A new approach to the reaction-diffusion systems modelling

Abstract

We consider a new methodology for modelling the reaction-diffusion systems based on systems of ordinary differential equations. In contrary to the specialised numerical methods like straight line method, this new methodology is positioned as a pure alternative at the model level for partial differential equations. In its description, the new method is largely similar to the finite volume method, but unlike the latter, it uses statistical simplifications and principles of geometric probability to describe diffusion. The main objectives of this approach are to simplify the qualitative analysis of reaction-diffusion systems and to improve the efficiency of numerical model implementation. The first objective is successfully addressed, as it becomes possible to use the apparatus of classical dynamical systems theory for a qualitative analysis of model dynamics based on the systems of ordinary differential equations. The second objective is only partially addressed, as the gain in efficiency while maintaining acceptable accuracy for numerical implementation will be significant only for certain simple initial distribution of molecules and for specific diffusion coefficients. Furthermore, to formulate criteria for practical applicability, we separately evaluate the modelling error using this new methodology.