_1^2\right)m^2+O({\epsilon }^{1/2})$$

(87)

for the roots ${\lambda }_m={\lambda }_m(\epsilon )$. Thus, infinitely many roots of Equation (86) tend to the imaginary axis as $\epsilon \to 0$. 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

$$u=\sum _{m=-\infty }^{\infty }{\xi }_m(t)exp(imy)+\epsilon u_1(t,y)+\dots ,$$

(88)

where $y=x+{\epsilon }^{1/2}{h}_{1}t$. We insert (88) into (79) and equate the coefficients at the same power of $\epsilon$. We obtain the infinite system of ordinary differential equations to determine the function ${\xi }_m(t)$. As it turns out, this system can be written as one complex parabolic equation of the Ginzburg–Landau type for the function

$$\xi (t,y)=\sum _{m=-\infty }^{\infty }{\xi }_m(t)exp(imy)$$

$${\displaystyle \frac{\partial \xi }{\partial t}}=\left(k+{\displaystyle \frac{1}{2}}{h}_1^2\right)\frac{{\partial }^2\xi }{\partial y^2}+r[1-\xi (t-T,y-\Delta )]\xi ,$$

(89)

$$\xi (t,y+2\pi )\equiv \xi (t,y),$$

(90)

where $\Delta =\Delta (\epsilon )={({\epsilon }^{-1/2}{h}_{1}T)}_{mod2\pi }$.

Theorem 9.

Let, for some fixed $\Delta ={\Delta }_0\in [0,2\pi ]$, the boundary value problem (89), (90) be bounded together with the time derivative solution ${\xi }_0(t,y)$ as $t\to \infty$. Then, as determined from the equality $\Delta (\epsilon )={\Delta }_0$, for the sufficiently small ${\epsilon }_n$, the function

$$N(t,x,{\epsilon }_n)={\xi }_0(t,x+{\epsilon }_n^{-1/2}{h}_{1}t)$$

satisfies the boundary value problem (79), (80) up to $O({\epsilon }_n^{1/2})$.

Theorem 10.

On conditions of Theorem 9, let ${\xi }_0(t,y)$ 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 ${\epsilon }_n$, the boundary value problem (79), (80) has a periodic solution $N_0(t,x,\epsilon )$ of the same stability as ${\xi }_0(t,y)$, and

$$N_0(t,x,\epsilon )={\xi }_0((1+o({\epsilon }_n^{1/2}))t,x+{\epsilon }_n^{-1/2}{h}_{1}t)+O({\epsilon }^{1/2}).$$

5.1.2. Rapidly Oscillating Structures

Here, we demonstrate that the decrease of diffusion coefficients (k and $h_1$) 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 $h_1$, 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.

Let

$$\mu ={\epsilon }^2{k}_1,h=\epsilon h_2$$

(91)

for some fixed positive $k_1$ and $h_2$.

We arbitrarily fix z as real, and let $\theta =\theta (\epsilon ,z)$ be the value from the semi-open interval $[0,1)$ that complements the expression $z{\epsilon }^{-1/2}$ to an integer. We consider the integer set

$$(z{\epsilon }^{-1/2}+\theta )m+n,m,n=0,\pm 1,\pm 2,\dots .$$

For these numbers, the characteristic Equation (86) has the set of roots similar to (87)

$${\lambda }_{m,n}=i{\epsilon }^{-1/2}{h}_{2}zm+i(\theta m+n)h_2-\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)z^2{m}^2+O({\epsilon }^{1/2}).$$

Then, the family of the boundary value problems depending on the parameter z

$${\displaystyle \frac{\partial \xi }{\partial t}}=z^2\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)\frac{{\partial }^2\xi }{\partial y^2}+h_2\left(\theta \frac{\partial \xi }{\partial y}+\frac{\partial \xi }{\partial v}\right)+r\xi [1-\xi (t-T,y-\delta ,v)],$$

(92)

$$\xi (t,y+2\pi ,v)\equiv \xi (t,y,v)\equiv \xi (t,y,v+2\pi )$$

(93)

plays a role in the boundary value problem (89), (90) in the considered case. Here, $\delta ={({\epsilon }^{-1/2}{h}_{2}z+h_2\theta )}_{mod2\pi }$.

Theorem 11.

Let, for some fixed $\delta ={\delta }_0$ and $z=z_0$, the boundary value problem (92), (93) be bounded together with a derivative with respect to t solution ${\xi }_0(t,y,v)$ as $t\to \infty$. Then, as determined from the equality $\delta (\epsilon )={\delta }_0$, for the sufficiently small ${\epsilon }_n$, the function

$$N(t,x,\epsilon )={\xi }_0\left(t,(z_0{\epsilon }^{-1/2}+\theta )x+({\epsilon }^{-1/2}{h}_2{z}_0+h_2\theta )t,x\right)$$

satisfies the boundary value problem (79), (80) up to $O({\epsilon }^{1/2})$.

