Stability and bifurcation analysis of a Lotka-Volterra time delayed system

Models that describe interactions between species are often used to study the dynamics of populations in an ecosystem. In this paper we focus on one such model, i.e. a time delayed version of the Lotka-Volterra dynamical system. In particular, we study the effects of time delays in, (i) the interspecies interactions and, (ii) the carrying capacity of the prey population, on the system dynamics. In these two cases, we first perform a local stability analysis, where we derive the necessary and sufficient condition for stability. It is shown that, as the time delay is varied, the system loses stability in the first case and exhibits a finite number of stability switches in the second case. We then explicitly show that the loss of stability, in the first case, happens through a Hopf bifurcation. Further, using Poincaré normal forms and center manifold theorem, we analyse the type of the Hopf bifurcation. To complement our analysis, stability charts and bifurcation diagrams are also presented.