Abstract
The reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diffusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. The result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. Then, we analyze the system equation with the diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system. Finally, we perform the numerical simulations for the obtained results which show that the Turing patterns are either spot or stripe patterns.
1. Introduction
As early as 1952s, the famous British mathematician Turing turned his attention to the field of biology and succeeded with a reaction diffusion system. Meanwhile, the principle of surface pattern generation in some organisms is illustrated in his well-known paper [1]. Turing also mathematically showed that in a reaction-diffusion system, the steady state is unstable under certain conditions and spontaneously generates spatial stationary patterns and the pattern is usually called the Turing patterns [2].
It is worth nothing that Turing patterns have long been widely found in nature and in many experimental systems, such as real chemical system [3–5], spiral galaxies in space [6], spiral wave electrical signals of myocardial tissue [7], biology systems [8, 9], hyper-points in nonlinear optical systems [10], etc. To this end, the exploration and analysis of the Turing patterns has attracted the attention of many scholars. For instance, in their seminal paper, Gierer and Meinhardt [8] proposed a kind of reaction diffusion system, which contains Turing patterns. For the Gierer–Meinhardt system, there are a lot of related research work on it. Ruan [11] has studied the instability of the homogeneous equilibrium and periodic solution under different diffusion coefficients. Kolokolnikov et al. [12] have analyzed the stability of a stripe for the Gierer–Meinhardt system and the effect of saturation. Ghergu [13] has considered steady state solutions in the Gierer–Meinhardt system with Dirichlet boundary condition. An et al. [14] have studied the explicit solution to the initial-boundary value problem of Gierer–Meinhardt model under certain conditions.
Although much work has been done in this research field, most researchers are mainly concerned with the Activator-Inhibitor model of Gierer–Meinhardt system. However, the Depletion model as a remarkable type of reaction diffusion system with its research value is also very intuitive. Therefore, we will mainly concentrate on Hopf bifurcation and Turing instability analysis for Depletion model in this paper. More specifically, Depletion model has the following form
where is the sources of distribution, and represent the density of the activator and consumed by activation, and are positive constants.
In addition, Gierer and Meinhardt set , then system becomes
In this paper, we will follow the same settings as above, and for notational convenience, we use instead of , thus the system is expressed as
The rest of paper is organized as follows: In Section 2, we analyze the existence and stability of the positive equilibrium and the Hopf bifurcation. Section 3 studies the Turing instability of the equilibrium and limit cycles, and theoretically give the sufficient conditions and periodic solutions for the equilibrium and spatial homogeneous Turing instability. Lastly, in Section 4, we give examples to illustrate the analytic conditions and the numerical simulations are presented to verify the theoretical analysis.
2. Stability Analysis of the Equilibrium
In this section, we mainly consider the system equation without diffusion. Thus, we can write system (4) as
Obviously, there is a unique equilibrium The Jacobian matrix of (4) is
Therefore, the Jacobian matrix at the equilibrium is
and the corresponding characteristic equation is
It is easy to note that characteristic equation (7) has two eigenvalues
and
Let
We have the following Theorem 1.
Theorem 1. System (4) has a unique positive equilibrium that is asymptotically stable if either condition or condition holds and is unstable if condition holds.
Furthermore. (i) The equilibrium is a stable node if one of the following conditions is satisfied: (ia) (ib) (ic) (ii) The equilibrium is a stable foucs if one of the following conditions is satisfied: (iia) (iib) (iii) The equilibrium is an unstable node if (iv) The equilibrium is an unstable foucs if
Proof. Assume condition holds, then if or . Assume holds, it is easy to verify that and we have if or .
From the above analysis, we note that the system (4) experiences a Hopf bifurcation at under condition .
Let then system (4) linearizes at equilibrium to obtain the following system
where
and represents the remaining terms with order greater than or equal to 4.
According to the Hopf bifurcation theorem [15], we need nonzero transversally condition in fact Nextly, we consider the Hopf bifurcation at the critical point For we can get and and the Jacobian matrix is
It can be easily obtained that the eigenvector corresponding to the eigenvalue of the matrix is
Setting
and the transformation
then (12) is converted into
where
In order to determine the type of the Hopf bifurcation at equilibrium according to [16], the type of bifurcation is determined by the following symbols.
where
To this end,
According to the above analysis, we can get the following Theorem 2.
Theorem 2. Assume then system (4) at the equilibrium experiences a Hopf bifurcation for . Since , the Hopf bifurcation is supercritical and the bifurcated limit cycle is stable.
3. Turing Instability Analysis
In this section, we will consider the system equation with diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system (3).
3.1. Instability Analysis of the Equilibrium
We first assume that condition is established. Obviously, the equilibrium is a stable for system (4) with Neumann boundary conditions
We consider the diffusion system (3) in the Banach space , where
It is easy to obtain that equilibrium is a stable solution of (3) and (30). The equilibrium is nonlinearly unstable for (3) and (30), if it is linearly unstable in .
Let then diffusion system (3) is transformed into
with the boundary conditions
Let be a solution of (32) and (33). Since (32) is linear, we can signify it as
where is the temporal spectrum, is the wave number and are real numbers for . Bring (34) into (32), we know
Considering the like terms with the wave number of , we have
where
It is easy to verify that (36) has a nonzero solution if and only if
We are searching conditions and to satisfy the equation
where
and
Under condition we have for all and
In addition, the equilibrium is still stable, if for all . The equilibrium will lose its stability and occur Turing pattern occurs, if for at least one makes , namely, if , then there exist at least one negative in Hence, the equilibrium is unstable for system (3).
