Verifying Life Science Models Containing Diffusion, Transport and Interaction of Species

Abstract

In deriving evolution models of the problems of life sciences we are usually dealing with the temporal evolutions of non-negative quantities like concentrations of nutrients and chemicals,population densities, temperature, pressure etc. Each model is some approximation of a real situation and therefore it is not clear beforehand, that the obtained models will obey the following crucial properties for the modellers: the solutions of continuous evolution equations remain nonnegative if the initial data are nonnegative. We emphasize that, numerical simulations cannot answer the question whether the model in this sense is valid or not, even though numerical simulations can provide empirical evidence we can never be sure. Hence it is extremely important for modellers to have such criteria, that is necessary and sufficient conditions for solutions of parabolic systems containing diffusion, transport (convection) and interaction of species (nonlinear term) to preserve positive cones. It is worth noting, that so far such a criterion, in full generality, has only been found for scalar equations, which criterion is based on the maximum principle. It is well-known that for a system of parabolic PDEs the maximum principle fails. At present in the mathematical literature there exist many sufficient conditions for parabolic systems preserving positive cones. However this is not satisfactory for modellers. They need to have an algorithm to know whether or not the derived realistic models satisfy this property before starting to do analysis and carrying out simulations. Therefore in Chap. 3 (which in turn consists of three sections) we derive a criterion for positivity of solutions for quite large classes of deterministic parabolic systems containing diffusion, transport (advection or convection) and interaction of species (nonlinear term).