Inﬁnite–Dimensional Bifurcations in Spatially Distributed Delay Logistic Equation

: This paper investigates the questions about the local dynamics in the neighborhood of the equilibrium state for the spatially distributed delay logistic equation with diffusion. The critical cases in the stability problem are singled out. The equations for their invariant manifolds that determine the structure of the solutions in the equilibrium state neighborhood are constructed. The dominant bulk of this paper is devoted to the consideration of the most interesting and important cases of either the translation (advection) coefﬁcient is large enough or the diffusion coefﬁcient is small enough. Both of this cases convert the original problem to a singularly perturbed one. It is shown that under these conditions the critical cases are inﬁnite–dimensional in the problems of the equilibrium state stability for the singularly perturbed problems. This means that inﬁnitely many roots of the characteristic equations of the corresponding linearized boundary value problems tend to the imaginary axis as the small parameter tends to zero. Thus, we are talking about inﬁnite–dimensional bifurcations. Standard approaches to the study of the local dynamics based on the application of the invariant integral manifolds methods and normal forms methods are not applicable. Therefore, special methods of inﬁnite–dimensional normalization have been developed which allow one to construct special nonlinear boundary value problems called quasinormal forms. Their nonlocal dynamics determine the behavior of the initial boundary value problem solutions in the neighborhood of the equilibrium state. The bifurcation features arising in the case of different boundary conditions are illustrated.


Introduction
We consider the spatially distributed delay logistic equation with the periodic boundary conditions The coefficient d > 0 is called the diffusion coefficient or the mobility coefficient when it comes to a biological population. The coefficient r > 0 is called the Malthusian coefficient and T > 0 is the delay time. The presence of the translation operator b∂u/∂x in the boundary value problem (1), (2) differs from the logistic equation with diffusion. The coefficient b in this operator can be considered positive. The function u(t, x) makes sense of the population density and therefore u(t, x) ≥ 0. The translation operator is irrelevant for the boundary value problem without delay It 'disappears' after the spatial variable replacement x → x + bt. An equation of the (1) type arises in many applied problems of mathematical ecology and mathematical biology (see, for example, [1][2][3][4][5][6][7]). The most complete research results are presented in [8][9][10].
In this paper, we study the local dynamics of the boundary value problem (1), (2) in a neighborhood of a positive equilibrium state, that is, the behavior of the (1), (2) solutions with initial conditions from some sufficiently small neighborhood of the equilibrium state u 0 ≡ 1. We fix the space C [−T,0] × W 2 2 [0,2π] as the space of initial conditions. We pay special attention to the study of cases when either the translation coefficient b is sufficiently large or the diffusion coefficient is sufficiently small. It is in these cases that the boundary value problem (1), (2) becomes singularly perturbed, which can lead to the appearance of new interesting dynamic effects.
We recall the well-known (see, for example, [11,12]) results for the delay logistic equatioṅ Under the condition rT ≤ π/2, the equilibrium state u 0 ≡ 1 is asymptotically stable, and it is unstable when rT > π/2 and there is a stable cycle in (4). The asymptotic behavior of this cycle under the condition 0 < rT − π/2 1 is given in [13]. Questions about the global stability of Equation (4) are studied in [11,12,14]. Under the condition rT 1, the asymptotic stability of the cycle is described in [15]. We recall a well-known result of the Andronov-Hopf bifurcation in (4) under the conditions rT ≈ π 2 . We fix the values r 0 and T 0 in (4) so that r 0 T 0 = π 2 . Let r = r 0 + εr 1 , T = T 0 + εT 1 (5) where ε is a small positive parameter: Then in some sufficiently small and ε independent neighborhood of Equation (4) solution u 0 there exists [16][17][18] a stable local invariant integral two-dimensional manifold M(ε) on which Equation (4) can be written in the form of a scalar complex ordinary differential equation to within O(ε). Here, τ = εt is a 'slow' time and Equation (6) is called the normal form for (1), (2) in the neighborhood of u 0 . The solutions (6) and (4) are related by the asymptotic equality Accordingly, the cycle in (6) corresponds to the stable cycle in (4) (as α > 0).
Under the conditions (5) and for b = 0, the same manifold M(ε) is a stable invariant manifold for the delay logistic equation with diffusion Therefore, the equilibrium state of u 0 is stable as rT ≤ π 2 and is unstable as rT > π 2 for this equation, and the same cycle as in Equation (4) exists. The cycle bifurcation for (4) and (8) is of the Andronov-Hopf type [10,19]. There is only one pair of pure imaginary roots, whereas other roots of the characteristic equations for the linearized on u 0 equations have negative real parts as ε = 0.
We get back to the boundary value problem (1), (2) consideration. Its local dynamics in the equilibrium state of u 0 neighborhood depend largely on the behavior of solutions of the linearized on u 0 boundary value problem In turn, the behavior of Equation (9) solutions is related to the location of the roots of its characteristic quasi-polynomial, which consists of the set of the equations λ = −dk 2 + ibk − r exp(−λT), k = 0, ±1, ±2, . . . . (10) In the case when the roots of (10) have negative real parts, all the solutions of (9) tend to zero as t → ∞ and the equilibrium state of u 0 is asymptotically stable in (1), (2). However, if a root with a positive real part exists in (10), then (9) has a solution that grows exponentially as t → ∞ and the solution of u 0 in (1), (2) is unstable.The critical case in the problem of u 0 stability takes place under the condition that (10) has no roots with positive real part, but a root with zero real part exists.
In this paper, we focus our attention on the determination of the parameters for which critical cases take place and on the study of the (1), (2) solutions in near-critical situations.
Below we show that the bifurcation phenomena are much more complicated and diverse for the boundary value problem (1), (2) than those that take place for the boundary value problem (8). In the case of singular perturbations when b 1 or d 1 some interesting situations may arise when infinitely many roots of the characteristic Equation (10) tend to the imaginary axis as the small parameter tends to zero. Thus, the critical case of infinite dimension is realized in the problem of the solutions stability. Note that singular perturbations in a nonlocal setting were studied, for example, in [20][21][22].
Special nonlinear equations that do not contain small parameters are constructed as the main results. Their nonlocal dynamics determine the behavior of the boundary value problem (1), (2) solutions in the neighborhood of the equilibrium state of u 0 . These equations are classical normal forms on invariant manifolds in finite-dimensional critical cases. There are no invariant manifolds in infinite-dimensional critical cases, but the formal method of normal forms allows us to construct special boundary value problems of the parabolic type, the so-called quasinormal forms, which play the role of normal forms. Asymptotic formulas that couple the solutions of the initial problem and the solutions of the quasinormal forms are given.
In the next section, the critical cases are defined on the basis of the characteristic Equation (10) roots analysis, and the bifurcations when the parameter b is changed are studied. Moreover, main attention is paid to the singularly perturbed case when b 1. The most interesting situations that arise at asymptotically small values of the diffusion coefficient d are considered in Section 3. The infinite-dimensional bifurcations for the Dirichlet boundary conditions with sufficiently large values of the delay coefficient are considered in Section 4. Finally, the conclusions are formulated in Section 5.

