Analysis of stationary droplets in a generic Turing reaction-diffusion system

Solitonlike structures called “droplets” are found to exist within a paradigm reaction-diffusion model that can be used to describe patterning in a number of biological systems, for example, on the skin of various fish species. They have also been found in many other systems that can be modeled with a complex Ginzburg-Landau system. These droplets can be analyzed in the biological paradigm model because the system has two nonzero stable steady states that are symmetric; however, the asymmetric case is more challenging. We first review the properties of the paradigm system and then extend a recently developed perturbation technique [D. Gomila et al., J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004)] to investigate the weakly asymmetric case. We compare the results of our mathematical analysis with numerical simulations and show good agreement in the region where the assumptions hold.

Figure 1

(a) An example of pattern produced when two BVAM models are coupled together. The domain is square with side length of 120. Reproduced from [10]. “Copyright 2009 by the American Physical Society.” (b) The skin pattern of the stingray Potamotrygon motoro. Scale bar of 10 cm.

Figure 2

(a) Cut away concentration of the stationary stable droplet, $u$, with the central profile emphasized in black. (b) Concentration of the stationary stable droplet, $v$, with central profile in black. $u$ and $v$ were found by solving Eqs. (3, 4) with parameters $C=0.005$, $D=0.516$, $\eta =0.35$, $h=-1.32$, $a=0.05775$, and $b=-0.30525$.

Figure 3

Patterns obtained when $h\ne -1$ in Eqs. (3, 4). (a) and (b) are stationary and found in region 2 of Fig. 5. (c) Traveling wave fronts, found in region 2. (d) Oscillating spiral patterns found on the border between regions 2 and 5 of Fig. 5. All pattern peaks and troughs are contained in the region $\left[-2,2\right]$ and in each case the domain is square with side length of 254. Reproduced from [10]. “Copyright 2009 by the American Physical Society.”

Figure 4

Stationary symmetric structure profiles of $u$ (solid line) and $v$ (dashed line), obtained from Eqs. (3, 4) in one dimension with parameters $C=0$, $D=0.516$, $\eta =0.35$, $h=-1.32$, $a=0.05775$, and $b=-0.30525$.

Figure 5

Phase plane, in the absence of diffusion, in Eqs. (3, 4). Parameters are $D=0$, $C=0$, and $h=-2.5$. The lines separating the regions are derived from the dispersion relation ${\omega }^2-\omega \text{Tr}\left(\mathbf{J}\right)+\text{Det}\left(\mathbf{J}\right)=0$, where $\omega ={\sigma }_j+i{\tau }_j$, $j=1$ and 2. In region 1, ${\sigma }_1<0$, ${\sigma }_2>0$, and $\tau =0$ (saddle points). In region 2, ${\sigma }_{1,2}<0$ and $\tau \ne 0$ (oscillating stable points). In region 3, ${\sigma }_{1,2}<0$ and $\tau =0$ (stable points). In region 4, ${\sigma }_{1,2}>0$ and $\tau =0$ (both points are unstable). In region 5, ${\sigma }_{1,2}>0$ and $\tau \ne 0$ (oscillating unstable points).

Figure 6

Illustration of the new coordinate variables given by Eq. (9). $\mathbf{x}$ is the position vector in the reference frame and $\mathbf{X}$ is the position vector of the line front. The vector $\widehat{\mathbf{n}}$ is the unit normal vector to the line front, $w$ is the normal coordinate to the front, and $s$ is the arc length along the front. The various $s_i$ are illustrative points along the front. Similarly, the $t_i$ denotes the evolution of the system at different time points.

Figure 7

(a) Profiles of droplet structures $u$ (solid line) and $v$ (dashed line). (b) Profiles of soliton structures $u$ (solid line) and $v$ (dashed line). (c) Phase plane plot. The dashed-dotted trajectory is the solution to Eq. (18). The solid trajectory is the droplet profile. The dashed trajectory is the soliton profile. The latter two solutions were obtained by solving Eq. (21) with different initial conditions. In (a) and (b) $C=0.005$, the values of $C$ in (c) are noted in the figure. Other parameters are $D=0.516$, $\eta =0.35$, $h=-1.32$, $a=0.05775$, and $b=-0.30525$.

Figure 8

Comparison of the computed steady-state radius of the perturbed system with the analytically derived radius attained from one-dimensional Eqs. (18, 21). The dashed line indicates instability to either radial or azimuthal perturbations. The solid line indicates a radius stable to radial and azimuthal perturbations. The dotted line is the theoretical prediction from Eq. (17) with $C=0.005$ and all other parameters as in Fig. 7. The stability of Eqs. (3, 4) with a two-dimensional Laplacian was found by using a finite element method on a mesh of 9440 elements. Through a bisection parameter search the stability disappears at $h_d=-1.326$, correct to three decimal places.

Figure 9

Time series for the evolution of a square initial condition. The left column has $h=-1.32$ and so is in the stable region; thus, the system reaches a final droplet steady pattern. The times shown are $t=0$, 100, ${10}^4$, and ${10}^5$. The right column has $h=-1.33$ and so is in the unstable region; thus, it produces a labyrinthine steady state. The times shown are $t=0$, ${10}^5$, $2\times {10}^5$, and ${10}^6$. Other parameters are $D=0.516$, $\eta =0.35$, $C=0.005$, $a=0.05775$, and $b=-0.30525$. The gray scale is from $-1.2$ (darkest) to 1.2 (lightest) and the domain is $\left[-50,50\right]\times \left[-50,50\right]$.

Figure 10

Time series for the evolution of pseudorandom initial conditions to stable droplets. The times shown are $t=0$, 100, ${10}^3$, and ${10}^4$. Parameters $D=0.516$, $\eta =0.35$, $C=0.005$, $h=-1.3$, $a=0.05775$, and $b=-0.30525$. The gray scale is from $-1.4$ (darkest) to 1.4 (lightest) and the domain is $\left[-50,50\right]\times \left[-50,50\right]$.