Diffusion Dynamics and Impact of Noise on a Discrete-Time Ratio-Dependent Model: An Analytical and Numerical Approach

Abstract

The paper deals with the dynamical behavior of a discrete-time ratio-dependent predator–prey system. The predator dependence is one of the main concerns of the system. The stability analysis of this 2-dimensional map was carried out analytically. Numerical simulation was carried out to verify the analytical results. We analyzed some specific features that could arise in discrete system. Basin of attraction was found for the endemic equilibrium state. We extended the numerical simulation for the maximal Lyapunov exponent. The presence of positive Lyapunov exponent indicated chaotic behavior of the map. The sensitive dependence on initial condition is one of the criteria for a discrete system. We showed that the system is sensitive on the initial conditions. We also carried out the analysis of diffusion and impact of noise.

A Ratio-Dependent Nonlinear Predator-Prey Model With Certain Dynamical Results

“…Moreover, this scheme of discretization is more effective and reliable as compared to Euler forward difference scheme used in [10]. It is observed that system (1) has two fixed points Ρ 1 = (1,0) and Ρ 2 = ( * , * ).…”

“…It is also shown that the normal form exhibit a chance of pitchfork bifurcation, see [8]. Here, we consider the following continuous-time ratio-dependent predator-prey model which was studied by several authors [6,9,10]:…”