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In this work, we investigate the dynamical properties of a reaction-diffusion system arising from tumor-therapy modelling that features both nonlinear interactions and nonlocal delay. By applying the Lyapunov-Schmidt reduction, we establish…
Robustness of spatial pattern against perturbations is an indispensable property of developmental processes for organisms, which need to adapt to changing environments. Although specific mechanisms for this robustness have been extensively…
A variety of simulation methodologies have been used for modeling reaction-diffusion dynamics -- including approaches based on Differential Equations (DE), the Stochastic Simulation Algorithm (SSA), Brownian Dynamics (BD), Green's Function…
The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction-diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying…
We explain the principles of gene expression pattern stabilization in systems of interacting, diffusible morphogens, with dynamically established source regions. Using a reaction-diffusion model with step-function production term, we…
Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual…
Straightforward method for the derivation of linearized version of stochastic stability analysis of the nonlinear differential equations is presented. Methods for the study of large time behavior of the moments are exposed. These general…
Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays traveling waves in its…
The analysis of non-equilibrium steady states of biochemical reaction networks relies on finding the configurations of fluxes and chemical potentials satisfying stoichiometric (mass balance) and thermodynamic (energy balance) constraints.…
In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and…
Spreading processes are often modelled as a stochastic dynamics occurring on top of a given network with edge weights corresponding to the transmission probabilities. Knowledge of veracious transmission probabilities is essential for…
We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns…
Diffusion Models are probabilistic models that create realistic samples by simulating the diffusion process, gradually adding and removing noise from data. These models have gained popularity in domains such as image processing, speech…
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…
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…
The advent and proliferation of social media have led to the development of mathematical models describing the evolution of beliefs/opinions in an ecosystem composed of socially interacting users. The goal is to gain insights into…
Physical systems with complex unsteady dynamics, such as fluid flows, are often poorly represented by a single mean solution. For many practical applications, it is crucial to access the full distribution of possible states, from which…
This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic…
Reaction-diffusion (Turing) systems are fundamental to the formation of spatial patterns in nature and engineering. These systems are governed by a set of non-linear partial differential equations containing parameters that determine the…
A practical introduction to stochastic modelling of reaction-diffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the…