Dynamical Behaviors Analysis For Several Classes Of Ecological Systems

The mathematicians and ecologists have done lots of important studies and obtained very rich results about the population dynamics models. In order to make the model more reasonable, we not only need to consider time factor, but also pay attention to the influ-ence of space. The time delay, impulse, stage structure, random disturbance should also be considered. We usually add two or more of the factors into the investigated systems to establish more accurate models. Therefore, four kinds of ecological models are studied in this dissertation. The main contents of this papers are arranged as follows:In the first part, we introduce the research background and significance on popu-lation dynamics, as well as the research status of population models with diffusion and random disturbance. Four population ecological models are studied in the dissertation:an impulsive predator-prey model with distributed time-delay, an impulsive diffusive dynam-ical model with stage structure, a reaction-diffusion ecological model with homogeneous Neumann boundary condition and two predator-prey systems with random disturbance.A predator-prey model with impulsive diffusion and distributed time-delay is pro-posed and investigated. In this model, prey species was restricted to a single patch, while the predator population can impulsively diffuse between two patches. The prey and preda-tor in the second patch will be periodic harvested at some fixed moments. At the other impulsive fixed times, the predator species can diffuse between two patches. By using the comparison theory and Floquet theory of impulsive differential equations, small ampli-tude perturbation skills, we obtained the sufficient conditions ensuring the global asymp-totical stability of the prey-eradication periodic solution and permanence of the system. Finally, some numerical simulations are carried out to consolidate the analytical findings.A two-prey and one predator model with impulsive diffusion and stage structure is analyzed. Suppose that two preys can establish their own territory and does not inter-act with each other, and predator can diffuse between patches and prey on the preys in own territory. In this model, we suppose the prey species are stage-structured. By using comparison theorem of impulsive differential equation and delayed differential equation as well as some analysis techniques, sufficient conditions for the ultimate boundedness and the existence of preys-extinction periodic solution as well as the permanence of the system are established. Numerical examples with hypothetical set of parameter values are carried out to consolidate the analytic conclusions.A diffusive Leslie-Gower predator-prey model under homogeneous Neumann boundary condition is investigated. Firstly, we prove the existence and uniqueness of global positive solutions of the system and obtain the sufficient conditions for guarantee-ing the permanence of the system. By using the linearization method, Lyapunov function method and comparison method, the local stability and global stability of the positive constant equilibrium point are discussed. The fine upper bounds and lower bounds of the positive steady state of the corresponding elliptical system are established by maxi-mum principle and Harnack inequality. The existence and nonexistence of non-constant positive steady states of the corresponding elliptical system are proved by using energy method and Leray-Schauder degree theory. Finally, numerical examples are presented to check the effectiveness of the stability criteria of the positive constant equilibrium point.Two kinds of stochastic predator-prey model are presented and studied. The exis-tence and uniqueness of the global positive solutions of the stochastic ecological model with Crowley-Martin functional response is proved. Based on Ito formula and Chebyshev inequality, the stochastic ultimate boundedness of the considered system and the asymp-totic property of moment are investigated. By constructing suitable Lyapunov function, the global asymptotic stability criteria for the positive equilibrium point is obtained. Sim-ilarly, we also obtain the global asymptotic stability criteria for the positive equilibrium point of other stochastic predator-prey system with stage structure for prey by constructing appropriate Lyapunov function and using Ito formula. Numerical examples are presented to illustrate the validity of our theoretical results.