Qualitative Permanence For 4-d LOTKA-VOLTERRA System And Anlysis Of A Single Population Dynamic Model

Lotka-Volterra system was proposed by the American Lotka and Italian mathematician Volterra independently. Hence, this system is called Lotka-Volterra system. Nowadays, Lotka-Volterra system has been applied to economics, physics, chemistry, population etc.One of the important challenges in ecology is to develop a theory which can predict the stability of an ecosystem when the relationship between species, i.e., community structure, is given. In order to tackle this problem, Hofbauer et al. focus on the investigation of the communicate matrix of Lotka-Volterra system, and raise a concept qualitative permanence which resolves the problem above to some extent. They obtain the necessary conditions, which are called (Cl) and (C2) conditions, for any dimensional Lotka-Volterra system. They also prove the sufficiency of (C1) and (C2) in low dimensional cases (i.e.n≤3). Furthermore, they conjecture that (C1) and (C2) conditions are still sufficient in higher dimensional cases (i.e. n≥4) for Lotka-Volterra system.In this paper, the sufficiency of (C1) and (C2) conditions will be proved if n=4, therefore, their conjecture is verified whenn=4.Impulsive differential equation is an important tool to describe phenomena in reality, especially in ecology. A stage-structured single population dynamic model with birth pulse and impulsive harvesting at different moments is constructed. By the stroboscopic map in discrete dynamical system, the importance of the impulsive releasing for the persistence of the system will be obtained.In the second Chapter, all matrices of order 4 belonging to 442 kinds of sign patterns up to permutation will be given. For convenience of proving, all 442sign patterns will be categorized into 42 sign patterns, therefore, a whole category of the sign patterns for matrices of order 4 is given.In the third Chapter, the 42 sign patterns will be put into two kinds of situations, and the sufficiency of (C1) and (C2) conditions will be proved in two different ways.In the fourth Chapter, a stage-structured single population dynamic model with birth pulse and impulsive harvesting at different moments is discussed.In the fifth Chapter, some concluding remarks for the future work are given.