Citation: Erin N. Bodine, K. Lars Monia. A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma[J]. Mathematical Biosciences and Engineering, 2017, 14(4): 881-899. doi: 10.3934/mbe.2017047
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Patterns, which represent a kind of organized yet heterogeneous macroscopic structure in time or space or space-time, are widely investigated using the reaction diffusion equations [24, 8, 3]. In the natural world, many animals have fascinating color patterns on their skins, exemplified by the coloration of zebrafish and tigers [20]. Such patterns are one of the most obvious traits of animals and serve a variety of functions. For example, patterns have been successfully used in camouflage, warning, social aggregation, mate choice, adaptive radiation, and other strategies [23, 14, 18]. Given patterns' prominence and ecological functions, zebrafish patterns often are determined by natural selection and are of particular interest to biologists [16]. These patterns elicited long-standing interest from developmental and cell biologists as well: their accessibility to observation and manipulation has made them a classic and enduring system for studying basic genetic and cellular mechanisms [16, 22, 19].
In recent years, some research results have been brought to bear on zebrafish patterns. In [4, 5], the authors established three different models to investigate the molecular mechanisms of zebrafish patterns, and discussed one-dimensional patterns. Schnackenberg [17] studied the following reaction-diffusion zebrafish model,
{∂u∂t=∇2u+γ(a−u+u2v)∂v∂t=d∇2v+γ(b−u2v), | (1) |
where
As is well known, the above existing zebrafish spatio-temporal models in ecological networks only concentrate on self-diffusion. However, zebrafish gene products can freely diffuse within a cell, apart from the random motion of individuals, i.e. self-diffusion, one production of genes tends to diffuse in the direction of the other. More precisely, the movement of a gene at any particular location is influenced by the gradient of the concentration of the other at that location. In biomathematics, such a scenario can be well described by reaction-diffusion systems with cross-diffusion terms [27], but it has not been applied to study zebrafish patterns. Therefore, considering the effect of cross-diffusions between zebrafishes will provide a new insight into pattern formation.
In light of the above discussions, based on system (1), the zebrafish pattern model can be formulated by the following equation with cross-diffusion terms:
{∂u(t,x,y)∂t=d11∇2u(t,x,y)+d12∇2v(t,x,y)+R(a−u2(t,x,y)+u2(t,x,y)v(t,x,y))∂v(t,x,y)∂t=d21∇2u(t,x,y)+d22∇2v(t,x,y)+R(b−u2(t,x,y)v(t,x,y)) | (2) |
for
∂u∂ν(t,x,y)=∂v∂ν(t,x,y)=0,t≥0,(x,y)∈∂Ω, |
and initial conditions
{u(0,x,y)=u0≥0v(0,x,y)=v0≥0 | (3) |
where
(1) Diffusion is a very important factor to selection of patterns, especially, cross-diffusions. In this work, to better describe the zebrafish patterns, we propose a new viewpoint to research zebrafish patterns, i.e. studying the spatial dynamics of model (2) with cross-diffusions.
(2) Through bifurcation theory analysis of the equilibrium points for the proposed model, we obtain the sufficient conditions of Hopf bifurcation and Turing instability, respectively. To determine the selection of Turing patterns, we deduce the amplitude equations based on multiple-scale analysis.
(3) It is easy to gain five categories of Turing patterns by analyzing amplitude equations. The associated characteristic equation is established based on random initial perturbation. Then through stability analysis, we obtain the corresponding stable range of these patterns.
(4) Numerical results show that the five categories of patterns are decided by the parameter of model (2) or the coefficients of cross-diffusions. It is important to note that cross-diffusions also can change the selection of patterns, which is one reason the coloration of zebrafish varies.
