Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model

Abstract

We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.

1. Introduction

In 1952, Turing published his pioneering paper describing the chemical process between signaling molecules that spread away from their source to form a concentration gradient within a series of cells [1]. In this paper, Turing suggested that diffusion could destabilize an otherwise stable equilibrium of the reaction-diffusion system and lead to nonuniform spatial patterns [2–10].

The following model (which we called Turing model here) is one of the famous models used by Turing to argue the above-mentioned mechanism: 𝑢_𝑡=𝑑_𝑢()𝑣Δ𝑢+𝑠𝑢𝑣−𝑢−𝑎,𝑥∈Ω,𝑡>0,_𝑡=𝑑_𝑣()𝑢()Δ𝑣+𝑠𝑏−𝑢𝑣,𝑥∈Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥)𝜕,𝑥∈Ω,_𝜈𝑢=𝜕_𝜈𝑣=0,𝑥∈𝜕Ω,𝑡>0, (1) where Ω is a bounded connected domain (the reactor) in 𝐑^𝑛 (𝑛≥1), with smooth boundary 𝜕Ω; the reactor is assumed to be closed; thus reflexive Neumann boundary condition is imposed (here 𝜕_𝜈𝑢 is the outer normal derivative of 𝑢); here 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) are the morphogen concentration at the time 𝑡 and the positions 𝑥, 𝑎, and 𝑏 can be thought of as being a decay and growth rate of 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡), respectively, and 𝑠 is the reaction rate.

Since the time when the model was proposed by Turing, many researchers have studied the model extensively (see [11] and the references therein for great details). However, they mostly focus on the pattern formations of Turing type.

It is well known that, apart from occurring Turing patterns, a reaction-diffusion equation system might also exhibit other types of patterns, say, oscillatory patterns.

In [12], the authors derived a simplified Hopf bifurcation theorem for the general semilinear reaction-diffusion equations on the one-dimensional domain and used the abstract theorem to prove the existence of oscillatory patterns emerging from Hopf bifurcations. Since then, more papers can be found in the existing literatures to study the existence of oscillatory patterns of the R-D systems by using Hopf bifurcation techniques, and we refer to [12–14] and the references therein for details.

To the best of our knowledge, in the existing literature, no papers were found to deal with the oscillatory pattern formation of this Turing model from purely analytical point of view. Thus, motivated by [12], we are interested in considering the existence of Hopf bifurcations and the corresponding periodic solutions by using the abstract bifurcation results in [12].

In order to set our problem in the framework of [12], we assume that the domain is one- dimensional spatial interval (0,𝜉𝜋) with 𝜉>0. In this case, the system (1) is reduced to 𝑢_𝑡=𝑑_𝑢𝑢_𝑥𝑥()()𝑣+𝑠𝑢𝑣−𝑢−𝑎,𝑥∈0,𝜉𝜋,𝑡>0,_𝑡=𝑑_𝑣𝑣_𝑥𝑥()()𝑢()+𝑠𝑏−𝑢𝑣,𝑥∈0,𝜉𝜋,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥)(),𝑢,𝑥∈0,𝜉𝜋_′(0,𝑡)=𝑢^′(𝜉𝜋,𝑡)=𝑣^′(0,𝑡)=𝑣^′(𝜉𝜋,𝑡)=0,𝑡>0. (2)

We are going to show that if we choose 𝑎 as the bifurcation parameter, then the system might exhibit a sequence of Hopf bifurcation points, at which oscillatory patterns exist. By using the center manifold theory, normal form methods, and the abstract results in [12], we are able to prove that the spatially homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point are unstable. From [12], all the spatially nonhomogeneous periodic solutions are all unstable. Thus, our bifurcation results suggest that all the Hopf bifurcating periodic solutions are all unstable.

The rest of this paper is structured in the following way. In Section 2, we use the standard Hopf bifurcation theorem to prove the existence of the Hopf bifurcations of the system. In Section 3, we prove the instability of the spatially homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point. In Section 4, we draw our conclusions.

2. Existence of Hopf Bifurcations

In this section, we consider the following PDEs subject to Neumann boundary condition on spatial domain Ω=(0,𝜉𝜋), with 𝜉∈ℝ⁺∶𝜕𝑢𝜕𝑡=𝑑_𝑢𝑢_𝑥𝑥+𝑠(𝑢𝑣−𝑢−𝑎),𝑥∈(0,𝜉𝜋),𝑡>0,𝜕𝑣𝜕𝑡=𝑑_𝑣𝑣_𝑥𝑥()()𝑢+𝑠𝑏−𝑢𝑣,𝑥∈0,𝜉𝜋,𝑡>0,_′(0)=𝑢^′(𝜉𝜋)=𝑣^′(0)=𝑣^′(𝜉𝜋)=0,𝑡>0. (3)

System (3) has a unique equilibrium solution (𝑢_𝑎,𝑣_𝑎), with 𝑢_𝑎=𝑏−𝑎,𝑣_𝑎=𝑏,𝑏−𝑎 (4) which is in the first quadrat if and only if 0<𝑎<𝑏. In the following, we always assume that 0<𝑎<𝑏 holds and choose 𝑎 as the bifurcation parameter.

