
In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates.
Citation: Weiyu Li, Hongyan Wang. Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions[J]. Electronic Research Archive, 2023, 31(7): 4139-4154. doi: 10.3934/era.2023211
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In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates.
In nonlinear chemical reaction systems, the three-molecule autocatalytic model shows abundant dynamical behaviors, many abundant research results have been obtained [1,2,3,4,5,6,7]. In 1979, Schnakenberg proposed a typical three-molecule autocatalytic reaction-diffusion model [8]. In [9], the authors studied the one-dimensional static Turing bifurcation of Schnakenberg model. In [10,11,12], the authors introduced the relevant research background of reaction-diffusion Schnakenberg system. However, most references [13,14,15,16] focus on whether the constant equilibrium solution has Turing instability, but pay little attention to whether the periodic solutions of system may also suffer from Turing instability. Therefore, by applying the theoretical methods in [17,18], we study the Turing instability of Hopf bifurcating periodic solutions for the Schnakenberg model.
On the basis of the Schnakenberg model [8], We introduce self-diffusion and cross-diffusion coefficients, and establish reaction-diffusion Schnakenberg model with cross-diffusion and self-diffusion:
{ut−d11Δu−d12Δv=a−u+u2v,x∈Ω,t>0,vt−d21Δv−d22Δv=b−u2v,x∈Ω,t>0,u(x,0)=u∗(x),v(x,0)=v∗(x),x∈Ω,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0, | (1.1) |
where Ω is a open bounded domain in n-dimensional Euclidean space, and its boundary ∂Ω is smooth. Δ is Laplace operator. The parameters a,b,d11,d12,d21,d22 are all positive constants. u=u(x,t) and v=v(x,t) indicate the concentrations of chemicals at position x∈Ω and time t>0, respectively, and the initial concentrations u∗(x),v∗(x) are nonnegative functions. d11,d22 denote self-diffusion coefficients of u and v, respectively. d12,d21 represent cross-diffusion coefficients of u and v, respectively. Simultaneously, we suppose that d11d22−d12d21>0 holds.
We consider the corresponding zero-dimensional dynamic system of system (1.1)
{dudt=a−u+u2v, t>0,dvdt=b−u2v, t>0,u(0)=u∗>0,v(0)=v∗>0. | (2.1) |
The equilibrium (u0,v0) of system (2.1) satisfies
{a−u+u2v=0,b−u2v=0. |
with u0=a+b,v0=b(a+b)2. By straightforward computation, we know (u0,v0) is the only equilibrium of system (2.1). For convenience, setting μ:=a+b, then (u0,v0)=(μ,bμ2). In the following, for the three-molecule autocatalytic Schnakenberg model, we discuss the stability of its Hopf bifurcating periodic solutions by taking μ as parameter.
Theorem 2.1. Let μH0=3√b+√b2+127+3√b−√b2+127, for the ODEs (2.1), the following statements are true:
(1) At (μ,bμ2)T, system (2.1) is unstable for μ∈(0,μH0), while locally asymptotically stable for μ∈(μH0,+∞).
(2) At λ=μH0, system has a family of periodic solutions (uT(t),vT(t))T bifurcating from (μ,bμ2)T. Supercritical Hopf bifurcation of system (2.1) occurs at (μ,bμ2)T, and the bifurcating periodic solutions are stable.
Proof. The Jacobian matrix of system (2.1) at (μ,vμ)T is J(μ)=(−1+2bμμ2−2bμ−μ2). The characteristic equation of J(μ) is
λ2−T(μ)λ+D(μ)=0, | (2.2) |
with
T(μ)=−μ2−1+2bμ,D(μ)=μ2. |
The eigenvalue λ(μ) of J(μ) is given by
λ(μ)=T(μ)±√T2(μ)−4D(μ)2. |
When μ≥2b, all the eigenvalues of J(μ) have strict negative real parts, according to the stability theory, the equilibrium (μ,bμ2)T is locally asymptotically stable. When 0<μ<2b, T′(μ)=−2μ−2bμ2<0, then T(μ) is monotonically decreasing for 0<μ<2b. Since limμ→0T(μ)=+∞,T(2b)=−4b−12b<0, then T(μ) has only zero point μH0∈(0,2b), namely, T(μH0)=0. For any μ∈(μH0,2b), we have T(μ)<0, then system (2.1) is locally asymptotically stable at (μ,bμ2)T, while for any μ∈(0,μH0), system (2.1) is unstable at (μ,bμ2)T. When μ=μH0, J(μ) has a pair of pure imaginary roots λ=±iω0 with ω0=μ. Let λ(μ)=α(μ)±iω(μ) be the roots of Eq (2.2) near μ=μH0, then we have
α(μ)=−μ22−12+bμ,dα(μ)dμ|μ=μH0<0. |
According to Poincaré-Andronov-Hopf bifurcation theorem, system (2.1) experiences a Hopf bifurcation at μ=μH0.
