Quantities Research For Three Kinds Of Ecological Models And Oscillation For Secong-order Ode Withimpulse

This paper investigates the stability and bifurcation of three kinds of models and oscillation for second-order ODE with impulses. Model 1 is a single species system with continuous delays. By changing the model into four order ordinary differential equation and using theorem in [1] and [2]and characteristic equation, we obtain the sufficient conditions for stability of positive equilibrium and bifurcation value. Model 2 is a differential equation with two discrete delays. We obtain sufficient conditions for stability of positive equilibrium by using algebra theory and we regard delay as parameter to get the sufficient conditions when bifurcated periodic solution apears. At the same time, we also analysis its stability. Model 3 is a prey-preydator system with multi-delays. By constructing permanent functions and Liapunov functianal. we obtain some results concerning permanence and stability. In model 4, we investigate the oscillation for a class of second-order nonlinear ODE with impulses. By use of th theory of differential inequility, we get the three sufficient conditions for its all solutions 'oscillation, our results improve and generalize some known ones...