Define the real Sobolev space in the following way: 𝑋∶=(𝑢,𝑣)^𝑇∈𝐻²(0,𝜉𝜋)×𝐻²(0,𝜉𝜋)∶𝑢_𝑥,𝑣_{𝑥𝑥=0,𝜉𝜋}=0, (5) and 𝑋_ℂ∶={𝑈=𝑈₁+𝑖𝑈₂∶𝑈₁,𝑈₂∈𝑋}.

The aim of the whole paper is just to study the existence and the stability of the spatial homogeneous and nonhomogeneous periodic solutions of system (3) in the real-valued Sobolev space 𝑋 by using the Hopf bifurcation techniques.

To that end, we calculate the linearized operator of system (3) evaluated at (𝑢_𝑎,𝑣_𝑎) which is given by 𝐿(𝑎)𝑑∶=^𝑢Δ+𝑠𝑎()−𝑏−𝑎,𝑠𝑏−𝑎𝑠𝑏𝑏−𝑎,𝑑_𝑣(),,Δ−𝑠𝑏−𝑎 (6) where 𝐷_𝐿(𝑎)=𝑋_ℂ.

One thing is to know more detailed information on the eigenvalues of the linearized operator 𝐿(𝑎) defined in (6). Notice that the eigenvalues of −Δ subject to the Neumann boundary condition in (0,𝜉𝜋) are given by 𝜌_𝑘=𝑘²𝜉², (7) with the corresponded eigenfunctions 𝜙_𝑘(𝑥)=cos(𝑘𝑥/𝜉), with 𝑘=0,1,2,….

We consider the characteristic problem of 𝐿(𝑎) taking in the following form: 𝐿(𝑎)𝜙𝜑(𝑎)𝜙𝜑=𝛽. (8)

Setting 𝜙𝜑=^∞𝑘=0𝑚_𝑘𝜙_𝑘(𝑥)_∞𝑘=0𝑛_𝑘𝜙_𝑘(𝑥), (9) and introducing it into (8), one has 𝐿_𝑘(𝑎)𝑚_𝑘𝑛_𝑘(𝑎)𝑚=𝛽^𝑘𝑛_𝑘, (10) where 𝐿_𝑘(𝑎)−𝑑∶=^𝑢𝑘²𝜉²+𝑠𝑎()−𝑏−𝑎,𝑠𝑏−𝑎𝑠𝑏𝑑𝑏−𝑎,−^𝑣𝑘²𝜉²().−𝑠𝑏−𝑎 (11)

This suggests that the eigenvalues of 𝐿(𝑎) can be given by the eigenvalues of the matrix 𝐿_𝑘(𝑎), 𝑘=0,1,2,…. For more details on how to derive 𝐿_𝑘(𝑎), we suggest the interested readers to refer to [9, 12] and more references therein for great details.

The characteristic equation of 𝐿_𝑘(𝑎) for certain fixed 𝑘 is given by 𝜌²−𝜌𝑇_𝑘(𝑎)+𝐷_𝑘(𝑎)=0,(𝑘=0,1,2,…), (12) where 𝑇_𝑘(𝛼)=𝑠−𝑎²+(1+2𝑏)𝑎−𝑏²−𝑑𝑏−𝑎^𝑢+𝑑_𝑣𝑘²𝜉²,𝐷_𝑘(𝛼)=𝑑_𝑢𝑑_𝑣𝑘⁴𝜉⁴+𝑠𝑘²𝜉²𝑑_𝑢()−𝑑𝑏−𝑎^𝑣𝑎𝑏−𝑎+𝑠²(𝑏−𝑎), (13) where we adopt the same notations of 𝑇_𝑘(𝑎) and 𝐷_𝑘(𝑎), which stand for the trace and determinant of 𝐿_𝑘(𝑎), respectively, as done in [12].

Before proceeding, we state the following useful lemma on the existence of local Hopf bifurcations of the system (3).

