# Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources

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## Abstract

We study the pattern *generating* mechanism of a generalized Gierer–Meinhardt model with diffusions. We show the existence and stability of the Hopf *bifurcation* for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we *use bifurcation analysis*, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer–Meinhardt system is classical, our system with more general settings has yet *to be* analyzed in the literature.

## Keywords

Supercritical Hopf bifurcation Turing instability Spatial pattern## 1 Introduction

*Under*given conditions, chemical compounds can interact with each other and spread in space in some ways, which result in heterogeneous spatial patterns of chemical compound or morphogen concentration [2]. This means that

*without*diffusions, the homogenous equilibrium maintains stability to small perturbations, whereas with diffusions, the homogenous equilibrium may lose its stability, and spatial inhomogeneous patterns can emerge due to the unequal spatial diffusions. Reaction–diffusion equations and systems can characterize a substantial number of pattern-related biology phenomena. In 1972, Gierer and Meinhardt [3] constructed a prototypical activator and inhibitor model of the form

*a*and an inhibitor

*h*acting on sources of activators and inhibitors have distributions \(\rho (x)\) and \(\rho'(x)\), respectively. At time \(t>0\) and spatial position

*x*, the concentrations of the activator and inhibitor are expressed by \(a=a(x,t)\) and \(h=h(x,t)\). The terms \(a^{r}/h^{s}\) and \(a^{T}/h^{u}\) represent the activation and inhibition of sources. The terms

*μa*and

*νh*represent the leakage, reuptake by source, enzyme degradation, or any of their combinations. So

*μa*and

*νh*are removed from

*a*and

*h*, and \(D_{a}\) and \(D_{h}\) denote the spatial diffusion coefficients. Taking into account the actual biological significance, all the parameters are positive constants. Furthermore, to form a gradient,

*r*,

*s*,

*T*, and

*u*must satisfy \(\frac{sT}{u+1}>r-1>0\), which means that

*r*must be at least 2 if it is an integer [3]. Reaction–diffusion

*system*(

*1*)

*is*seen as one of

*the*most important systems characterizing the formation of patterns [4, 5].

Over the years, system (1) has attracted considerable attention. In [6], the author investigated system (1), subjected to Neumann boundary conditions on the interval \((0,\pi )\), by taking \(r=T=2,s=1,u=0\) and showed that if the diffusion coefficients are selected suitably, then the homogeneous steady state and the time-dependent periodic solution can undergo Turing instability. The same system was considered but subjected to Dirichlet boundary conditions on the interval \((0,l\pi)\), \(l\in\mathbb{R}^{+}\). We refer to [7] for more detail. Furthermore, *many researchers studied the system by considering the saturation of activator production*. In this situation, the activated area and the total structure size are proportional [8], since the activator concentration has a maximum value. In [9], the authors investigated the same system but with saturated activator production under Neumann boundary conditions in the interval \((0,\pi)\) and showed the existence of Turing instabilities of the positive spatial homogeneous equilibrium and homogenous periodic solution. They found that there are at least two limit cycles. Spectral analysis and Floquet exponent as in [10, 11] are significant in analyzing the dynamical behavior of reaction–diffusion systems. In [12], the authors studied the global attractivity of equilibrium and gene expression time delays in the same system with production saturation. In [13], the authors investigated system (1) with \(r=s=2,T=1,u=0\) and obtained the parameter range for the system to become diffusively unstable. By taking \(u=s=4\) and \(r=T=2\) system (1) can be extended to the generalized or modified Gierer–Meinhardt model as in [14, 15, 16, 17]. A meaningful result of the generalized system is the spike solutions for the related elliptic system in regions in \(\mathbb{R}^{n}\). When \(r=T,s=u\), and \(\rho =\rho'\), system (1) is called the general form of activator–inhibitor system with common sources [3], which was studied in [18]. They obtained a precise parametric condition for the presence of Turing instability. For more results about reaction diffusion systems of activator–inhibitor type, see [19, 20, 21]. See [22, 23, 24] and the references *therein* for more real-world models on Turing instability.

