Chemical Turing Patterns and Diffusive Instabilities

Abstract

The Brusselator activator-inhibitor reaction-diffusion model is considered and conditions deduced by a normal-mode linear stability analysis for the development of chemical Turing instabilities over a parameter range for which the dynamical system in the absence of diffusion would exhibit a stable homogeneous distribution. The effect the introduction of an immobilizer would have on such diffusive instabilities is also examined. The limitations of linear stability predictions of this sort are discussed and the results of a nonlinear stability analysis which will be treated in detail in later chapters are sketched for the Brusselator. In the problems similar normal-mode linear stability analyses of the Schnackenberg simplification of the Brusselator and a simplified version of the so-called CDIMA (Chlorine Dioxide Iodine Malonic Acid) chemical reaction-diffusion system are considered.

Problems

8.1.

The Schnackenberg simplification of the Brusselator replaces its second reaction with $B{{\mathop {\rightarrow }\limits ^{k_2}}}Y$. Consider the nondimensionalized Schnackenberg chemical reaction–diffusion model system given by

$$\begin{aligned} \frac{\partial x}{\partial t} = 1 - \frac{\alpha }{2} - x + \alpha \frac{x^2y}{2} + \mu \frac{\partial ^2 x}{\partial r^2}, \frac{\partial y}{\partial t} = \beta (1 - x^2y) + \frac{\partial ^2 y}{\partial r^2} \end{aligned}$$

(P1.1a)

where x and y represent the concentrations of the activator and inhibitor species, respectively, which are dynamical variables of space r and time t, while the quantities

$$\begin{aligned} \alpha = \frac{2k_2B}{k_1A + k_2B}, \, \beta = \frac{k_3(k_1A + k_2 B)^2}{k_4^3}, \, \mu = \frac{D_x}{D_y} \end{aligned}$$

(P1.1b)

are parameters since the pool species concentrations (A, B), reaction rates $(k_1,k_2,k_3,k_4)$, and diffusion coefficients $(D_x, D_y)$ have been assumed to remain constant.

• (a) Show that system (P1.1) possesses a single community equilibrium point

  $$\begin{aligned} x(r,t)\equiv x_e, \, y(r, t) \equiv y_e \text{ satisfying } x_e = y_e = 1. \end{aligned}$$

  (P1.2)

• (b) Seeking a normal-mode solution of system (P1.1) of the form

  $$\begin{aligned} x(r, t) = 1 + \varepsilon _1 x_{11}\cos {(qr)}e^{\sigma t} + O(\varepsilon _1^2), \, y(t, y) = 1 + \varepsilon _1 y_11 \cos {(qr)}e^{\sigma t} + O(\varepsilon _1^2) \end{aligned}$$

  (P1.3a)

  where

  $$\begin{aligned} |\varepsilon _1|<< 1, \, |x_{11}|^2 + |y_{11}|^2 \not = 0, \, q \ge 0, \end{aligned}$$

  (P1.3b)

  demonstrate that $\sigma$ satisfies the same quadratic secular equation as the Brusselator

  $$\begin{aligned} \sigma ^2 + [(1+\mu )q^2 + \beta + 1 - \alpha ]\sigma + \mu q^4 + (\mu \beta + 1 - \alpha )q^2 + \beta = 0, \end{aligned}$$

  (P1.3c)

  and hence conclude that there exists a Turing-type diffusive instability provided

  $$\begin{aligned} 0< \alpha - 1< \beta< \frac{(\sqrt{\alpha } - 1)^2}{\mu } \text{ for } 0< \mu < 1. \end{aligned}$$

  (P1.3d)

• (c) Particularizing the result of (P1.3d) to $\alpha = 2$ obtains the instability criteria

  $$\begin{aligned} 1< \beta< \frac{\mu _c}{\mu } \text{ and } 0< \mu < \mu _c = \frac{1}{3 + 2\sqrt{2}}, \end{aligned}$$

  (P1.4)

  represent this instability region of (P1.5) graphically in the $\mu$-$\beta$ plane, and discuss the physical significance of $c_{\text {min}}=1/\mu _c$ with reference to the parameter c defined by

  $$\begin{aligned} c = \frac{1}{\mu } = \frac{D_y}{D_x}. \end{aligned}$$

  (P1.5)

8.2.

Consider the simplified chemical reaction–diffusion model system

$$\begin{aligned} \frac{\partial X}{\partial \tau } = k_1 - k_2X - \frac{4k_3Y}{X} + D_x \frac{\partial ^2 X}{\partial s^2}, \end{aligned}$$

(P2.1a)

$$\begin{aligned} \frac{\partial Y}{\partial \tau } = k_2X - \frac{k_3Y}{X} + D_y \frac{\partial ^2 Y}{\partial s^2}, \end{aligned}$$

(P2.1b)

where X and Y represent iodide and chlorite concentrations, respectively, which are functions of space s and time $\tau$, while the reaction rates $(k_1,k_2,k_3)$ and diffusion coefficients $(D_x, D_y)$ have been assumed to remain constant.

