Study On The Harvest Model Of Fishery Resources In A Eutrophic Water Body

For a eutrophic water body, the dynamical properties of a kind of fishery resource model with Holling type II functional response and different harvesting strategies are investigated in this thesis which is divided into 4 chapters.In Chapter 1, as an introduction, the biological background of the topics including the current situation of domestic water resources environment, the causes and treatment measures of water eutrophication, as well as some related preliminary knowledge, are introduced.In Chapter 2, for the eutrophic water body, the dynamic behaviors of the fishery resource model with continuous harvest are studied. The boundary equilibrium state and stability of the system are discussed by using the Hurwitz criterion and the Jacobian matrix. By means of Shengjin formulas and the method of geometric analysis, the existence and number of the positive equilibrium state are discussed, the results shows that the number of positive equilibrium number may be 0,1,2 and 3 under different parameters. The numerical simulation graph with three positive equilibria is given. Furthermore, the saddle-node bifurcation is discussed based on the bifurcation theory. Here, the existence of positive equilibrium shows that the fish and the algae can coexist under certain conditions.In Chapter 3, the eco-economical model with continuously harvesting the algae is mainly studied. Six nonnegative equilibria, eco-economic balance state and the corresponding harvest effort are discussed, respectively. Finally, by using the Pontryagin maximum principle, the Hamiltonian function is construct to investigate and obtain the necessary condition for the existence of the optimal harvesting policy. The mathematical results show that when the discount rate tends to zero, the social income will reach the maximum.In Chapter 4, the dynamic behavior of the fishery model with fixed time impulsive harvest and the state impulsive feedback control are analyzed, respectively. For the model with fixed time impulsive harvest, the boundary of the system is given by using comparison theorem and scaling analysis. The local asymptotic stability of the boundary periodic solution is discussed by using the Floquet multiplier theory of impulsive differential equations, and the sufficient conditions for the extinction of the system are obtained. For the model with impulsive state feedback control, the existence of order-1 periodic solutions is discussed by constructing the Poincare mapping and successor function, and the sufficient conditions for the stability of positive periodic solutions are obtained. When the density of the algae reaches a threshold, harvesting suitable amount of the algae can prevent the fish from extinction, some numerical simulations are also given. Similarly, for harvesting the fish, the threshold amount of harvesting the fish is also obtained. In this case, the algae can avoid to be extinct.