So far, most work on Turing instability of system (1) has been carried out by taking specific parameter values. In contrast, few work is done directly from the general system (1). The increasing number of parameters in the general system (1) make *the mechanism of spatial pattern formation more difficult to understand*. System (1) with \(s \neq u\) is called the general form of activator–inhibitor system with different sources [3]. In comparison with the activator–inhibitor system with common sources, a system with different sources has stronger nonlinearity. Therefore, the analysis is more involved.

*want*to understand the dynamics responsible for the spot and stripe patterns.

*We*drop the asterisks and use lowercase letters for algebraic convenience. Then system (2) becomes

*c*,

*r*,

*μ*,

*ν*, \(d>0\), \(a\geq0\), \(h>0\), and \(r\geq2, r \in\mathbb{N}^{+}\). We also impose the Neumann boundary conditions

The remaining parts of the paper are organized as follows. In Sect. 2, we proceed with a detailed study on the dynamical behavior of the corresponding kinetic system, such as the existence and stability of positive homogeneous solutions and a steady-state and time-dependent periodic solution bifurcated from Hopf bifurcation. In Sect. 3, we derive sufficient analytic conditions for homogeneous solutions to undergo diffusion-driven instability. Numerical examples to illustrate the analytic results are presented in Sect. 4. From the simulations we see spot and stripe spatial patterns.

## 2 Homogeneous equilibria and stability

### Lemma 2.1

*For*\(c, \mu, \nu>0\)

*and*\(r\geq2, r\in\mathbb{N}^{+}\), the auxiliary function \(\phi(a)=c-a\mu+a^{r-r^{2}}\nu^{r}\), \(a \in(0,+\infty )\) has

*the following properties*:

- (1)
\(\phi(a)\)

*is a decreasing function in*\((0,+\infty)\); - (2)
\(\lim_{a \to0}{\phi(a)}=+\infty\), \(\lim_{a \to+\infty}{\phi (a)}=-\infty\);

- (3)
\(\phi(a)\)

*is a concave function in*\((0,+\infty)\); - (4)
\(\phi(a)\)

*has a unique zero point in*\((\frac{c}{\mu},+\infty)\).

### Proof

Conclusions (1), (2), and (3) are obvious. The fact that \(\phi(\frac{c}{\mu})=(\frac{c}{\mu})^{r-r^{2}}\nu^{r}>0\), along with (1) and (2), leads to (4). □

The following proposition gives the existence and uniqueness of the positive equilibrium for system (5).

### Proposition 2.1

*For*\(a \geq0, h > 0\), *system* (5) *has a unique positive equilibrium*\((a_{*},h_{*})\)*with*\(\phi(a_{*})=0\), \(h_{*}=a_{*}^{r}/{\nu}\), *and*\(a_{*}\in(\frac{c}{\mu},+\infty)\).

### Proof

The equilibrium \((a_{*},h_{*})\)*of system* ( *5* ) *is the solution of equation* ( *7* ). Then \(a_{*}>0\) and \(h_{*}>0\). The remaining conclusions follow from Lemma 2.1. □

On the stability of the equilibrium \((a_{*},h_{*})\), we have the following:

### Proposition 2.2

*The unique positive equilibrium*\((a_{*},h_{*})\)

*of system*(5)

*is stable if one of the following conditions holds*:

- (H1)
\(0< (r-1)\mu\leq\nu\), \(c>0\);

- (H2)
\((r-1)\mu> \nu>0\), \(c > c_{h}\).

*It is unstable if*

- (H3)
\((r-1)\mu>\nu>0\), \(0< c < c_{h}\),

*where*\(c_{h}=\frac{a_{0}}{r}[(r-1)\mu-\nu]\)

*and*\(a_{0}= ( \frac{r\nu ^{r}}{\mu+\nu} )^{\frac{1}{1-r+r^{2}}}\).

### Proof

*one or more*eigenvalues

*have*a positive real part. This implies that the equilibrium is unstable. □

The next corollary gives detailed information about the equilibrium.

