Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type

e reaction diusion 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 diusion system changing with the system parameters. erefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. e result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. en, we analyze the system equation with the diusion and study the impacts of diusion coecients 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.


Introduction
As early as 1952s, the famous British mathematician Turing turned his attention to the eld of biology and succeeded with a reaction di usion system. Meanwhile, the principle of surface pattern generation in some organisms is illustrated in his wellknown paper [1]. Turing also mathematically showed that in a reaction-di usion 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][4][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 di usion 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 di erent diffusion coe cients. Kolokolnikov et al. [12] have analyzed the stability of a stripe for the Gierer-Meinhardt system and the e ect 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 eld, most researchers are mainly concerned with the Activator-Inhibitor model of Gierer-Meinhardt system. However, the Depletion model as a remarkable type of reaction di usion system with its research value is also very intuitive. erefore, we will mainly concentrate on Hopf bifurcation and Turing instability analysis for Depletion model in this paper. More speci cally, 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. (1)

Let
We have the following eorem 1.
Theorem 1. System (4) has a unique positive equilibrium that is asymptotically stable if either condition ( 1) or condition ( 2) holds and is unstable if condition ( 3) holds.
Furthermore. (i) e equilibrium * , * is a stable node if one of the following conditions is satis ed: (ii) e equilibrium * , * is a stable foucs if one of the following conditions is (iii) e equilibrium * , * is an unstable node if 0 < < 1 and 0 < ≤ 1 . (iv) e equilibrium * , * is an unstable foucs if 0 < < 1 and 1 < ≤ 0 .
Let = − * , = − * , then system (4) linearizes at equilibrium * , * to obtain the following system where and (4) represents the remaining terms with order greater than or equal to 4.
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 eorem 2. ☐ Theorem 2. Assume 0 < < 1, then system (4) at the equilibrium * , * experiences a Hopf bifurcation for = 0 . Since ὔ < 0, the Hopf bifurcation is supercritical and the bifurcated limit cycle is stable.

Turing Instability Analysis
In this section, we will consider the system equation with di usion and study the impacts of di usion coe cients on the stability of equilibrium * , * and the limit cycle of system (3).

Instability Analysis of the Equilibrium.
We rst assume that condition ( 2) is established. Obviously, the equilibrium * , * is a stable for system (4) with Neumann boundary conditions We consider the di usion system (3) in the Banach space It is easy to obtain that equilibrium * , * is a stable solution of (3) and (30). e equilibrium * , * is nonlinearly unstable for ) be a solution of (32) and (33). Since (32) is linear, we can signify it as (32) Discrete Dynamics in Nature and Society 4

Instability Analysis of the Limit Cycle.
In this subsection, we discuss the stability of the limit cycle in eorem 3 under spatially inhomogeneous perturbations. Assume condition ( 3) is satis ed, then the supercritical Hopf bifurcation appears at = 0 . erefore, the limit cycle is stable under spatially homogeneous perturbation.

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 , 0 , 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. Furthermore, we set = 0.5, then 0 = 0.1481, the equilibrium * , * = (1.500, 0.4444). Let = 0.155 (here > 0 ), so that condition ( 2) in eorem 1 is satis ed, and We summarize the above analysis and get the following eorem 4. (86)  Conflicts of Interest e author declares no con icts of interest.
Furthermore, we introduce the e ect of di usion on the equilibrium and the spatial homogeneous periodic solution. Let

Conclusions
Turing pattern dynamics of Gierer-Meinhardt model of the Depletion type is demonstrated in this paper. rough the mathematical analysis, we note that the system (4) undergoes a Hopf bifurcation at the equilibrium * , * for = 0 and the Hopf bifurcation is supercritical. Under the conditions of di usion, some conditions of the Turing instability are obtained. To this end, the system (3) will have di usion-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 di erent and the patterns formed will be di erent. 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 e data in this paper are generated in numerical simulations.