Complex Dynamics And Control For Several Ecological Models

Ecosystem offers material products and natural environment which the mankind depends on for living.In recent years,ecological complexity,mainly aiming to explore the dynamic behavior of ecosystem,has been a focus for ecology research.Ecosystem is a typical complex system,whose internal mechanism would complicate it.In ecology,bifurcation and chaos often bring disaster to population under study.Moreover,mutation events caused by the complexity and dynamic features of ecosystem would lead to the blindness of ecological management and even result in the invalidation and failure.On the other hand,the construction of dynamic models based on ecological process has been the main objective on which ecologists make their efforts for long time.In China,the study of bioeconomic models in fishery resources is still at the stage of theoretical exploration at present.In particular,few researches are done on its application.Compared with developed countries,our country's research on this field still lags behind relatively.The discussion on the application in development,utilization and management of bioeconomic models would provide reference for the sustainable exploitation and scientific management of fishery resources as well as development strategy evaluation.The interaction between the predator and prey is the important component of ecosystem.Moreover,biological resources in predator-prey systems will be most likely to be harvested to obtain economic benefits.So,our study focuses on the predator-prey dynamical systems.Two classes of predator-prey models are formulated and studied,one is normal system and the other one is singular system.Some preliminary theoretic principles on dynamic behaviors of these systems are derived.The analysis in the dissertation results from dynamical systems theory and mathematical analysis techniques.The methods of bifurcation and chaos control are discussed in detail by means of control theory.On the other hand,two dynamic bioeconomic models are established.The stability of equilibrium points and bioeconomic equilibrium are analyzed.Different harvesting and management policies are simulated by optimal control theory.In addition,the effect of the parameter uncertainties on results is also discussed.The main works are carried out as follows:A discrete-time predator-prey model with modified Leslie-Gower and Holling's type II schemes is formulated.The existence and stability of fixed points for the model are discussed.The existence conditions of Neimark-Sacker bifurcation and Marotto's chaos are obtained on the basis of normal form method and bifurcation and chaos theory.Furthermore,to delay or eliminate the bifurcation and chaos phenomena that exist objectively in this system,we try to design two control strategies to delay the appearance of bifurcation and stabilize chaotic orbit,respectively.The two methods also can be applied to the research on discrete-time Kolmogorov model,which is of the most general form.A discrete-time singular predator-prey model is formulated.In order to explore its dynamic behaviors,the tangent space local parameterization method for continuoustime singular systems is generalized,and the corresponding result for discrete-time singular systems is also derived.The new local parameterization method is sufficiently general.By the new parameterization method for discrete singular systems,we obtain a parameterized system,which is normal and topologically equivalent to the original singular system.According to equivalence relationship,the complex dynamic behaviors of the original singular system are explored in detail by using stability theory,center manifold theorem and bifurcation and chaos theory,such as Neimark-Sacker bifurcation,flip bifurcation,chaotic attractors.At the end of the chapter,we discuss the effect of the different integration step size on results when the continuous model is discretized.A predator-prey taxation model with Holling's type III schemes incorporating a reasonable catch-rate function is constructed.Firstly,the existence of equilibrium of the system is discussed.Then the sufficient condition for the local asymptotic stability of the positive equilibrium point is reached by Routh-Hurwitz criterion.By constructing a suitable Lyapunov function,sufficient condition for the global asymptotic stability of the positive equilibrium point is obtained.Finally,the optimal taxation policy and optimal solution of the model are derived by means of Pontryagin maximum principle.Because of the complexity of ecological systems,certain parameters in mathematics models cannot be accurately quantified.To ensure the reliability of the modeling results,we consider a bioeconomic model in fishery resources under impreciseness and introduce a parametric functional form of an interval which differs from those of models with precise biological parameters.The existence of the positive equilibrium point is discussed.Based on an eigenvalue analysis,the sufficient condition for the local asymptotic stability of the system is given,and by Bendixson-Dulac theorem,the global stability condition is derived.Also the bioeconomic equilibrium of the model is analyzed.Next dynamical optimization of the harvesting policy is carried out by invoking Pontryagin's maximum principle,and the optimal harvesting policy is derived.Some logical mistakes of the main results in recent researches have been modified.Finally,the effects of the parameter uncertainties on the dynamic behavior of ecosystem and the optimal harvesting policy are discussed.