Branch Analysis And Numerical Simulation Of Two Types Of Single-species Models

In nature, the single-species is the basic unit of the entire ecosystem, so the study on a variety of inertia of the single-species model laid the foundation for the discussion on the complex ecological model. In this paper, two single population with piecewise constant variables and time-delay are discussed, including the local asymptotic stability of the positive equilibrium, Neimark-Sacker bifurcation and Flip bifurcation of the model etc. Through the research on dynamic behaviors of these ecological models, people can understand the population growth rule in nature indirectly, and that has certain theoretical guiding significance for making use of the natural resources better reasonably and scientifically and protecting ecological environment.On the one hand, the number(density) of the population is usually related to the factors such as fertility, mortality, human harvest, changes on environment and so on, and most of the influences that these factors on the population have certain hysteresis effect, that influences generally performance for the delay in the mathematical models which are built. On the other hand, by the interference of seasonal breeding, migration and so on, the number (density) of the majority population will make discontinuous periodic changes in a certain range, which usually use mathematics model with piecewise constant variables to depict. In chapter2of this paper, the local asymptotic stability of the positive equilibrium, the existence and stability of Neimark-Sacker bifurcation of a single population harvest model with time-delay and piecewise constant variables dx(T)/dt=rx(t)[1-ax([t-k])]-bx([t-l]) are investigated. The sufficient conditions for the local asymptotic stability of the positive equilibrium and the existence of bifurcation of this model are derived by using the theory of characteristic value and the Jury criterion; Furthermore, the direction and stability of Neimark-Sacker bifurcation are discussed by using the bifurcation theory and the center manifold theorem; Finally, some examples and numerical simulations are presented to illustrate the correctness and realizability of our theoretical results and the complex dynamical behaviors of this model. As well known, the growth of any individual organisms can not get away from the supply of food. However, their survival environment is food-limited. Therefore, it is particularly important to research the dynamical behaviors of food-limited mathematical models. In recent years, the food-limited models are favored by ecologists, and get a very wide range of applications. In chapter3of this paper, a food-limited model with time-delay and piecewise constant variables dx(t)/dt=rx(t)[k-xθ({t-m])/1+cxθ([t-m])] is put forward. Based on the theoretical basis of the second chapter, it is shown that there exists Flip bifurcation and Neimark-Sacker bifurcation for this model by using the standard theory and the center manifold theorem, and the conditions for the existence and stability of bifurcation are analyzed. Finally, numerical simulations are presented to illustrate the absolute consistency that our results with the theoretical analysis, and to deduce the complex dynamical behaviors of this model, such as period-doubling bifurcation with period2,4,8,16,and even chaos.