The Study Of Biological Systems Of Population Dynamics With Impulsive Perturbations

The problem about the dynamics complexity of population dynamicsis one of the hot research fields of mathematical ecology which hasattracted many momestic and foreign scholars. The development of thenonlinear dynamics has accelerated the pace of the population dynamics.By reading lots of literature, the problem about the dynamicscomplexity of population dynamics with impulsive perturbation has beenstudied deeply on the basis of the previous research. First, the researchbackgrounds and application prospects of the subject has been presentedbriefly. Some basic definition and important lemma are provided in thepaper which displays the basics of the later research. In section2, amathematical model for the dynamics of a consumption model withimpulsive control strategy is studied. The critical conditions for thepermanence and the globally asymptotically stable of the algae extinctionperiodic solution are given by using the theory of impulsive differentialequation. The long-term dynamics behavior of the biological system hasbeen simulated, the results show that the system exihibits rich dynamicalbehaviors which verify the validity of the mathematical theory. Theresearch results on the problems of controlling the subtropical reservoireutrophication may be helpful. In section3, the dynamics of the twospecies population with biological control with chemical contro isinvestigated. During the research process, the author makes qualitativeanalysis by using mathematical theory and skills. Then the correctness ofthe results derived from mathematical theory is verified throughnumerical simulations. Furthermore, the existence of chaos is verified bycaculating the Largest Lyapunov Exponent, the biological explanationsand the practical application is given. In section4, the problems of thesemi-continuous dynamical system with rotation vector field areinvestigated. The existence of the homclinic hoop is proved by usinggeometric theory. The existence and uniqueness of the periodic solutionsare also proved for when the homoclinic loop disappears. Finally,numerical simulations of the given example verify the feasibility and correctness of the theory.