Determined by Translation Coefficient b Bifurcations
In this section, we first focus on the analysis of the characteristic Equation (10) roots, and then we consider the bifurcation problem of constructiong a normal form. In the final part, we investigate the dynamics of the boundary value problem (1), (2) for large values of b.

Linear Analysis
We assume λ = iω where ω > 0 to construct the boundaries of the stability domain in the space of parameters of Equation (10) for some k = k 0 . We obtain from (10) that This equation is equivalent to the system of two equations 2.1.1. Case of k 0 = 0 From (10) we obtain the equation which is a well-known characteristic equation for the classical delay logistic equation. Therefore, we conclude that there is a root with a positive real part in (14) and hence in (10) under the condition rT > π 2 .
Below we assume that the inequality holds.
There is a pair of the complex conjugate roots λ 1,2 = ±iπ(2T) −1 for rT = π 2 and the other roots of (14) have negative real parts.
2.1.2. Case of k 0 = 1 We state one simple proposition first.
hold. Then the roots of Equation (10) have negative real parts.
Indeed, it follows from (16) that Equations (12) and (13) are unsolvable. We consider the case where k = 1 and r = d. It then follows from (12) that the equality ωT = π(2n + 1) holds for some integer n, and from (13) we obtain that π(2n + 1)T −1 = b. Below, let T k (r, b) (k = 0, 1, 2, . . .) stand for such a value of the parameter T that for 0 < T < T k (r, b) the roots of (10) have negative real parts for the given k, and there is a root on the imaginary axis as T = T k (r, b). Thus, T 0 (r, b) = T 0 (r) = π(2r) −1 , and under the condition r = d the equality T 1 (d, b) = πb −1 holds.
Further, we consider the case where Under this condition and for all values of the parameters b and T, the roots of all of Equation (10) have negative real parts as |k| > 1. From (12), (13) we conclude that We note that T 1 (r, 0) > T 0 (r) and lim In this case, each element of Equation (10) has pure imaginary roots as k = 0 and k = ±1. For example, the equality holds. Acting in accordance with the previous scheme, we obtain that Let k 0 stand for the largest integer k > 0 for which the inequality dk 2 0 ≤ r holds. We assume that

Lemma 2.
Under the condition 0 < T < T min (r, b) the roots of the characteristic Equation (10) have negative real parts, and Equation (10) has no roots with positive real part but the root on the imaginary axis as T = T min (r, b).
It is important to note that the parameter T increment from T min (r, b) to T 0 (r) in the boundary value problem (9) can lead to several alternations of stability and instability of solutions.
The following statement is more interesting.