• (a) Show that system (P2.1) possesses a community equilibrium point of the form

  $$\begin{aligned} X(s,\tau ) \equiv X_e, \, Y(s,\tau ) \equiv Y_e \text { such that } Y_e = \frac{k_2X_e^2}{k_3} \end{aligned}$$

  (P2.2)

  by explicitly finding $X_e > 0$.

• (b) Introducing the nondimensional variables and parameters

  $$\begin{aligned} r = \frac{s}{(D_y/k_2)^{1/2}}, \, t = k_2\tau , \, x = \frac{X}{X_e}, \, y = \frac{Y}{Y_e}; \end{aligned}$$

  (P2.3a)
  $$\begin{aligned} \beta = \frac{5k_3}{3k_1}, \, \mu = \frac{D_x}{D_y}; \end{aligned}$$

  (P2.3b)

  transform system (P2.1) into

  $$\begin{aligned} \frac{\partial x}{\partial t} = F(x, y) + \mu \frac{\partial ^2 x}{\partial r^2}, \end{aligned}$$

  (P2.4a)
  $$\begin{aligned} \frac{\partial y}{\partial t} = \beta G(x, y) + \frac{\partial ^2 y}{\partial r^2}, \end{aligned}$$

  (P2.4b)

  where

  $$\begin{aligned} F(x, y) = 5 - x - \frac{4y}{x}, \, G(x, y) = 3\left( x - \frac{y}{x}\right) ; \end{aligned}$$

  (P2.4c)

  and show that $F(1,1) = G(1,1) = 0$.

• (c) Seeking a linear perturbation solution of system (P2.4) of the form

  $$\begin{aligned} x(r, t) = 1 + \varepsilon _1 x_1(r, t) + O(\varepsilon _1^2), \, y(r, t) = 1 + \varepsilon _1 y_1(r, t) + O(\varepsilon _1^2) \text{ for } |\varepsilon _1|<< 1, \end{aligned}$$

  (P2.5)

  deduce that

  $$\begin{aligned} \frac{\partial x_1}{\partial t} = F_1(1,1)x_1 + F_2(1,1)y_1 + \mu \frac{\partial ^2 x_1}{\partial r^2}, \end{aligned}$$

  (P2.6a)
  $$\begin{aligned} \frac{\partial y_1}{\partial t} = \beta [G_1(1,1)x_1 + G_2(1,1)y_1] + \frac{\partial ^2 y_1}{\partial r^2}, \end{aligned}$$

  (P2.6b)

  where

  $$\begin{aligned} F_1(1,1) = 3,\, F_2(1,1) = -4,\, G_1(1,1) = 6, \, G_2(1,1) = -3. \end{aligned}$$

  (P2.6c)

• (d) Then assuming a normal-mode resolution for the perturbation quantities in system (P2.6) by letting

  $$\begin{aligned}{}[x_1,y_1](r, t) = [x_{11}, y_{11}] \cos {(qr)}e^{\sigma t} \text{ where } |x_{11}|^2 + |y_{11}|^2 \not = 0 \text{ and } q \ge 0, \end{aligned}$$

  (P2.7)

  obtain the following quadratic secular equation satisfied by $\sigma$

  $$\begin{aligned} \sigma ^2 + [(1 + \mu )q^2 + 3(\beta -1)]\sigma + \mu q^4 + 3(\beta \mu -1)q^2 + 15 \beta = 0. \end{aligned}$$

  (P2.8)

• (e) Show that the community equilibrium point is stable to homogeneous pertubations ($q^2=0$) provided

  $$\begin{aligned} \beta > 1 \end{aligned}$$

  (P2.9)

  and that there then exists a diffusive instability when $q^2 > 0$ for

  $$\begin{aligned} \beta < \beta _0 (q^2) = \frac{3q^2 - \mu q^4}{3(\mu q^2 + 5)}\le \beta _c = \beta _0(q_c^2) \end{aligned}$$

  (P2.10a)

  where

  $$\begin{aligned} q_c^2 = \frac{2\sqrt{10}-5}{\mu } \end{aligned}$$

  (P2.10b)

  and

  $$\begin{aligned} \beta _c = \frac{13 - 4\sqrt{10}}{3\mu } = \frac{3}{\mu (13 + 4\sqrt{10})}. \end{aligned}$$

  (P2.10c)

• (f) Deduce, from (P2.9) and (P2.10), the following criteria for the occurrence of Turing diffusive instabilities

  $$\begin{aligned} 1< \beta< \frac{\mu _c}{\mu } \text{ and } 0< \mu < \mu _c = \frac{3}{13 + 4\sqrt{10}}. \end{aligned}$$

  (P2.11)

• (g) Defining the parameter

  $$\begin{aligned} c = \frac{1}{\mu } = \frac{D_y}{D_x}, \end{aligned}$$

  (P2.12a)

  demonstrate that (P2.11) is equivalent to

  $$\begin{aligned} 1< \beta < \frac{c}{c_{\text {min}}} \text { and } c > c_{\text {min}} = \frac{1}{\mu _c} = \frac{13 + 4\sqrt{10}}{3}, \end{aligned}$$

  (P2.12b)

  represent the region of (P2.12b) graphically in the c-$\beta$ plane, and discuss the physical significance of $c_{\text {min}}$.