Dynamic Transition Analysis for Activator-Substrate System

The main objective of this article is to study the dynamic transition associated with the activator-substrate system. Two criteria are derived to describe the transition from real eigenvalues or complex eigenvalues and the types of transition. Notably, we get two parameters b1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{1}$$\end{document} and b2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{2}$$\end{document}, which can determine the the types of transitions for the two criteria respectively. The analysis is carried out using dynamic transition theory developed recently by Ma and Wang (Phase transition dynamics, Springer, New York, 2013, Bifurcation Theory and Applications, World Scientific, Singapore, 2005, Stability and Bifurcation of Nonlinear Evolutions Equations, Science Press, Beijing, China, 2007).


Introduction
Natural patterns are various in shape and form.The development processes of such patterns are complex, and also interesting to researchers.To understand the underlying mechanism for patterns of plants and animals, Turing [4] first proposed the coupled reaction-diffusion equations.It was shown that the stable process could evolve into an instability with diffusive effects.He showed that diffusion could destabilize spatially homogeneous states and cause nonhomogeneous spatial patterns, which accounted for biological patterns in plants and animals.Such instability is frequently called the Turing instability, also known as diffusion-driven instability.Gierer and Meinhardt [5] presented the Gierer-Meinhardt model( Activation-inhibition diffusion system [6,7]) and activator-substrate system (depletion model) [8,9], and which was used to describe the Turing instability.
In this article, we consider the bifurcation of activator-substrate system, which was used to describe pigmentation patterns in sea shells [10,11] and the ontogeny of ribbing on ammonoid shell [12], and the model could be written as follows where u(x, t) and v(x, t) represent the population densities of the activator and the substrate at time t > 0 and spatial location x respectively.Here, the substrate with concentration v(x, t) could be consumed by activation or some indirect effect of activation, and is supplied at a constant rate.d and D are the diffusion constants of the activator and the substrate respectively.is the source concentration for the substrate, and the activator-substrate system of interest in present work is confined in a rectangle: There are extensive studies from the mathematical point of view for activatorsubstrate system, and we refer in particular to [13,14], and the references therein for studies related to the steady-state solutions, Hopf bifurcation, global structure.Motivated by the above papers, what we are concerned in this paper is to study the dynamical transition for the system (1).The technical method for the analysis is the dynamical transition theory, which has been developed by Ma and Wang [1][2][3].It is worth noticing that the dynamical transition theory is recently developed to identify the transition states and classify them both dynamically and physically, see [15][16][17][18].
With this method in our disposal, we derive in this article a characterization of dynamic transition of the activator-substrate system.In particular, the analysis in this article shows that the activator-substrate system always undergoes a dynamic transition either to multiple equilibria or to periodic solution, dictated by the sign of the parameters 1 and 2 , where 1 and 2 are related to the diffusion constants of the substrate D, the source concentration for the substrate and the k-th eigenvalue of the Laplacian k : (1) For the case of transitions to multiple equilibria, for ∫ Ω e 3 k 2 dx ≠ 0 , the transition is either continuous or jump based on the sign of parameter b 1 .In the periodic case (complex eigenvalues ), the types of transitions are determined again by another parameter b 2 .b 1 , b 2 are related to the diffusion constants of the substrate D, the source concentration for the substrate and the k-th eigenvalue of the Laplacian k , k-th eigenvector of the Laplacian e k , see ( 24) and (45).
This article is organized as follows: Sect. 2 introduces the abstract operator form and the principle of exchange of stabilities (PES), Sect. 3 studies the dynamic transitions of the activator-substrate system and presents the main results.In Sect.4, we summarize the conclusions and give the example derived from previous calculations.

Basic State and Abstract Operator Form
The equation (1) admits three physically realistic constant steady-state solutions: In this paper, we mainly focus on the bifurcation and transition problem of (1) at the steady-state solution U 3 in (5).For this purpose, we take the transition Omitting the prime, the system (1) is written as Since we will study the influence of the diffusion constants of the activator-d on the stability of bifurcation, so we select d = as the control parameter. (5) For system (6), there are two types of physically-sound boundary conditions: the Dirichlet boundary condition and the Neumann boundary condition Ω is as in Equ.(2) .
Define the function spaces

Define the operators
Then the Equ.( 6) with (7) or ( 8) can be written in the following abstract form

Linear Theory and Principle of Exchange of Stabilities(PES)
The linearized eigenvalue Equ. ( 6) are given by with the boundary condition (7) or (8) 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:956-979 Where Thus we introduce two critical numbers Obviously, the following theorem holds true.
Theorem 2 Let 1 and 2 be the two numbers given by (15) and (16).Then we have the following assertions: 1. Let  2 <  1 , and k 2 > 0 be the integer such that the minimum is achieved at k 2 in the definition (16) ( ) is the first real eigenvalue of (11) near = 2 satisfying that 2. Let  1 <  2 , and k 1 > 0 be the integer such that the minimum is achieved at k 1 in the definition (15) ( ) are a pair of first complex eigenval- ues of (11) (15) , and in general, if ( ) are also simple at  2 (<  1 ).