Next, we study the properties of Hopf bifurcating periodic solutions of system (2.1). Here, we still use the notations and computation in [20] to deduce the expression of cubic term coefficient c1(μH0) in the norm form. By [21], we can rewrite the Poincaré normal form of the abstract form of system (2.1) in the small neighborhood of p0 as follows:
dUdt=J(μH0)U+F(μ,U)|μ=μH0. | (2.3) |
Let the eigenvector of J(μH0) corresponding to the eigenvalue iω0 be q=(a0,b0)T satisfying
J(μH0)q=iω0q,q=(a0,b0)T=(−μH0,μH0−i)T. |
Define inner product in XC:
⟨U1,U2⟩=∫lπ0(ˉu1u2+ˉv1v2)dx, |
where Ui=(ui,vi)T∈XC,i=1,2. Note that ⟨λU1,U2⟩=ˉλ⟨U1,U2⟩, denote the adjoint operator of J(μH0) by J∗(μH0), then the eigenvector of J∗(μH0) corresponding to the eigenvalue −iω0 be q∗=(a∗0,b∗0)T∈XC satisfying
J∗(μH0)q∗=−iω0q∗,<q∗,q>=1,<q∗,ˉq>=0. |
Therefore, q∗=(a∗0,b∗0)T=(−1−iμH02μH0lπ,−i2lπ)T. Performing the spatial decomposition X=Xc⊕Xs, where Xc={zq+ˉzˉq|z∈C},Xs={u∈X|<q∗,u>=0}, then for any U=(u,v)T∈X, there exist z∈C and ω=(ω1,ω2)∈Xs such that (u,v)T=zq+ˉzˉq+(ω1,ω2)T. Thus, system (2.3) can be transformed into the following system with (z,ω) as the coordinate:
{dzdt=iωz+<q∗,F(p,U)|p=p0>,dωdt=L(p0)ω+H(z,ˉz,ω), | (2.4) |
with
{H(z,ˉz,ω)=F(p,U)|p=p0−<q∗,F(p,U)|p=p0>q−<ˉq∗,F(p,U)|p=p0>ˉq,F(p,U)|p=p0=F0(zq+ˉzˉq+ω). | (2.5) |
Writing F0 as
F0(U)=12Q(U,U)+16C(U,U,U)+O(|U|4), | (2.6) |
here, Q,C is a symmetric multilinear form. For convenience, denoting QXY=Q(X,Y),CXYZ=C(X,Y,Z), we calculate Qqq, Qqˉq and Cqqˉq, where
Qqq=(c0d0),Qqˉq=(e0f0),Cqqˉq=(g0h0), |
here, denoting f(u,v)=a−u+u2v,g(u,v)=b−u2v, with
c0=fuua20+2fuva0b0+fvvb20=μH0−3μH03+4μH02i,d0=guua20+2guva0b0+gvvb20=−c0=−μH0+3μH03−4μH02i,e0=fuu|a0|2+fuv(a0¯b0+¯a0b0)+fvv|b0|2=μH0−3μH03,f0=guu |a0|2+guv (a0¯b0+¯a0b0)+gvv|b0|2=−e0=3μH03−μH0,g0=fuuu |a0|2a0+fuuv (2|a0|2b0+a20¯b0)+fuvv(2|b0|2a0+b20¯a0)a=6μH03−2μH02i,h0=guuu |a0|2a0+guuv (2|a0|2b0+a20¯b0)+guvv(2|b0|2a0+b20¯a0)=−g0=−6μH03+2μH02i, |
here, all the partial derivatives of f(u,v) and g(u,v) are evaluated at the bifurcation point (μH0,bμH02)T,
Let
H(z,ˉz,ω)=H202z2+H11zˉz+H022ˉz2+o(|z|3)+o(|z||ω|), | (2.7) |
from (2.5) and (2.6), we can obtain
{H20=Qqq−<q∗,Qqq>q−<ˉq∗,Qqq>ˉqH11=Qqˉq−<q∗,Qqˉq>q−<ˉq∗,Qqˉq>ˉq |
Because system (2.4) has normal manifold, which can be written as follows
