The Asymptotic Behavior Of Two Ecological Systems And The Unconditionally Stability Of A Class A-Order Delay Differential Equations

Ecological mathematics is a bridge which is to be laid two things. Ecological math-ematics studies all kinds of natural phenomenon through mathematical modeis. The asymptoticbehavior of two ecological systems and the unconditionally stability and oscillations in kinds of 4-order delay difference equations are investigated by using analysis method, constructing Lyapunovfunctions, the logarithmic norm, charateristic roots theory and the bifurcation theory. Here, theasymptotic properity includes the uniform persistence, the global attractability of the solutions,the asymptotic stability of the positive solution local or global and the oscillations of solutions.Blood is lifelines of the human body mainly. Then, the hemopoiesis of stem cells is a focuswhich the medical field pay close attention to. Ecological mathmatics can help the medical fieldto analysis all factors which effect on the hemopoiesis of stem cellsand the influence degree. Inthe chapter 2, a hematopoiesis model with time delay is considered. Firstly, The sufficient con-ditions of the existence and uniquity of the positive equlibria by applying functional derivative isobtained; Secondly, the global attractability of the positive equlibria is investigated by charateristicroots theory and oscillation theory; Thirdly, regarding the delay as a parameter, conditions of theexistence of Hopf bifurcation and the peridic solution, furthermore, the form of the approximateperidic solution are obtained; Finally, Some specific examples are given and the solution diagrameappears by Matlab. these examples illustrate these results. Every parameter has different effectson the hemopoiesis of stem cells. The hemopoiesis of stem cells can permanence by controling allparameters.population's persistence is one of important problems which people pay close attention to.The predator-prey, competition and reciprocity coexistence between population and populationwill gradually become violent. We konw that the study of the co-exist, stability and persistenceof the spices has very important practical meaning to keep ecological equilibration and protectecological environment, even to save valuable and rare biologics which are on the verge of becomingextinct. To keep persistence through introducing dispersal, however, In real ecological environment,feedback controls can keep persistence also. Of course, we can lead into both dispersal and feedbackcontrols to keep population's persistence. In the chapter 3, the persistence and global stability ofthe n-species Lotka-Volterra competition system with feedback controls, dispersal and aα_i-kindof functional response is investigated. We prove uniform persistence and global attractability bythe comparable theory and constructing Lyapunov function method. Finally. the correctness of the conditions of the theorem is shown by example and using Matlab software.Difference equations have vert important practical significance. For making clear the variativeregulars of the practical system with time, we require to discuss the form of the solutions ofdifference equations. There are three motheds generally: to seek equational solutions to analysis;to seek the numerical solutions of equtions; solutions' qualitative analysis. In the chapter 4,The unconditionally stability and oscillations in kinds of 4-order delay difference equations areinvestigated by using analysis method. Oscillations of all solutions of examples are obtained byusing the logarithmic norm. Some specific examples are given and the solution diagrames aredrown by Matlab.