Stability And Hopf Bifurcation Of A Delayed Ratio-dependent Predator-prey System

Time delay is ubiquitous in natural science and humankind science.The phenomenon of "time delay" or "delay" means that the tendency of the current evolution of the system depends greatly on it's past history.The effect of delay on dynamics of the dynamical systems is usually essential and critical.In recent years because of the growing evidences, the ratio-dependent theory is an very important theory put forward by biologists.Although it is focused by many scholars,the effect of the long production time delay are never considered to the best of our knowledge.The continuous Mackey-Glass model is the important model which describes the process of the production of neutrophils.But the effect of the time delay on it's dynamics is never taken into account.The delay is a long time between the production of immature cells in the bone marrow and their maturation for release in circulation bloodstreams.Therefore,it's very necessary to consider the effects of time delays on the dynamics of the above two dynamical systems.Supported by the National Natural Science Foundation of China under Grant NO.10702065,the problems of the stability of the positive equilibria and their local Hopf bifurcation in the above two delayed dynamical systems in the fields of ecology and physiology are studied in this dissertation.The results of the numerical simulation employing the dynamical software WinPP are in good agreement with those of the theoretical analysis.It verifies the correctness of theoretical analysis methods in this dissertation.The results of this dissertation may have very important revelations for the biological control and clinical treatment.In Chapter 1,some fundamental theories which are employed in this dissertation are presented,i.e.,delay,the theories of stability and Hopf bifurcation.In Chapter 2,the stability and the local Hopf bifurcation of the positive equihbrium in a ratio-dependent predator-prey model with the production time delays of predators and preys are considered.By the Nyquist criteria,the stability of the positive equilibrium is considered.The existence of the local Hopf bifurcation is analyzed by the theory of Hopf bifurcation.By the center manifold and the normal form theories,explicit formulae which determine the stability,direction and other properties of bifurcating periodic solutions are calculated.Employing the dynamical software WinPP,the correctness of the above theoretical results are verified.The results show if both the production delays are small enough,their sizes will keep stable in the long run,but if one of them is big enough,their sizes will periodically fluctuate in the long term.In Chapter 3,in the similar way,the stability and the local Hopf bifurcation of the positive equilibrium in the continuous Mackey-Glass model which describes the dynamical process of the production of neutrophils in the blood are considered and the correctness of the theoretical results is verified.The results show that the supercritical Hopf bifurcation exists in the production of neutrophils,i.e.,if the delay is small enough,the concentration of neutrophils will stay stable in the long run,but if the delay is big enough,the concentration of neutrophils will periodically fluctuate.It is very important to predict the dynamical law of the concentration of neutrophils in the blood,and it has very important revelations to monitor,control and cure leukemia.There are two innovation and features in this dissertation.(1)In Chapter 2,for the predator-prey system based on the ratio-dependent theory, we take the fact into account in the biological meaning that the production delays are usually bigger,and thus their effect on the system's dynamics perhaps can be important. Therefore,the production delays are introduced for the first time to the best of our knowledge.Thus the ratio-dependent predator-prey model turns into the ratio-dependent predator-prey model with production delays T₁ and T₂.It makes the model more accurate to describe the dynamical evolution of their predator-prey relationship and the rising and falling of their sizes.(2)In Chapter 3,the continuous Mackey-Glass model of neutrophils is analyzed.The result shows if the delay is small enough,for example,(?) = 6＜(?)_c = 12.092,the concentration of neutrophils in the blood will stay stable in the long run but will not become chaotic.It seems to correct a mistake in the past research,and needs further experiment verification.