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 $m_0$ and real numbers $z_1,\dots ,z_{m_0}$. We consider the set of integer numbers

$$\sum _{j=1}^{m_0}(z_j{\epsilon }^{-1/2}+{\theta }_j)m_j+n_j,m_j,n_j=0,\pm 1,\pm 2,\dots .$$

For modes with these numbers, the equation

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \xi }{\partial t}}=& {\left[z_1\frac{\partial }{\partial y_1}+\dots +z_{m_0}\frac{\partial }{\partial y_{m_0}}\right]}^2\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)\xi +h_2({\theta }_1\frac{\partial \xi }{\partial y_1}+\dots +{\theta }_{m_0}\frac{\partial \xi }{\partial y_{m_0}}+\hfill \\ \hfill & \frac{\partial \xi }{\partial v_1}+\dots +\frac{\partial \xi }{\partial v_{m_0}})+r[1-\xi (t-T,y_1-{\delta }_1,\dots ,y_{m_0}-{\delta }_{m_0},v_1,\dots ,v_{m_0})]\xi \hfill \end{array}$$

plays the role of the boundary value problem (92), (93) with $2\pi$-periodic boundary conditions with respect to each spatial variable, and

$${\delta }_j={({\epsilon }^{-1/2}{h}_2{z}_j+h_2{\theta }_j)}_{mod2\pi }.$$

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 $h_1$ in (89) are also asymptotically small.

In this part, the case of $k=\epsilon k_1$ and $h_1={\epsilon }^{1/2}{h}_2$ is considered. Of course, one can investigate a more general case, when for some positive $\alpha ,k_1$ and $h_2$, the equalities

$$k={\epsilon }^{1+\alpha }{k}_1,h_1={\epsilon }^{\frac{1+\alpha }{2}}{h}_2$$

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,

$$\mu ={\epsilon }^2{k}_1.$$

We consider the integer set $(2\pi h_1^{-1}{\epsilon }^{-1/2}+\theta )m+n$, where $\theta \in [0,1)$ is such that the expression in brackets is an integer, $m,n=0,\pm 1,\pm 2,\dots$. The characteristic Equation (86) has the set of roots ${\lambda }_{m,n}$ for which

$${\lambda }_{m,n}=i{h}_1(\theta m+n){\epsilon }^{-1/2}-\left(\frac{4{\pi }^2}{h_1^2}{k}_1{m}^2+{\displaystyle \frac{1}{2}}{h}_1^2{(\theta m+n)}^2\right)+O({\epsilon }^{1/2}).$$

We apply the above construction and obtain the resulting boundary value problem

$${\displaystyle \frac{\partial \xi }{\partial t}}=4{\pi }^2{h}_1^{-2}{k}_1\frac{{\partial }^2\xi }{\partial y^2}+\frac{h_1^2}{2}{\left(\theta \frac{\partial }{\partial y}+\frac{\partial }{\partial v}\right)}^2\xi +r[1-\xi (t-T,y-\delta ,v-\kappa )]\xi ,$$

(94)

$$\xi (t,y+2\pi ,v)\equiv \xi (t,y,v)\equiv \xi (t,y,v+2\pi ),$$

(95)

where

$$\delta ={(h_{1}T\theta {\epsilon }^{-1/2})}_{mod2\pi },\kappa ={(h_{1}T{\epsilon }^{-1/2})}_{mod2\pi }.$$

We introduce some notation in order to formulate an analogue of Theorem 9 in the considered situation. We fix an arbitrary $\kappa ={\kappa }_0\in [0,1)$ and the sequence ${\epsilon }_p\to 0$ of the roots of equation $\kappa ({\epsilon }_p)={\kappa }_0$. Let ${\theta }_0$ be the arbitrary limit point of the sequence $\theta ({\epsilon }_p)$. Let $\theta ({\epsilon }_{p_q})\to {\theta }_0$ for the sequence ${\epsilon }_{p_q}$. Finally, ${\sigma }_0\in [0,1)$ denotes the arbitrary limit point of the sequence $\sigma ({\epsilon }_{p_q})$, and let ${\epsilon }_R\subset {\epsilon }_{p_q}$ and $\sigma ({\epsilon }_R)\to {\sigma }_0$.

Theorem 12.

Let, for some ${\theta }_0,{\sigma }_0$, and ${\kappa }_0$, the boundary value problem (94), (95) be bounded together with a derivative with respect to t solution ${\xi }_0(t,y,v)$. Then, for $\sigma ={\sigma }_0$, $\kappa ={\kappa }_0$, $\theta ={\theta }_0$, and $\epsilon ={\epsilon }_R\to 0$, the function

$$N(t,x,\epsilon )={\xi }_0\left(t,(2\pi h_1^{-1}{\epsilon }_R^{-1/2}+{\theta }_0)x+h_1{\theta }_0{\epsilon }_R^{-1/2}t,x+h_1{\epsilon }^{-1/2}t\right)$$

satisfies the boundary value problem (79), (80) up to $O({\epsilon }^{1/2})$.

