An influence of random disturbances on the pattern formation in reaction–diffusion systems is studied. As a basic model, we consider the distributed Brusselator with one spatial variable. A coexistence of the stationary nonhomogeneous spatial structures in the zone of Turing instability is demonstrated. A numerical parametric analysis of shapes, sizes of deterministic pattern–attractors, and their bifurcations is presented. Investigating the corporate influence of the multistability and stochasticity, we study phenomena of noise-induced transformation and generation of patterns.

We study a stochastic spatially extended population model with diffusion, where we find the coexistence of multiple non-homogeneous spatial structures in the areas of Turing instability. Transient processes of pattern generation are studied in detail. We also investigate the influence of random perturbations on the pattern formation. Scenarios of noise-induced pattern generation and stochastic transformations are studied using numerical simulations and modality analysis. This article is part of the theme issue ‘Patterns in soft and biological matters’.

In this paper, a distributed Brusselator model with diffusion is investigated. It is well known that this model undergoes both Andronov-Hopf and Turing bifurcations. It is shown that in the parametric zone of diffusion instability the model generates a variety of stable spatially nonhomogeneous structures (patterns). This system exhibits a phenomenon of the multistability with the diversity of stable spatial structures. At the same time, each pattern has its unique parametric range, on which it may be observed. The focus is on analysis of stochastic phenomena of pattern formation and transitions induced by small random perturbations. Stochastic effects are studied by the spatial modality analysis. It is shown that the structures possess different degrees of stochastic sensitivity.

In this paper, a spatially extended stochastic reaction-diffusion model is studied.Due to Turing instability, stable nonhomogeneous stationary patterns are generated in such models. A theoretical approach to estimating the mean-square deviation of random solutions from the stable deterministic pattern-attractor is demonstrated. Stochastic sensitivity functions for stable stationary patterns are introduced. Theoretical evaluations are compared with statistically obtained data. Based on this approach, we investigate stochastic properties of different patterns in Brusselator. Variations in pattern sensitivity to noise and phenomenon of stochastic preference are discussed.

In this paper, a distributed stochastic Brusselator model with diffusion is studied. We show that a variety of stable spatially heterogeneous patterns is generated in the Turing instability zone. The effect of random noise on the stochastic dynamics near these patterns is analysed by direct numerical simulation. Noise-induced transitions between coexisting patterns are studied. A stochastic sensitivity of the pattern is quantified as the mean-square deviation from the initial unforced pattern. We show that the stochastic sensitivity is spatially non-homogeneous and significantly differs for coexisting patterns. A dependence of the stochastic sensitivity on the variation of diffusion coefficients and intensity of noise is discussed.