General Reaction-diffusion Model Research Articles

Dynamics for a type of general reaction–diffusion model

In this paper, we discuss the following reaction–diffusion model which is a general form of many population models (∗) ∂ u ( t , x ) ∂ t = △ u ( t , x ) − δ u ( t , x ) + f ( u ( t − τ , x ) ) . We study the oscillatory behavior of solutions about the positive equilibrium K of system (∗) with Neumann boundary conditions. Sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of K . These results improve and complement existing results for system (∗) without diffusion. Moreover, when these results are applied to the diffusive Nicholson’s blowflies model and the diffusive model of Hematopoiesis, some new results are obtained for the latter.

Multinonlinear interactions in quasi-linear reaction-diffusion equations with nonlinear boundary flux

This paper deals with multinonlinearity interactions for more general quasi-linear reaction-diffusion models, where three nonlinear mechanisms-nonlinear diffusion, nonlinear reaction (or absorption), and nonlinear boundary flux-are included. The absorption/ boundary flux balance is studied to get the global solvability conditions as well as the blow-up criterion for the model with boundary flux and absorption. Specifically, the global solvability conditions established in this paper for the model with boundary flux and reaction are both necessary and sufficient.

A minimal model for vorticity and gradient banding in complex fluids

A general phenomenological reaction-diffusion model for flow-induced phase transitions in complex fluids is presented. The model consists of an equation of motion for a nonconserved composition variable, coupled to a Newtonian stress relation for the reactant and product species. Multivalued reaction terms allow for different homogeneous phases to coexist with each other, resulting in banded composition and shear rate profiles. The one-dimensional equation of motion is evolved from a random initial state to its final steady state. We find that the system chooses banded states over homogeneous states, depending on the shape of the stress constitutive curve and the magnitude of the diffusion coefficient. Banding in the flow gradient direction under shear rate control is observed for shear-thinning transitions, while banding in the vorticity direction under stress control is observed for shear-thickening transitions.

On the Unique Solvability of a Nonlinear Reaction-Diffusion Model with Convection

Nonnegative solutions of a general reaction-diffusion model with convection are known to be unique if the reaction, convection, and diffusion terms are all Lipschitz continuous with respect to their dependence on the solution variable. However, it is also known that such a Lipschitz condition is not necessary for the unique solvability of the model if either convection or reaction is not present. We introduce monotonicity conditions which, when imposed on the reaction and convection, are sufficient for the uniqueness of all nonnegative solutions of the general model. Consideration of the model where reaction, diffusion, and convection are governed by power laws also reveals the extent to which these conditions are necessary.

Stationary states in a reaction-diffusion system with albedo boundary conditions

We study the existence and stability of stationary states in an exactly solvable reaction-diffusion model, under conditions of partial absorption and reflection at a boundary. The rich variety of time-independent solutions-which, if stable, are candidates to describe the long-time asymptotic state of the system-suggests that such mixed boundary conditions can strongly affect the solution set of more general reaction diffusion models. This conclusion is relevant in many applications, in particular, in nucleating systems.

Vortices with linear cores in mathematical models of excitable media

Most naturally occurring excitable media exhibit vortices in the form of spirals rotating around circular cores. Here we demonstrate that quite another type of vortices - vortices whose cores resemble a long line - may result from decreasing a small parameter in the different mathematical models of excitable media (general reaction-diffusion model, Fitz-Hugh-Nagumo equations, Beeler-Reuter model of cardiac tissue, Wiener axiomatic model, and cellular automata model).

The behavoir of packed bed electrode reactor

A generalized reaction-diffusion model for packed-bed electrode reactor is developed. Model equations include an elliptical PDE and a parabolic PDE describing two dimensionnal potential and concentration distributions and an algebraic equation showing the relations between reaction and mass transfer rates. This model can be used for heterogeneous reaction systems of mass transfer controlling, reaction controlling and for those affected by both. Strategies for numerical solution are also discussed.Cathodic reduction of K3Fe(CN)6 is carried out in a 0.03m ×0.06m ×0.15m rectangular electrode reactor packed with graphite particles. Eight laterally movable potential probes, each mounted at a certain altitude, are installed for potential distribution measurement. Model prediction agrees well with the experimental data at various operating conditions.This model has been applied for designing a pilot reactor for electrochemical reduction of nitrobenzene to produce p-aminophenol.

A generalized reaction diffusion model for spatial structure formed by motile cells

A non-linear stability analysis using a multi-scale perturbation procedure is carried out on a model of a generalized reaction diffusion mechanism which involves only a single equation but which nevertheless exhibits bifurcation to non-uniform states. The patterns generated by this model by variation in a parameter related to the scalar dimensions of domain of definition, indicate its capacity to represent certain key morphogenetic features of multicellular systems formed by motile cells.