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 $\pi$. This means, that for some relatively prime integers $m_1$ and $m_2$,

$$h={\displaystyle \frac{\pi m_1}{m_2}}-{\epsilon }^{1/2}{h}_1.$$

(96)

We assume that $m=Mn$, where $n=0,\pm 1,\pm 2,\dots$, $M=2m_2$ if $m_1$ is even, or $M=2m_2$ if $m_1$ is odd. Here, we repeat the constructions from Section 5.1.1 and obtain, similar to (89), (90), the boundary value problem

$${\displaystyle \frac{\partial \xi }{\partial t}}=M^2\left(k+{\displaystyle \frac{1}{2}}{h}_1^2\right)\frac{{\partial }^2\xi }{\partial y^2}+r[1-\xi (t-T,y-\Delta )]\xi ,$$

$$\xi (t,y+2\pi )\equiv \xi (t,y).$$

The formula

$$N(t,x,\epsilon )=\xi \left(t,M(x-{\epsilon }^{-1/2}{h}_{1}t)\right)+O({\epsilon }^{1/2})$$

establishes a relation between its solutions and the solutions of problem (79), (80). Accordingly, for (91), we arrive at the boundary value problem

$$\begin{array}{c}\frac{\partial \xi }{\partial t}=z^2{M}^2\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)\frac{{\partial }^2\xi }{\partial y^2}+h_{2}M\frac{\partial \xi }{\partial v}+r[1-\xi (t-T,y-\delta ,v)]\xi ,\\ \xi (t,y+2\pi ,v)\equiv \xi (t,y,v)\equiv \xi (t,y,v+2\pi ).\end{array}$$

(97)

The solutions of this boundary value problem and of the boundary value problem (79), (80) are coupled by the equality

$$N(t,x,\epsilon )=\xi \left(t,M(z{\epsilon }^{-1/2}+\theta )x+M({\epsilon }^{-1/2}{h}_{2}z+h_2\theta ),Mx\right)+O({\epsilon }^{1/2}).$$

Satisfaction of only one condition $\mu ={\epsilon }^2{k}_1(h={\epsilon }^{1/2}{h}_1)$ is a more interesting situation. Let $\theta =\theta (\epsilon )\in [0,M)$ be a value which complements the expression $\pi m_1{(m_2{h}_1{\epsilon }^{1/2})}^{-1}$ to an integer multiple of M. We examine the set of integers $K=\left\{(\pi m_1{(m_2{h}_2{\epsilon }^{1/2})}^{-1}+\theta )p\right\}$, $p=0,\pm 1,\pm 2,\dots$. Then, for the roots of Equation (86) with numbers m from K and $n=0,\pm 1,\pm 2,\dots$, we obtain

$${\lambda }_{m,n}=i{h}_1(\theta p+n){\epsilon }^{-1/2}-\left[\frac{{\pi }^2{m}_1^2}{m_2^2}{k}_1{p}^2-{\displaystyle \frac{1}{2}}{h}_1^2{(\theta p+n)}^2\right]+O({\epsilon }^{1/2}).$$

Here, the boundary value problem

$$\begin{array}{c}\frac{\partial \xi }{\partial t}=\frac{{\pi }^2{m}_1^2}{m_2^2}{k}_1\frac{{\partial }^2\xi }{\partial y^2}+\frac{1}{2}{h}_1^2\left(\theta \frac{\partial }{\partial y}+\frac{\partial }{\partial v}\right)\xi +r[1-\xi (t-T,y-\delta ,v-\kappa )]\xi ,\\ \xi (t,y+2\pi ,v)\equiv \xi (t,y,v)\equiv \xi (t,y,v+2\pi )\end{array}$$

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 $\gamma$-Induced Bifurcations

Upon condition (83), we study the behavior of solutions of the boundary value problem (79), (80) in a sufficiently small equilibrium state $N_0\equiv 1$ neighborhood. The characteristic quasipolynomial of a boundary value problem linearized on $N_0$ has the form

$$\epsilon \lambda =-\epsilon rexp(-\lambda T)+exp(ihm-\mu m^2)-1,0<\epsilon \ll 1,m=0,\pm 1,\pm 2,\dots .$$

(98)

As in Section 5.1, we restrict ourselves to the most interesting and important situations depending on the relationship between the parameters $\epsilon ,h$, and $\mu$. In Section 5.2.1, we assume that condition (84) holds: $h=\sqrt{\epsilon }{h}_1$, $\mu =k\epsilon$. Below, we demonstrate that Andronov–Hopf bifurcation occurs in (79), (80) even for small values of the delay parameter $T\sim {\epsilon }^{1/2}$. The periodic solution bifurcating from the equilibrium state $N_0$ turns out to be rapidly oscillating in time. In Section 5.2.2, the relations $h=\epsilon h_2$, $\mu =k_1{\epsilon }^2$ 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 $\epsilon \to 0$. 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 $N_0$ for small $\epsilon$. 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 $\mu =k_1{\epsilon }^2$. 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 $2rT=\pi$ 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 $h={\epsilon }^{1/2}{h}_1$, $\mu =k\epsilon$

