Topological Dynamic Consistency of Numerical Methods for Dynamical Models in Biosciences

Abstract

Qualitative analysis of dynamical systems modelling biological processes makes a valuable contribution to understanding these processes and elucidating relevant properties. These include, invariant sets (including equilibria) and their stability properties, existence of bifurcation thresholds, shape of trajectories, etc. Due to the high level of complexity of the biological systems, the models often involve large number of parameters with unknown or only estimated values. In such setting, qualitative analysis tends to be more important than high precision computations. Hence, the importance of numerical methods which accurately replicate the properties of the model. There are several attempts to capture this property of numerical methods in concepts like qualitative stability, dynamic consistency, structural stability under numerics. In this paper we present the concept of topological dynamic consistency of numerical methods, which aligns the model and the method in such a way that all properties of the model which are of topological nature, like the mentioned above, are replicated. Further, we present some practical approaches of designing such methods. Let us remark that topological dynamic consistency cannot be derived through the standard tools of numerical analysis based on consistency, stability, order of approximation and convergence. The theory is based on the concepts topological conjugacy of maps, topological equivalence of flows as well as the structural stability of flows and maps.