Studies On Impulsive Semi-Dynamical Systems And Its Applications On Population Dynamical Systems

Continuous dynamical system, discrete dynamical system and impulsive dynamical system are three main kinds of dynamical systems. The solutions of impulsive systems are continuous between two impulses and discontinuous at the impulses, which makes the theory of impulsive systems richer and more complicated than that of the corresponding continuous systems. There are impulsive phenomena in all kinds of applied fields, especially in population dynamics. Consequently, studying population dynamical systems involving impulse effects is of great practical importance. In this dissertation, based on the classical Lotka-Volterra system, we systematically study the effects of impulses on the dynamical behavior of the models brought up here, such as the existence and stability of periodic solutions, the persistence and extinction of the populations etc. The mathematical theory used here include the theories of discrete dynamical system, continuous dynamical system and impulsive dynamical systems, nonlinear functional analysis and numerical analysis etc. Some impulsively exploited population systems are also investigated by control theory, giving the optimal harvesting strategies, which supplies reliable theoretical foundation to the exploitation of biological resources.In Chapter 2, based on a special predator-prey system, we establish and study the predator- prey system involving impulses at fixed moments and state-dependent impulses respectively. For the system with impulses at fixed moments, we obtain the existence and global stability of its trivial periodic solution, and analyze the exitance and local stability of its positive periodic solutions. For the state-dependent impulsive system, by the property of Poincare map and the Analogue of Poincare Criterion, we show the existence and orbital asymptotic stability of the order-1 periodic solution. Moreover, we prove the existence of the order-2 periodic solution, and verify that the existence of order-2 periodic solution implies the existence of the order-1 periodic solution. Finally, we discuss the persistence and extinction of the impulsive system.In Chapter 3, we mainly investigate population patchy system involving impulsive effects and diffusion simultaneously. First, we generalize an existed result by simplifying its conditions. Based on this, we discuss the effect of impulses on the existence of periodic solutions of the original system. Therefore, we establish the corresponding impulsive differential system, transfer the problem of the existence of its periodic solutions into the problem of the existence of solutions of an operator equation, and prove theexistence of periodic solutions by the theory of topological degree. Second, we study the impulsive control of a Lotka-Volterra diffusion system. A set of effective algorithms to stabilize a certain positive point are obtained. Several examples are given to illustrate the effectiveness of the algorithms. This result bears very good practical value. For an extinct population system, if we could stabilize a certain positive point by impulsive controls, the system can be changed into persistent system by impulsive controls, which can be used to prevent some species from going extinct. In addition, we show that, under some conditions, no positive point can be stabilized for certain systems.In Chapter 4, we construct more accurate and pertinent models to describe real world phenomena, and analyze the optimal exploitation strategies for some special renewable resources by mathematical theory. First, noting that most species do not reproduce continuously throughout the whole year but at some fixed time every year, and they are also harvested at some special moments every year, we establish the single-species stage-structured model involving impulsive birth and seasonal harvesting, discuss the existence and stability of its periodic solutions and the complexity of the model, and obtain its annual yield, which is numerically optimized. Second, we investigate the exploitation strategies for the periodic Gompertz system. It is assumed that the system can be exploited both continuously and impulsively. By the theory of optimization and cybernetics, we obtain the maximum annual-sustainable yields, the optimal harvesting efforts and the optimal population levels for such a system subjecting to different ways of harvesting, compare the obtained results and analyze their relation. Finally, we study the exploitation of fishery resources in the ocean. The different densities, population levels and habits of the same kind of fish in the in-shore and off-shore areas cause the diffusion between the fish subpopulations in this two areas. We therefore establish models to study the exploitation of the in-shore and off-shore fish resources with impulsive diffusions. By treating the maximum annual-sustainable yield as the objective function, the harvesting effort is optimized. The Lyapunov exponents graph is given. The complexity of the system is investigated.