Rigorous Mapping of Data to Qualitative Properties of Parameter Values and Dynamics: A Case Study on a Two-Variable Lotka-Volterra System.

Abstract

In this work, we describe mostly analytical work related to a novel approach to parameter identification for a two-variable Lotka-Volterra (LV) system. Specifically, this approach is qualitative, in that we aim not to determine precise values of model parameters but rather to establish relationships among these parameter values and properties of the trajectories that they generate, based on a small number of available data points. In this vein, we prove a variety of results about the existence, uniqueness, and signs of model parameters for which the trajectory of the system passes exactly through a set of three given data points, representing the smallest possible data set needed for identification of model parameter values. We find that in most situations such a data set determines these values uniquely; we also thoroughly investigate the alternative cases, which result in nonuniqueness or even nonexistence of model parameter values that fit the data. In addition to results about identifiability, our analysis provides information about the long-term dynamics of solutions of the LV system directly from the data without the necessity of estimating specific parameter values.