In (98), we assume

$$\lambda =i\omega (\omega >0),\omega ={\epsilon }^{-1/2}{h}_1+{\omega }_2,T={\epsilon }^{1/2}{T}_1.$$

Then, we obtain

$$\begin{array}{c}rcos(h_1{T}_{1m})=-\left(k+{\displaystyle \frac{1}{2}}{h}_1^2\right)m^2,\\ rsin(h_1{T}_{1m})={\omega }_2.\end{array}$$

(99)

up to $O({\epsilon }^{1/2})$. If it exists, let $T_{1m}$, $m=1,2,\dots$ be the least positive root of Equation (99). Obviously, there is a finite number of such roots. Let $T_1^0$ be the least of them. Let $T_1^0=T_{1m_0}$. We formulate two statements about the roots of Equation (98) that are simple but cumbersome proofs that are omitted.

Lemma 6.

Let $T_1<T_1^0$. Then, for all sufficiently small ε, the Equation (98) roots separate from zero negative real parts as $\epsilon \to 0$.

Lemma 7.

Let $T_1>T_1^0$. Then, for all sufficiently small ε, Equation (98) has a root separate from the zero positive real part as $\epsilon \to 0$.

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 $N_0$ neighborhood close to the critical case condition

$$T_1=T_1^0+{\epsilon }^{1/2}{T}_{01}.$$

Then, the characteristic Equation (98) has the coupling of the roots ${\lambda }_{1,2}(\epsilon )$ of the form

$${\lambda }_{1,2}(\epsilon )=\pm i({\epsilon }^{-1/2}{h}_1+{\omega }_2)+O({\epsilon }^{1/2}).$$

The real parts of the remaining roots of Equation (98) are negative and zero-separated as $\epsilon \to 0$. In this case, for small $\epsilon$, the boundary value problem (79), (80) has a two-dimensional stable local invariant integral manifold in the (small) $N_0$ neighborhood, on which this boundary value problem can be represented in the form of the scalar complex equation up to the summands of the $O({\epsilon }^{1/2})$ order:

$${\displaystyle \frac{\partial \xi }{\partial \tau }}=\alpha \xi +d{\left|\xi \right|}^2\xi ,$$

(100)

where $\tau ={\epsilon }^{1/2}t$ is a ‘slow’ time. The solution $N(t,x,\epsilon )$ on this manifold is coupled to solutions of Equation (100) by the relation

$$\begin{array}{cc}\hfill N(t,x,\epsilon )=& {\epsilon }^{1/4}[\xi (\tau )exp(i{m}_{0}x+i({\epsilon }^{-1/2}{h}_1{m}_0+{\omega }_2)t)+\overline{\xi }(\tau )exp(-i{m}_{0}x-\hfill \\ \hfill & i({\epsilon }^{-1/2}{h}_1{m}_0+{\omega }_2)t)]+{\epsilon }^{1/2}{u}_2(t,\tau ,x)+{\epsilon }^{3/4}{u}_3(t,\tau ,x)+\dots .\hfill \end{array}$$

(101)

Here, the functions $u_j(t,\tau ,x)$ are periodic with respect to first and third arguments, with periods $2\pi {({\epsilon }^{-1/2}{h}_1{m}_0+{\omega }_2)}^{-1}$ and $2\pi$, respectively.

Regarding the dynamics of Equation (100) (and hence the boundary value problem (79), (80) in the $N_0$ neighborhood), one needs to find the coefficients $\alpha$ and d. To accomplish this, we insert the formal series (101) into (79) and equate the coefficients at the same powers of $\epsilon$. At the second step, we obtain

$$\begin{array}{cc}\hfill u_2(t,\tau ,x)=& u_{20}{\left|\xi \right|}^2+u_{21}{\xi }^{2}exp(2i{m}_{0}x+2i({\epsilon }^{-1/2}{h}_1{m}_0+{\omega }_2)t)+\hfill \\ \hfill & {\overline{u}}_{21}{\overline{\xi }}^{2}exp(-2i{m}_{0}x-2i({\epsilon }^{-1/2}{h}_1{m}_0+{\omega }_2)t)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill u_{20}& =& 2cos(T_1^0{h}_1{m}_0),\hfill \\ \hfill u_{21}& =& -r{\left(exp(-2i{T}_1^0{h}_1{m}_0)(2i{\omega }_2+4m_0^{2}k+rexp(-2i{T}_1^0{h}_1{m}_0))\right)}^{-1}.\hfill \end{array}$$

From the solvability condition obtained at the third-step equation with respect to $u_3(t,\tau )$, we obtain

$$\begin{array}{ccc}\hfill d& =-& r[1+exp(-2i{T}_1^0{h}_1{m}_0)-rexp(-i{T}_1^0{h}_1{m}_0)(2i{\omega }_2+4m_0^{2}k+\hfill \\ \hfill & & rexp(-2i{T}_1^0{h}_1{m}_0){)}^{-1}],\hfill \\ \hfill \alpha & =& ir{h}_1{T}_1^0{m}_{0}exp(-i{T}_1^0{h}_1{m}_0)-\left(k+{\displaystyle \frac{1}{2}}{h}_1^2\right)m_0^2.\hfill \end{array}$$

We note that each of the quantities $\Re \alpha$ and $\Re d$ can be either positive or negative, depending on the values of the parameters $T_1^0$, $m_0$, r, k, and $h_1$. As an example, we make one statement about the dynamics of problem (79), (80).

