This paper deals with the effects of partial differential equations on development spatiotemporal patterns in reaction–diffusion systems. These systems describe how concentration a certain substance is distributed space or time under effect two phenomena: chemical reactions different substances into some other product and diffusion which results dispersion over surface space. Mathematical repre...

The replication and differentiation of spots in reaction diffusion equations are studied by extending the Gray-Scott model with self-replicating spots to include many degrees of freedom needed to model systems with many chemicals. By examining many possible reaction networks, the behavior of this model is categorized into three types: replication of homogeneous fixed spots, replication of oscil...

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize homotopy analysis transformation method to obtain approximate solutions for This gives more realistic series of that converge rapidly exact solution....

Abstract Nowadays, in the Scientific Machine Learning (SML) research field, traditional machine learning (ML) tools and scientific computing approaches are fruitfully intersected for solving problems modelled by Partial Differential Equations (PDEs) science engineering applications. Challenging SML methodologies new computational paradigms named Physics-Informed Neural Networks (PINNs). PINN ha...

It is well known that simple reaction-diffusion systems can display very rich pattern formation behavior. Here we have studied two examples of such systems in three dimensions. First we investigate the morphology and stability of a generic Turing system in three dimensions and then the well-known Gray-Scott model. In the latter case, we added a small number of morphogen sources in the system in...

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss numerical scheme for solving the GS system. diffusion coefficients are on surfaces and they depend space time. regard, first adopt an implicit difference stepping method semi-discretize in time direction. Then, implement hybrid advanced meshless disc...

We consider a nonlocal eigenvalue problem which arises in the study of stability of point-condensation solutions in the Gierer-Meinhardt system and generalized Gray-Scott system. We give some suucient conditions for stability and instability. The conditions are new and can be applied to the study of stability of single point-condensation solutions.

In this article, we prove the existence of a random attractor of the stochastic three-component reversible Gray-Scott system on infinite lattice with additive noise. We use a transformation of addition involved with Ornstein-Uhlenbeck process, for proving the pullback absorbing property and the pullback asymptotic compactness of the reaction diffusion system with cubic nonlinearity.