Dynamic Behaviors Of Four Types Of Two Species Interaction Models

This article contains four parts:A two species competitive plankton allelopathy model was proposed by Bandyopadhyay. By constructing a suitable Lyapunov function, a set of sufficient conditions was obtained to ensure global asymptotic stability of the positive equilibrium. However there is a gap in his proof and the result does not make logical sense.The first section of this paper aims at generalizing the model of Bandyopadhyay to the non-autonomous case. Firstly, by using the comparison theorem of differential equation, we obtain a set of sufficient conditions to ensure the permanence of the system. Secondly, through constructing a suitable Lyapunov function, we obtain sufficient conditions to guarantee the global attractivity of positive solution of the system. The conclusion shows that the toxic inhibition rate y for the first species by the second have an effect on the global attractivity of positive solutions, however, if y meets certain condition, then the global attractivity of positive solution is not affected by toxin.In the second section we try to give some more insight to the dynamic behaviors of the model proposed by Bandyopadhyay. Firstly, by using the comparison theorem of differential equation, we obtain a set of condition which ensures the permanence of the system. Secondly, by supplying the gap of the Bandyopadhyay’s proof, we obtain a set of sufficient conditions which guarantees the global attractivity of positive equlibrium of the system. Finally, by constructing some suitable Lyapunov functions, we obtain sufficient conditions which guarantee the global attractivity of boundary equilibrium points of the system.The third section based on the model proposed by Bandyopadhyay, we proposed a more complicated model which is non-autonomous two species allelopathic phytoplankton. Firstly, by using the comparison theorem of differential equation and constructing a suitable Lyapunov function, we obtain some sufficient conditions to guarantee permanence of the system and global attractivity of positive solution in the system of general nonautonomous case. Secondly, similarly by constructing a suitable Lyapunov function, a set of sufficient conditions which ensure the existence of a unique globally asymptotically stable almost periodic solution is obtained.In the fourth section we study a stage-structured Leslie-Gower predator-prey model (stage structure for both predator and prey). Using the iterative method and fluctuation lemma, sufficient conditions which guarantee the global stability of the positive equilibrium and boundary equilibria are obtained. Conclusions in this section suggest that for a predator-prey community stage structure of the predator and the death rate of the mature species are two of the important factors which cause permanence and extinction of the system.