ω=ω202z2+ω11zˉz+ω022ˉz2+o(|z|3). | (2.8) |
By (2.7), (2.8) and J(μH0)ω+H(z,ˉz,ω)=dωdt=∂ω∂zdzdt+∂ω∂ˉzdˉzdt, we have
ω20=(2iω0I−J(μH0))−1H20,ω11=−J−1(μH0)H11. |
By calculation, we have
<q∗,Qqq>=−c02μH0=−μH0+3μH03−4μH02i2μH0,<q∗,Qqˉq>=−e02μH0=−μH0+3μH032μH0,<ˉq∗,Cqqˉq>=−g02μH0=−6μH03+2μH02i2μH0,<ˉq∗,Qqq>=−c02μH0=−μH0+3μH03−4μH02i2μH0,<ˉq∗,Qqˉq>=−e02μH0=−μH0+3μH032μH0, |
we can also get H20=0,H11=0, this implies ω20=ω11=0, then
<q∗,Qω11q>=<q∗,Qω20ˉq>=0. |
Thus, we have
c1(μ)=i2ω0<q∗,Qqq>⋅<q∗,Qqˉq>+12<ˉq∗,Cqqˉq>. |
The real part and imaginary part of c1(μH0) are as follows
Rec1(μH0)=Re{i2ω0<q∗,Qqq>⋅<q∗,Qqˉq>+12<ˉq∗,Cqqˉq>}=−12,Imc1(μH0)=Im{i2ω0<q∗,Qqq>⋅<q∗,Qqˉq>+12<ˉq∗,Cqqˉq>}=18μH0−μH04+9μH038. | (2.9) |
By Rec1(μH0)<0, we know that the Hopf bifurcating periodic solutions of system (2.1) are stable at μ=μH0. Additionally, because transversality condition dα(μ)dμ|μ=μH0<0, so the direction of Hopf bifurcation is subcritical.
We introduce the following perturbed system model on the basis of ODEs (2.1)
(1+εd11εd12εd211+εd22)(dudt,dvdt)T=(a−u+u2vb−u2v), | (2.10) |
here, ε is sufficiently small such that (1+εd11εd12εd211+εd22) is reversible. System (2.10) is equivalent to the following system
(dudt,dvdt)T=1N(ε)(1+d22ε−d12ε−d21ε1+d11ε)(a−u+u2vb−u2v), | (2.11) |
where
N(ε):=|(1+d22ε−d12ε−d21ε1+d11ε)|=(d11d22−d12d21)ε2+(d11+d22)ε+1>0. |
Then at (μ,vμ), the Jacobian matrix of system (2.11) is
J(μ,ε):=1N(ε)(a11(μ,ε)a12(μ,ε)a21(μ,ε)a22(μ,ε)), | (2.12) |
with
a11(μ,ε):=(1+d22ε)(−1+2bμ)+d12ε2bμ,a12(μ,ε):=(1+d22ε)μ2+d12εμ2,a21(μ,ε):=−(1+d11ε)2bμ−d21ε(−1+2bμ),a22(μ,ε):=−μ2(1+d11ε)−d21εμ2. | (2.13) |
The characteristic equation corresponding to the jacobian matrix J(μ,ε) is
λ2−H(μ,ε)λ+D(μ,ε)=0, | (2.14) |
where
H(μ,ε)=1N(ε)[(2bμ−1−μ2)+ε(d22(2bμ−1)−μ2d11+d12⋅2bμ−d21⋅μ2)],D(μ,ε)=μ2N(ε). | (2.15) |
Notice that H(μH0,0)=T(μH0)=0 and ∂μH(μ,ε)=T′(μH0)≠0. According to the implicit function existence theorem, there exist a sufficiently small ε0>0 and a continuously differentiable function μHε=μH(ε) such that when ε∈(−ε0,ε0), H(μHε,ε)=0 and μH(0)=μH0 hold. Let λ(με)=β(με)±iω(με) be the characteristic root of Eq 2.14, then when μ→με, we have
β(με)=12H(μ,ε),ω(με)=12√4D(μ,ε)−H2(μ,ε). | (2.16) |
By [18], we have the following lemma.
