Dynamically consistent numerical methods for general productive–destructive systems

Abstract

The dynamic consistency of a class of non-standard finite-difference schemes is analysed for general 2D and 3D productive–destructive systems (PDS). Based on those results a methodology for construction of positive and elementary stable non-standard numerical methods is developed. The numerical techniques are based on a non-local modelling of the right-hand side function and a non-standard treatment of the time derivative. This discretization approach leads to significant qualitative improvements in the behaviour of the numerical solutions. The explicit form of the proposed new schemes makes them a computationally effective tool in simulations of the dynamics of systems of biological, chemical and physical interactions that are naturally modelled by PDS. Applications to several specific biological systems are presented.