The Study of Stability Analysis of Modified Leslie-Gower Herbivore Model with Allee Effect in Plants

Abstract

A modified Leslie-Gower plant-herbivore model is studied under the Allee effect. Holling-type II functional response is used to modify the model. Delay differential equations play an essential role in making this model more realistic and complicated. The non-trivial equilibrium E* (P* ≠ 0, H* ≠ 0) of the proposed model is calculated. Moreover, the stability and instability of the state variables, which include plant population P and herbivore population H are described graphically. It is shown that the system represents absolute stability when it has no time parameter (τ). When the time parameter is less than the threshold value, then the system exhibits asymptotic stability. In addition, the system surrenders its stability, and Hopf-bifurcation occurs when the time parameter surpasses the threshold value. The timeseries graphs are also represented. It is demonstrated that the system becomes more stable with the maximum rate of predation. MATLAB software is used to perform the graphs to justify the theoretical results.