Lemma 2.1. Assume μ is sufficiently close to μHε, T is the minimum positive period of the stable periodic solution (uT(t),vT(t)) of system (2.1) bifurcating from (μ,vμ), then there exists ε1>0 such that for any ε∈(−ε1,ε1), system (2.10) has a periodic solution (uT(t,ε),vT(t,ε)) depending on ε. Its minimum positive period is T(ε), simultaneously, it satisfies
1) When ε→0, (uT(t,ε),vT(t,ε))→(uT(t),vT(t)) and T(ε)→T.
2) T(ε)=2πω(μHε)(1+(β′(μHε)Im(c1(μHε))ω(μHε)Re(c1(μHε))−ω′(μHε)ω(μHε))(μ−μHε)+O((μ−μHε)2) with
c1(μHε)=i2ω(μHε)(g20(ε)g11(ε)−2|g11(ε)|2−13|g02(ε)|2)+g21(ε)2. |
Theorem 2.2. Suppose that μ is sufficiently close to μHε, (uT(t),vT(t)) is the stable periodic solution of system (2.1), then when ε→0, we have
T′(0)=πμH0(L1(μH0)d11−L2(μH0)d22−L3(μH0)d12−L4(μH0)d21), |
with
L1(μH0)=54−μH202+9μH404,L2(μH0)=(14μH20+9μH204−1)(2bμH0−1)−1,L3(μH0)=(14μH0+9μH304−μH0)2bμH20,L4(μH0)=(μH0−14μH0−9μH304)μH0. |
Proof. According to Lemma 2.1, we have
T′(ε)=−2πω2(μHε)dω(μHε)dε−2πω(μHε)(β′(μHε)Im(c1(μHε))ω(μHε)Re(c1(μHε))−ω′(μHε)ω(μHε))dμHεdε+O(μ−μHε). |
If μ→μHε, then O(μ−μHε)→0, so the sign of T′(ε) is mainly determined by the sign of the first two at ε=0. Next, we calculate the expressions of dμHεdε|ε=0 and dω(μHε)dε|ε=0.
At μ=μHε, by (2.15), we can derive
(2bμHε−1−μHε2)+ε(d22(−1+2bμHε)−μHε2d11+d122bμHε−d21μHε2)=0. | (2.17) |
Differentiating (2.17) with ε, we have
dμHεdε|ε=0=b(μH0)2bμH02+2μH0, | (2.18) |
with
b(μH0)=d22(−1+2bμH0)−μH02d11+d122bμH0−d21μH02. | (2.19) |
According to (2.16),
Differentiating it with , we know
When , and . Hence,
(2.20) |
From and , differentiating them with , we can obtain
(2.21) |
(2.22) |
When , we get
(2.23) |
Thus, according to (2.22) and (2.23), we have
(2.24) |
where is defined in (2.19). From (2.21) and (2.24), we can obtain
(2.25) |
By (2.18), (2.20) and (2.25), we can deduce
(2.26) |
At last, substituting (2.19) into (2.26), we can obtain
with
In this section, applying the theory elaborated in [17], we study stable spatially homogeneous Hopf bifurcating periodic solutions of system (2.1) will become Turing unstable in reaction-diffusion system (1.1) with cross-diffusion. According to the previous discussion, we give the following theorem.
Theorem 3.1. Suppose that is sufficiently close to , is stable spatially homogeneous Hopf bifurcating periodic solution of system (2.1) bifurcating from . If and is large enough, then will become Turing unstable in system (1.1) with cross-diffusion.
