Stability and Bifurcation Analysis for a Nitrogen-Fixing Evolutionary Game with Environmental Feedback and Discrete Delays

In this paper, a nitrogen fixation game system with two time delays under nitrogen limitation is investigated. Firstly, we discuss the existence and local stability of the equilibrium points for the nondelay system. Then the nitrogen fixation delay and strategy-dependent delay are used as bifurcation parameters to analyze the local stability and the Hopf bifurcation. In addition, we obtain the direction of Hopf bifurcation, the change of the period for periodic solution and the stability of periodic solution via center manifold theory and normal form method. Finally, numerical simulation is employed to visualize the theoretical analysis results and we find that the nitrogen fixation strategy is the dominant strategy when the values of the two time delays are large enough. This study promotes the investigation of the effects of two time delays on the dynamics of evolutionary games with environmental feedback, especially on stability and bifurcation.