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

Authors: Lianchao Gu; Peiliang Gong; Hongqing Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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.


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## Body

## 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(1)∂a∂t=ρaa2s−μaa+σa+Da∂2a∂x2,∂s∂t=ρsa2−μss+σs+Ds∂2s∂x2,whereρx is the sources of distribution, ax,  t and sx,  t represent the density of the activator and consumed by activation, and Da,Ds,μa,μs,ρa,ρs,σa,σs are positive constants.In addition, Gierer and Meinhardt setμs=σs=ρs,ρa=μa=Ds=1, then system becomes(2)∂a∂t=sa2−a+σa+Da∂2a∂x2,∂s∂t=−μssa2+μs+∂2s∂x2.In this paper, we will follow the same settings as above, and for notational convenience, we useD,σ,μ instead of Da,σa,μs, thus the system is expressed as(3)∂a∂t=sa2−a+σ+D∂2a∂x2,∂s∂t=μ1−sa2+∂2s∂x2.The rest of paper is organized as follows: In Section2, 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(4)dadt=sa2−a+σ,dsdt=μ1−sa2.Obviously, there is a unique equilibriuma∗,s∗=1+σ,1/1+  σ2. The Jacobian matrix of (4) is(5)J=2as−1a2−2asμ−a2μ.Therefore, the Jacobian matrix at the equilibriuma∗,s∗ is(6)Jμ=1−σ1+σ1+  σ2−2μ1+σ−1+σ2μ,and the corresponding characteristic equation is(7)λ2−1−σ−μ1+  σ31+σλ+μ1+  σ2=0.It is easy to note that characteristic equation (7) has two eigenvalues(8)λ1,2=−Tr±Tr2−4Det2=−μ1+σ3+σ−1±μ1+  σ3+σ−  12−4μ1+  σ421+  σ,and(9)Δ=μ21+  σ6−21+  σ33+σμ+1−  σ21+σ2.Let(10)μ0=1−σ1+  σ3,μ1=3+  σ−221+σ1−σμ0,μ2=3+  σ+221+σ1−σμ0.We have the followingTheorem 1.Theorem 1.
System (4) has a unique positive equilibrium that is asymptotically stable if either condition H  1 or condition H  2 holds and is unstable if condition H  3 holds.(11)
H  1
σ
≥
1
.
H  2
μ
>
0
,
0
<
σ
<
1
;
H  3
μ
<
μ
0
,
0
<
σ
<
1
.Furthermore. (i) The equilibriuma∗,s∗ is a stable node if one of the following conditions is satisfied: (ia) σ≥1,0<μ≤μ1; (ib) σ≥1,μ≥μ2; (ic) 0<σ<1,μ≥μ2. (ii) The equilibrium a∗,s∗ is a stable foucs if one of the following conditions is satisfied: (iia) σ≥1,μ1<μ<μ2; (iib) 0<σ<1,μ0<μ<μ2.(iii) The equilibrium a∗,s∗ is an unstable node if 0<σ<1and0<μ≤μ1. (iv) The equilibrium a∗,s∗ is an unstable foucs if 0<σ<1andμ1<μ≤μ0.Proof.