Lemma 3.
Let the solutions of (9) be unstable for some value of the parameter b. Then stability and instability of the (9) solutions alternate infinitely as b → ∞.
If 0 ≤ T < T + 1 then the roots of (10) have negative real parts as k = k 0 ; 2.
If ω 2 < 0 and T > T + 1 then Equation (10) has a root with positive real part as k = k 0 ; 3.
If ω 2 > 0 and T + 3 < T − 2 then Equation (10) has a root with positive real part for all T > T + 1 as k = k 0 ; 5.
If ω 2 > 0 and T − 2 < T < T + 3 then the roots of (10) have negative real parts as k = k 0 . More generally, under the conditions T − 2n < T < T + 2n+1 the roots of (10) have negative real parts as k = k 0 . The resulting domain of instability (in the space of parameters) of the characteristic Equation (10) is an union of the instability domains of each of the equations that make up (10). Thus, a situation is possible when this domain consists of one or several (because their number is finite) intervals.

Andronov-Hopf Bifurcation
Let for some T = T 0 and k = k 0 (the case of k = 0 is studied in [19]) the characteristic Equation (10) has one pure imaginary root λ = iω, whereas all the other roots have negative real parts (as k ≥ 0). We assume ω = ω 1,2 and let the equalities (5) hold. We consider the behavior of the (8) solutions with initial conditions from some rather small (ε-independent) neighborhood of the equilibrium state N ≡ 1. According to the general theory (see, for example, [16][17][18]) in this neighborhood there is a local invariant two-dimensional stable integral manifold on which (8) can be presented as a normal form to within O(ε). We put z = ωt + k 0 x to obtain explicit expressions for the coefficients α 1 and β 1 , and introduce the formal series Substituting (18) into (1) and collecting the coefficients at the equal powers of ε we obtain in the second step that u 20 = 2r −1 0 cos ωT 0 , From the solvability condition of the resulting equation with respect to u 31 (and u 31 ), we arrive at a relation for the unknown value ξ(τ), which has the form of (17) in which We note that the sign of the value (α 1 ) coincides with the sign of the expression ω(ω − b). The stability of the equialibrium state of u 0 of the boundary value problem (1), (2) for small values of ε is obviously determined by the sign of (α 1 ), and the existence and stability of the cycle in (1), (2) are related to the existence of the cycle in (17), i.e., to the signs of the values (α 1 ) and (β 1 ).

Local Dynamics in the Case of Large Translation Coefficient
Here we assume that the parameter b is large enough: It then follows from equality (13) that the quantity ω is of the order of ε −1 , and the corresponding values of T at which the stability of the equilibrium state can change are of the order of ε. In this regard, it is natural to set T = εT 1 and change the time t = εt 1 in (1). Below, it is convenient to replace u with u − 1 in (1). Then the corresponding boundary value problem with respect to u 1 = u − 1 after multiplying by ε of the left and right parts can be written in the form Formally assuming ε = 0, we arrive at a linear equation whose entire stability spectrum is pure imaginary. Thus, the critical case of infinite dimension is realized in the problem of the equilibrium state of (20), (21) stability. An algorithm for studying the dynamic properties of solutions in such situations is developed in [23,24]. We apply the corresponding constructions for (20), (21). We introduce the formal expression u = ∞ ∑ n=−∞ ξ n (τ) exp ik(t 1 + x) + εv(τ, y) + . . . = ξ(τ, y) + εv(τ, y) + . . . , Substituting this expression into (20), (21) and performing standard operations, we obtain the boundary value problem with respect to ξ(τ, y) ξ(τ, y + 2π) = ξ(τ, y). We note that the boundary value problem (22), (23) plays the role of a quasinormal form for (20), (21) and does not contain time delay but contains a deviation of the spatial variable.
Further, we consider the issue of the (22), (23) local dynamics in the equilibrium state of ξ ≡ 0 neighborhood. The characteristic equation of the linearized at zero problem has the form In the case when the roots of this equation have negative real parts, the equilibrium states of ξ 0 = 0 in (22), (23) and of u 0 ≡ 0 in (20), (21) are asymptotically stable for small ε, and the solutions from some ε independent neighborhood of these equilibrium states tend to zero as t → ∞. If (24) has a value of λ with positive real part, then ξ 0 and u 0 are unstable, and the problem of dynamic behavior in the equilibrium state neighborhood becomes nonlocal. Below, we assume that for some integer k 0 > 0 and T 1 = T 10 , Equation (24) has the pure imaginary root λ = iσ. All the other roots of (24) have negative real parts as k = ±k 0 .
We introduce another small parameter µ, which characterizes the T 1 deviation from T 10 : T 1 = T 10 + µT 11 , 0 < µ 1. In this case, a two-dimensional local invariant integral stable manifold exists in a small neighborhood of zero in (20), (21) and in (22), (23), on which the boundary value problem (22), (23) can be presented as a normal form to within O(µ).
Repeating the constructions of the previous section, we introduce into consideration a formal expression of the form (18): (26) where z = στ + k 0 y. We substitute (26) into (22), (23) and consecutively find that We summarize what has been said. (25) have the bounded solution η(s) as s → ∞. Then the function
It remains to be noted that the stability of the zero equilibrium state in (22), (23) is determined by the sign of the quantity (α 2 ), and the existence and stability of the nonzero cycle in (25) and in (20), (21) are determined by the signs of the quantities (α 2 ) and (β 2 ).
We dwell on some of the conclusions. The presence of advection in the distributed logistic equation with diffusion significantly complicates the dynamic properties of the solutions. Bifurcation phenomena (which are based on the Andronov-Hopf bifurcation) begin to occur at lower values of the delay coefficient. The possibility of stabilization of the equilibrium state as delay increases is shown. In the problem of the stability of a positive equilibrium state an infinite-dimensional critical case can be realized for sufficiently large values of the advection (translation) coefficient. This critical case can occur even at asymptotically small values of delay. It is shown that the corresponding bifurcations occur at high frequencies and on asymptotically large modes. Thus, rapid oscillations arise both with respect to the spatial variable and with respect to time. A special nonlinear parabolic equation with the deviation of the spatial variable that does not contain large and small parameters is constructed. Its nonlocal dynamics determine the behavior of the initial equation solutions in a small neighborhood of the equilibrium state.

