Note that the boundary conditions 2. In the presence of diffusion, we require the emergence of spatial growth. Solving 2.
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The proof is a direct consequence of conditions 2. Assuming that conditions 2. This entails that the system can no longer exhibit spatially inhomogeneous solutions. The only two conditions left to hold true are 2.
This case corresponds to the classical standard two-component reaction—diffusion system which requires that for details, see for example [ 9 ]. If condition 2. If conditions 2.
Similar to classical reaction—diffusion systems, conditions 2. In order for diffusion-driven instability to occur, the bulk and surface diffusion coefficients must be greater than the values of the critical diffusion coefficients. Next, we investigate under what assumptions on the reaction-kinetics do conditions 2.
Then, it follows that condition 2.
It follows then that none of the conditions 2. However, condition 2. This implies that none of the conditions 2. The above cases clearly eliminate conditions 2.
Finite difference reaction diffusion equations with nonlinear boundary conditions
We are now in a position to state our main result. The necessary conditions for diffusion-driven instability The necessary conditions for diffusion-driven instability for the coupled system of BSRDEs 2. From the analytical results, we state the following theoretical predictions to be validated through the use of numerical simulations. For this case, the uniform steady state is the only stable solution for the coupled system of BSRDEs 2.
Here, all the conditions 2. Similarly, all the conditions 2. All the conditions 2. Here, we present bulk-surface finite-element numerical solutions corresponding to the coupled system of BSRDEs 2. Here, we omit the details of the bulk-surface finite-element method as these are given elsewhere see [ 19 ] for details.
Initial conditions are prescribed as small random perturbations around the equilibrium values. We briefly outline how the bulk-surface triangulation is generated. For further specific details, please see reference [ 19 ]. The bulk triangulation induces the surface triangulation as illustrated. Example meshes for the bulk a and surface system b. Part of the domain has been cut away and shown on the right to reveal some internal mesh structure.
Online version in colour. We present patterns corresponding only to the chemical species u and r in the bulk and on the surface, respectively. Those corresponding to v and s are degrees out of phase to those of u and r and are therefore omitted.
It must be noted however that this is not always the case in general, Robin-type boundary conditions may alter the structure of the solution profiles depending on the model parameter values and the coupling compatibility boundary parameters. We observe that no patterns form in complete agreement with theoretical predictions. Similar to classical reaction—diffusion systems, diffusion coefficients must be greater than one.
In particular, the diffusion coefficients must be greater than their corresponding respective critical diffusion coefficients in the bulk and on the surface.
SIAM Journal on Applied Mathematics
An example is shown next. The uniform steady-state solutions are converged to and no patterns form. Columns 1 and 2: solutions in the bulk representing u. Columns 3 and 4: solutions on the surface representing r. Second and fourth columns represent cross sections of the bulk and the surface, respectively. Spots are observed to form on the surface, whereas in the bulk, small balls form inside.
Far away from the surface, no patterns form, because the necessary conditions for diffusion-driven instability are not fulfilled in the bulk. These results confirm our theoretical predictions. We note that this particular example describes realistically pattern formation in biological systems. We expect skin patterning to manifest in the epidermis layer as well as on the surface.
Spot patterns form on the surface, whereas small balls form in the vicinity of the surface inside the bulk. On the surface, small amplitude patterns occur consistent with theoretical predictions. Although the patterns for the u species columns one and two appear uniform on the surface this is simply owing to the colour scale, with the amplitude of the patterns in the bulk larger than those on the surface.
This difference in the amplitude of the pattern of the bulk solution in the bulk and on the surface is due to the Robin-type boundary conditions.
Elliptic Partial Differential Equations Volume by Vitaly Volpert
Unlike zero-flux also known as homogeneous Neumann , boundary conditions for standard reaction—diffusion systems which imply that no species enter or leave the domain, here, there is deposition or removal of chemical species through the flux on the surface, resulting in differences in amplitude between the bulk and surface solutions. Spectacular patterning occurs in the bulk exhibiting spots, stripes and circular patterns. The surface dynamics produce uniform patterning. In the bulk, we observe the formation of balls which can be seen as spots through cross sections and these translate to spots on the surface.
The surface dynamics themselves induce spot pattern formation. We observe spot pattern formation both in the bulk and on the surface. We have presented a coupled system of BSRDEs whereby the bulk and surface reaction—diffusion systems are coupled through Robin-type boundary conditions.
Nonlinear reaction-kinetics are considered in the bulk and on the surface and for illustrative purposes, the activator-depleted model was selected because it has a unique positive steady state. By using linear stability theory close to the bifurcation point, we state and prove a generalization of the necessary conditions for Turing diffusion-driven instability for the coupled system of BSRDEs. Our most revealing result is that the bulk reaction—diffusion system has the capability of inducing patterning under appropriate model and compatibility parameter values for the surface reaction—diffusion model.
On the other hand, the surface reaction—diffusion is not capable of inducing patterning everywhere in the bulk; patterns can be induced in only regions close to the surface membrane. For skin pattern formation, this example is consistent with the observation that patterns will form on the surface as well as within the epidermis layer close to the surface. We do not expect patterning to form everywhere in the body of the animals.
Our studies reveal the following observations and research questions still to be addressed:. However, these boundary conditions do not appear explicitly in the conditions for diffusion-driven instability and this makes it difficult to theoretically analyse their role and implications to pattern formation. Further studies are required to understand the role of these boundary conditions as well as the size of the boundary layer;. We are currently studying the implications of relaxing the compatibility condition;.
Such studies might reveal more interesting properties of the coupled bulk-surface model and this forms part of our current studies. We have presented a framework that couples bulk dynamics three-dimension to surface dynamics two-dimension with the potential of numerous applications in cell motility, developmental biology, tissue engineering and regenerative medicine and biopharmaceutical where reaction—diffusion-type models are routinely used [ 5 , 6 , 11 — 14 , 16 , 17 ]. We have restricted our studies to stationary volumes. In most cases, biological surfaces are known to evolve continuously with time.
This introduces extra complexities to the modelling, analysis and simulation of coupled systems of BSRDEs. In order to consider evolving bulk-surface partial differential equations, evolution laws geometrical should be formulated describing how the bulk and surface evolve.
Related Elliptic Partial Differential Equations: Volume 2: Reaction-Diffusion Equations
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