Analysis of Bogdanov–Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington–DeAngelis-type response function

Abstract

In this article, we consider a predator–prey system with constant rate of harvesting, which exhibits Hopf and Bogdanov–Takens bifurcations under certain parametric conditions. The parametric space under which the system enters into Hopf bifurcation is investigated. By constructing suitable Lyapunov function, global stability results are obtained. Here, death rate and harvesting rate are taken as the Bogdanov–Takens bifurcation parameters. The canonical form of Bogdanov–Takens bifurcation is derived with the use of repeated nonlinear analytic transformation of coordinates. Later, we include the spatiotemporal effect on the same system and observed some relevant outcomes like Turing pattern, Turing–Bogdanov–Takens bifurcation, Turing–Hopf bifurcation and asynchrony of predator and prey in the space. The present study renders important tools for investigations of the dynamics of biotic organisms (predator and prey) for the management and control of overharvesting. Extensive numerical examples are given in support of the physical existence of the model system under consideration.