Assume conditionH  1 holds, then Δ<0≥  0 if μ1<μ<μ20<μ  ≤μ1 or μ≥μ2. Assume 0<σ<1 holds, it is easy to verify that μ1<μ0<μ2 and we have Δ<0≥  0 if μ1<μ<μ0 or μ0<μ<μ20<μ≤μ1orμ≥μ2.From the above analysis, we note that the system (4) experiences a Hopf bifurcation at μ=μ0 under condition 0<σ<1.Letx=a−a∗,y=s−s∗, then system (4) linearizes at equilibrium a∗,s∗ to obtain the following system(12)ddtxy=Jμxy+f1x,y,  μg1x,y,  μ,where(13)f1x,y,  μ=21+  σxy+x2y+x21+  σ2+O4,(14)g1x,y,  μ=−21+  σxyμ−x2yμ−x2μ1+  σ2+O4,andO4 represents the remaining terms with order greater than or equal to 4.According to the Hopf bifurcation theorem [15], we need nonzero transversally condition d/dμReλ1,2μ=μ0, in fact d/dμReλ1,2μ=μ0=1+  σ2/2>0. Nextly, we consider the Hopf bifurcation at the critical point μ=μ0. For μ=μ0, we can get λ1,2μ0=±iω0 and ω0=1−  σ/1+  σ>0 and the Jacobian matrix is(15)Jμ0=1−σ1+σ1+σ2−21−σ1+σ4−1−σ1+σ.It can be easily obtained that the eigenvector corresponding to the eigenvalueiω0 of the matrix Jμ0 is ξ=−1+  σ3/  2−1+  σ3/2ω0i,1T.Setting(16)P=−1+σ32ω0−1+σ3201,and the transformation(17)xy=Puv,then (12) is converted into(18)u˙v˙=0−ω0ω00uv+f2u,v,μ0g2u,v,μ0,where(19)f2u,v,μ0=−1+  σ3ω041−  σu2+121+  σ2uv+341+  σ2ω0v2−1+  σ5ω041−  σu2v−121+  σ4−1+  σ4ω04v3+O4,(20)g2u,v,μ0=−141+  σ2u2+121+  σ2ω0uv+341−σ2v2−141+  σ4u2v−121+  σ4ω0uv2−141−  σ1+  σ3v3+O4.In order to determine the type of the Hopf bifurcation at equilibriuma∗,s∗, according to [16], the type of bifurcation is determined by the following symbols.(21)σ′=116f2uuu+g2uuv+f2uvv+g2vvv+116ω0f2uvf2uu+f2vv−g2uvg2uu+g2vv−f2uug2uu+f2vvg2vv,where(22)f2uu=−1+  σ321−σ2,g2uu=−121+  σ2,(23)f2uv=121+σ2,g2uv=121+σ1−σ2,(24)f2vv=−3−1+σ1+σ221−σ2,g2vv=−32−1+σ1+σ,(25)f2uuu=0,g2uuu=0,(26)f2uuv=−1+σ521−σ2,g2uuv=−121+σ4,(27)f2uvv=−1+σ4,g2uvv=−1+σ31−σ2,(28)f2vvv=3−1+σ1+σ421−σ2,g2vvv=32−1+σ1+σ3.To this end,(29)σ′=116g2uuv+f2uvv+g2vvv+116ω0f2uvf2uu+f2vv−g2uvg2uu+g2vv−f2uug2uu+f2vvg2vv=−1+σ4161−σ<0.According to the above analysis, we can get the following Theorem2.Theorem 2.
Assume0<σ<1, then system (4) at the equilibrium a∗,s∗ experiences a Hopf bifurcation for μ=μ0. Since σ′<0, 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 equilibriuma∗,s∗ and the limit cycle of system (3).


### 3.1. Instability Analysis of the Equilibrium

We first assume that conditionH2 is established. Obviously, the equilibrium a∗,s∗ is a stable for system (4) with Neumann boundary conditions(30)∂a∂x0,t=∂a∂xπ,t=0,∂s∂x0,t=∂s∂xπ,t=0.We consider the diffusion system (3) in the Banach space H20,π×H20,π, where(31)H20,π=w·,t∂iw∂xi·,t∈L20,π,i=0,1,2.It is easy to obtain that equilibriuma∗,s∗ is a stable solution of (3) and (30). The equilibrium a∗,s∗ is nonlinearly unstable for (3) and (30), if it is linearly unstable in H20,π×H20,π.Letu1=a−a∗,u2=s−s∗, then diffusion system (3) is transformed into(32)u1tu2t=1−σ1+σ+d∂xx1+σ2−2μ1+σ−μ1+σ2+∂xxu1u2≐Lu1u2,with the boundary conditions(33)u1x0,t=u2x0,t=u1xπ,t=u2xπ,t=0.Letu1,u2∈H20,π×H20,π be a solution of (32) and (33). Since (32) is linear, we can signify it as(34)u1x,tu2x,t=∑k=0∞AkSkeλt+ikx,whereλ∈C is the temporal spectrum, k is the wave number and Ak,Sk are real numbers for k=0,1,2…,n,…. Bring (34) into (32), we know(35)∑k=0∞λAkSkeλt+ikx=∑k=0∞1−σ1+σ+d∂xx1+σ2−2μ1+σ−μ1+σ2+∂xxAkSkeλt+ikx.Considering the like terms with the wave number ofk, we have(36)λI−JkμAkSk=00,k=0,1,2,…,n,…,where(37)Jkμ=1−σ1+σ−dk21+σ2−2μ1+σ−μ1+σ2−k2.It is easy to verify that (36) has a nonzero solution if and only if(38)DetλI−Jkμ=0.We are searching conditionsReλk>0 and λk to satisfy the equation(39)λ2−Trkλ+Detk=0,k=0,1,2,…,n,…,where(40)Trk=−1+dk2−μ1+σ3+σ−11+σ,and(41)Detk=k2dk21+σ−1−σ+dk2μ1+σ3+μ1+σ31+σ.Under conditionH2, we have Trk<0 for all k=0,1,2,…,n,… and Det0>0.In addition, the equilibriuma∗,s∗ is still stable, if Detk>0 for all k=0,1,2,…,n,…. The equilibrium a∗,s∗ will lose its stability and occur Turing pattern occurs, if for at least one k∈N makes Detk<0, namely, if   m2<1−σ/d1+σ≤m+12andd>Dm , then there exist at least one negative in Det1,Det2...Detm. Hence, the equilibrium a∗,s∗ is unstable for system (3).Basing on the above analysis, we have Theorem3.Theorem 3.