Theorem 13.

Let $\Re \alpha >0$ and $\Re d<0$. Then, for all sufficiently small ε, the boundary value problem (79), (80) has a stable periodic solution $N_0(t,x,\epsilon )$ in the $N_0$ neighborhood, for which

$$N_0(t,x,\epsilon )=1+2{\epsilon }^{1/4}{\rho }_{0}cos(m_{0}x+{\epsilon }^{-1/2}{h}_1{m}_0+{\omega }_2+{\epsilon }^{1/2}{\varphi }_0+O({\epsilon }^{3/4}))t+O({\epsilon }^{1/2}),$$

where

$${\rho }_0={\left(\Re \alpha {(-\Re d)}^{-1}\right)}^{1/2},{\phi }_0=\Im \alpha +{\rho }_0^2\Im d.$$

Remark 2.

The results presented here could be obtained by studying the bifurcations from the equilibrium state ${\xi }_0\equiv 1$ in the boundary value problem (89), (90).

5.2.2. High-Mode Bifurcations upon Condition $\mu ={\epsilon }^2{k}_1,h=\epsilon h_2$

It is assumed here that the conditions $h=\epsilon h_2$, $\mu =k_1{\epsilon }^2$ are satisfied. Under these conditions, we study the behavior of the solutions of the problem (79), (80) from a small $N_0$ neighborhood. First, we consider the characteristic Equation (98). For $T=0$, its roots have negative real parts (separated from zero as $\epsilon \to 0$). Let us demonstrate that there are roots with a close to zero real part as $T\sim {\epsilon }^{1/2}$, and the numbers (modes) $m\sim {\epsilon }^{-1/2}$ correspond to them. In (98), we assume

$$T={\epsilon }^{1/2}{T}_1,m=c{\epsilon }^{-1/2}+{\theta }_c,\lambda =i{\epsilon }^{-1/2}{h}_{2}c+i\omega (c),$$

where ${\theta }_c={\theta }_c(\epsilon )\in [0,1)$ is the value that complements the previous summand $c{\epsilon }^{-1/2}$ to an integer. Then, we obtain the equality

$$i\omega (c)=-rexp(-i{h}_2{T}_{1}c)-\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)c^2$$

(102)

(up to $O({\epsilon }^{1/2})$). Let us determine the least value of $T_1=T_1^0$ for which this equation is solvable with respect to c, i.e., for some $c=c_0$, the relation

$$rcos(h_2{T}_1^0{c}_0)=-\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)c_0^2$$

(103)

holds. We denote by $x_0$ the least positive root of the equation

$$2x=-x.$$

Then, the following simple statements hold.

Lemma 8.

For $T_1^0$ and $c_0$, the equalities

$$\begin{array}{ccc}\hfill T_1^0& =& {\displaystyle \frac{1}{2}}{x}_0{\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)}^{1/2}{(2-rcos{x}_0)}^{-1/2},\hfill \\ \hfill c_0& =& {(-rcos{x}_0)}^{-1/2}{\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)}^{-1/2}\hfill \end{array}$$

hold.

Lemma 9.

Let $T_1<T_1^0$. Then, for all sufficiently small ε, the roots of Equation (98) have negative real parts (separated from zero as $\epsilon \to 0$).

Lemma 10.

Let $T_1>T_1^0$. Then, for all sufficiently small ε, Equation (98) has a root with a positive real part (separated from zero as $\epsilon \to 0$).

In the context of Lemmas 9 and 10, the behavior of the solutions of the problem (79), (80) in small ($\epsilon$-independent) neighborhood of $N_0$ is determined in the standard way.

Below, we assume that a case close to critical in the stability problem of $N_0$ is realized. Let, for some arbitrary constant $T_{11}$, the equality

$$T_1=T_1^0+{\epsilon }^{1/2}{T}_{11}$$

hold. In this case, infinitely many roots of Equation (98) tend to the imaginary axis as $\epsilon \to 0$, and there are no roots with a positive zero-separated as $\epsilon \to 0$ real part. Thus, the critical case of an infinite dimension is realized in the $N_0$ stability problem. Let us apply the technique of [26,28] to study the local dynamics of problem (79), (80) for small $\epsilon$.

