The reaction diffusion Gray-Scott model with time delay is put forward the assumption of Neumann boundary condition satisfied. Based on Turing bifurcation condition, curves two parameter plane are discussed without delay. normal form computed via applying Lyapunov-Schmidt reduction method in system PDE, and bifurcating direction pitchfork underlying codimension-1 singularity point computed. con...

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for the Sel'kov model, the Gray-Scott system and angiogenesis. We give some suucient and explicit conditions for stability.

Among reaction-diffusion systems showing Turing patterns, the diffusive Gray-Scott model [Pearson, J. A., 1993, Science 261, 189] stands out by showing self-replicating patterns (spots), which makes it the ideal simple model for developmental research. A first study of the influence of noise in the Gray-Scott model was performed by Lesmes et al. [2003 Phys. Rev. Lett. 91, 238301] concluding tha...

This article concerns annular ring solutions of the Gray–Scott model. In the monostable regime, annular rings are far-fromequilibrium patterns supported on annuli inside of which the activator is concentrated. The diffusive flux of inhibitor over long length scales toward such an annulus feeds the production of activator there, and the interaction is semi-strong. Numerical and experimental obse...

In this paper, we prove the local controllability to positive constant trajectories of a nonlinear system two coupled ODE equations, posed in one-dimensional spatial setting, with nonlocal nonlinearities, and using only one localized control moving support. The model deal is derived from well-known reaction–diffusion Gray–Scott when diffusion coefficient first chemical species \(d_u\) tends 0 s...

Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion in bounded domain and the Neumann boundary conditions are studied case particular nonlinearities from Brusselator model, Gray-Scott Oregonator model certain predator-prey model. It is shown that considered systems have both smooth discontinuous stationary solutions, however, only ones can be stable.