Equations with Small Diffusion Coefficient
The dynamic features of equations with low diffusion are even more interesting and varied. The assumption that the values of the diffusion coefficient are small is natural. In mathematical ecology, it is the mobility coefficient divided by the length of the habitat, which often has relatively large dimensions. In many problems of physics and mechanics, the values of the diffusion coefficient are also quite small in normalized units.
Therefore, we consider the delay logistic equation with diffusion with the periodic boundary conditions We assume 0 < ε 1 (29) i.e., the diffusion coefficient is small enough. We investigate the dynamic properties of these boundary value problem solutions in some small enough and ε independent neighborhood of the equilibrium state of u 0 ≡ 1.
The structure of the solutions may differ significantly depending on the value of the translation coefficient. Four cases are considered separately. In the first of them, the coefficient b is of the same order as the diffusion coefficient, i.e., for some fixed value This case is covered in Section 3.1. A much more complicated situation is considered next in Section 3.2 when the parameter b is sufficiently small again but is greater that the diffusion coefficient with respect to the ε order, i.e., We note at once that under this condition, the biffurcations occur on the modes with asymptotically large numbers. Section 3.3 considers the case when the parameter b does not depend on ε. The peculiarity of this case is that bifurcations occur at high modes as well as in Section 3.2, but the delay coefficient is asymptotically small in this case. Section 3.4 reveals the features of the case when the condition b 1 (32) holds together with condition (29).
In each of these cases, the bifurcation values of the parameters are determined and quasinormal forms are constructed to analyze the dynamics of solutions.

Quasinormal Forms Construction under Condition
plays the role of the characteristic equation for the linearized in u 0 boundary value problem We formulate several simple statements regarding the roots of (33). We omit their simple but cumbersome proofs.

Lemma 4.
Let the condition 0 < rT 0 < π 2 hold. Then, for all sufficiently small ε, the real parts of the (33) roots are negative and separated from zero as ε → 0.

Lemma 5.
Let the condition rT 0 > π 2 hold. Then, for all sufficiently small ε, there exists a root with positive real part separated from zero as ε → 0.
Then, there are no roots with positive real part separated from zero in (33) as ε → 0, but there are infinitely many roots λ ± k (ε) (k = 0, ±1, ±2, . . .), the real parts of which tend to zero for each k, and the asymptotic representations We note that for rT 0 < π 2 and for rT 0 > π 2 the situation for (33) is the same as for the characteristic equation Only one pair of (36) roots lies on the imaginary axis as rT 0 = π 2 , and for (33) infinitely many (35) roots tend to the imaginary axis as ε → 0. Thus, the critical case is realized in the problem of the stability of solutions of (34) of infinite dimension.
The solutions v ± k (t, x, ε) = exp(ikx + λ ± k (ε)t) of the boundary value problem (34) correspond to the roots λ ± k (ε). Therefore, the boundary value problem (34) has the set of solutions where ξ k are the arbitrary constants. This expression can be written as Applying the methodology from [23], we find the solutions of (27), (28) in the form The function ξ(τ, x) is the unknown amplitude, u j (t, τ, x) are 2π/ω 0 -periodic with respect to t and 2π-periodic with respect to x.
We substitute the formal expression (37) into (27) and sequentially equate the coefficients at equal powers of ε in the resulting formal identity. We obtain the correct equality for ε 1 . Collecting the coefficients at ε 2 we obtain the equation for u 2 determining. From here where At the next step, we collect the coefficients at ε 3 and obtain the equation for u 3 . From the condition of its solvability in the indicated class of functions, we obtain the boundary value problem for ξ(τ, x) determining: The formula holds for the Lyapunov quantity σ, and σ < 0. We state the basic result of this section. (29), (30), rT 0 = π 2 hold and the boundary value problem (39) has the bounded solution ξ(τ, x) as τ → ∞, x ∈ [0, 2π]. Then, for τ = ε 2 t the function u(t, x, ε) = 1 + ε(ξ(τ, x) exp(iω 0 t) + cc) + ε 2 Aξ 2 exp(2iω 0 t) + cc satisfies the boundary value problem (27), (28) to within O(ε 3 ).