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 $c_0{\epsilon }^{-1/2}$. In this regard, we consider all modes with numbers $m=m(\epsilon )$ for which

$$m(\epsilon )=c_0{\epsilon }^{-1/2}+{\theta }_0+b{\epsilon }^{-1/4}+{\theta }_1,$$

(104)

where ${\theta }_0={\theta }_{c_0}$, b is arbitrarily fixed, and ${\theta }_1=\theta (b,\epsilon )\in [0,1)$ complements the value of $b{\epsilon }^{-1/4}$ to an integer. The roots $\lambda ={\lambda }_{m(\epsilon )}$ of Equation (98) with numbers $m(\epsilon )$ satisfy the asymptotic equalities

$$\lambda =i{h}_2{c}_0{\epsilon }^{-1/2}+i{h}_{2}b{\epsilon }^{-1/4}+i\omega (c_0)+i{\epsilon }^{1/4}\Delta b+{\epsilon }^{1/2}{\lambda }_2+O({\epsilon }^{3/4}).$$

Here, the following designations are accepted:

$$\Delta =r{h}_2{T}_1^{0}exp(-i{h}_2{T}_1^0{c}_0)+i2c_0\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right),\Im \Delta =0,$$

$${\lambda }_2=-\sigma b^2-2c_0{\theta }_0\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)+rexp(-i{h}_2{T}_1^0{c}_0)\left(i\omega (c_0)T_1^0+i{h}_2{c}_0{T}_{11}\right),$$

where

$$\sigma =\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)\left(1+{\displaystyle \frac{1}{2}}{c}_0^2{h}_2^2{(T_1^0)}^2\right)+i\omega (c_0){\displaystyle \frac{1}{2}}{h}_2^2{(T_1^0)}^2.$$

Another notation is used below. Let $\Omega =\Omega (\epsilon )$ be the set of all such values of b for which the values of $b{\epsilon }^{-1/4}$ are integers from $-\infty$ to ∞.

Let us introduce the formal series

$$\begin{array}{cc}\hfill N=& 1+{\epsilon }^{1/4}[exp((i{h}_2{\epsilon }^{-1/2}{c}_0+i\omega (c_0))t+\hfill \\ \hfill & (i{\epsilon }^{-1/2}{c}_0+{\theta }_0)x)\sum _{b\in \Omega }{\xi }_b(\tau )expiby+exp(-(i{h}_2{\epsilon }^{-1/2}{c}_0+i\omega (c_0))t-\hfill \\ \hfill & i({\epsilon }^{-1/2}{c}_0+{\theta }_0)xby)\sum _{b\in \Omega }{\overline{\xi }}_b(\tau )exp(-iby)+{\theta }_{0}x]+{\epsilon }^{1/2}{u}_2(t,\tau ,x,y)+\hfill \\ & {\epsilon }^{3/4}{u}_3(t,\tau ,x,y)+\dots ,\hfill \end{array}$$

(105)

where

$$\tau ={\epsilon }^{1/2}t,y={\epsilon }^{-1/4}x+({\epsilon }^{-1/4}{h}_2+{\epsilon }^{1/4}\Delta )t.$$

The dependence on the first, third, and fourth arguments of function $u_j(t,\tau ,x,y)$ is periodic, with periods $2\pi {(h_2{\epsilon }^{-1/2}{c}_0+\omega (c_0))}^{-1}$, $2\pi ({\epsilon }^{-1/2}{c}_0+{\theta }_0$, and $2\pi$, respectively. We insert (105) into (79) and perform standard operations. At the second step, we obtain

$$u_2(t,\tau ,x,y)=u_{20}(\tau ,y){\left|\xi \right|}^2+u_{21}(t,\tau ,x,y){\xi }^2+{\overline{u}}_{21}(t,\tau ,x,y){\overline{\xi }}^2$$

and

$$\begin{array}{ccc}\hfill u_{20}(\tau ,x)& =& 2(cos{x}_0){\left|\xi \right|}^2,\hfill \\ \hfill u_{21}& =& -rexp(-2i{x}_0){\left[2i\omega (c_0)+rexp(2i{x}_0)+\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)4c_0^2\right]}^{-1}\times \hfill \\ \hfill & & exp\left(2i[(h_2{\epsilon }^{-1/2}{c}_0+\omega (c_0))t+({\epsilon }^{-1/2}{c}_0+{\theta }_0)x]\right).\hfill \end{array}$$

At the third step, from the solvability condition of the resulting equation with respect to $u_3$, we obtain the equation for determining the unknown amplitude

$$\begin{array}{c}\xi (\tau ,y)=\left({\displaystyle \sum _{b\in \Omega }}{\xi }_{b}expiby\right)exp(i[r\omega (c_0)T_1^0+h_2{c}_0{T}_{11}cos{x}_0-\\ 2c_0{\theta }_0\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)]),\\ {\displaystyle \frac{\partial \xi }{\partial \tau }}=\sigma \frac{{\partial }^2\xi }{\partial y^2}+\beta \xi +d{\left|\xi \right|}^2\xi ,\end{array}$$