Lemma 1 (see [12]). Suppose that, at a certain 𝑎₀∈𝑅, the following condition holds.(ℍ)There exists a neighborhood 𝒪 of 𝑎₀, such that, for any 𝑎∈𝒪, 𝐿(𝑎) has a pair of simple, conjugate, and complex eigenvalues 𝜌(𝑎)=𝛼(𝑎)+𝜔(𝑎), continuously differentiable in 𝑎, with 𝛼(𝑎₀)=0, 𝜔(𝑎₀)>0, and 𝛼^′(𝑎₀)≠0; all the other eigenvalues of 𝐿(𝑎) have nonzero real parts. Then, system (3) has a family of real-valued 𝑇(𝑠)-periodic solutions (𝑎(𝑠),𝑢(𝑠)(𝑥,𝑡),𝑣(𝑠)(𝑥,𝑡)), 𝑠 sufficiently small, bifurcating from (𝑎₀,𝑎₀,𝑏₀). More precisely, there exists a unique 𝑘∈{0,1,2,…}, such that the bifurcating periodic solutions (𝑎(𝑠),𝑢(𝑠)(𝑥,𝑡),𝑣(𝑠)(𝑥,𝑡)) can be expressed by 𝑢(𝑠)(𝑥,𝑡)=𝑠ℎ_𝑘𝑒^{2𝜋𝑖𝑡/𝑇(𝑠)}+ℎ_𝑘𝑒^{−2𝜋𝑖𝑡/𝑇(𝑠)}cos𝑘𝑥𝜉+𝑜𝑠²,𝑣(𝑠)(𝑥,𝑡)=𝑠𝑔_𝑘𝑒^{2𝜋𝑖𝑡/𝑇(𝑠)}+𝑔_𝑘𝑒^{−2𝜋𝑖𝑡/𝑇(𝑠)}cos𝑘𝑥𝜉+𝑜𝑠², (14) where 𝐿_𝑘𝑎₀𝑚_𝑘𝑛_𝑘=𝑖𝜔𝑎₀𝑚_𝑘𝑛_𝑘. (15)

In the following, we will use Lemma 1 to prove the existence of the Hopf bifurcations and its corresponding periodic solutions, spatially homogeneous and nonhomogeneous. Clearly, the condition (ℍ) stated in Lemma 1 holds at a certain critical point 𝑎^𝐻 if and only if one of the following conditions holds: 𝑇_𝑘𝑎^𝐻=0,𝐷_𝑘𝑎^𝐻𝑇>0;_𝑗𝑎^𝐻≠0,𝐷_𝑗𝑎^𝐻≠0for𝑗≠𝑘;𝑇^′_𝑘𝑎^𝐻≠0 (16) or 𝑇_𝑘𝑎^𝐻=0,𝐷_𝑘𝑎^𝐻𝑇>0;_𝑗𝑎^𝐻=0,𝐷_𝑗𝑎^𝐻<0for𝑗≠𝑘;𝑇^′_𝑘𝑎^𝐻≠0. (17)

We define 𝐴(𝑎)=(−𝑎²+(1+2𝑏)𝑎−𝑏²)/(𝑏−𝑎). Then it is direct to check that, for all 𝑎∈(0,𝑎₀), we have 𝐴(𝑎)<0; and, for all 𝑎∈(𝑎₀,𝑏), we have 𝐴(𝑎)>0, where 𝑎₀√∶=2𝑏+1−4𝑏+12. (18) This implies that, for all 𝑎∈(0,𝑎₀), 𝑇_𝑘(𝑎)<0 for all 𝑘∈ℕ. Thus, Hopf bifurcation is impossible.

Then, any potential Hopf bifurcation point 𝛼^𝐻 must be in the interval [𝑎₀,𝑏). For any Hopf bifurcation point 𝑎^𝐻 in [𝑎₀,𝑏), let 𝛼(𝑎)±𝑖𝜔(𝑎) be the eigenvalues of 𝐿_𝑘(𝑎) near 𝑎=𝑎^𝐻; then we have 𝛼(𝑎)=𝑠−𝑎²+(1+2𝑏)𝑎−𝑏²2()−𝑑𝑏−𝑎^𝑢+𝑑_𝑣𝑛²2𝜉²,𝜔(𝑎)=𝐷_𝑘(𝑎)−𝛼²(𝑎),𝛼^′𝑎^𝐻=𝑠(𝑏−𝑎)²+𝑠𝑏2(𝑏−𝑎)²_𝑎=𝑎^𝐻>0. (19)(20) This implies that 𝛼^′(𝑎^𝐻)≠0 is always satisfied for any 𝑎^𝐻∈[𝑎₀,𝑏).

