Existence and the dynamical behaviors of the positive solutions for a ratio-dependent predator-prey system with the crowing term and the weak growth

This paper deals with a ratio-dependent predator-prey system with the crowing term and the weak growth in the prey equation. Under the condition that the coefficient λ is less than a critical value λ1D(Ω0), we obtain existence of multiple positive steady state solutions of the predator-prey system and the dynamical behaviors of its positive solutions. Our results show that the predator and the prey possess not only the common coexistence, but also the very weak coexistence which both of the predator and the prey are very low. Meantime, the persistence of the positive solutions for the corresponding parabolic type system sometime depends strictly on the ratio of its initial data. Therefore, our results may be used to explain some special phenomena which under some bad environment, the predator and the prey may still coexist.