Research On Question Of Dynamics In Biological System Under Impulse Control

Addressed in this dissertation is the dynamic complexity ofbiological system under impulse control. Several relevant problemsattracting more and more attention in this field are studied, and a series ofimportant results are obtained.The backgrounds and research status on impulsive dynamicalsystems are introduced. It also provides the readers with somepreliminaries, including important definitions and lemmas that arefrequently used in the following chapters.Firstly, the effect of seasonal disturbance on non-linear question ofdynamic in biological system under impulsive is studied. On the basis ofthe mathematical theories and biological significance, a biological model,including seasonal disturbance, impulsive perturbation and interspecificcompetition, is established. According to the theory of impulsiveequations, small amplitude perturbation skills, and comparison techniques,the influences of seasonal disturbance on the dynamical behavior ofbiological system under impulsive control are studied. These show to beconsistent with the theoretical analysis and rich complex populationdynamics, such as chaos, species extinction and permanence. Moreover,numerical results verified the theoretical results.Secondly, the dynamical behavior of a biological system withimpulsive state feedback control are studied analytically and numerically.On the basis of the theories and methods of ecology, mathematicaltheoretical works have been pursuing the investigation of the existenceand orbitally asymptotically stable for the semi-trivial periodic andorder-1periodic solution. Furthermore, the bifurcation diagrams andphase portraits are investigated by means of numerical simulations, which illustrate the feasibility of our main results. In addition, the largestLyapunov exponent is computed. This computation demonstrates thechaotic dynamic behavior of the biological system.Finally, using rotated vector fields, the homoclinic bifurcation insemi-dynamical systems is studied. We first discuss the existence oforder-1periodic solution using Successor function and fixed point theory.And then the homoclinic cycle of semi-continual biological system isobtained. Furthermore, we study rotated vector fields of the system. Andthen, based on previous analysis, the homoclinic cycle and the homoclinicbifurcation are described. And numerical simulations illustrate thefeasibility of our main results.