Diffusion Population Dynamics Model

Our main purpose in this paper is to obtain much detailed investigation of dy-namic behavior for a class of nonautonomous predator-prey dispersing systemand nonautonomous competitive dispersal system, i.e., nonautonomous periodicpredator-prey system with prey dispersing and predator density-independentin a multi-patches environment and its corresponding delayed system; nonau-tonomous periodic predator-prey system with predator and prey both dispersalin two patches; multi-species nonautonomous periodic competitive dispersal sys-tem with time delay in a patchy environment. The ultimate boundedness, persis-tence, extinction, existence of periodic solution for above systems were obtained.Some results found in our work is innovative because we solved an open questionthat is the dynamic behavior for a predator-prey system with prey dispersingand predator density-independent in a patchy environment, which enriched thefield of theoretical research of dispersal population dynamics.Firstly, in first section, we presented our motivation, gist and significationfor our paper and made a summarization of prior work on dispersal populationdynamics. Secondly, in section two, some useful lemmas, definitions and funda-mental theorems were given in the preliminaries.In the third section, we investigated a class of predator-prey system withprey dispersing and predator density-independent in a patchy environment. Com-paring with studied models in predator-prey system with prey dispersal, the au-thors always assume that the predator is strictly density-dependent, which istoo realistic to be true in the sound ecosystem. To our best knowledge, up tonow, there is no work on such investigation for the situation of predator density-independent in patchy environment. Moreover, the predator species always beassumed to be confined in a fixed patch in studied models of dispersing predator-prey system, which is also realistic and not fit for all of predators. On the otherhand, the e?ect of periodically varying environment plays an important role in thepermanence and extinction of ecological systems (e.g., seasonal e?ects of climate,food supply, mating habits, hunting or harvesting seasons, etc.). Based on aboveconsideration, we established a class nonautonomous periodic predator-prey sys- tem with prey dispersal and predator density-independent in n patches. In thissystem, prey species can dispersal among n patches, the density-independentpredator is confined in some of patches, maybe one, two or all patches, more-over, we assume the predator can not disperse among these patches. By usinganalytic method, comparison theorem of cooperative system, integral-form cri-teria of ultimate boundedness, persistence, permanence, extinction, existence ofperiodic solution were founded.Furthermore, for the sake of objectivity, the prey species may be losingduring the course of dispersion, some authors introduced the loss rate of preyspecies during the course of dispersal into their research models. On the otherhand, The e?ects of time delay play an important role in the permanence andextinction of population dynamic systems. Therefore, in the fourth section,we introduce the dispersal loss rate, time delay of di?usion and conversion ofnutrients for reproduction of the predator into the above model. Based on anew series of assumptions, we obtained the ultimate boundedness, permanence,existence of periodic solution.In many investigated two species predator-prey models with dispersal ina heterogeneous environment considered so far, it had been assumed that thepredator population have no barriers between patches, i.e., the predator speciescan arrive to any patches to prey on without restriction by the environment, orthe predator species is confined on a fixed patch and can not disperse. Obviously,this assumption is too realistic to be true in many circumstance. In fact, in reallife, the prey species will be usually a?ected by the patchy environment, oncethe food in their patch is exhausted or there are abundant natural enemies intheir patch, they will leave in search for new habitats or refuges, at the sametime, the predator species also will be confined in a isolated patch, when thereis little food to feed on or the last prey is eaten the predators will depart andbecome predator dispersers. To our best knowledge, the work on the two speciespredator-prey system with both the predator and prey dispersing in a patchyenvironment is very few and far between. Therefore in section five, we studieda class nonautonomous predator-prey system with predator and prey both dis-persal between two patches. As an elementary work, integral-form criteria of ultimate boundedness, persistence, permanence, extinction, existence of periodicsolution were founded.There are lots of work on competitive dispersal system, no matter au-tonomous, nonautonomous, with time delay or not, dispersal loss rate or not.However, we found the work involve all of above situation is few, therefore, insection six, we considered a class multi-species, multi-patches, nonautonomousperiodic competitive dispersal system with time delay. In this system, the com-petitive species contribute among many patches, we discuss the permanence andthe existence of periodic solution in two situation: existing dispersal loss rateor without dispersal loss rate. Particularly, in the second situation, we give theintegral-form criteria.