Jacobi stability analysis of predator-prey models with holling-type II and III functional responses

Abstract

In this study, we perform the Jacobi stability analysis of the Lotka Volterra type predator-prey models with Holling-type II and III functional responses. The Jacobi stability analysis is based on the geometry of Finsler spaces and is generally known as the Kosambi, Cartan and Chern (KCC) theory. In the KCC theory, one associates a non-linear connection, and a Berwald type connection to the dynamical systems, and five geometrical invariants were obtained. The second invariant known as the curvature deviation tensor gives the Jacobi stability of the system which is a measure of the robustness of the system to small perturbations of the whole trajectory. Particularly in this study, we review the linear stability of the models and perform a full Jacobi stability analysis of the models via the KCC theory. The Jacobi stability of equilibrium points of the models was studied and a comparative study of the linear stability and Jacobi stability was done to determine the special regions where they both overlap. Conclusively, the time evolution of the components of the deviation near each equilibrium point of the predator-prey models was also considered. We observed that the Jacobi stability of equilibrium points for the Holling-type II and III model guarantees linear stability. Also, parameter regions were the Jacobi and Linear stability overlaps were presented in the form of phase diagrams. The torsion tensor components which geometrically characterizes the chaotic behaviour of dynamical systems are all equal to zero for the models. This eliminates the possibility of the onset of chaos in the studied models.