Main Results and Proofs
In this section, we will give the main results and proofs which based on the the dynamical transition theory for nonlinear dissipative systems developed by Ma and Wang [1][2][3].Then, the following theorems will show the types of transitions that the system undergos basing on Theorem 2 .

Transition from Real Eigenvalues
Here after, we always assume that the eigenvalues 17) are simple.Based on theorem 3.1, as  2 <  1 the transition of (10) occurs at = 2 , which is from real eigenvalues.Let k 2 be as in theorem 3.1, and e k 2 the eigenvector of (12) corresponding to e k 2 satisfying Then, under the condition (19), for the system (6) with boundary condition (7) or (8) we have the following transition theorem.Theorem 4 Let  2 <  1 , Then the system (10) has a transition at = 2 , which is mixed.In particular, the system bifurcates on each side of = 2 to a unique branch U of steady state solution, such that the following assertions hold true: 1. On  <  2 , the bifurcated solution U is a saddle, and the stable manifold sepa- rates the space H into two disjoint open sets W 1 and W 2 , such that U = 0 ∈ W 1 is an attractor, and the orbits of (10) in W 2 are far from U = 0.

On 𝜆 > 𝜆
where U(t, U 0 ) is the solution of (10) with U(0, U 0 ) = U 0 .3. The bifurcated solution U can be expressed as Proof We apply Theorem 2.3.2 in [1] to prove this theorem.Let Φ be the center manifold function of (10) at = 2 .We need to simplify the following expression: where y ∈ R 1 , G is the operator defined by ( 9), + k 2 is the eigenvector corresponding to + k 2 ( 2 ) = 0 , and + * k 2 is the conjugate eigenvector, which is the eigenvector of adjoint equation for Eq.(11).And the adjoint equation are given by By (13) By definition of 2 and k 2 , we infer from (23) and ( 24) that 20) is rewritten as By ( 22)and (25) we see that Thus, we deduce from ( 22) and ( 25), (26) that Therefore the function (20) is given by (25) 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:956-979 Where and the theorem follows from Theorem 2.3.2 [1], The proof is completed.
dx ≠ 0 , then the local topological structure of the transitions of ( 10) is schematically shown in the center manifold in Fig. 1.Now, we consider the case where (19) is not true, i.e., We introduce the following parameter where = ( 1 , 2 ) satisfies Theorem 5 Let (27) hold true,  2 <  1 , and b 1 is the number given by (28).Then the transition of (10) at = 2 is continuous if b 1 < 0 , and is jump if b 1 > 0 .Moreover, the following assertion hold true: (10) has no bifurcation on  >  2 , and has exact two bifurcated solu- tions U + and U − on  <  2 , which are saddles.Moreover, the stable manifolds of the two bifurcated solutions divide the space H into three disjoint open sets U + , U 0 , U − , such that U = 0 ∈ U 0 is an attractor, and the orbits of (10) in U ± are far from U = 0. 2. If b 1 < 0 , (10) has no bifurcation on  <  2 , and has exact two bifurcated solu- tions U + and U − on  >  2 , which are attractors.In addition, there is a neigh- bourhood O ⊂ H of U = 0 , such that the stable manifold of Proof We use Theorem 2.3.1 [1] to prove this theorem.To get the function g(y), we need to calculate the center manifold function Φ(y) .By Theorem A1.1 [1], Φ(y) satisfies (29) , ) ∈ H . Hence. it follows from (30) and (32) that Which is an equivalent form of (29).By (32), we have Hence, we deduce from ( 22), ( 25) and (34) that (30) ), ) T .
(34) Thus, the function g(y) in (20) can be written as where b 1 is as in (28).Hence the theorem follows from Theorem 2.3.1 in [1].For example, we consider the case where Ω = (0, L) .The eigenvalues k and eigenvectors e k of ( 12) are given by It is clear that k 2 ≥ 2 , and (27) hold true.We see that Hence, by (29), we have where It is readily to see that Inserting (52) and 1 , 2 , 1 , 2 into (28), we can get the explicit expression of b 1 .