(106)

where

$$\begin{array}{ccc}\hfill \beta & =& r{h}_2{c}_0{T}_{11}sin{x}_0\hfill \\ \hfill d& =& -r\{2(cos{x}_0)(1+exp(-i{x}_0))-\hfill \\ \hfill & & (exp(i{x}_0)+exp(-2i{x}_0))rexp(-2i{x}_0)[2i\omega (c_0)+\hfill \\ \hfill & & rexp(-2i{x}_0)+\left(k_1+{\displaystyle \frac{1}{2}}{h}_2^2\right)4c_0^2{]}^{-1}\}.\hfill \end{array}$$

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 $\xi (\tau ,y)$. Then, for $\epsilon \to 0$, the function

$$\begin{array}{ccc}\hfill N(t,x,\epsilon )& =& 1+{\epsilon }^{1/4}[\xi ({\epsilon }^{1/2}t,({\epsilon }^{-1/4}+{\theta }_R)x+({\epsilon }^{-1/4}{h}_2+{\epsilon }^{1/4}\Delta )t)\times \hfill \\ \hfill & & exp(i[h_2{\epsilon }^{-1/4}{c}_0+\omega (c_0)]t+i[{\epsilon }^{-1/2}{c}_0+{\theta }_0]x)+\overline{\xi }({\epsilon }^{1/2}t,({\epsilon }^{-1/4}+{\theta }_R)x+\hfill \\ \hfill & & ({\epsilon }^{-1/4}{h}_2+{\epsilon }^{1/4}\Delta )t)exp(-i[h_2{\epsilon }^{-1/4}{c}_0+\omega (c_0)]t-i[{\epsilon }^{-1/2}{c}_0+{\theta }_0]x)]+\hfill \\ \hfill & & {\epsilon }^{1/2}{u}_2(t,\tau ,y,x)\hfill \end{array}$$

satisfies the boundary value problem (79), (80) up to $O({\epsilon }^{3/4})$, where the expression ${\theta }_R={\theta }_R(\epsilon )\in [0,2\pi /R)$ complements the summand ${\epsilon }^{-1/4}$ to an integer multiple of $2\pi /R$.

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 $\epsilon ,h$, and $\mu$. 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 $N_0$ neighborhood.

We consider a model situation where

$$h=\epsilon h_2,\mu =k_0{\epsilon }^{7/2},T={\epsilon }^{1/2}{T}_1.$$

(107)

Let $m=m(\epsilon )$ be such asymptotically large modes for which

$$m(\epsilon )={\displaystyle \frac{2\pi h}{h_2}}{\epsilon }^{-1}+c{\epsilon }^{-1/2}+{\theta }_1+b{\epsilon }^{-1/4}+{\theta }_2,$$

(108)

where $n=0,\pm 1,\pm 2,\dots$, the parameter c is determined below, the values of b are arbitrary, the value ${\theta }_1={\theta }_1(\epsilon )\in [0,1)$ complements the sum of two previous terms to an integer, and ${\theta }_2={\theta }_2(\epsilon )\in [0,1)$ complements the expression $b{\epsilon }^{-1/4}$ to an integer. For the roots of the characteristic Equation (98) with numbers (108), the asymptotic formulas

$$\lambda =-rexp(-\lambda {\epsilon }^{-1/2}{T}_1)+i{h}_{2}c{\epsilon }^{-1/2}+i{h}_{2}b{\epsilon }^{-1/4}+i({\theta }_1+{\theta }_2)h_2-{\displaystyle \frac{1}{2}}{h}_2^2{c}^2+\dots$$

hold. In order to determine the stability boundary of $N_0$ for the problem (79), (80) in the range of parameter $T_1$, we assume

$$\lambda =i{h}_{2}c{\epsilon }^{-1/2}+i{h}_{2}b{\epsilon }^{-1/4}+i\omega +{\epsilon }^{1/4}{\lambda }_1+{\epsilon }^{1/2}{\lambda }_2+\dots .$$

(109)

At the first step of the asymptotic analysis, from (109), we arrive at the equality

$$i\omega =-rexp(-i{h}_{2}c{T}_1)-\frac{1}{2}{h}_2^2{c}^2.$$

(110)

The least value of $T_1=T_1^0$ for which Equation (110) is solvable for some $c=c_0$ is determined from the relations (103) as $k_1=0$, i.e.,

$$\begin{array}{ccc}\hfill c_0& =& {(-rcos{x}_0)}^{1/2}{2}^{-1/2}{h}_2,\hfill \\ \hfill T_1^0& =& 2^{-3/2}{x}_0{h}_2{(-rcos{x}_0)}^{1/2}.\hfill \end{array}$$

