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Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion

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  • In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.

    Mathematics Subject Classification: 35K57, 35B36, 35R01, 35R37, 65M60.

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  • Figure 1.  Phase-diagrams corresponding to the non-autonomous system of ordinary differential equations (3.1)-(3.2) with exponential growth. A stable limit cycle or spiral point exists depending on the choice of the parameter values $a$ and $b$. (a) $a = 0.1$, $b = 0.75$, (b) $a = 0.15$, $b = 0.6$, (c) $a = 0.15$, $b = 0.5$, (d) $a = 0.1$, $b = 0.1$, (e) $a = 0.1$, $b = 0.75$, and (f) $a = 0.12$, $b = 0.5$. Other parameter values are fixed as follows: $r = 0.01$, $\gamma = 200$, $m = 2$, $\kappa = 2$ and $A = 1$

    Figure 5.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = d_v = 0$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.75$ from the green parameter space in (a). (b)- (e) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development. In the absence of domain growth, these patterns are non-existent

    Figure 6.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.75$ from the green parameter space in (a). (b)- (h) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development. Note that when the domain is not sufficiently large enough, no patterns are observed

    Figure 7.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = 0, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.6$ from the green parameter space in (a). (b)- (e) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development

    Figure 8.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 10, $ linear cross-diffusion coefficients $d_u = 1, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.5$ from the green parameter space in (a). (b)- (f) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development

    Figure 9.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.5$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.1$ from the green parameter space in (a). (b)- (e) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development

    Figure 10.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.5$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.75$ from the green parameter space in (a). (b) - (f) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development

    Figure 11.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = -0.8, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.25$ from the green parameter space in (a). (b) - (f) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development

    Figure 12.  (a) Snap-shots of the evolving parameter space for exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.8$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.12$ and $b = 0.5$ from parameter space in $(a)$. (b) - (g) Finite element numerical simulations exhibiting the formation of spatial structure corresponding to the chemical specie $u$ during growth development

    Figure 13.  (a)-(d) Snap-shots of the finite element numerical simulations on the evolving unit sphere under exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = -0.8, d_v = 1$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.15$ and $b = 0.2$ from the domain-induced parameter space shown in Figure 11(a)

    Figure 14.  (a)-(d) Snap-shots of the finite element numerical simulations on the evolving saddle like surface under exponential growth rate with diffusion coefficient $d = 1, $ linear cross-diffusion coefficients $d_u = 1, d_v = 0.5$, with $\gamma = 200$ and growth rate $r = 0.01$. We select the model kinetic parameter values $a = 0.1$ and $b = 0.1$ from the domain-induced parameter space shown in Figure 9(a)

    Figure 2.  Snap-shots of continuously evolving parameter spaces for an exponential evolution of the domain with diffusion coefficient $d = 10$ with varying linear cross-diffusion coefficients $d_u$ and $d_v$ and the growth rate $r$. Row-wise: (a)-(c) $d_u = d_v = 0$, (d)-(f) $d_u = 0$ and $d_v = 1$, (g)-(i) $d_u = 1$ and $d_v = 0$, (j)-(l) $d_u = d_v = 1$. Column-wise: First column: $r = 0.01$, second column, $r = 0.04$ and third column: $r = 0.08$. All plots are exhibited at times $t = 0$ (no growth), $t = 0.1$ and $t = 0.3$

    Figure 3.  Snap-shots of continuously evolving parameter spaces for an exponential evolution of the domain with diffusion coefficient $d = 1$ with variable linear cross-diffusion coefficients and growth rates: Row-wise: (a)-(c) $d_u = 1$, $d_v = 0.8$, (d)-(f) $d_u = 0.8$, $d_v = 1$, and (g)-(i) $d_u = -0.8$, $d_v = 1$. Column-wise: First column: $r = 0.01$, second column, $r = 0.04$ and third column: $r = 0.08$. All plots are exhibited at times $t = 0$ (no growth), $t = 0.1$ and $t = 0.3$. Such spaces do not exist if linear cross-diffusion is absent (with or without domain growth)

    Figure 4.  Super imposed parameter spaces with no growth, linear, logistic and exponential growth profiles of the domain with diffusion coefficient $d = 1$ and linear cross-diffusion coefficients $d_u = 1$ and $d_v = 0.8$. The growth rate $r$ is varied row-wise accordingly: (a)-(b) $r = 0.01$, (c)-(d) $r = 0.04$, and (e)-(f) $r = 0.08$. Snap-shots of the continuously evolving parameter spaces at times $t = 0$ and $t = 0.1$ (first column) and $t = 0$ and $t = 0.3$ (second column), respectively. Without linear cross-diffusion, these spaces do not exist

    Table 1.  Table illustrating the function $h(t)$ for linear, exponential and logistic growth functions on evolving domains. Similar functions can be obtained for evolving surfaces. $\kappa$ is the carrying capacity (final domain size) corresponding to the logistic growth function. In the above $h (t)$ as defined by (2.6) (or (2.7)) on planar domains (or on surfaces)

    Type of growthGrowth Function $\rho (t)$$h(t)= \nabla \cdot \mathit{\boldsymbol{v}}$$q(t)=e^{-\int_{t_0}^t h(\tau) d\tau }$
    Linear$\rho(t)=rt+1$ $h(t)=\frac{m\, r}{rt+1}$$q(t)= \left(\frac{1}{rt+1}\right)^m$
    Exponential$\rho(t)=e^{rt}$$h(t)=m\, r$$q(t)=e^{-mrt}$
    Logistic$\rho (t) = \frac{\kappa Ae^{\kappa rt}}{1+Ae^{\kappa rt}}, \, A = \frac{1}{\kappa-1}$$h(t)= m\, r\, \Big(\kappa-\rho (t)\Big)$$q(t) = \left(\frac{e^{-\kappa r t}+A}{1+A}\right)^m$
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