Qualitative behavior of three species food chain around inner equilibrium point: spectral analysis

Abstract

This work deals with analytical investigation of local qualitative temporal behavior around inner equilibrium point of a model for three species food chain, studied earlier by Hastings and Powel and others. As an initial step towards the spectral analysis of the model, the governing equations have been split into linear and nonlinear parts around arbitrary equilibrium point. The explicit parameter dependence of eigenvalues of Jacobi matrix associated to the linear part have been derived. Analyzing these expressions in conjunction with some pedagogical analysis, a lot of predictions on stable, unstable or chaotic change of species have been highlighted. Agreement of predictions of this work with available numerical or semi-analytical studies suggest the utility of analytical results derived here for further investigation/analysis of the model as desired by earlier works.

Keywords

• Hastings and Powell’s model
• spectral analysis
• local stability
• bifurcation point
• steady state-, Hopf-, Takens–Bagdanov bifurcations
• Sil’nikov chaos