Asymptotic Property Research For Two Kinds Of Ecological Models And Oscillation For Solutions Of A Neutral Differential Equations

Asymptotic behavior of mathematical ecological models is an important conception with rich connotation,which mainly includes attractability of the solutions,stability,periodicity and oscillation and so on.In this paper,the asymptotic property for two kinds of ecological systems and oscillation for solutions of a neutral differential equations is studied by employing differential inequalities and using eigenvalue analysis,iterative algorithm,induction and comparison theorem etc.Here, the system's asymptotic property includes the uniform persistence,the global attractability of the solutions,the asmptotic stability of the solutions(local and global) and the oscillation of solutions.In some specific ecological questions,it is necessary to change the species equilibrium by mankind for the practical reasons. As we all know ,it is an effective method to introduce feedback controls into the model. Otherwise, the growth of population is disturbed by environment,for example. temperature,humidity,air pollution .In chapter 2,we study a generalized logistic growth model with feedback regulation.The sufficient conditions of the globally asymptotical stable positive equilibrium are derived by differential inequations. Some known results are improved.In the classical predator-prey systein.it is assumed that each individual predator admits the same ability to attack prey. However in natural world.almost all animals have a life history that takes them through two stages:immature and mature.And different stges have evident difference about their physiological characters(birthrate.deathrat.predation ability etc).For instance,the immature species cannot have reproductive ability and predation ability while the mature species not only have reproductive ability but also have more powerful survival capacity and predation ability.And ecological systems are usually affected by season variation,food's resource and the habit of animal's pregnancy etc.On the other hand,from the point of view of ecology and economics,the exploitation and management of renewable resources has been a very important investigated problem.In chapter 3,we study a predator-prey system with harvesting and stage structure.The positive invariant set of system is discussed.By using eigenvalue analysis and iterative algorithm,the local asymptotic property of non-negative equilibrium and sufficient conditions for global asymptotic stability of the unique positive equilibrium are obtained.An example illustrates these results.The theory of impulsive differential equations is now being recognized to be not only richerthan the correspondinging theory of differential equations without impulses but also provids a more adequate mathematical model for numerous processes and phenomena studied in physics,biology, economy,etc.However.the theory of impulsive functional differential equations is developing comparatively slowly due to numerous theoretical and technical difficulties caused by their peculiarities.In particular, there is little in the way of results for the oscillation of impulsive delay differential equations of neutral type despite the extensive development of the oscillatory and nonoscillatory properties of neutral differential equations without impulses.In chapter 4,we study the impulsive neutral delay differential equations with variable coefficients.Using differential inequalities and induction,some oscillation criteria for solutions of this equation are estabished.An example is presented to illustrate the feasibility of our main results.