A Class Of Time-delay Of Prey And Predators Of Infectious Disease Model Stability Analysis

On the population dynamics，the predator-prey system is a kind of basic structure,predator-prey system for understanding the real world had very important direct sense.Therefore, this paper studies the branching problem and stability of predator-preysystem.The first chapter, this chapter introduces a mathematical and biological edgesubject that mathematical biology this discipline, it consists of two branches (infectiousdisease dynamics and population dynamics), and describes the two branches of theresearch background and current situation, gives the ecological model of classificationand the common theoretical tools (usually tools and theories include: system dynamics,integral equation, differential equation, functional differential equation, differentialequation, semi group theory and linear algebra).Given the frequently used two maindelay model(discrete model and the continuous time delay).and explains the modelstudies the background and some preparation knowledge.The second chapter, with a class of delayed predator prey of infectious diseasemodels. Models of delay predator that healthy pregnancy. Making use of thecharacteristic equation of correlation method, Hopf bifurcation theory and Rouche’stheorem theory analysis of the non-negative asymptotically stability of equilibriumpoint of the infectious disease model. Linear stability of the system, in variousequilibrium point stability conditions and a bifurcating periodic solution when thecondition, in certain conditions gives the system a column of Hopf branch.The third chapter, this chapter uses Li Zi’s representation theorem, boundedvariation function, the Dirac function, the center manifold theorem variation function,the Dirac function, the center manifold theorem and the normal form method ad well assome operator calculating nature, has decided to branch out the nature of the accurateexpression of periodic solution. First with Taylor expansion will delay of system at theequilibrium point to start, with bilinear integrable properties calculation, through thenormal form method and the center manifold theorem time-delay systems in equilibrium bifurcation dimension and bifurcation, through the normal form method and the centermanifold theorem time-delay systems in equilibrium bifurcation direction andbifurcation stability and bifurcation direction each volume precision expressions.