Hopf Bifurcation of a Delay Commensalism System with Density Dependent Birth Rates

In ecology, commensalism and amensalism models are two kinds of important and interesting models. They have attracted much attention of ecologists and mathematicians in recent years. In this paper, we consider a two-species commensalism system with a discrete delay and density dependent birth rates. First, we investigate the characteristic equation of the proposed system and study the distribution of its roots. We obtain that, when the delay is sufficiently small, the positive equilibrium is locally asymptotically stable, when increases to a critical value, the positive equilibrium loses its stability and a Hopf bifurcation occurs, as continues to increase, a family of periodic solutions bifurcate from the positive equilibrium. Then, by using the normal form theory and the center manifold theorem, we derive the precise formulae to determine the Hopf bifurcation direction and the stability of the bifurcating periodic solutions. Numerical simulation results are included to support our theoretical analysis. We plot the trajectory graphs on , plane respectively. We also plot phase graphs to illustrate the change of stability of the positive equilibrium and arise of periodic solution. In order to fully validate the occurrence of Hopf bifurcation, we use the numerical continuation package DDE-Biftool to generate bifurcation diagrams and accurately track stability changes of the positive equilibrium and periodic solution with respect to the delay parameter . The commensalism model we propose considers both density dependent birth rate and time delay, it is of great practical and theoretical significance. The theoretical and numerical analysis that we do on the proposed system can make a supplement to the literature on the dynamics of delay commensalism systems. 2020 MSC: 34K18, 34K20, 92D25