Bifurcation And Control Of Some Singular Plankton Models

In order to obtain more interests,the phenomenon of overfishing is very serious in recent years,which brings certain threat to the steady-state of marine system.Therefore,it becomes more meaningful to study the impact of economic benefits and commercial harvesting on the marine system and how to take corresponding measures to deal with this threat.In this paper,three singular plankton models are proposed.By investigating the stability,bifurcation and control of the three models,some instructive conclusions are obtained.In Chapter 2,a singular plankton model with the Beddington-DeAngelis functional response function,nonlinear commercial harvest and double time delay is presented.In the case of this model without time delay,the conditions of the existence of singularity induced bifurcation and control are analyzed.Inverse,in the case of such system with two delays,the existence of Hopf bifurcation are discussed in four different situations.Of course,simulations are given to demonstrate obtained theoretical results.In Chapter 3,in order to observe the effect of taxation on the dynamical behavior of system,taxation and nonlinear fish harvesting are introduced into a delayed differential algebraic phytoplankton-zooplankton-fish model.First,the conditions of the existence of singularity induced bifurcation and Hopf bifurcation are given,and appropriate state feedback controllers are designed to eliminate the corresponding bifurcations.Next,properties of Hopf bifurcation are investigated based on normal form theory and center manifold theorem.Furthermore,Pontryagin's maximum principle is used to obtain optimal tax policy as well as conservation of the ecosystem.Finally,some numerical simulations are given to verify theoretical analysis.In Chapter 4,a singular phytoplankton-zooplankton model with fuzzy parameters,refuge,fishery protection and harvesting is studied by regarding the imprecise biological parameters as a form of triangular fuzzy number in nature.By using the utility function method,such model is reduced to a differential system.The conditions of stability and bifurcations of the system are studied respectively.In addition,appropriate state feedbacks are designed to make this system be stable.Furthermore,Pontryagin's maximum principle is used to obtain optimal harvesting policy and conservation of the ecosystem.Finally,some numerical simulations are given to demonstrate our theoretical results.Results show that imprecise parameters not only affect the interior equilibrium of the system,but also affect the critical value of bifurcation and branch range.