Dynamic Behavior Studies For The Models With Impulsive Diffusion And Dispersal Delay

In most models of population dynamics, difusion between two patches is assumed tobe either continuous or discrete. In fact, the real difusion behavior is very complex, but itoften occurs in the form of regular pulse. The pulse difusion is to describe the populationof certain motions to rapid change at a fxed time. For example, artifcial breeding pondfsh, fshing in a certain time interval, number of big fsh can decrease instant, put in thefsh, small fsh will instantly increase quantity.To more naturally refect the change law of things, population difusion model shouldfurther consider the time during the migration process, namely, difusion lag. To our bestknowledge, up to now, there is no work on such investigation for the dispersal delay inthe impulsive diferential systems.Therefore, in this paper we obtain much detailed investigation of dynamic behaviorfor two single species models with bidirectional impulsive difusion and dispersal delayand a predator-prey model with prey bidirectional impulsive difusion and dispersal delayin two patches.The main contents and organization in this paper as follows:The frst section is introduction, in which we present research background, purposeand signifcance of impulsive difusion, and then we generalize the whole difusion dy-namics behavior of population biology research both at home and abroad. Finally, theorganization of this paper is also presented.In section2, due to predation, disease or for other reasons in the process of migrationof population, may cause the number of losses. In view of this facts, we discuss theLotka-Volterra single species model with dissymmetric bidirectional impulsive difusionand dispersal delay. Discrete dynamic system theory is used to get the stroboscopic map ofthe system, then apply the comparison theorem of impulsive diferential equation and some analysis methods analyzing the stroboscopic map and get the permanence, the extinction,the existence, the uniqueness and global asymptotic stability of positive periodic solutions.Finally, numerical simulations and discussion are presented to illustrate the usefulness ofthe proposed result.In section3, we study a logarithmic growth single species model with symmetricbidirectional impulsive difusion and dispersal delay. Using discrete dynamic system the-ory and some analysis methods, we get the sufcient conditions for the permanence, theexistence, the uniqueness and global asymptotic stability of positive periodic solutions ofsystem. Finally, numerical simulations is given.In section4, we discuss two species predator-prey model with impulsive dispersaland delays, in which the prey species can disperse between two patches, but the predatorspecies is confned to the second patch and cannot disperse. On the basis of compari-son theorem of impulsive diferential equation and other analysis methods, a set of easilyverifable sufcient conditions ensuring the global asymptotic stability of the predator-extinction periodic solution and the permanence of species is established. By the numer-ical simulations, our results are set up.