Asymptotic Behavior Of Three Ecological Systems

In this paper, we consider the asymptotic behavior of three ecological systems by establishing region of permanence, using Liapunov function, algebraic theory and characteristic equation.In the study of ecology, discrete model is more significant in practice than differential model as for these species which are short in life and non-overlapping in generations or long in life and overlapping in generations but fewer in quantity. In this paper, first, we consider a discrete nonautonmous food-chain system with delays. The sufficent condition and necessary condition are obtained by means of constructing the region of permanence and necessary and sufficient condition for permanence is obtained when the system is autonomous. As a result the mathematic methods in this paper can be used to research food-chain systems with multi-species.One of the most important questions in population ecology is to find the permanence conditions for the species. Disperse is used making the extinct species migrate so that they are saved for protecting species. In recent years, many ecological systems have been investigated by ecological scientists. But we find few studies on the systems with delayed effect in disperse. In this paper, second, we consider a nonautonomous predator-prey system with dispersal delays in two habitats. The persistence of system and the sufficient condition of persistence is obtained by means of using the differential inequality and when the coefficents of system are positive periodic functions with common periodic u> and delays are integro-multiple of w, the w-periodic positive solution of solutions are exist by means of using of Brouwer's fixed point theorem and sufficient condition is obtained for the global attracticity of w-periodic positive solution of system by means of constructing suitable Lyapunov function.Individual's growth of many species have two stage that are juvenile stage and adult stage. In each stage of its development, it always shows different characteristic. For instance, the immature species cannot have reproductive ability and predative ability while the mature species not only have reproductive ability but also have more powerful survival capacity. Therefore studing of stage-structured systems have much practical significance. Mathematic models have been used to analysis and control epidemic models with thefurther investigating epidemic. Epidemic models have been studied by many scientist. But they always propose that the individuals have same infection conversing in different stage. However this is not the thing for some disease transmission. For instance, some epidemic (measles, smallpox) always transmite in juvenile stage while some epidemic (typhoid fever, paratyphoid fever)are transmite in adult stage. Therefore, it has much practical significance towards investgating epidemic models with stage-structured. In this paper, final, we investigate a SI epidemic model with stage-structured. We propose that individual of species has two stage that are immature and mature stage and the mature rate of immature species is proportional to existing immature specie density in proportion to the constant d and the death rate of immature species is proportional to existing immature specie density, the number of mature specie depents on the density of exiting mature specie and we assume that immature species dosn't infect disease but only mature species infect disease and disease has the latent period r. In this paper we investigated the local stability of non-negative equilibria and the sufficient condition of locally asymptotically stable is obtaied by means of using algebra theory and characteristic equations. We show that positive equilibrium will loss of stabillity with the delay increased and a Hopf biffur-cation will occur and the global attractivity of disease-free equilibrum is surveyed and the threshold of disease disppearing is obtained by means of using Liapunov function.