Exploring a Simple Stochastic Mathematical Model Including Fear with a Linear Functional Response

Abstract

This paper concentrates on a simple population model incorporating fear. Firstly, positivity and steady state analysis are performed, where the theoretical investigations show that change in the level of fear in prey population does not effect the local stability of the system around each equilibria (either stable or unstable). For the deterministic model, the numerical simulations are plotted for the density of prey species as a function of various system parameters. The stability analysis of the coexisting state shows that only transcritical bifurcation, where the steady states intersect, is observed. Secondly, the model is analysed with Gaussian noise term incorporated in the prey's death rate. The model comprising noise term turns the system into stochastic differential equations and irregular noise related oscillations are observed in the densities of both species.