Assume conditionH2 holds, let(42)
D
m
=
min
1
≤
k
≤
m
k
2
1
−
σ
−
μ
1
+
σ
3
k
4
1
+
σ
+
k
2
μ
1
+
σ
3
,thena∗,s∗ is a stable equilibrium for system (3) if condition H4 holds and is an unstable equilibrium for system (3) if condition H5 holds.(43)H4d>1−σ1+σ;H5m2<1−σd1+σ≤m+12,d>Dm.


### 3.2. Instability Analysis of the Limit Cycle

In this subsection, we discuss the stability of the limit cycle in Theorem3 under spatially inhomogeneous perturbations. Assume condition H3 is satisfied, then the supercritical Hopf bifurcation appears at μ=μ0. Therefore, the limit cycle is stable under spatially homogeneous perturbation.According to [17], let u1=a−a∗,u2=s−s∗,μ=μ0, and U=u1,u2T, then system (3) becomes(44)Ut=Jμ0+D∂xx00∂xxU+FU,μ0,Ux0,t=Uxπ,t=0,0T,where(45)Jμ0=1−σ1+σ1+σ2−21−σ1+σ4−1−σ1+σ,D=d001,(46)FU,μ0=f1u1,u2,μ0,g1u1,u2,μ0T,andf1 and g1 are defined in (13) and (14), respectively.According to [18], the form of FU,μ0 is as follows(47)FU,μ0=12QU,U+16CU,U,U+OU4,whereQ and C are in the following form(48)QU,U=Q1U,U,Q2U,UT,CU,U,U=C1U,U,U,C2U,U,UT,with(49)Q1U,V=f2uuu1v1+f2uvu1v2+f2vuu2v1+f2vvu2v2=21+σ2u1v1+1+σ3u1v2+1+σ3u2v1,(50)Q2U,V=g2uuu1v1+g2uvu1v2+g2vuu2v1+g2vvu2v2=2σ−1u1v1+u1v21+σ3+1+σ3u2v11+σ5,(51)C1U,V,W=f2uuuu1v1w1+f2uuvu1v1w2+f2uvuu1v2w1+f2uvvu1v2w2+f2vuuu2v1w1+f2vuvu2v1w2+f2vvuu2v2w1+f2vvvu2v2w2=2u1v1w2+u1v2w1+u2v1w1,(52)C2U,V,W=g2uuuu1v1w1+g2uuvu1v1w2+g2uvuu1v2w1+g2uvvu1v2w2+g2vuuu2v1w1+g2vuvu2v1w2+g2vvuu2v2w1+g2vvvu2v2w2=2σ−1u1v1w2+u1v2w1+u2v1w11+σ3,for anyU=u1,u2T, V=v1,v2T, W=w1,w2T, and U,V,W∈H20,π×H20,π.The linear operatorL defined in (30) for μ=μ0 is(53)LU=Jμ0+D∂xx00∂xxU,forU∈H20,π×H20,π.LetL∗ be the conjugate adjoint operator of L defined in H20,π×H20,π, then(54)L∗U=J∗μ0+D∂xx00∂xxU,where(55)J∗μ0=1−σ1+σ2−1+σ1+σ41+σ2−1+σ1+σ.It is easy to note thatL∗U,V=U,LV for any U,V∈H20,π×H20,π and define the inner product of U,V=1/π×∫0πU¯TVdx on H20,π×H20,π.The linearized system of (44) at the equilibrium 0,0 is(56)u1tu2t=Lu1u2,with the boundary conditions(57)Ux0,t=Uxπ,t=0,0T.LetU=u1,u2T∈H20,π×H20,π is a solution of (56) and (57). Since (56) is linear, we can denote it as(58)u1x,tu2x,t=∑k=0∞akskeλt+ikx,whereλ∈C is the temporal spectrum, k is the wave number and ak, sk are real numbers for k=0,1,2…,n,…. Plugging (58) into (56), we have(59)∑k=0∞λakskeλt+ikx=∑k=0∞Lkakskeλt+ikx.Nextly, we consider like terms with the wave number ofk, we have(60)λI−Lkaksk=00,k=0,1,2,…,n,…,where(61)Lk=1−σ1+σ−dk21+σ2−21−σ1+σ4−1−σ1+σ−k2.It is obvious to note that (60) has a nonzero solution if and only if(62)detλI−Lk=0.We are searching conditions such thatReλk>0 and λk satisfies equation(63)λ2−Tkλ+Dk=0,k=0,1,2,…,n,…,where(64)Tk=−1+dk2,and(65)Dk=k2dk21+σ−1−σ1+σ+dk2+1ω02.we haveT0=0 and Tk<0 for all k=1,2,…,n,… and D0=ω02>0, if condition μ=μ0 holds. Then, it follows that for k=0,L has eigenvalues with zero real parts. We need to proceed to the center manifold reduction.Firstly, ifd>1−σ/1+σ then Dk>d+1ω02>0 for all k=1,2,…,n,…. In addition, if m2<1−σ/d1+σ≤m+12andd>D¯, there exists at least one negative in D1,D2…Dm.LetLq=iω0q and L∗q∗=−iω0q∗, by calculation we can get q=−1+σ31−σ−iω01+σ4/21−σ,1T and q∗=1−σ/−ω01+σ4i,ω01+σ−1−σi/2ω01+σT, respectively. It is easy to verify that q∗,q=1 and q∗,q¯=0.According to [18], let(66)U=zq+zq¯+w,z=q∗,U,w=w1,w2T,and(67)u1=−1+σ31−σ−iω01+σ421−σz+−1+σ31−σ+iω01+σ421−σz¯+w1,u2=z+z¯+w2.The coordinate of the system (44) on z,w is converted to(68)z˙=iω0z+q*,f˜,w˙=Lw+Hz,z¯,w,where(69)f˜=Fzq+z¯q¯+w,μ0,Hz,z¯,w=f˜−q∗,f˜q−q¯∗,f˜q¯,and(70)f˜=12Qq,qz2+Qq,q¯zz¯+12Qq¯,q¯z¯2+Oz3,z·w,w2,(71)q∗,f˜=12q∗,Qq,qz2+q∗,Qq,q¯zz¯+12q∗,Qq¯,q¯z¯2+Oz3,z·w,w2,(72)q¯∗,f˜=12q¯∗,Qq,qz2+q¯∗,Qq,q¯zz¯+12q¯∗,Qq¯,q¯z¯2+Oz3,z·w,w2.SoHz,z¯,w=1/2z2H20+zz¯H11+1/2z¯2H02+Oz3,|z|·|w|,|w|2,where(73)H20=Qq,q−q∗,Qq,qq−q¯∗,Qq,qq¯,H11=Qq,q¯−q∗,Qq,q¯q−q¯∗,Qq,q¯q¯,H02=Qq¯,q¯−q∗,Qq¯,q¯q−q¯∗,Qq¯,q¯q¯.Besides, we have(74)H20=H11=H02=0,0T.Hence,Hz,z¯,w=Oz3,|z|·|w|,|w|2.According to [18], the system possesses a central manifold, which we write as(75)w=12z2w20+zz¯w11+12z¯2w02+Oz3.SinceLw+Hz,z−,w=w˙=∂w/∂zz˙+∂w/∂z¯z¯˙, then(76)w20=2iω0-L-1H20=2iω0-Jμ0-1H20=0,0T,w11=-L-1H11=-J-1μ0H11=0,0T,w02=-2iω0-L-1H02=-2iω0-Jμ0-1H02=0,0T,w=Oz3.In consequence, the diffusion system restricted to the center manifold is(77)z˙=iω0z+q∗,f˜=iω0z+∑2≤i+j≤3giji!j!ziz¯j+Oz4,where(78)g20=q∗,Qq,q,g11=q∗,Qq,q¯,g02=q∗,Qq¯,q¯,g21=q∗,Cq,q,q¯.The dynamics of the system (68) are determined by (77), and the normal form of Poincaré of (44) is expressed as(79)z˙=αμ+iωμz+z∑j=1Mcjμzz¯j,wherez is a complex variable, M≥1, and cjμ0 is a complex-valued coefficient. So we can obtain(80)c1μ0=g20g113αμ0+iωμ02α2μ0+ω2μ0+g112αμ0+iωμ0+g0222αμ0+3iωμ0+g212.Owing toαμ0=0 and ωμ0=ω0>0, then(81)Rec1μ0=Reg20g112ω0i+g212.Since(82)g20=q∗,Qq,q=1+σ3ω0−i2ω0,(83)g11=q∗,Qq,q−=2σ2+σ−1i−ω02ω0,(84)g21=q∗,Cq,q,q−=1+σ3i−3ω02ω0,we haveRec1μ0=−1+σ4/41−σ<0. As a consequence, the supercritical Hopf bifurcation occurs at μ=μ0.We summarize the above analysis and get the following Theorem4.Theorem 4.
