Stability and Bifurcation for a Single-Species Model with Delay Weak Kernel and Constant Rate Harvesting

In this paper, we consider the effect of constant rate harvesting on the dynamics of a single-species model with a delay weak kernel. By a simple transformation, the single-species model is transformed into a two-dimensional system. The existence and the stability of possible equilibria under different conditions are carried out by analysing the two-dimensional system. We show that there exists a critical harvesting value such that the population goes extinct in finite time if the constant rate harvesting u is greater than the critical value, and there exists a degenerate critical point or a saddle-node bifurcation when the constant rate harvesting u equals the critical value. When the constant rate harvesting u is less than the critical value, sufficient conditions about the existence of the Hopf bifurcation are derived by topological normal form for the Hopf bifurcation and calculating the first Lyapunov coefficient. The key results obtained in the present paper are illustrated using numerical simulations. These results indicate that it is important to select the appropriate constant rate harvesting u .