In this section, based on Hopf bifurcation and Turing bifurcation theory, we will discuss the dynamical behavior of model (2). Before our discussion, according to thermodynamics theroy one should consider the following fact
d11d22−d12d21≥0, |
which implies that all eigenvalues of the diffusion matrix
For simplicity, one can rewrite model (2) as the following form
{∂u∂t=d11∇2u+d12∇2v+f(u,v)∂v∂t=d21∇2u+d22∇2v+g(u,v), | (4) |
where
The steady state of this system is
J=(a11a12a21a22), | (5) |
where
Then we get the following linear equation
{∂u∂t=d11∇2u+d12∇2v+a11u+a12v∂v∂t=d21∇2u+d22∇2v+a21u+a22v. | (6) |
Expanding the variables in the Fourier space
(uv)=∞∑k=0(c1kc2k)exp(λkt+ikr), | (7) |
where
λ2−trkλ+Δk=0. | (8) |
According to (8), it is easy to show that the eigenvalues
λk=trk±√tr2k−4Δk2, | (9) |
where
trk=a11+a22−(d11+d22)k2, | (10) |
Δk=(d11d22−d12d21)k4+(a12d21+a21d12−a11d22−a22d11)k2+a11a22−a12a21. | (11) |
Hopf bifurcation occurs when
b3+3ab2+(3a2−1)b+a3+a=0. | (12) |
The unbalance changes of phases, corresponding to Turing branches, are the transitions of model from the uniform state to the oscillatory state. After the process, the formed patterns are called Turing patterns. From above discussion, we can obtain the necessary conditions for causing Turing instability. For some
{tr0=−a−ba+b−(a+b)2<0Δ0=(a+b)2>0Δk=(d11d22−d12d21)k4+R((a+b)2d21−2ba+bd12+a−ba+bd22+(a+b)2d11)k2+R2(a+b)2<0. | (13) |
(13) indicates that system (2) is unstable for some perturbations to the wave number. So getting
(a+b)3(d11+d21)−2bd12+(a−b)d22+2√d(a+b)2=0, | (14) |
where
When Turing patterns come into being, the wave number
k2T=−(a+b)2d21−2ba+bd12+a−ba+bd22+(a+b)2d112dR. | (15) |
From the above analysis, it can be obtained that the simplified conditions of Turing instability for model (2)
{a−b+(a+b)3>0(a+b)3(d11+d21)−2bd12+(a−b)d22+2√d(a+b)2<0. | (16) |
Remark 1. Eq. (16) shows that diffusion can damage stability. Moreover, if the basic production rate of the activator is greater than the basic production rate of the substrate, namely,
At the Turing bifurcation threshold, the spatial symmetry of system (2) is broken, and the patterns are stationary in time and oscillatory in space with the corresponding wavelength
According to Hopf and Turing bifurcation conditions, we could get the Hopf bifurcation region and Turing instability region. They are shown in the Figure 1.
Now let us observe the real parts of eigenvalues when
In this section, we perform extensive numerical simulations of the spatially extended model (2) in two-dimensional spaces, and the qualitative results are shown here. All our numerical simulations employ the non-zero initial and zero-flux boundary conditions with a system size of
Take
Initially, the entire system is placed in the stationary state
Example 1. Let
Example 2. Take
Example 3. Choose
Example 4. Let
Example 5. Take
Example 6. To observe the impact of different cross-diffusions on the selection of patterns, we consider system (2) with
Example 7. Take
Example 8. Choose