In the following we fix 𝑑_𝑢,𝑑_𝑣,𝑏,𝑠>0 but choose 𝜉 appropriately. We notice that lim_{𝑎→𝑏−}𝐴(𝑎)=lim_{𝑎→𝑏−}−𝑎²+(1+2𝑏)𝑎−𝑏²𝑏−𝑎=+∞. (21)

Then, for any 0≤𝑗, 𝐴(𝑎)=(𝑑_𝑢+𝑑_𝑣)𝑗²/𝑠𝜉² has positive roots in [𝑎₀,𝑏). We define 𝛼^𝐻_𝑗 to be the roots of 𝐴(𝑎)=(𝑑_𝑢+𝑑_𝑣)𝑗²/𝑠𝜉² satisfying 0<𝑎^𝐻₀<𝑎^𝐻_𝑗<𝛽 and lim_𝑗→∞𝑎^𝐻_𝑗=𝑏. These points satisfy 0<𝑎₀=𝑎^𝐻₀<𝑎^𝐻₁<𝑎^𝐻₂<⋯<𝑎^𝐻_𝑛<⋯<𝑏. (22) Clearly 𝑇_𝑗(𝑎^𝐻_𝑗)=0 and 𝑇_𝑖(𝑎^𝐻_𝑗)≠0 for 𝑖≠𝑗. Now we only need to verify whether 𝐷_𝑖(𝑎^𝐻_𝑗)≠0 for all 𝑖∈{0,1,2,…} and, in particular, 𝐷_𝑗(𝑎^𝐻_𝑗)>0.

In fact, if the discriminant of 𝐷_𝑖(𝑎) is less than 0, then we can get 𝐷_𝑖(𝑎)>0 for all 𝑖∈{0,1,2,…} and, in particular, 𝐷_𝑗(𝑎^𝐻_𝑗)>0. The discriminant of 𝐷_𝑖(𝑎) is given by Δ=𝑠²𝑑_𝑢()−𝑑𝑏−𝑎^𝑣𝑎𝑏−𝑎²−4𝑑_𝑢𝑑_𝑣(𝑏−𝑎), (23) which is negative when 𝛼∈[𝑎^𝐻₀,𝑎_𝑐), where 𝑎_𝑐∶=2𝑑_𝑢𝑏−𝑑_𝑣−2𝑑_𝑢𝑑_𝑣𝑏+𝑑²_𝑣+4𝑑₂𝑑_𝑢𝑑_𝑣𝑏2𝑑_𝑢. (24)

Summarizing our analysis above, we obtain our main result in this subsection.

Theorem 2. Let 𝑎^𝐻_𝑗, 𝑗∈{0,1,2,…} be defined above. Then, at any 𝑎^𝐻_𝑗 satisfying 𝑎^𝐻_𝑗∈[𝑎^𝐻₀,𝑎_𝑐), the system (1) undergoes a Hopf bifurcation at 𝛼=𝛼^𝐻_𝑗. Moreover, (1)the bifurcating periodic solutions from 𝑎=𝑎^𝐻₀ are spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system;(2)the bifurcating periodic solutions from 𝑎=𝑎^𝐻_𝑗, 𝑗∈{1,2,…}, are spatially nonhomogeneous.

Proof. By the analysis above, we know that the condition (17) holds true (while (16) does not). This implies that the condition (ℍ) in Lemma 1 is satisfied. Thus, by Lemma 1, the local Hopf bifurcation occurs near (𝑎^𝐻_𝑗,𝑢_𝑎^𝐻_𝑗,𝑣_𝑎^𝐻_𝑗) at 𝑎=𝑎^𝐻_𝑗, with 𝑗∈{0,1,2,…}. By the expression of (14), if 𝑘=0, then the solution depends only on the time variable 𝑡; thus, the periodic solution is exactly the same as that of the periodic solutions in the corresponding ODEs system. However, if 𝑘≥1, then the periodic solutions depend not only on 𝑥 but also on 𝑡 and thus are spatially nonhomogeneous.

3. Stability of the Spatially Homogeneous Periodic Solutions Bifurcating from 𝑎=𝑎^𝐻₀

In this section, we consider the Hopf bifurcation direction and the stability of the bifurcating spatially homogeneous periodic solutions from the Hopf bifurcation points 𝑎^𝐻₀.

Theorem 3. For the system (1), the Hopf bifurcation at 𝑎=𝑎^𝐻₀ is subcritical, and the bifurcating (spatial homogeneous) periodic solutions are unstable.