Transition from Complex Eigenvalues
As  1 <  2 , the transition of (10) occurs at = 1 , and the system bifurcates to a periodic solution.
In this case, Then we define the following parameter b 1 as in (49).Here M k is the matrix defined by and we have the following theorem Theorem 6 Let b 2 be the number given by (49) and  1 <  2 .For the problem (10), the following assertions hold true The problem undergoes a dynamic transition at = 1 , which is the Hopf bifurca- tion.2. When b 2 < 0 , the transition bifurcates to a stable periodic solution on  >  1 , and when b 2 > 0 , transition bifurcates to an unstable periodic solution on  <  1 .3. The bifurcated periodic solution U = (U 1 , U 2 ) can be expressed as 13) the eigenvalues and eigenvectors of ( 11) with ( 7) or ( 8) at are determined by the matrices M k given by (37).It is clear that M k 1 has a pair of imaginary eigenvalues 2 ), Journal of Nonlinear Mathematical Physics (2023) 30:956-979 Then, by (13) the eigenvectors of ( 11) corresponding to ± is the transpose of M k 1 , Direct calculation shows that It is easy to see that Let U = x + y + Φ(x, y) ∈ H be a solution of (6) at = 1 , and Φ be the center manifold function.By (42), the reduced Equ.( 6) read where the operator G is given by We are now in a position to derive the center manifold function Φ .By Theorem A1.1 [1] Where L is as in (9), where P 2 ∶ H → E 2 is the canonical projection, and are given by (22).Hence, we obtain from (38,39,44 and 47). (45)

3
Journal of Nonlinear Mathematical Physics (2023) 30:956-979 Direct calculation shows that Thus we have where Where a ij and b ij (0 ≤ i, j ≤ 3) are as in (46), and ã30 are as in Appendix B. Then we give the number Thus, Assertions (1) and ( 2) of this theorem follow from Theorem 2.3.7 [1] It is known that the bifurcated periodic solution near = 1 takes form where , are as in (38) and (39), and x(t), y(t) are the solutions of the following equation (48) where , are eigenvectors of L corresponding to the complex eigenvalues ± k 0 ( ) , and * , * the conjugate eigenvectors.The solution (x(t), y(t)) near 1 is of the form where b 1 is as in (49).Therefore, assertion (3) holds true from (50), (51).The proof is completed.◻

Conclusions and examples
In this work, we study the dynamical transition for a activator-substrate system from the perspective of dynamic transition recently developed by Ma and Wang.By using the Principle of Exchange of Stabilities condition for activator-substrate system, we note that the system is in a static state in space patterns for  < min{ 1 ,  2 } , where 1 and 2 are determined by parameters ( , d) ∈ R 2 and = d is the diffusion constant of the activator.However, when  > min{ 1 ,  2 } , i.e., the diffusion constants are greater than the specified value, the stability is broken.Then we have the following characteristics: First, when  1 >  2 , it was demonstrated that chaotic coexistence bifurcates from the periodic when ∫ Ω e 3 k 0 dx ≠ 0 .If ∫ Ω e 3 k 0 dx = 0 , we show that the permanent coexist- ence was existed for activator-substrate system.
Second, when  1 <  2 , the first eigenvalues are complex, and we show that the sys- tem undergoes a dynamic transition, which is Hopf bifurcation.
2 ), For example, we consider the case where Ω = (0, L) .The eigenvalues k and eigen- vectors e k of ( 12) are given by It is clear that k 2 ≥ 2 , and (27) hold true.We see that Hence, by (29), we have where It is readily to see that Inserting (52) and 1 , 2 , 1 , 2 into (28), we can get the explicit expression of b 1 .

k 2 dx
= 0 , then the local topological structure of the transition of (10) is schematically shown in the center manifold in Fig2 and 3.

k 2 dx y 3 + o(y 3 ), 1 3 Remark 3
Journal of Nonlinear Mathematical Physics (2023) 30:956-979 When the domain Ω is a rectangle, i.e.Ω = n Π j=1 (0, L j ) , the b 1 in (28) for the Neumann condition can be explicitly expressed in terms of the parameters D, and L j . k

k 1 ( k 1 )
are given by = e k 1 , and = e k 1 .It is readily to check that We consider the conjugate eigenvectors * = * e k 1 and * = * e k 1 with where M * k 1 Based on (38-42) and (44,45, 43) can be rewritten as where a 20 , , a 11 , a 02 , a 03 , a 21 , a 12 , a 30 and b 20 , , b 11 , b 02 , b 03 , b 21 , b 12 , a 30 are given in Appendix A

Remark 4
When the domain Ω is a rectangle, i.e.Ω = n Π j=1 (0, L j ) , the b 1 in Theo- rem 3.2 for the Neumann condition can be explicitly expressed in terms of the parameters D, and L j . k

Proposition 1 With the above calculation, eigenvectors can be derived and are given in the following two groups
. Let k and e k be the k-th eigenvalue and eigenvector of the Laplacian with either the Dirichlet or the Neumann condition: Journal of Nonlinear Mathematical Physics (2023) 30:956-979where P 2 ∶ H → E 2 is the canonical projection, L is as in (9), + 1 3