Remark 1.
It can be shown that if the boundary value problem (39) has a periodic with respect to τ solution and certain conditions of nonsingularity type hold, then the initial boundary value problem has an almost periodic solution of the same stability with the asymptotic behavior indicated in Theorem 3.

Construction of Quasinormal Forms under Condition
The results of this section are the most complicated and interesting. First, we dwell on the linear analysis.

Linear Analysis
In this section, we fix arbitrarily the positive values b 0 and r and write out the characteristic Equation (33) in the form where z = εk, k = 0, ±1, ±2, . . . . Let λ(z) stand for the root of this equation with the largest real part. We recall that the equality λ(z) = −z 2 − r holds as T = 0, therefore λ(z) < 0 for all z ∈ (−∞, ∞). At the first step, we find the smallest positive value T 0 of the parameter T for which λ(z) ≤ 0 (z ∈ (−∞, ∞)), and there exists z 0 > 0 that λ(z 0 ) = 0. We show below that z 0 is uniquely defined. We put ω = λ(z 0 ). Further, we write out the system of equations for the unknown quantities z 0 , ω and T 0 . Initially, from the condition λ(z 0 ) = iω and from Equation (40) we obtain that From the condition dλ(z) dz z=z 0 = 0 we arrive at the equality Taking this into consideration, we obtain from (42) that Then, from here we obtain the equation with respect to the quantity T 0 : Now, we find that the equality holds for the positive root T 0 = T 0 (z 0 ) of the equation above. Finally, taking into account (42) and the first of the equalities (41), we obtain the equation to determine z 0 : After the roots of this equation have been found for those r and b 0 for which they exist, we obtain the desired value T 0 = T 0 (z 0 ). Figure 1 shows the graphs of the left and right sides of Equation (45). The main difference between the results of this and the previos sections is that T 0 < π 2r here, and the value z 0 = εk ε , at which the critical case is realized, is positive. We consider a set of integers where the quantity Θ = Θ(ε) ∈ [0, 1) complements the expression z 0 ε −1 to an integer value. We assume in (40) that z = z 0 + ε(Θ + m) and let λ + m (ε) and λ − m (ε) = λ + m (ε) stand for those roots of (40), the real parts of which tend to zero as ε → 0. The following simple statement holds.
It is important to note that infinitely many roots of the characteristic Equation (40) tend to imaginary axis as ε → 0. This gives grounds to say that the critical case under consideration has an infinite dimension in the stability problem.
The root λ + m (ε) corresponds to the solution v m (t, x, ε) of the linearized equation and which means that the same equation has a set of solutions where ξ m are arbitrary complex constants. Let τ = ε 2 t 0 . Then (46) can be presented in the form Here, we have the equality for the Fourier coefficients of the function ξ(t, y). Further constructions are based on the representation (47).

Construction of Quasinormal Form
For fixed r and b 0 and under the conditions (29), (31) we define ω, ω 1 , z 0 , and T 0 . We assume that (27), (28) and let E = E(t, x, ε) stand for the function We introduce into consideration the formal asymptotical series Here, ξ(τ, y) are the unknown complex amplitudes. The functions u j = u j (t, τ, y) are 2πω −1 -periodic with respect to t and 2π-periodic with respect to y. We search for solutions of the nonlinear boundary value problem (27), (28) in the form of (48). For this purpose we substitute (48) into (27) and equate the coefficients at the same powers ε in the resulting formal identity. At the first step, we obtain the correct equality by collecting the coefficients at the first power of ε. Collecting the coefficients at ε 2 we obtain the equation for u 2 . We search for u 2 in the form Then, we immediately get that At the next step, we obtain the equation for u 3 : The explicit form of the functions A 0,2,3 (τ, x) is inessential, so we do not write them out. We obtain the formula The satisfaction of the equality is the condition of the solution of the equation for u 3 existence in the indicated class of functions, i.e., ∂ξ ∂τ In order to formulate the basic result of this section, we introduce one more notation. Let ε n (Θ 0 ) > 0 stand for the sequence that tend to zero as n → ∞, and the equality Θ(ε n (Θ 0 )) = Θ 0 holds for all n. (29) and (31) be satisfied. Let Θ = Θ 0 , and let ξ(τ, y) be the bounded solution of the boundary value problem (49) as τ → ∞, y ∈ [0, 2π]. Then for ε = ε n (Θ 0 ) the function This statement means that in the considered infinite dimensional critical case, the local dynamics of the initial boundary value problem (27), (28) for small ε is determined by the nonlocal behavior of the quasinormal form (49) solutions.

