Dynamical Analysis And Region Of Attraction Estimation Of Population Models

Population dynamics is an important subject on a quantitative research of population ecology.Mathematical models play a important role in understanding population dynamics.By the study on dynamical behavior and numerical simulation of mathematical models,it analyzes the development of population,reveals the laws of populations' survival,predicts the trends as well as providing the theoretical and quantity basis for the protection and utilization of population.In recent years,domain of attraction has attracted extensive attention of many scholars abd become a hot topic because it was widely applied in many fields such as prevent the extinction of population,control the spread of disease,maintain the balance of ecological,ect.In this thesis,based on the analysis of a differential equations model of plankton allelopathy and the research of a tumor propagation model,the author researches the dynamical behaviors and domain of attraction for population models by employing the stability theory,bifurcation theory and the theory of manifold.The paper is organized as follows:In the first Chapter,the author elaborates the signfiance,current staus and progress of population systems and the domain of attraction.The main contents and origialities of this article are expounds.In the second Chapter,the author elaborates some preliminaries.The third chapter studies a differential equations model of plankton allelopathy.According to numerical simulations we find the greater the competitiveness of a population,the greater the domain of attraction corresponding to its survival and the extinction of the other is.furthermore,a population that reproduces faster have more chances of survival.In addition,if the rates of toxic inhibition of the first species by the second is greater than the rates of toxic inhibition of the second species by the first,the second species will have the greater domain of attraction.The forth chapter talks about a tumor propagation model,studying stability,bifurcation and domain of attraction for this system.Firstly,selecting delay as the bifurcation parameter,disscusses locally asymptotical stability and bifurcation of equilibria through the characteristic roo method.Then,Construct a suitable Lyapunov functional and the region of attraction has been estimated by using the SOS method.Finally,numerical simulation verify the conclusions and estimate the domain of attraction for the boundary equilibrium point and the positive equilibrium point.The fifth chapter summarizes the research work of this dissertation.Furthermore,the future work is made.