Proof. Assume that the stable periodic solution of system (2.1) is with minimum positive period , then the linearized system of (1.1) evaluated at is
(3.1) |
where , is Laplace operator, the Jacobian matrix of system (2.1) at is . Setting and be the eigenvalue and eigenfunction of in with Neumann boundary condition. Let , then
(3.2) |
in which, . Assume , then system (3.2) can be rewritten as
(3.3) |
Let be the fundamental solution matrix of system (3.3) satisfying . Denote as the eigenvalue of , the corresponding characteristic function is , i.e.,
then is the Floquet multiplier corresponding to the periodic solution of system (3.2). Define
apparently,
In system (2.1), differentiating with , we have
Then 1 is the eigenvalue of , the corresponding eigenvector is . We might as well assume , . Since is stable, then . Let be the fundamental solution matrix of system (3.2), and , namely,
For sufficiently small , is continuously differentiable with respect to and , and . Define mapping , where , we have
clearly, and
here, is the Fréchet derivative of with respect to , and is the Fréchet derivative of with respect to . Setting is the single eigenvalue of , then we have
where represents the kernel of , and represents the range of , then is an isomorphic mapping. By the implicit function theorem, there exist , and continuously differentiable functions such that
(3.4) |
where are the Floquet multipliers corresponding to . Define
(3.5) |
by and (3.5), we can obtain
(3.6) |
From (3.4) and (3.6), we have
Specifically, by (3.5), we have
(3.7) |
By the definition of in (3.5), we can obtain
(3.8) |
Differentiating (3.8) with respect to and setting , we have
(3.9) |
here, . On the other hand, from (3.4) and (3.5), we can derive
(3.10) |
Differentiating (3.10) with , we get
(3.11) |
In (3.11), setting , from (3.6)and , we have
(3.12) |
According to Lemma 2.1, is the periodic solution of system (2.10), that is,
(3.13) |
Differentiating (3.13) with respect to and setting , we have
(3.14) |
where . Since is the periodic solution with period , thus,
(3.15) |
In (3.15), differentiating with respect to and setting , then
(3.16) |
here, . Define
From (3.9), (3.12), (3.14) and (3.16), we have
(3.17) |
(3.18) |
Let be the general solution of (3.17), where any vector . Since and are linearly independent, then there exit constants and so that
(3.19) |
Substituting (3.19) into (3.18), we can obtain . Assume that , which is equivalent to , if is large enough so that the minimum positive eigenvalue of is small enough, then there exists at least one eigenvalue of such that . Thus, becomes Turing unstable in reaction-diffusion system (1.1) with cross-diffusion.
In this section, we shall select several groups of data for numerical simulations to support theoretical analysis. In system (1.1), Fix parameters , initial values , then the equilibrium is , , , . According to Theorem 2.1, system (2.1) produces a spatially homogeneous Hopf bifurcating periodic solution at the equilibrium, which is stable and subcritical. By calculation, we can obtain
For different diffusion coefficients, Turing instability of system (1.1) at the periodic solution is different. Hence, we give diffusion coefficients in four cases and carry out corresponding numerical simulations.
(1) If we select , at this moment, Turing instability of does not exist in system (1.1) (Figure 1). That is, the same diffusion rates will not cause Turing instability of periodic solution ([19]).
(2) If we select , then
According to Theorem 3.1, system (1.1) is Turing unstable at periodic solution . Through simulation simulation, it can be observed that the stable periodic solution produces Turing bifurcation (Figure 2), which is consistent with the theoretical analysis.
(3) If we choose , from Theorem 3.1, we have
then system (1.1) is Turing unstable at periodic solution . Through simulation simulation, it can be verified that the stable periodic solution generates Turing bifurcation (Figure 3). Therefore, cross-diffusion causes the stable periodic solution of system (1.1) to become Turing unstable. self-diffusion induces Turing instability of periodic solution.
(4) If we choose , then
By Theorem 3.1, we can derive that diffusion causes the stable periodic solution of system (1.1) to become Turing unstable (Figure 4). This conforms to the theoretical analysis.
In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is considered. From both theoretical and numerical perspectives, we investigated how cross-diffusion causes the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions.
The theoretical results indicate that in the Schnakenberg model, when the parameters satisfy certain conditions and is sufficiently large, once the instability of periodic solutions induced by diffusion occurs, new and rich spatiotemporal patterns may emerge. For the reaction-diffusion Schnakenberg model, we can derive the precise conditions of diffusion rates, under which the periodic solutions may experience the instability caused by diffusion.
By numerical simulations, the Turing instability of periodic solution can be observed. Figure 1 shows that without cross-diffusion coefficients, the identical self-diffusion coefficients will not cause the stable periodic solution to produce Turing bifurcation. Figures 2 and 4 illustrate that with appropriate diffusion coefficients, the stable periodic solution of system (1.1) generates Turing instability. Figure 3 indicates that if we select appropriate cross-diffusion coefficients, even for the models with identical self-diffusion rates, cross-diffusion can also cause stable periodic solution to produce Turing bifurcation. Thus, the Turing instability of stable periodic solution is actually induced by cross-diffusion.
The authors declare there is no conflicts of interest.
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