Application of stability index in characterizing the basin of attraction in ecological modelling

An approach called the “stability index” was proposed by Podvigina and Ashwin in the year 2011 is very useful to characterize the local geometry of riddled basins of attraction for dynamical systems. With the stability index, one can study the behaviour of a dynamical system. It would be interesting to understand how the stability index behaves on the basin boundary between multiple basins of attraction. In this paper, the stability index is applied in the case of the ecological model. A two-species of competition system is considered, in which this system contains two attractors. To characterize the geometry of the basin of attraction, the stability index is applied. The results show that as a parameter in the system increases, the stability indices vary from infinity down to positive values to minus infinity. The changes in the indices indicate that the attractor loses its stability from being asymptotically stable to riddled basin attractor to totally unstable.