### Corollary 2.1

*Suppose*\(r \geq2\)

*and let*

- (1)
*For*\(\mu\neq\nu\),*the equilibrium*\((a_{*},h_{*})\)*of system*(5)*is a stable node if one of the following conditions holds*:- (SN1)
\(0< \mu\leq\mu_{01}, c > 0\);

- (SN2)
\(\mu>\mu_{01},c>\bar {M}_{01}\).

*It is an unstable node if*- (UN)
\(\mu>\mu_{02},0< c<\bar{M}_{02}\).

- (SN1)
- (2)
*For*\(\mu\neq\nu\),*the equilibrium*\((a_{*},h_{*})\)*of system*(5)*is a stable focus if one of the following conditions holds*:- (SF1)
\(\mu_{01}<\mu\leq\frac{\nu}{r-1},0<c<\bar{M}_{01}\);

- (SF2)
\(\mu>\frac{\nu}{r-1}, c_{h}< c<\bar{M}_{01}\).

*It is an unstable focus if one of the following conditions holds*:- (UF1)
\(\frac{\nu}{r-1}<\mu\leq\mu_{02},0<c<c_{h}\);

- (UF2)
\(\mu>\mu _{02},\bar{M}_{02}< c< c_{h}\).

- (SF1)
- (3)
*For*\(\mu=\nu\),*the equilibrium*\((a_{*},h_{*})\)*of system*(5)*can only be a focus*.*It is stable if*- (SF3)
\(c>c_{h}\)

*and is unstable if*- (UF3)
\(0< c< c_{h}\).

- (SF3)

*c*close to \(c_{h}\). This is a necessary condition for the occurrence of the Hopf bifurcation. We further prove this and

*analyze*the direction and stability of the Hopf bifurcation. Suppose that \((r-1)\mu>\nu>0\) and

*c*close to \(c_{h}\), and let \(x=a-a_{*}\), \(y=h-h_{*}\). Then system (5) becomes

*the eigenvector corresponding to*\(i\omega_{0}\)

*is*\(\xi=(\nu(\mu+\nu )a_{0}^{1-r}, \nu-i\omega_{0})^{T}\)

*, where*\(\omega_{0}= \sqrt{\nu(r\mu-\nu+r\nu )}\)

*.*Furthermore, since \(r \geq2, r \in\mathbb{N}^{+} \), we have

*η*is positive or negative [25, 26]. We calculate:

## 3 Analysis on the full reaction–diffusion model

From [1] we know that the diffusion can influence the stability of the homogeneous solutions. Turing instability occurs when the homogeneous solutions, which should be stable for system (5), become unstable due to diffusions. In this section, we focus on the *full reaction–diffusion model* ( *3* )*–*( *4* ) to obtain the parametric ranges in which the homogeneous solutions undergo Turing instability. The analysis of Turing instabilities for the *equilibrium*\((a_{*},h_{*})\) and the periodic solution bifurcated from Hopf bifurcation is carried out respectively in Sects. 3.1 and 3.2.

### 3.1 Turing instability of the equilibrium for *the full reaction–diffusion model*

In this section, we follow the standard treatment for this type of problems as in [9]. First, we assume that one of conditions (H1) and (H2) holds and study system (3) with the Neumann boundary conditions (4) in the Banach space \(\mathbb {H}^{2}((0,\pi))\times\mathbb {H}^{2}((0,\pi))\), where \(\mathbb {H}^{2}((0,\pi ))=\{w(\cdot)\mid\frac{\partial^{i}w}{\partial x^{i}}(\cdot)\in\mathbb {L}^{2}((0,\pi)),i=0,1,2\}\). Obviously, \((a_{*},h_{*})\) is a steady state for system (3)–(4).