On the one hand. The amplitude equations for the active modes are established, which determines the stability of amplitudes towards uniform and inhomogeneous perturbations. It presents all five categories of Turing patterns by analyzing amplitude equations, which indicates that the model dynamics exhibits complex pattern. In addition, system (2) perfectly simulates the pattern on the body of zebrafish. More specifically, from Figure 3 to Figure 12, we can find: in the range
On the other hand, we also discuss the effect of cross-diffusions on pattern selection. From Figure 13, Figure 14, and Figure 16, it is obvious that
The work is partially supported by National Natural Science Foundation of China under Grant 11571170 and 61174155. The work is also sponsored by Funding of Jiangsu Innovation Program for Graduate Education \break KYZZ
Based on the above discussion, we still cannot determine the selection and competition of Turing patterns. In the following, we will discuss the pattern selection close to the onset
Let
{∂u∂t=d11∇2u+d12∇2v+a11u+a12v+R(v∗u2+2u∗uv+u2v)∂v∂t=d21∇2u+d22∇2v+a21u+a22v−R(v∗u2+2u∗uv+u2v), | (17) |
transforming the positive constant steady state
(uv)=3∑i=1(AuiAvi)eikir+c.c, | (18) |
where
Setting
c=(uv),N=(N1N2). |
Model (17) can be converted to the following system
∂c∂t=Lc+N, | (19) |
where
L=(a11+d11∇2a12+d12∇2a21+d21∇2a22+d22∇2), | (20) |
N=(R(v∗u2+2u∗uv+u2v)−R(v∗u2+2u∗uv+u2v)). | (21) |
We only analyse the behavior of the controlled parameter which is close to the onset
bT−b=ε2b2+O(ε3), | (22) |
where
By expanding the variable
c=(uv)=ε(u1v1)+ε2(u2v2)+⋯, | (23) |
N=ε2h2+ε3h3+O(h4), | (24) |
where
L=LT+(bT−b)M, | (25) |
where
LT=(a∗11+d11∇2a∗12+d12∇2a∗21+d21∇2a∗22+d22∇2), | (26) |
M=(b11b12b21b22), | (27) |
where
The core of the standard multiple-scale analysis is separating the dynamical behavior of system (19) according to different time scale or spatial scale. It is needed to separate the time scale for model (17). Each time scale
∂∂t=ε2∂∂T2+O(ε3). | (28) |
We substitute (20)
LT(u1v1)=0. | (29) |
LT(u2v2)=(−R(v∗u21+2u∗u1v1)R(v∗u21+2u∗u1v1)). | (30) |
LT(u3v3)=∂∂T2(u1v1)−b2M(u1v1)−(R(2v∗u1u2+2u∗u1v2+2u∗u2v1+u21v1)−R(2v∗u1u2+2u∗u1v2+2u∗u2v1+u21v1)). | (31) |
For the first order of
(u1v1)=(S11)(W1eik1r+W2eik2r+W3eik3r)+c.c, | (32) |
where
For the second order of
LT(u2v2)=(−R(v∗u21+2u∗u1v1)R(v∗u21+2u∗u1v1))=(FuFv). | (33) |
According to the Fredholm condition, to guarantee the existence of the nontrivial solution of the equation, the vector function of the right-hand side of (31) must be orthogonal with the zero eigenvectors of operator
(1S2)e−ikjr+c.c(j=1,2,3), | (34) |
where
The orthogonality condition is
(1,S2)(FiuFiv)=0, | (35) |
where
We might as well take
(1,S2)(−R(2v∗S21¯W2¯W3+4u∗S1¯W2¯W3)R(2v∗S21¯W2¯W3+4u∗S1¯W2¯W3))=0, | (36) |
Solving the above equation and obtaining
We suppose
(u2v2)=(X0Y0)+3∑i=1(XiYi)eikir+3∑i=1(XiiYii)e2ikir+(X12Y12)ei(k1−k2)r+(X23Y23)ei(k2−k3)r+(X31Y31)ei(k3−k1)r+c.c, | (37) |
the coefficients of (37) can be obtained by solving the set of linear equations about
(X0Y0)=(C1C2)(|W21|+|W22|+|W23|), | (38) |
where
C1=−R(2v∗S21+4u∗S1)(a∗12+a∗22)a∗11a∗22−a∗12a∗21,C2=R(2v∗S21+4u∗S1)(a∗11+a∗21)a∗11a∗22−a∗12a∗21.Xi=S1Yi(i=1,2,3), | (39) |
(XiiYii)=(E1E2)W2i, | (40) |
where
E1=−R(a∗12+a∗22−4d12k2T−4d22k2T)(v∗S21+2u∗S1)(a∗11−4d11k2T)(a∗22−4d22k2T)−(a∗12−4d12k2T)(a∗21−4d21k2T),E2=R(a∗11+a∗21−4d11k2T−4d21k2T)(v∗S21+2u∗S1)(a∗11−4d11k2T)(a∗22−4d22k2T)−(a∗12−4d12k2T)(a∗21−4d21k2T).(XjkYjk)=(F1F2)W2jˉW2k, | (41) |