Assume conditionH3 holds, the spatially homogeneous periodic solution of system (4) bifurcated from the equilibrium is stable. Let(85)
D
¯
=
min
1
≤
k
≤
m
−
1
+
σ
+
k
2
1
−
σ
k
4
1
+
σ
+
k
2
1
−
σ
.The spatially homogeneous periodic solution for system (3) is stable if condition H6 holds and unstable if condition H7 holds.(86)H6d>1−σ1+σ;H7m2<1−σd1+σ≤m+12,d>D¯.


## 3.1. Instability Analysis of the Equilibrium

We first assume that conditionH2 is established. Obviously, the equilibrium a∗,s∗ is a stable for system (4) with Neumann boundary conditions(30)∂a∂x0,t=∂a∂xπ,t=0,∂s∂x0,t=∂s∂xπ,t=0.We consider the diffusion system (3) in the Banach space H20,π×H20,π, where(31)H20,π=w·,t∂iw∂xi·,t∈L20,π,i=0,1,2.It is easy to obtain that equilibriuma∗,s∗ is a stable solution of (3) and (30). The equilibrium a∗,s∗ is nonlinearly unstable for (3) and (30), if it is linearly unstable in H20,π×H20,π.Letu1=a−a∗,u2=s−s∗, then diffusion system (3) is transformed into(32)u1tu2t=1−σ1+σ+d∂xx1+σ2−2μ1+σ−μ1+σ2+∂xxu1u2≐Lu1u2,with the boundary conditions(33)u1x0,t=u2x0,t=u1xπ,t=u2xπ,t=0.Letu1,u2∈H20,π×H20,π be a solution of (32) and (33). Since (32) is linear, we can signify it as(34)u1x,tu2x,t=∑k=0∞AkSkeλt+ikx,whereλ∈C is the temporal spectrum, k is the wave number and Ak,Sk are real numbers for k=0,1,2…,n,…. Bring (34) into (32), we know(35)∑k=0∞λAkSkeλt+ikx=∑k=0∞1−σ1+σ+d∂xx1+σ2−2μ1+σ−μ1+σ2+∂xxAkSkeλt+ikx.Considering the like terms with the wave number ofk, we have(36)λI−JkμAkSk=00,k=0,1,2,…,n,…,where(37)Jkμ=1−σ1+σ−dk21+σ2−2μ1+σ−μ1+σ2−k2.It is easy to verify that (36) has a nonzero solution if and only if(38)DetλI−Jkμ=0.We are searching conditionsReλk>0 and λk to satisfy the equation(39)λ2−Trkλ+Detk=0,k=0,1,2,…,n,…,where(40)Trk=−1+dk2−μ1+σ3+σ−11+σ,and(41)Detk=k2dk21+σ−1−σ+dk2μ1+σ3+μ1+σ31+σ.Under conditionH2, we have Trk<0 for all k=0,1,2,…,n,… and Det0>0.In addition, the equilibriuma∗,s∗ is still stable, if Detk>0 for all k=0,1,2,…,n,…. The equilibrium a∗,s∗ will lose its stability and occur Turing pattern occurs, if for at least one k∈N makes Detk<0, namely, if   m2<1−σ/d1+σ≤m+12andd>Dm , then there exist at least one negative in Det1,Det2...Detm. Hence, the equilibrium a∗,s∗ is unstable for system (3).Basing on the above analysis, we have Theorem3.Theorem 3.
