The reaction-diffusion master equation is a stochastic model often utilized in the study of biochemical reaction networks in living cells. It is applied when the spatial distribution of molecules is important to the dynamics of the system. A viable approach to resolve the complex geometry of cells accurately is to discretize space with an unstructured mesh. Diffusion is modeled as discrete jump...

In order to model fundamental cell biological processes including the transcription, translation, and nuclear membrane transport of biological molecules within a eukaryotic cell it is necessary to be able to approximate the stochastic reaction and diffusion of a small number of molecules in the complex three dimensional geometry of a cell. For this reason a method is developed that incorporates...

Bi-stable chemical systems are the basic building blocks for intracellular memory and cell fate decision circuits. These circuits are built from molecules, which are present at low copy numbers and are slowly diffusing in complex intracellular geometries. The stochastic reaction-diffusion kinetics of a double-negative feedback system and a MAPK phosphorylation-dephosphorylation system is analys...

at present paper, we establish the existence of pullback $mathcal{d}$-attractor for the process associated with non-autonomous partly dissipative reaction-diffusion equation in $l^2(mathbb{r}^n)times l^2(mathbb{r}^n)$. in order to do this, by energy equation method we show that the process, which possesses a pullback $mathcal{d}$-absorbing set, is pullback $widehat{d}_0$-asymptotically compact.

We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the soluti...

We consider positive, radial and exponentially decaying steady state solutions of the general reaction-diffusion and Klein-Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction-diffusion equation and a precise center-stable manifold theorem for th...

Abstract. We study the phase field method for the volume preserving mean curvature flow. Given an initial data which is a measure-theoretic boundary of a Caccioppoli set with a suitable density bound, we prove the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula for the re...