*k*is the wave number, and \(A_{k},H_{k} \in\mathbb{R}\) for \(k=0,1,2 ,\ldots\) . Substituting (12) into system (10), we have

*k*, we have

For any wave number *k*, there exists a pair of temporal eigenvalues \(\lambda_{k}^{\pm}= \frac{\operatorname{Tr}(J_{k}(c))\pm\sqrt{\operatorname {Tr}^{2}(J_{k}(c))-4\operatorname{Det}(J_{k}(c))}}{2}\). If (H1) or (H2) holds, then, for \(k=0\), \(\operatorname{Tr}(J_{k}(c))=\operatorname{Tr}J(c)<0\) and \(\operatorname {Det}(J_{k}(c))=\operatorname{Det}J(c)>0\), which implies that the real parts of the related temporal spectra \(\lambda_{0}^{\pm}\) are negative. Since \(\operatorname{Tr}(J_{k}(c))=\operatorname{Tr}J(c)-(1+d)k^{2} < 0\) for \(k=1,2,\ldots \) , we need further consider the sign of \(\operatorname{Det}(J_{k}(c))\). If \(r\nu^{r}a_{*}^{-1+r-r^{2}}-\mu\leq1\), then we have \(a_{*} \geq(\frac{r\nu ^{r}}{\mu+1})^{\frac{1}{1-r+r^{2}}}\) since \(-1+r-r^{2} <0\). Denoting \(\bar {a}= (\frac{r\nu^{r}}{\mu+1})^{\frac{1}{1-r+r^{2}}}\), from Lemma 2.1 and Proposition 2.1 we have \(\phi(\bar{a}) \geq0\), that is, \(c \geq\frac{\bar{a}}{r}[\mu(r-1)-1]\). Therefore \(k^{2}-(r\nu ^{r}a_{*}^{-1+r-r^{2}}-\mu)\geq0\), which means that \(\operatorname{Det}(J_{k}(c)) >0\). Moreover, if \(m^{2} < r\nu^{r}a_{*}^{-1+r-r^{2}}-\mu\leq(m+1)^{2}, m \in \mathbb {N}^{+}\), then we have \([\frac{r\nu^{r}}{\mu+(m+1)^{2}}]^{\frac {1}{1-r+r^{2}}} \leq a_{*} < (\frac{r\nu^{r}}{\mu+m^{2}})^{\frac{1}{1-r+r^{2}}}\) since \(-1+r-r^{2} <0\). Denoting \(a_{m+1}=[\frac{r\nu^{r}}{\mu +(m+1)^{2}}]^{\frac{1}{1-r+r^{2}}}\) and \(a_{m}=(\frac{r\nu^{r}}{\mu+m^{2}})^{\frac {1}{1-r+r^{2}}}\), from Lemma 2.1 and Proposition 2.1 we have \(\phi(a_{m})<0\leq\phi(a_{m+1})\), that is, \(\frac{a_{m+1}}{r}[\mu (r-1)-(m+1)^{2}] \leq c < \frac{a_{m}}{r}[\mu(r-1)-m^{2}]\). In this case, for any \(k=m+1,m+2,\ldots \) , we have \(k^{2}-(r\nu^{r}a_{*}^{-1+r-r^{2}}-\mu)\geq0\), which indicates that \(\operatorname{Det}(J_{k}(c)) >0\), whereas for any \(k=1,2,\ldots,m\), we have \(k^{2}-(r\nu^{r}a_{*}^{-1+r-r^{2}}-\mu)<0\). Further computations reveal that if \(0< d < \min_{1\leq k \leq m}\frac{(k^{2}+\mu )\nu+(r-1)r\nu^{1+r}a_{*}^{-1+r-r^{2}}}{k^{2}(r\nu^{r}a_{*}^{-1+r-r^{2}}-\mu -k^{2})}\), then \(\operatorname{Det}(J_{k}(c))>0\).

The above discussion implies *that*, for any spatial spectrum \(k=1,2,\ldots \) , the real parts of the corresponding temporal spectrum \(\lambda_{k}^{\pm}\) are negative. This means the real parts of the solutions of (16) are negative for all \(k=0,1,2,\ldots \) , so that \((a_{*},h_{*})\) is asymptotically stable for (3). According to [28], because of the sectorial of *L*, \((a_{*},h_{*})\) is also locally uniformly stable, and therefore there is no Turing pattern for system (3) under the above conditions. However, if \(d > \min_{1\leq k \leq m}\frac{(k^{2}+\mu)\nu+(r-1)r\nu ^{1+r}a_{*}^{-1+r-r^{2}}}{k^{2}(r\nu^{r}a_{*}^{-1+r-r^{2}}-\mu-k^{2})}\), then at least one of \(\operatorname{Det}(J_{1}(c)), \operatorname{Det} (J_{2}(c)), \ldots, \operatorname{Det}(J_{m}(c))\) is negative. Thus the equilibrium is unstable for (3). In this case, Turing patterns will occur. Summarizing the discussion, we have the following:

### Theorem 3.1

*Let*(H1)

*or*(H2)

*hold*.

