Spatio-temporal patterns and Turing–Hopf bifurcation in a spatially extended prey–predator model with ratio-dependent interactions

Abstract

In this manuscript, we investigate the (2+1)-dimensional ratio-dependent prey–predator system. Prey–predator dynamics are a vital component of the eco-system. It provides the basic food for the living organisms. So, we considered the extended prey–predator model. The underlying model has 2 equilibrium points and stability analysis is carried out about the coexistence equilibrium. The condition for the Hopf bifurcation and Turing instabilities are derived. These conditions help to analyze the formation of patterns in the prey–predator system. The dispersion relation shows the changing behavior of Hopf bifurcation and Turing instability from stable to unstable. The bifurcation and Turing instability simulation divide the parametric space into 4 Regions. In Region I, the solution is stable, in Region II there is purely Turing instability, in Region III, there is only Hopf instability and in Region IV there is Hopf as well as Turing instability. Different types of Turing patterns are produced to capture rate parameters. The numerical solution of the model is obtained by positivity preserving finite difference scheme. The applied scheme is von Neumann stable, and consistent with the model. The bounded behavior of a given scheme is established. Mainly, we are focused on the graphical simulations for pattern formation and steady-state analysis. The 3D and 2D visualization for the Turing pattern and numerical solution are drawn for the various parameter values. The numerical simulations endorsed the analytical results.