Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting

In this paper, we investigate a class of population model with seasonal constant-yield harvesting and discussthe effect of the seasonal harvesting on the survival of the population. It is shown that the population can be survival if and only if the model has at least a periodic solution and the initial amount of population is not lower than the minimum periodic solution. And if the population goes to extinction then it must be in the finite time. As an application of the conclusion, we systemically study the global dynamics of a logsticequation with seasonal constant-yield harvesting, and prove that there exists a threshold value $h_{MSY}$ of the intensity of harvesting, called the maximum sustainable yield,which is strictly greater than the maximum sustainable yield of logstic equation with constant-yield harvesting,such that the model has exactly two periodic solutions: one is attracting and the other is repelling if $0 h_{MSY}$. Hence, the logstic equation with seasonal constant-yield harvesting undergoes saddle-node bifurcation of the periodic solution as $h$ passes through $h_{MSY}$. Biologically, these theoretic results reveal that the seasonal constant-yield harvesting can increase the maximum sustainable yield such that the ecological system persists comparing to the constant-yield harvesting.