Theorem 4. Let the conditions
We note that the dynamic properties of (49) may vary for different values of Θ. This means that an infinite alteration of straight and reverse bifurcations can occur in the initial boundary value problem (27), (28) as ε → 0.

Quasinormal Forms for Fixed Value b = 0 and for Sufficiently Small ε
In this section, we first define the smallest positive value of the delay coefficientT such that the zero equilibrium state in (27), (28) is asymptotically stable for T ∈ (0,T), but unstable for T >T. At the next stage, in the critical case of T ≈T, we construct a quasinormal form for the local dynamics study.
It is convenient to perform a change in (27), (28). As a result, we obtain the boundary value problem with delay and deviation of the spatial variable u(t, y + 2π) ≡ y(t, y).

Nonlinear Analysis
In the case under consideration, the formal representation of the nonlinear boundary value problem (27), (28) solutions is based on formula (58) for the linearized problem solutions. Therefore, we introduce into consideration the asymptotic expression to construct a quasinormal form. As in Section 3.2.2, we obtain here Substituting (59) into (27) and performing standard operations, we obtain the equalities u 20 = −2 cos(bT 0 z 0 )|ξ(τ, y)| 2 , first. At the next step, we get the equation for u 3 . Expressions for u 30 , u 32 , and u 33 are simply defined, and the condition of solvability of the equation for u 31 leads to the relations For the value δ the equality δ = −ru 20 (1 + exp(−ibT 0 z 0 )) − ru 21 (exp(−2ibT 0 z 0 ) + exp(ibT 0 z 0 )) holds. The next statement follows from the above constructions.

Quasinormal Form in the Case of Low Diffusion and Large Translation Coefficient
This case is simpler than the one discussed in the previous section. Let b 1, i.e., the parameter µ = b −1 satisfies the condition 0 < µ 1.
In this case, the threshold value of the parameter T is determined by the condition This means that it is an order of magnitude less than in (53). Then, the characteristic equation has the form The equation of first approximation defines the behavior of the roots (63) with higher precision (compared to (54)) near the imaginary axis. The formulas (56) and (57) in which the parameter b should be replaced by 1 are correct. The resulting quasinormal form coincides with (60), (61) for b = 1, T 1 = z 1 = 0.

On Dynamics of Delay Logistic Equation with Small Diffusion and Classical Boundary Conditions of General Form
We consider the problem of the local dynamics of the delay logistic equation with small coefficients of diffusion and advection with the boundary conditions All coefficients in (64), (65) are real, r > 0, T > 0, and ε is a small positive parameter: The construction of the characteristic equation for the linearized at zero boundary value problem ∂v ∂t is related to the eigenvalues of the stationary boundary value problem All eigenvalues µ j (j = 0, 1, . . .) of this boundary value problem are real and can be arranged in descending order. The corresponding to µ j eigenfunctions ϕ j (x) are also real. We note that they form a complete set in the corresponding space. We consider the question of the roots of the quasipolynomial for each number j. Here are some standard statements.
Below we assume that the equality holds for an arbitrarily fixed value r 1 . The linear boundary value problem (67), (68) has a set of solutions where c j are arbitrary, and the Fourier coefficients c j (τ, x) of the function ξ(τ, x) have the form c j (τ, x) = c j exp (λ j1 + o(ε))τ . Based on this representation of 'critical' solutions of the linear problem (67), (68), we look for the nonlinear boundary value problem (64), (65) solutions in the form Here and below, let cc stand for the expression that is a complex conjugate to the previous term. The unknown function ξ(τ, x) is sufficiently smooth and satisfies the boundary conditions (65). The dependence on the argument t on the right-hand side of (72) is 4T-periodic. We substitute expression (72) into (64) and collect the coefficients at the same powers of ε. We obtain the correct equality for the first degree of ε. At the next step, we obtain the equation for u 2 . From this we find that However, the boundary conditions (65) for the function u 2 , generally speaking, are not satisfied. In order to satisfy these boundary conditions for the terms of ε 2 order, we look for the expression for ε 3 in (72) in the form where All functions are 4T-periodic with respect to the variable t, and each of the functions w 31 and w 32 is exponentially decreasing with respect to its third argument: for some p 0 > 0 and c 0 > 0 the evaluations are satisfied. We substitute into (64), (65). Then, we obtain the relations for the degree ε 2 in the boundary conditions (65). Taking into account equality (73) here, we find that We take one more step. We write down the relation for the coefficients of ε 3 that is obtained after substituting (74) into (64): Here the following notation is adopted: It is natural to look for the functions appearing in (80) in the form Then, from the Equation (80) we arrive at the system of four equations We conclude from (81) that We obtain from (82) that u 33 = Cξ 3 and From (83) and conditions (75), (78), (79) we obtain that We denote by δ 0 one of iπT −1 + r exp −iπT −1 1/2 roots, whose real part is negative. We summarize with the following statement. We make one remark. The algorithm presented here for constructing the asymptotics of the boundary value problem (64), (65) solution can be continued indefinitely.