Assume conditionH2 holds, let(42)
D
m
=
min
1
≤
k
≤
m
k
2
1
−
σ
−
μ
1
+
σ
3
k
4
1
+
σ
+
k
2
μ
1
+
σ
3
,thena∗,s∗ is a stable equilibrium for system (3) if condition H4 holds and is an unstable equilibrium for system (3) if condition H5 holds.(43)H4d>1−σ1+σ;H5m2<1−σd1+σ≤m+12,d>Dm.


## 3.2. Instability Analysis of the Limit Cycle

In this subsection, we discuss the stability of the limit cycle in Theorem3 under spatially inhomogeneous perturbations. Assume condition H3 is satisfied, then the supercritical Hopf bifurcation appears at μ=μ0. Therefore, the limit cycle is stable under spatially homogeneous perturbation.According to [17], let u1=a−a∗,u2=s−s∗,μ=μ0, and U=u1,u2T, then system (3) becomes(44)Ut=Jμ0+D∂xx00∂xxU+FU,μ0,Ux0,t=Uxπ,t=0,0T,where(45)Jμ0=1−σ1+σ1+σ2−21−σ1+σ4−1−σ1+σ,D=d001,(46)FU,μ0=f1u1,u2,μ0,g1u1,u2,μ0T,andf1 and g1 are defined in (13) and (14), respectively.According to [18], the form of FU,μ0 is as follows(47)FU,μ0=12QU,U+16CU,U,U+OU4,whereQ and C are in the following form(48)QU,U=Q1U,U,Q2U,UT,CU,U,U=C1U,U,U,C2U,U,UT,with(49)Q1U,V=f2uuu1v1+f2uvu1v2+f2vuu2v1+f2vvu2v2=21+σ2u1v1+1+σ3u1v2+1+σ3u2v1,(50)Q2U,V=g2uuu1v1+g2uvu1v2+g2vuu2v1+g2vvu2v2=2σ−1u1v1+u1v21+σ3+1+σ3u2v11+σ5,(51)C1U,V,W=f2uuuu1v1w1+f2uuvu1v1w2+f2uvuu1v2w1+f2uvvu1v2w2+f2vuuu2v1w1+f2vuvu2v1w2+f2vvuu2v2w1+f2vvvu2v2w2=2u1v1w2+u1v2w1+u2v1w1,(52)C2U,V,W=g2uuuu1v1w1+g2uuvu1v1w2+g2uvuu1v2w1+g2uvvu1v2w2+g2vuuu2v1w1+g2vuvu2v1w2+g2vvuu2v2w1+g2vvvu2v2w2=2σ−1u1v1w2+u1v2w1+u2v1w11+σ3,for anyU=u1,u2T, V=v1,v2T, W=w1,w2T, and U,V,W∈H20,π×H20,π.The linear operatorL defined in (30) for μ=μ0 is(53)LU=Jμ0+D∂xx00∂xxU,forU∈H20,π×H20,π.LetL∗ be the conjugate adjoint operator of L defined in H20,π×H20,π, then(54)L∗U=J∗μ0+D∂xx00∂xxU,where(55)J∗μ0=1−σ1+σ2−1+σ1+σ41+σ2−1+σ1+σ.It is easy to note thatL∗U,V=U,LV for any U,V∈H20,π×H20,π and define the inner product of U,V=1/π×∫0πU¯TVdx on H20,π×H20,π.The linearized system of (44) at the equilibrium 0,0 is(56)u1tu2t=Lu1u2,with the boundary conditions(57)Ux0,t=Uxπ,t=0,0T.LetU=u1,u2T∈H20,π×H20,π is a solution of (56) and (57). Since (56) is linear, we can denote it as(58)u1x,tu2x,t=∑k=0∞akskeλt+ikx,whereλ∈C is the temporal spectrum, k is the wave number and ak, sk are real numbers for k=0,1,2…,n,…. Plugging (58) into (56), we have(59)∑k=0∞λakskeλt+ikx=∑k=0∞Lkakskeλt+ikx.Nextly, we consider like terms with the wave number ofk, we have(60)λI−Lkaksk=00,k=0,1,2,…,n,…,where(61)Lk=1−σ1+σ−dk21+σ2−21−σ1+σ4−1−σ1+σ−k2.It is obvious to note that (60) has a nonzero solution if and only if(62)detλI−Lk=0.We are searching conditions such thatReλk>0 and λk satisfies equation(63)λ2−Tkλ+Dk=0,k=0,1,2,…,n,…,where(64)Tk=−1+dk2,and(65)Dk=k2dk21+σ−1−σ1+σ+dk2+1ω02.we haveT0=0 and Tk<0 for all k=1,2,…,n,… and D0=ω02>0, if condition μ=μ0 holds. Then, it follows that for k=0,L has eigenvalues with zero real parts. We need to proceed to the center manifold reduction.Firstly, ifd>1−σ/1+σ then Dk>d+1ω02>0 for all k=1,2,…,n,…. In addition, if m2<1−σ/d1+σ≤m+12andd>D¯, there exists at least one negative in D1,D2…Dm.LetLq=iω0q and L∗q∗=−iω0q∗, by calculation we can get q=−1+σ31−σ−iω01+σ4/21−σ,1T and q∗=1−σ/−ω01+σ4i,ω01+σ−1−σi/2ω01+σT, respectively. It is easy to verify that q∗,q=1 and q∗,q¯=0.According to [18], let(66)U=zq+zq¯+w,z=q∗,U,w=w1,w2T,and(67)u1=−1+σ31−σ−iω01+σ421−σz+−1+σ31−σ+iω01+σ421−σz¯+w1,u2=z+z¯+w2.The coordinate of the system (44) on z,w is converted to(68)z˙=iω0z+q*,f˜,w˙=Lw+Hz,z¯,w,where(69)f˜=Fzq+z¯q¯+w,μ0,Hz,z¯,w=f˜−q∗,f˜q−q¯∗,f˜q¯,and(70)f˜=12Qq,qz2+Qq,q¯zz¯+12Qq¯,q¯z¯2+Oz3,z·w,w2,(71)q∗,f˜=12q∗,Qq,qz2+q∗,Qq,q¯zz¯+12q∗,Qq¯,q¯z¯2+Oz3,z·w,w2,(72)q¯∗,f˜=12q¯∗,Qq,qz2+q¯∗,Qq,q¯zz¯+12q¯∗,Qq¯,q¯z¯2+Oz3,z·w,w2.SoHz,z¯,w=1/2z2H20+zz¯H11+1/2z¯2H02+Oz3,|z|·|w|,|w|2,where(73)H20=Qq,q−q∗,Qq,qq−q¯∗,Qq,qq¯,H11=Qq,q¯−q∗,Qq,q¯q−q¯∗,Qq,q¯q¯,H02=Qq¯,q¯−q∗,Qq¯,q¯q−q¯∗,Qq¯,q¯q¯.Besides, we have(74)H20=H11=H02=0,0T.Hence,Hz,z¯,w=Oz3,|z|·|w|,|w|2.According to [18], the system possesses a central manifold, which we write as(75)w=12z2w20+zz¯w11+12z¯2w02+Oz3.SinceLw+Hz,z−,w=w˙=∂w/∂zz˙+∂w/∂z¯z¯˙, then(76)w20=2iω0-L-1H20=2iω0-Jμ0-1H20=0,0T,w11=-L-1H11=-J-1μ0H11=0,0T,w02=-2iω0-L-1H02=-2iω0-Jμ0-1H02=0,0T,w=Oz3.In consequence, the diffusion system restricted to the center manifold is(77)z˙=iω0z+q∗,f˜=iω0z+∑2≤i+j≤3giji!j!ziz¯j+Oz4,where(78)g20=q∗,Qq,q,g11=q∗,Qq,q¯,g02=q∗,Qq¯,q¯,g21=q∗,Cq,q,q¯.The dynamics of the system (68) are determined by (77), and the normal form of Poincaré of (44) is expressed as(79)z˙=αμ+iωμz+z∑j=1Mcjμzz¯j,wherez is a complex variable, M≥1, and cjμ0 is a complex-valued coefficient. So we can obtain(80)c1μ0=g20g113αμ0+iωμ02α2μ0+ω2μ0+g112αμ0+iωμ0+g0222αμ0+3iωμ0+g212.Owing toαμ0=0 and ωμ0=ω0>0, then(81)Rec1μ0=Reg20g112ω0i+g212.Since(82)g20=q∗,Qq,q=1+σ3ω0−i2ω0,(83)g11=q∗,Qq,q−=2σ2+σ−1i−ω02ω0,(84)g21=q∗,Cq,q,q−=1+σ3i−3ω02ω0,we haveRec1μ0=−1+σ4/41−σ<0. As a consequence, the supercritical Hopf bifurcation occurs at μ=μ0.We summarize the above analysis and get the following Theorem4.Theorem 4.
