Ecological Model With Time Delay And Piecewise Constant Variables Of Qualitative Analysis

Stability and Hopf bifurcation of a predator-prey model with delay, stability and bifurcation analysis of a single-species with interference and piecewise constant variables are discussed. Mainly including the equilibrium is locally asymptotically stable, global asymptotic stabilityFlip branch and Hopf bifurcation. Population growth law of the ecological model to learn how to protect biodiversity, maintain the sustainable development of the ecosystem has a very important biological significance.In the chapter2of this paper, stability and bifurcation of a single-species with interference and piecewise constant variables is discussed. Necessary and Sufficient conditions of the local asymptotically stability andsufficient conditions of the global stability are obtained, based on the theory of characteristic value and Lyapunov functional; Using center manifold argument and bifurcation theory, the existence of Flip bifurcation and Hopf bifurcation are investigated; Some numerical simulations are performed by Matlab to verify the theoretical results.Predator-Prey is a common relationship of populations in nature with each other to select the system of the population, the prey and predator, influence each other. Promote material cycling and energy flow between populations through predator-prey interactions. In the chapter3of this paper, stability and Hopf bifurcation of a predator-prey model with discrete and distribute delays is discussed. First, based on Lyapunov functional and the theory of characteristic value, sufficient conditions of the asymptotically stability are obtained; Second, using the normal form theory and center manifold argument, direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated; Last, some numerical simulations are performed by Matlab to verify the theoretical results.