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 $N_0$ is realized. Let, for some constant $T_{11}$, the equality

$$T_1=T_1^0+{\epsilon }^{1/2}{T}_{11}$$

hold. Similar to the previous part, infinitely many roots of Equation (98) tend to the imaginary axis as $\epsilon \to 0$, and there are no roots with a positive zero-separated as $\epsilon \to 0$ real part. However, there are ‘essentially’ more such roots in this case. Let us explain the above. For this purpose, we obtain expressions for ${\lambda }_1$ and ${\lambda }_2$:

$$\begin{array}{ccc}\hfill {\lambda }_1& =& i{\Delta }_{0}b,{\Delta }_0=r{h}_2{T}_1^{0}exp(-x_0)+i{c}_0{h}_2^2,\Im \Delta =0,\hfill \\ \hfill {\lambda }_2& =& -\sigma b^2-k_0{h}_2^{-2}4{\pi }^2{n}^2+{\beta }_0,\hfill \\ \hfill {\beta }_0& =& -c_0{\theta }_1(c_0)h_2^2+rexp(-x_0)\left(i{\omega }_0{T}_1^0+i{h}_2{c}_0{T}_{11}\right),\hfill \\ \hfill i{\omega }_0& =& -rexp(-x_0)-{\displaystyle \frac{1}{2}}{h}_2^2{c}_0^2,\hfill \\ \hfill {\sigma }_0& =& {\displaystyle \frac{1}{2}}{h}_2^2\left(1+{\displaystyle \frac{1}{2}}{c}_0^2{h}_2^2{(T_1^0)}^2\right)+i{\omega }_0{\displaystyle \frac{1}{2}}{h}_2^2{(T_1^0)}^2.\hfill \end{array}$$

(111)

The expression (111) differs from the similar formula for ${\lambda }_2$ in the previous part by the presence of parameter n, which takes all integer values $n=0,\pm 1,\pm 2,\dots$. Hence, we obtain that the quantity ${\xi }_b$ is also an $h_2$-periodic function with respect to x in the basic formula of the form (105) in the considered case: ${\xi }_b={\xi }_b(\tau ,x)$. Thus, we repeat the Equation (106) construction technique and obtain the more complicated equation

$${\displaystyle \frac{\partial \xi }{\partial \tau }}={\sigma }_0\frac{{\partial }^2\xi }{\partial y^2}+k_0\frac{{\partial }^2\xi }{\partial x^2}+{\beta }_0\xi +d_0{\left|\xi \right|}^2\xi$$

(112)

with $h_2$-periodic boundary conditions with respect to x, and $d_0$ differs from the value d appearing in (106) only by the presence of $k_1=0$ 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 $\mu$

To complete the picture, we briefly consider the simplest situation when both parameters h and $\mu$ are ‘intelligently’ small:

$$h={\epsilon }^2{h}_0,\mu ={\epsilon }^2{k}_0.$$

(113)

The bifurcation value of the delay coefficient $T=T_0$ satisfies the equality $T=\pi {(2r)}^{-1}$. Let $T=T_0+\epsilon T_{01}$ in (79), (80). Then, infinitely many roots ${\lambda }_m={\lambda }_m(\epsilon )$ of Equation (98) tend to the imaginary axis as $\epsilon \to 0$, and there are no roots with a positive zero-separated as $\epsilon \to 0$ real part. Therefore, the critical case of an infinite dimension is realized here, too.

Let us introduce the formal series

$$\begin{array}{cc}\hfill N=1+& {\epsilon }^{1/2}\left[exp\left(i{\displaystyle \frac{\pi }{2T_0}}t\right)\xi (\tau ,x)+exp\left(-i{\displaystyle \frac{\pi }{2T_0}}t\right)\overline{\xi }(\tau ,x)\right]+\hfill \\ \hfill +& \epsilon u_2(t,\tau ,x)+{\epsilon }^{3/3}{u}_2(t,\tau ,x)+\dots ,\hfill \end{array}$$

(114)

where $\tau =\epsilon t$, and $u_j(t,\tau ,x)$ are periodic with respect to first and third arguments with $4T_0$ and $2\pi$ 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 $\xi (\tau ,x)$:

$${\displaystyle \frac{\partial \xi }{\partial \tau }}={\left(1+i\frac{\pi }{2}\right)}^{-1}\left[k_0\frac{{\partial }^2\xi }{\partial x^2}+h_0\frac{\partial \xi }{\partial x}+r_0^2{T}_{11}\right]+g{\left|\xi \right|}^2\xi ,$$

(115)

$$\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x),$$

(116)

where

$$g=-r[3\pi -2+i(\pi +6)]{\left(10\left(1+\frac{{\pi }^2}{4}\right)\right)}^{-1}.$$

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 ${\xi }_0(\tau ,x)=Const·exp(i\omega \tau +ikx),$ 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

$${\epsilon }^{1/2}\left[{\xi }_0(\tau ,x)exp\left(i\left({\displaystyle \frac{\pi }{2T_0}}+O(\epsilon )\right)t\right)+{\overline{\xi }}_0(\tau ,x)exp\left(-i\left({\displaystyle \frac{\pi }{2T_0}}+O(\epsilon )\right)t\right)\right]$$

as $\epsilon \to 0$.

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 $\theta =\theta (\epsilon )\in [0,1)$. We recalled that the dynamic properties of the initial system are determined by the QNF (49), (50), which includes the parameter $\theta$. 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 $\epsilon \to 0$.

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 $\sigma$ of the function $F(s,\epsilon )$ ((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 $·exp(i\omega t+ikx)$. 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.