About Infinite-Dimensional Bifurcations in the Case of Large Delay and Dirichlet Boundary Conditions
We note that the zero solution of the boundary value problem (1), (2) is unstable for sufficiently large values of the delay parameter T. However, the relaxation cycle is stable [15] in this case. Its asymptotic behavior is given in [15].
The local behavior of the (1), (2) solutions under other classical boundary conditions is determined by the roots of its characteristic equation for the linearized at zero boundary value problem. Some results for such cases are presented in [25].
We analyse its roots. The roots of this equation have negative real parts for T > 0 as 0 < r < d. Let the condition r = r 0 be satisfied where The basic assumption of this section is that T 1, i.e., The dynamics of the solutions of the delay equations under the condition T 1 was studied in [26,27].
It is convenient to make the substitution t = Tt 1 in (94). Consequently, we obtain the singularly perturbed boundary value problem: Here, we omit the index 1 for t 1 , and the characteristic equation for the boundary value problem (96), which is linearized on u 0 ≡ 1 takes the form We investigate the behavior of the boundary value problem (96) solutions in the zero equilibrium neighborhood under the condition (95) as where r 1 is arbitrarily fixed. The next statement shows that the critical case of infinite dimension is realized in the boundary value problem (96).
According to the technique from the previous sections, we seek the solutions of the nonlinear boundary value problem (97) in the neighborhood of u 0 ≡ 0 in the form For the sequential finding of the elements of the formal series (99), we substitute (99) into (96) and perform standard actions. First, we obtain the equation for u 2 . It follows that u 2 (τ, x, y) = dπ 2 ξ 2 (τ, y)p(x), At the next step, we obtain the boundary value problem for u 3 . For the existence of this boundary value problem solution, it is necessary and sufficient that where γ = d 1 0 p(x) sin 2 πxdx. We do not present an explicit formula for γ due to its inconvenience. We only note that γ < 0. Hence the statement follows. Thus, the boundary value problem (101) is a quasinormal form for the boundary value problem (96). In contrast to the previously presented quasinormal forms, its coefficients here are real.

Case of b = 0
After the replacement (95) in the boundary value problem we obtain the following boundary value problem To obtain the characteristic equation, we first linearize this boundary value problem at zero and then set u = v exp λt. Then we obtain the equation After the replacement in (103) we obtain the equation with the Dirichlet boundary conditions Hence, we conclude that Now we formulate an assertion about the roots of this equation.
where τ = ε 2 t, y = 1 − ε(dr 0 ) −1 t. The function ξ(τ, y) is 1-antiperiodic with respect to y: The functions u 2,3 (τ, x, y) are periodic with respect to x and y. We substitute (106) into (102). Performing standard actions, we obtain the boundary value problem for u 2 . For simplicity, we assume that 4dr 0 = b. It follows that the equation dλ 2 + bλ + r 0 = 0 has simple roots λ 1 and λ 2 . From (108), (109) we obtain that where At the next step, we obtain the boundary value problem The equality to zero of the integral with respect to x from 0 to 1 from the right-hand side is the condition for the existence of a solution of this boundary value problem with respect to u 3 . Hence, we conclude that the function ξ(τ, y) is a solution of the boundary value problem p(x) sin 2 πx dx. Theorem 7 holds for this boundary value problem.

Extending the Results to Other Boundary Conditions
As an example, we consider the boundary value problem with the boundary conditions Here all the coefficients are real. We agree to assume that the notation γ 1 = ∞ corresponds to the boundary condition u| x=0 = 0, and the notation γ 2 = ∞ corresponds to the condition u| x=1 = 0. We note that the eigenvalues δ j (j = 0, 1, . . .) of the linear boundary value problem with the boundary conditions (111) are real. They can be numbered in descending order δ 0 > δ 1 > δ 2 > . . . , and the eigenvalue δ 0 corresponds to the eigenfunction ϕ 0 (x), which is positive on the interval (0, 1). Let r = r 0 + ε 2 r 1 in (110). We consider the equation Here are some simple statements.
The behavior of the roots of (113) in the critical case is described by the following statement.