*Denote*

*Then the equilibrium*\((a_{*},h_{*})\)

*of system*(3)

*persists the stability if one of the following conditions holds*:

- (H4)
\(c\geq\frac{\bar{a}}{r}[\mu(r-1)-1]\),

- (H5)
\(A_{m+1}\leq c < A_{m},0<d < \hat{d}\).

*It is unstable if*

- (H6)
\(A_{m+1}\leq c < A_{m}, d>\hat{d}\),

*where*\(\bar{a}=(\frac{r\nu^{r}}{\mu+1})^{\frac{1}{1-r+r^{2}}}\), \(A_{j}=\frac {a_{j}}{r}[\mu(r-1)-j^{2}]\), \(a_{j}=(\frac{r\nu^{r}}{\mu+j^{2}})^{\frac {1}{1-r+r^{2}}}, j=m, m+1, m \in\mathbb {N}^{+}\).

### Remark 3.1

It is worth pointing out that in Theorem 3.1 if \(d=\hat{d}\), then from the discussion it follows that there exists at least one \(k, k=1,2,\ldots,m\), \(m \in\mathbb {N}^{+}\), such that the linearized system of (3) at \((a_{*},h_{*})\) has a zero eigenvalue. At this time, the stability of \((a_{*},h_{*})\) for system (3) cannot be determined by the linearized system.

### 3.2 Turing instability of the limit cycle for the *full reaction–diffusion model*

From Theorem 2.1 note that if \(0<\nu<(r-1)\mu\), then there is a limit cycle bifurcated from \((a_{*},h_{*})\) for *c* sufficiently close to \(c_{h}\). We investigate the stability of the limit cycle obtained in Theorem 2.1. Throughout this section, we assume condition (H3), so that the limit cycle is stable to homogeneous perturbation.

*L*defined in (10) is

*evaluated at*\((0,0)\) is

*k*is the wave number, and \(a_{k},h_{k} \in\mathbb{R}\) for \(k=0,1,2,\ldots\) . Substituting (20) into (18) and comparing the like terms about

*k*, we have

*k*, a sufficient and necessary condition for (21) to have a nonzero solution \((a_{k},h_{k})^{T}\) is \(\operatorname{Det}(\lambda(k) I-L_{k})=0\), that is,

With \(0<\nu<(r-1)\mu\) and \(c=c_{h}\), we have \(\operatorname{Tr}(L_{0})=0\), \(\operatorname{Det}(L_{0})=\nu(r\mu-\nu+r\nu)>0\), and \(\operatorname{Tr}(L_{k})<0\) for \(k=1,2,\ldots \) . This means that, for \(k=0\), the real parts of the eigenvalues of *L* are zero. We then have to do the center manifold reduction.

First of all, for \(k=1,2,\ldots \) , if \(0<\nu\leq1\), then \(\operatorname {Det}(L_{k})>0\). Furthermore, if \(m^{2}<\nu\leq(m+1)^{2}\) and \(0< d<\bar {d}\), where \(\bar{d}=\min_{1 \leq k\leq m} \frac{k^{2}\nu+\nu(r\mu-\nu +r\nu)}{-k^{2}(k^{2}-\nu)}, m \in\mathbb{N}^{+}\), then we have \(\operatorname {Det}(L_{k})>0\) for \(k=1,2,\ldots \) , and if \(m^{2}<\nu\leq(m+1)^{2}\) and \(d>\bar{d}\), then at least one of \(\operatorname{Det}(L_{1}),\operatorname {Det}(L_{2}),\ldots,\operatorname{Det}(L_{m})\) is negative.