where
F1=−R(a∗12+a∗22−3d12k2T−3d22k2T)(v∗S21+2u∗S1)(a∗11−3d11k2T)(a∗22−3d22k2T)−(a∗12−3d12k2T)(a∗21−3d21k2T),F2=R(a∗11+a∗21−3d11k2T−3d21k2T)(v∗S21+2u∗S1)(a∗11−3d11k2T)(a∗22−3d22k2T)−(a∗12−3d12k2T)(a∗21−3d21k2T). |
For the third order of
LT(u3v3)=∂∂T2(u1v1)−b2M(u1v1)−(R(2v∗u1u2+2u∗u1v2+2u∗u2v1+u21v1)−R(2v∗u1u2+2u∗u1v2+2u∗u2v1+u21v1))=(FuFv). | (42) |
According to the orthogonality condition
(1,S2)(FiuFiv)=0, | (43) |
we have
(S1+S2)∂W1∂T2=b2(S1b11+S1S2b21+b12+S2b22)W1+2RS1(1−S2)(v∗S1+2u∗)(ˉY2ˉW3+ˉY3ˉW2)−(G1|W21|+G2(|W22|+|W23|))W1, | (44) |
where
G1=R(S2−1)(2v∗S1(E1+C1)+2u∗S1(E2+C2)+2u∗(E1+C1)+3S21),G2=R(S2−1)(2v∗S1(F1+C1)+2u∗S1(F2+C2)+2u∗(F1+C1)+6S21). |
We expand amplitude
A1=εW1+ε2Y1+⋯. | (45) |
Substituting
τ0∂A1∂t=μA1+hˉA2ˉA3−[g1|A1|2+g2(|A2|2+|A3|2)]A1, | (46) |
where
μ=bT−bbT,τ0=(S1+S2)bT(S1b11+S1S2b21+b12+S2b22),h=2RS1(1−S2)(v∗S1+2u∗)bT(S1b11+S1S2b21+b12+S2b22),g1=G1bT(S1b11+S1S2b21+b12+S2b22),g2=G2bT(S1b11+S1S2b21+b12+S2b22). |
Another two equations can be obtained through the transformation of the subscript of
{τ0∂A1∂t=μA1+hˉA2ˉA3−[g1|A1|2+g2(|A2|2+|A3|2)]A1τ0∂A2∂t=μA2+hˉA3ˉA1−[g1|A2|2+g2(|A1|2+|A3|2)]A2τ0∂A3∂t=μA3+hˉA1ˉA2−[g1|A3|2+g2(|A1|2+|A2|2)]A3. | (47) |
(47) can be decomposed to mode
{τ0∂φ∂t=−h(ρ21ρ22+ρ21ρ23+ρ22ρ23)sinφρ1ρ2ρ3τ0∂ρ1∂t=μρ1+hρ2ρ3cosφ−g1ρ31−g2(ρ22+ρ23)ρ1τ0∂ρ2∂t=μρ2+hρ1ρ3cosφ−g1ρ32−g2(ρ21+ρ23)ρ2τ0∂ρ3∂t=μρ3+hρ1ρ2cosφ−g1ρ33−g2(ρ21+ρ22)ρ3, | (48) |
where
System (48) has four kinds of solutions.
(ⅰ) The stationary state
ρ1=ρ2=ρ3=0. | (49) |
(ⅱ) Stripes (S) are given by
ρ1=√μg1,ρ2=ρ3=0. | (50) |
(ⅲ) Hexagons
ρ1=ρ2=ρ3=|h|±√h2+4(g1+2g2μ)2(g1+2g2), | (51) |
and exist in the following condition
μ>μ1=−h24(g1+2g2), | (52) |
where
(ⅳ) The mixed states are given by
ρ1=|h|g2−g1,ρ2=ρ3=√h−g1ρ21g1+g2, | (53) |
with
In the following, we will give a discussion about the stability of the above four stationary solutions.
To stripes, we give a perturbation at stationary solution
∂ρ∂t=L1⋅ρ, | (54) |
where
L1=(μ−3g1ρ20000μ−g2ρ20|h|ρ00|h|ρ0μ−g2ρ20),ρ=(Δρ1Δρ2Δρ3). | (55) |
The characteristic equation of matrix
λ3+P1λ2+P2λ+P3=0, | (56) |
where
The eigenvalues are
λ1=μ−3g1ρ20,λ2=μ+hρ0−g2ρ20,λ3=μ−hρ0−g2ρ20. | (57) |
Substituting
λ1=−2μ,λ2=μ(1−g2g1)+|h|√μg1,λ3=μ(1−g2g1)−|h|√μg1. | (58) |
It is all known to us that Eq.(58) has stable solutions when the eigenvalues
μ>μ3=h2g1(g2−g1)2. | (59) |
In the following, we discuss the stability of the hexagons, similar to the above process, we perturb Eq.(51) at the point
ρi=ρ0+Δρi, | (60) |
where
∂ρ∂t=L2⋅ρ, | (61) |
where
L2=(μ−(3g1+2g2)|h|ρ0−2g2ρ20|h|ρ0−2g2ρ20|h|ρ0−2g2ρ20μ−(3g1+2g2)|h|ρ0−2g2ρ20|h|ρ0−2g2ρ20|h|ρ0−2g2ρ20μ−(3g1+2g2)),ρ=(Δρ1Δρ2Δρ3). | (62) |
The characteristic equation of
λ3+Q1λ2+Q2λ+Q3=0, | (63) |
where
Solving the characteristic equation (63) and we have
λ1=λ2=μ−|h|ρ0−3g1ρ20,λ3=μ−3g1ρ20−63g2ρ20+2|h|ρ0. | (64) |
Substituting
For the stationary solution
ρ−=|h|−√h2+4(g1+2g2μ)2(g1+2g2), |
For the stationary solution
ρ+=|h|+√h2+4(g1+2g2μ)2(g1+2g2) |
μ<μ4=2g1+g2(g2−g1)2h2. | (65) |
Based on the above analysis, we can conclude
(Ⅰ) The stationary state (0, 0, 0) is stable for
(Ⅱ) The stripe is stable when
(Ⅲ) The hexagon
(Ⅳ) the mixed states can exist for
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