Lemma 16.
Let r 0 = |δ 0 |. Then Equation (113) has no roots with positive and separated from zero real parts as ε → 0, but has infinitely many roots λ ± m (ε) m = 0, 1, . . . , λ + (ε) = λ − m (ε) for which the following asymptotic equalities hold: The construction of quasinormal forms in each of the cases (114) and (115) is based on the formal asymptotic equality where τ = ε 2 t, y = 1 + εδ −1 0 t. The function ξ(τ, y) is 1-periodic with respect to y in the case of (114), and is 1-antiperiodic with respect to y in the case of (115). The functions u 3 are 1-periodic with respect to y in the case of (114). The function u 3 is 1-antiperiodic with respect to y in the case of (115).
We introduce several notations before formulating the resulting statements. Let ψ 0 (x) stand for the solution of the conjugate to Equation (112) Let the normalization requirement holds. We note that the satisfaction of the equality 1 0 p(x)ψ 0 (x)dx = 0 is the condition for the existence of the boundary value problem solution. Let K(p(x)) stand for this solution. An explicit formula for this expression is not given here.
The case of δ 0 < 0 is more interesting.
The boundary value problem (112), (111) is self-adjoint (see, for example, [28]). The situation can be much more complicated for not self-adjoint boundary value problems. We briefly demonstrate it with one example.
We consider the question of local dynamics of the boundary value problem with cubic nonlinearity The characteristic equation for the boundary value problem linearized at zero has the form ελ + r exp(−λ) = −dk 2 + ibk where k takes all odd values k = (2m + 1), m = 0, ±1, ±2, . . . . For k = 1 we obtain the equation We formulate several simple statements.
This boundary value problem plays the role of a normal form for (121), (122). Thus, the leading terms of the asymptotics of the solutions of (121), (122) with small enough initial conditions with respect to the norm (in the space C [−1,0] × W 2 2 [0,1] ) are reconstructed from its solutions with the help of the formula (126).

Remark 2.
In Sections 4.1 and 4.2, the 'critical' values of the parameter r are determined by the equality r = |δ 0 |. In this section, the role of the eigenvalue δ 0 is played by the quantity δ 0 = −d + ib. Here, the critical value of the parameter r is determined by the equality r = | δ 0 | = d. If in Sections 4.1 and 4.2 the solutions of the initial boundary value problem (110), (111) are formed according to the formula (116) at relatively low frequencies, then in this section the corresponding frequencies are relatively large of the order of bε −1 .
We note also that if we have the periodicity condition instead of antiperiodic boundary conditions, then δ 0 = 0. Therefore, for each r > 0, the solutions of (110) are unstable for small ε.

Conclusions
The bifurcation problems for the delay logistic equation with diffusion and advection are considered. The most important results relate to the cases of singular perturbations when either the diffusion coefficient is small enough, the translation coefficient is large enough, or the delay coefficient is large enough. A distinctive feature of these situations is that the critical cases in the problem of the stability of the equilibrium state have infinite dimension. This leads to the fact that the constructed quasinormal forms (infinite-dimensional analogs of classical normal forms) are the distributed equations with an infinite-dimensional phase space.
For example, in a problem with a large translation coefficient, such equations are the equations with diffusion and with deviation of the spatial variable. In the problems with small diffusion or in problems with large delay, they are parabolic equations of the Ginzburg-Landau type.
The algorithm for constructing the asymptotics of solutions developed is related to the algorithm for the quasinormal form construction. It is possible to pose a question of finding exact solutions of the initial boundary value problem that have the pointed asymptotics. If the quasinormal form has a periodic with respect to τ solution and certain conditions like nondegeneracy type are satisfied, we can justify the result about the existence of an exact almost periodic solution with the constructed asymptotics and answer the question of its stability.
The threshold values T 0 of the delay coefficient at which the bifurcation phenomena occur are found. In the case when the translation coefficient is b 1, this threshold value is of the order T 0 = O(b −1 ), i.e., the bifurcations occur even at small values of the delay.
The cases of a small diffusion coefficient are considered. Table 1 illustrates the changes of the T 0 values depending on the coefficient b. We consider the diffusion coefficient is equal to ε 2 (0 < ε 1).
Thus, as the coefficient b increases, the values of T 0 decrease. Moreover, we can conclude that the parameter b increase leads to a complication of the problem dynamic properties.
It is important to note that if b ≈ ε 2 , then bifurcations occur on small modes of the order of 1. In other cases they occur on asymptotically large modes of the order of ε −1 .
There is the parameter Θ in many quasinormal forms that infinitely many times runs through all values from 0 to 1 as ε → 0. The sequences ε n → 0 are eliminated on which the coefficient Θ does not change. An unlimited process of straight and reverse bifurcations alternation can occur [29] as ε → 0.
In the infinite-dimensional critical case, the quasinormal form of parabolic type is constructed for the Dirichlet boundary conditions in the case of a large delay.
Because the quasinormal forms are complex evolutionary equations of the Ginzburg-Landau type, we can formulate a general conclusion that complex dynamic behavior is typical for the infinite-dimensional bifurcation problems under consideration [30]. For example, irregular dynamic processes and multistability phenomena can be observed.
The solutions of quasinormal forms allow one to determine the leading terms of asymptotic expansions of the initial boundary value problem solutions. Among them, one can differ the situations when these expansions contain rapidly and slowly oscillating components with respect to spatial and time variables.
The influence of various boundary conditions on the dynamic properties of the initial problem is illustrated.