### Theorem 3.2

*Let*(H3)

*hold*,

*and let*

*Then the spatially homogenous periodic solution for*(3)

*is stable if either*(H7)

*or*(H8)

*holds and is unstable if*(H9)

*holds*,

*where*

- (H7)
\(0<\nu\leq1 \),

- (H8)
\(m^{2} <\nu\leq(m+1)^{2}\), \(0< d< \bar{d}\),

*and* - (H9)
\(m^{2} <\nu\leq(m+1)^{2}\), \(d>\bar{d}\).

## 4 Numerical simulation

In this section, numerical examples are used to illustrate the main conclusions in Sects. 2 and 3. In this section, we assume that \(r=2\), \(c>0\), \(\nu>0\), and \(\mu>0\).

First of all, we illustrate the results in Proposition 2.2 and Theorem 2.1. For the dynamics of the equilibrium of the temporal model (5), we just show the simulations when (H2) and (H3) are satisfied. The simulations when condition (H1) is satisfied are omitted since the dynamic behavior is similar with the case where condition (H2) holds.

In the following, we illustrate the results obtained in Theorem 3.1.

*The initial state we choose is*\((0.53801,0.57879)\)

*, which is a small perturbation of the equilibrium*\((a_{*},h_{*})=(0.538,0.5788)\)

*.*The dynamics are shown in Fig. 7. The simulation shows that the stable equilibrium \((a_{*},h_{*})\) does not change its stability under diffusions.

*The equilibrium is*\((a_{*},h_{*})=(1.0013,0.6684)\)

*, and the initial condition is*\((1.00129,0.66839)\)

*. The corresponding dynamics are shown in Fig.*

*8*

*. The equilibrium*\((a_{*},h_{*})\)

*is stable for system*(

*3*)

*.*

In both cases the equilibrium is locally uniformly stable, and hence Turing patterns do not occur.

*The initial condition is*\((a_{*}-0.00001\cos(x),h_{*}-0.00001\sin(x)), x\in (0,300)\). The dynamics are shown in Fig. 9. In this case, Turing patterns occur. From the projections of the dynamics for the activator

*a*and inhibitor

*h*, respectively, shown in Fig. 10, we see the stripe patterns.

*These stripe patterns are formed because the equilibrium*\((a_{*},h_{*})\)

*undergoes Turing instability.*

We next illustrate the results in Theorem 3.2. In the simulation, we use the criterion in [31] for the choice of the space-time scaling parameter. From Theorem 3.2 we find that the stable limit cycle does not undergo Turing instability under (H3) and (H7) or under (H3) and (H8). Here we just illustrate the case where (H3) and (H8) hold.

*According to the simulation about the limit cycle, we find a point*\((0.8614,0.7902)\)

*on it. The initial condition is a small spatial heterogeneous perturbation of this point, namely*\((0.8614+0.000001\sin (x),0.7092+0.00001\cos(x)),x\in(0,150)\)

*. The dynamics are shown in Fig.*

*11*

*which implies that the stable limit cycle of kinetic system*(

*5*)

*is still stable for the spatial temporal system*(

*3*)

*. Figure*

*12*

*is the projection of Fig.*

*11*

*in*\((x,t)\)

*coordinates. In this case the limit cycle does not undergo Turing instability.*

*The initial condition is also*\((0.8614+0.000001\sin (x),0.7092+0.00001\cos(x)),x\in(0,150)\)

*. We can see that the stable limit cycle of system*(

*5*)

*becomes unstable for the spatial temporal system*(

*3*)

*because of the diffusions. In this case, the Turing instability of the limit cycle takes place; see Fig.*

*13*

*. Figure*

*14*

*is a local enlargement of the projection of Fig.*

*13*

*. Moreover, from the simulation we can easily see the spot patterns. These spot patterns are formed because the limit cycle undergoes Turing instability.*

## Notes

### Acknowledgements

The authors are grateful to the anonymous reviewers for the helpful suggestions and comments.

### Authors’ contributions

All authors contributed equally to writing this paper. All authors read and approved the final manuscript.

### Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 10971009, 10771196, 11771033), the National Scholarship Fund (Grant Nos. 201303070222, 201706020203, 201706020094), and the Fundamental Research Funds for the Central Universities.

### Competing interests

The authors declare that they have no competing interests.

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