Dynamical analysis, linear feedback control and synchronization of a generalized Lotka-Volterra system

In this paper, a generalized Lotka-Volterra (GLV) system, which is modeled by a set of equations that represents the dynamical behaviors of two-predator and one prey species, is analysed. The boundedness, existence and uniqueness of the solutions of the generalized Lotka-Volterra system are studied. Continuous dependence on initial conditions in the model is also studied to determine the parameter values’ range of the model and time T where the system does not exhibit chaotic behaviors. In addition, extinction scenarios of solutions of the model are discussed. The necessary and sufficient conditions for the asymptotic stability of the equilibrium points of the GLV system are derived. It is also shown that the GLV system undergoes a saddle-node bifurcation and generalized Hopf (Bautin) bifurcation around one of its equilibrium points. The coexisting periodic solutions and chaotic attractors in the system are numerically investigated by phase diagrams and Lyapunov exponent spectrum. Linear feedback control technique is utilized to control the GLV model to its equilibrium points. Moreover, the Lyapunov direct method is used to synchronize two identical chaotic GLV models via linear feedback control technique.