Dynamical Analysis And Control In Two Kinds Of Multiple Delayed Bio-economic Systems

In recent decades,many research efforts have been put into investigation in population dynamics of ecosystem.Bio-dynamics is the basis for studying the relationship between population and environment as well as inter-specific competition and coexistence among different species.Time delay factors play an important role in the bio-dynamics system,which is particularly important in ecology.The time delay is considered to be an important factor that causes variations in the density stability of prey-predator system.In recent years,researchers have extensively investigated the Allee effect on dynamical behavior of prey-predator system and reveal that Allee effect has significant effects on the growth of prey population and the qualitative stability of the population dynamics.On the other hand,with the introduction of commercial harvesting factor,the general differential equation has not been able to characterize the relationship between prey-predator system and commercial harvesting.Hence,it is necessary to investigate dynamic effects of Allee effect and commercial harvesting on population dynamics.This paper is divided into four chapters.Firstly,the background of the predator-predator system and its related concepts are introduced,and its applications in practical engineering as well as some related research are also introduced results in recent years.Next,the differential-algebraic prey-predator system with commercial harvesting on predator is proposed,where maturation delay for prey and gestation delay for predator are considered.Since commercial harvesting is dynamically influenced by variation of economic interest,we will investigate combined dynamic effects of double time delays and economic interest on population dynamics.Positivity of solutions and uniform persistence of system are studied.In the absence of time delay,by taking economic interest as bifurcation parameter,existence of singularity induced bifurcation is investigated based on differential-algebraic system theory.State feedback controllers are designed to eliminate singularity induced bifurcation and stabilize the proposed system around corresponding interior equilibrium.In the presence of double time delays,by analyzing associated characteristic transcendental equation,it is found that interior equilibrium loses local stability when double time delays cross corresponding critical values.According to Hopf bifurcation theorem for functional differential equation,existence of Hopf bifurcation is investigated as local stability switches.Based on normal form theory and center manifold theorem,directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied.Numerical simulations are carried out to show consistency with theoretical analysis.Then,the multiple delayed differential-algebraic prey-predator system is established,where commercial harvesting on predator and strong Allee effect in prey growth are considered.Three time delays are introduced to represent maturation delay for prey,reaction delay to changes of prey surviving environment due to strong Allee effect and gestation delay for predator,respectively.Positivity of solutions and uniform persistence of system are discussed.In absence of time delay,existence of singularity induced bifurcation and local stability analysis are investigated due to variation of economic interest of commercial harvesting.In presence of multiple time delays,existence of Hopf bifurcation and local stability analysis are discussed by analyzing associated characteristic equation.By using new normal form of multiple delayed differential-algebraic system and center manifold theorem,properties of Hopf bifurcation are studied.Numerical simulations are carried out to show consistency with theoretical analysis.Finally,the whole paper is summarized and some the unresolved problems in the study are proposed.