Related papers: Taking the reaction-diffusion master equation to t…
We propose certain approach of solving two-dimensional non-stationary and stationary advection-diffusion-reaction boundary value problems through their reduction to the set of corresponding one-dimensional problems. This method leverages…
We investigate the multi-chain version of the Chemical Master Equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a…
By discretising space into compartments and letting system dynamics be governed by the reaction-diffusion master equation, it is possible to derive and simulate a stochastic model of reaction and diffusion on an arbitrary domain. However,…
Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid…
The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with…
We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology.…
Diffusion-limited association reactions are ubiquitous in nature. They are particularly important for biological reactions, where the reaction rates are often determined by the diffusive transport of the molecules on two-dimensional…
Morphogenesis is central to biology but remains largely unexplored in chemistry. Reaction-diffusion (RD) mechanisms are, however, essential to understand how shape emerges in the living world. While numerical methods confirm the incredible…
A combination of reaction-diffusion models with moving-boundary problems yields a system in which the diffusion (spreading and penetration) and reaction (transformation) evolve the system's state and geometry over time. These systems can be…
We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
The macroscopic behavior of dissipative stochastic partial differential equations usually can be described by a finite dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic…
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class…
The dynamics of ecological as well as chemical systems may depend on heterogeneous configurations. Heterogeneity in reaction-diffusion systems often increase modelling and simulating difficulties when non-linear effects are present. One…
In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such…
Models invoking the chemical master equation are used in many areas of science, and, hence, their simulation is of interest to many researchers. The complexity of the problems at hand often requires considerable computational power, so a…
In this paper, we study a parabolic reaction diffusion system with constraints that model biofilm growth. Within a unified framework encompassing multiple numerical schemes, we derive the first general convergence rates for approximating…
Open biochemical systems of interacting molecules are ubiquitous in life-related processes. However, established computational methodologies, like molecular dynamics, are still mostly constrained to closed systems and timescales too small…
We present a multiscale approach to model diffusion in a crowded environment and its effect on the reaction rates. Diffusion in biological systems is often modeled by a discrete space jump process in order to capture the inherent noise of…
Motivated by models of signaling pathways in B lymphocytes, which have extremely large nuclei, we study the question of how reaction-diffusion equations in thin $2D$ domains may be approximated by diffusion equations in regions of smaller…