Qualitative Nature Of Two Predator-prey System With Time Lags

With the development of technology and the progress of human civilization, environmental problems become more and more intimate with human beings. Researchers have built various types models which are important academic significance and application value. In all known models, predator-prey model is most widely used, and the dynamical nature (for example persistence, stability, the periodic solution) has also become a lesson and has been concerned by mathematicians and biologists. In this paper, we research on qualitative nature of two predator-prey system with time lags. It is divided into second parts,In the first part, we introduced researches of predator-prey system and put forward issues to be discussed in this article. In the second part, we discussed a class of infinite-time lagged ratio-dependent predator-prey system, and by applying the comparison theorem of differential equation, we has educed the sufficient conditions to permanence; we discussed a class of delayed discrete ratio-dependent predator-prey system and educed the sufficient conditions to permanence; by using the theorem of coincidence degree theory, we discussed a class of delayed discrete ratio-dependent predator-prey system and educed the sufficient conditions to existence of positive periodic solution.