Bifurcation Analysis for Two-Species Commensalism (Amensalism) Systems with Distributed Delays

This paper is devoted to studying the dynamics of two-species commensalism (amensalism) systems with delays. We first study the system with a distributed delay but without the discrete delay, investigate the local stabilities of equilibria and prove the existence of transcritical bifurcation. Then, we study the system with a discrete delay and a distributed delay. By analyzing the characteristic equation of the positive equilibrium and regarding the discrete delay as the bifurcation parameter, we show the existence of periodic solutions bifurcating from the positive equilibrium. Also, we derive the precise formulae to determine the Hopf bifurcation direction and the stability of the bifurcating periodic solutions by using the normal form theory and the center manifold theorem. Numerical simulation results are also included to support our theoretical analysis.