Assume conditionH3 holds, the spatially homogeneous periodic solution of system (4) bifurcated from the equilibrium is stable. Let(85)
D
¯
=
min
1
≤
k
≤
m
−
1
+
σ
+
k
2
1
−
σ
k
4
1
+
σ
+
k
2
1
−
σ
.The spatially homogeneous periodic solution for system (3) is stable if condition H6 holds and unstable if condition H7 holds.(86)H6d>1−σ1+σ;H7m2<1−σd1+σ≤m+12,d>D¯.


## 4. Numerical Simulations

In this section, we will use numerical simulations to illustrate the results in Sections2 and 3.A Hopf bifurcation occurs when a periodic solution or limit cycle, surrounding an equilibrium, arises or goes away as a parameterμ 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 σ,μ0, 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.Figure 1
(a) The supercritical Hopf bifurcation diagrams of system (4) and (b) Hopf bifurcation on a two-dimensional system in polar coordinates of system (4).
(a)
(b)Furthermore, we setσ=0.5, then μ0=0.1481, the equilibrium a∗,s∗=1.500,0.4444. Let μ=0.155 (here μ>μ0), so that condition H2 in Theorem 1 is satisfied, and the equilibrium a∗,s∗ is asymptotically stable. Let μ=0.14 (here μ<μ0), so that condition H3 in Theorem 1 is satisfied, and the equilibrium a∗,s∗ is unstable and phase orbit converges to the stable limit cycle. Figure 2(a) shows the equilibrium a∗,s∗ is asymptotically stable from initial values a0,s0=1.600,0.5444. Figure 2(b) shows that phase orbit starting from a0,s0=1.550,0.4944 converges to the stable limit cycle.Figure 2
(a) The phase orbit starting froma0,s0=1.600,0.5444 coverages to equilibrium   a∗,s∗=1.500,0.4444 for σ=0.5,μ=0.155 and (b) equilibrium is unstable and phase orbit starting from a0,s0=1.550,0.4944 coverages to stable limit cycle for σ=0.5,μ=0.14.
(a)
(b)Furthermore, we introduce the effect of diffusion on the equilibrium and the spatial homogeneous periodic solution.Letσ=0.5,μ=0.155,d=0.38, then a∗,s∗=1.5000,0.4444,μ0=0.1481,1−σ/1+σ=0.3333, the conditions H2 and H4 of Theorem 3 hold. Thus, the stable equilibrium a∗,s∗=1.500,0.4444 of system (4) is still stable for system (3). Please see Figures 3(a) and 3(b). Let σ=0.5,μ=0.155,d=Dm+0.05, then a∗,s∗=1.500,0.4444,μ0=0.1481,Dm=0, the conditions H2 and H5 of  Theorem 3 hold. As a result, the stable equilibrium a∗,s∗=1.500,0.4444 of system (4) becomes unstable for system (3) because of diffusion. Please see Figures 4(a) and 4(b).Figure 3
(a) The stable equilibriuma∗,s∗=1.500,0.4444 of (4) is still stable for (3) for σ=0.5,μ=0.155 and d=0.38 and (b) the projection of (a) in x,t coordinates.
(a)
(b)Figure 4
(a) The stable equilibriuma∗,s∗=1.500,0.4444 of (4) becomes unstable for (3) for σ=0.5,μ=0.155 and d=Dm+0.05 and (b) the projection of (a) in x,t coordinates.
(a)
(b)Letσ=0.5,μ=0.14,d=0.35, then the equilibrium a∗,s∗=1.500,0.4444,μ0=0.1481,1−σ/1+σ=0.3333, the conditions H3 and H6 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 σ=0.5,μ=0.14,d=D¯+0.05, then the equilibrium a∗,s∗=1.500,0.4444,μ0=0.1481,D¯=0, the conditions H3 and H7 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).Figure 5
(a) The stable limit cycle of (4) is still stable for (3) for σ=0.5,μ=0.14, and d=0.35 and (b) the projection of (a) in x,t coordinates.
(a)
(b)Figure 6
(a) The stable limit cycle of (4) becomes unstable for (3) for σ=0.5,μ=0.14, and d=D¯+0.05 and (b) the projection of (a) in x,t coordinates.
(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 a∗,s∗ for μ=μ0 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.



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