Basing on the above analysis, we have Theorem 3.
Theorem 3. Assume condition holds, let
then is a stable equilibrium for system (3) if condition holds and is an unstable equilibrium for system (3) if condition holds.
3.2. Instability Analysis of the Limit Cycle
In this subsection, we discuss the stability of the limit cycle in Theorem 3 under spatially inhomogeneous perturbations. Assume condition is satisfied, then the supercritical Hopf bifurcation appears at Therefore, the limit cycle is stable under spatially homogeneous perturbation.
According to [17], let and then system (3) becomes
where
and and are defined in (13) and (14), respectively.
According to [18], the form of is as follows
where and are in the following form
with
for any , , , and .
The linear operator defined in (30) for is
for
Let be the conjugate adjoint operator of defined in , then
where
It is easy to note that for any and define the inner product of on
The linearized system of (44) at the equilibrium is
with the boundary conditions
Let is a solution of (56) and (57). Since (56) is linear, we can denote it as
where is the temporal spectrum, is the wave number and , are real numbers for . Plugging (58) into (56), we have
Nextly, we consider like terms with the wave number of , we have
where
It is obvious to note that (60) has a nonzero solution if and only if
We are searching conditions such that and satisfies equation
where
and
we have and for all and if condition holds. Then, it follows that for has eigenvalues with zero real parts. We need to proceed to the center manifold reduction.
Firstly, if then for all . In addition, if there exists at least one negative in .
Let and by calculation we can get and respectively. It is easy to verify that and
According to [18], let
and
The coordinate of the system (44) on is converted to
where
and
So
where
Besides, we have
Hence, .
According to [18], the system possesses a central manifold, which we write as
Since then
In consequence, the diffusion system restricted to the center manifold is
where
The dynamics of the system (68) are determined by (77), and the normal form of Poincaré of (44) is expressed as
where is a complex variable, and is a complex-valued coefficient. So we can obtain
Owing to and then
Since
we have As a consequence, the supercritical Hopf bifurcation occurs at
We summarize the above analysis and get the following Theorem 4.
Theorem 4. Assume condition holds, the spatially homogeneous periodic solution of system (4) bifurcated from the equilibrium is stable. Let
The spatially homogeneous periodic solution for system (3) is stable if condition holds and unstable if condition holds.
4. Numerical Simulations
In this section, we will use numerical simulations to illustrate the results in Sections 2 and 3.
A Hopf bifurcation occurs when a periodic solution or limit cycle, surrounding an equilibrium, arises or goes away as a parameter varies. When a stable limit cycle surrounds an unstable equilibrium, the bifurcation is called a supercritical Hopf bifurcation [19].
Firstly, we draw the supercritical Hopf bifurcation diagram of (4) in parameter space , and we also draw a Hopf bifurcation on a two-dimensional system in polar coordinates of (4), please see Figures 1(a) and 1(b), respectively.
(a)
(b)
Furthermore, we set then the equilibrium . Let (here ), so that condition in Theorem 1 is satisfied, and the equilibrium is asymptotically stable. Let (here ), so that condition in Theorem 1 is satisfied, and the equilibrium is unstable and phase orbit converges to the stable limit cycle. Figure 2(a) shows the equilibrium is asymptotically stable from initial values Figure 2(b) shows that phase orbit starting from converges to the stable limit cycle.
(a)
(b)
Furthermore, we introduce the effect of diffusion on the equilibrium and the spatial homogeneous periodic solution.
Let , then the conditions and of Theorem 3 hold. Thus, the stable equilibrium of system (4) is still stable for system (3). Please see Figures 3(a) and 3(b). Let then , the conditions and of Theorem 3 hold. As a result, the stable equilibrium of system (4) becomes unstable for system (3) because of diffusion. Please see Figures 4(a) and 4(b).
(a)
(b)
(a)
(b)
Let then the equilibrium , the conditions and of Theorem 4 hold. Hence, the stable limit cycle of system (4) is still stable for system (3). Please see Figures 5(a) and 5(b). Let then the equilibrium , the conditions and of Theorem 4 hold. Thus, the stable limit cycle of system (4) becomes unstable for system (3) because of diffusion.Please see Figures 6(a) and 6(b).
(a)
(b)
(a)
(b)
5. Conclusions
Turing pattern dynamics of Gierer–Meinhardt model of the Depletion type is demonstrated in this paper. Through the mathematical analysis, we note that the system (4) undergoes a Hopf bifurcation at the equilibrium for and the Hopf bifurcation is supercritical. Under the conditions of diffusion, some conditions of the Turing instability are obtained. To this end, the system (3) will have diffusion-driven instability and some spot or stripe patterns will be possibly formed. In addition, to further verify the validity of theoretical analysis, the numerical simulation methods are employed. From the outcome of numerical simulation, the complex dynamics does happen in Gierer–Meinhardt model of the Depletion type. In particular, we can note that the parameters are different and the patterns formed will be different. To sum up, the results will help to understand the formation of biological patterns and the method provides us with an understanding of the dynamical complexity of space and time in the Depletion model. More interesting and complex behavior about such model will further be explored in the future.
Data Availability
The data in this paper are generated in numerical simulations.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This work was supported by National Natural Science Foundation of China (No.71403022) and Beijing Municipal Social Science Foundation (No.18YJB011).