Sensitivity Equation for Competitive Model: Derivation, Numerical Realization and Parameter Estimation

Abstract

Ecological systems can be quite complex, consisting of an interconnected system of plants and animals, predators and prey, flowering plants, seed dispersers, insects, parasites, pollinators, and so on. In the case of the existence of a species affecting the survival of other species and vice versa, it can derive a competitive model in the form of a system of differential equations. A competitive model involves a number of parameters which grow in proportion to the number of interacting species. The resistance of a state variable to tiny disturbances of some parameter is referred to as sensitivity. The competitive model of size N consists of N parameters for intrinsic growth, N parameters for carrying capacity, N 2 − N parameters for species interaction, and N parameters for initial conditions. As a result, there will be N 2 (N + 2) distinct values of sensitivity. The purpose of this paper is to derive a general formulation of the sensitivity equations of dynamical system and then apply it to the competitive model. This study also encompasses the formulation of some algorithms and the implementation for solving the sensitivity equation numerically. Finally, the sensitivity functions are employed as qualitative instruments in the optimal design of measurement for parameter estimation through a series of numerical experiments. The results of this study are the ordinary and the generalized sensitivity functions for interacting species. Based on numerical experiments, each group of data provides different information about the existing parameters.