Dynamics Of Some Reaction-Diffusion Models With Harvesting

In recent years, many scholars have achieved important results about the sustainable development of the ecological system.In this paper, we mainly discussed the dynamics of some reaction-diffusion models with harvesting. Firstly, a kind of reaction diffusion equation model with delay for the interaction of phytoplankton and zooplankton was studied. Our discussion focused on the following contents:the existence and prior estimate of a solution for the model without delay, the existence and stability of three equilibriums and a Hopf bifurcation. We established a sufficient criterion. The main methods of this paper are the theory of upper and lower solutions, the comparison principle and the stability theory of partial differential equations. We used a range of E in the capture level of phytoplankton to divide the model into several conditions. The effect of capture on the system was studied in detail. The results indicate that excessive harvesting can lead to the extinction of two species of phytoplankton and zooplankton. When the harvest effort is controlled in a certain rang, the phytoplankton and zooplankton can coexist. Secondly, the optimal harvesting problem of a prey-predator system with Holling-I functional response was studied by using the variational method. We obtained the optimal harvesting policy. We could find the optimal capture effort and the quantity of the population to reach the optimal benefit.