Proof. By Theorem 2.1 of [12], in order to determine the stability and bifurcation direction of the bifurcating periodic solution, we need to calculate Re(𝑐₁(𝑎^𝐻₀)). When 𝑎=𝑎^𝐻₀, we put 𝑚𝑞∶=⁰𝑛₀=1−1+𝑖𝜔₀𝑠𝑏−𝑎^𝐻₀,𝑞^∗𝑚∶=^∗0𝑛^∗₀=1+2𝜉𝜋𝑖𝑠𝑏−𝑎^𝐻₀2𝜔₀𝜉𝜋𝑖𝑠𝑏−𝑎^𝐻₀2𝜔₀,𝜉𝜋 (25) where 𝜔₀=𝑠𝑏−𝑎^𝐻₀.
As in Section 2 of [12], we define 𝑓(𝑎,𝑢,𝑣)=𝑠𝑢+𝑢_𝑎𝑣+𝑣_𝑎−𝑢+𝑢_𝑎,𝑔()−𝑎𝑎,𝑢,𝑣=𝑠𝑏−𝑢+𝑢_𝑎𝑣+𝑣_𝑎; (26) then we have, by (2.19) of [12], (we use the same notations of 𝑐₀, 𝑑₀, 𝑒₀, 𝑓₀, 𝑔₀ as those appeared in [12]) 𝑐₀=−𝑑₀=2𝑠−1+𝑖𝜔₀𝑠𝑏−𝑎^𝐻₀,𝑒₀=−𝑓₀=−2𝑠,𝑔₀=ℎ₀=0. (27) Thus, as in [12], we need to calculate ⟨𝑞^∗,𝑄_𝑞𝑞⟩, ⟨𝑞^∗,𝑄_𝑞𝑞⟩, ⟨𝑞^∗,𝑄_𝑞𝑞⟩, and ⟨𝑞^∗,𝐶_𝑞𝑞𝑞⟩. They are ⟨𝑞^∗,𝑄_𝑞𝑞⟩=₀^𝜉𝜋𝑞^∗𝑇𝑐₀,𝑑₀^𝑇𝑑𝑥=𝑖𝜔₀𝑏−𝑎^𝐻₀−𝑠,⟨𝑞^∗,𝑄_𝑞𝑞⟩=₀^𝜉𝜋𝑞^∗𝑇𝑒₀,𝑓₀^𝑇⟨𝑑𝑥=−𝑠,𝑞^∗,𝑄_𝑞𝑞⟩=₀^𝜉𝜋𝑞^∗𝑇𝑐₀,𝑑₀^𝑇𝑑𝑥=𝑖𝜔₀𝑏−𝑎^𝐻₀−𝑠,⟨𝑞^∗,𝐶_𝑞𝑞𝑞⟩=₀^𝜉𝜋𝑞^∗𝑇𝑔₀,ℎ₀^𝑇𝑑𝑥=0. (28)
Hence it is straightforward to calculate 𝐻₂₀=𝑐₀𝑑₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑚₀𝑛₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑚₀𝑛₀𝐻=0,₁₁=𝑒₀𝑓₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑚₀𝑛₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑚₀𝑛₀=0, (29) which implies that 𝑤₂₀=𝑤₁₁=0. So, ⟨𝑞^∗,𝑄_𝑤11𝑞⟩=⟨𝑞^∗,𝑄_𝑤20𝑞⟩=0. (30)
Therefore, after the steps of calculations, we obtain Re𝑐₁𝑎^𝐻₀𝑖=Re2𝜔₀⟨𝑞^∗,𝑄_𝑞𝑞⟩⋅⟨𝑞^∗,𝑄_𝑞𝑞1⟩+2⟨𝑞^∗,𝐶_𝑞𝑞𝑞⟩=𝑠2𝑏−𝑎^𝐻₀>0. (31)
From (20), it follows that 𝛼^′(𝑎^𝐻₀)>0, and then by Theorem 2.2 of [12] the bifurcation is subcritical. On the other hand, from (13), 𝑇_𝑘(𝑎^𝐻₀)<0 and 𝐷_𝑘(𝑎^𝐻₀)>0 for any 𝑛≥1, so the bifurcating periodic solutions are unstable since Re(𝑐₁(𝑎^𝐻₀))>0.

4. Conclusions

In this paper, by using the standard Hopf bifurcation theorem, we are able to prove the existence of periodic solutions of the system. And we also proved that the spatial homogeneous periodic solution bifurcating from the smallest Hopf bifurcation point 𝑎^𝐻₀ is unstable. By [12], the spatially nonhomogeneous periodic solutions bifurcating from 𝑎^𝐻_𝑗, 𝑗∈{1,2,…}, are all unstable. This together with our results shows that all the Hopf bifurcating periodic patterns are unstable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the anonymous referee for the comments and suggestions which definitely led to an improved presentation of the paper.