摘要
In this paper,a class of three-species multi-delay Lotka-Volterra ratio-dependent predator-prey model with feedback controls and shelter for the prey is considered.A set of easily verifiable sufficient conditions which guarantees the permanence of the system and the global attractivity of positive solution for the predator-prey system are established by developing some new analysis methods and using the theory of differentim inequalities as well as constructing a suitable Lyapunov function.Furthermore,some conditions for the existence,uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques.In addition,some numerical solutions of the equations describing the system are given to show that the obtained criteria are new,general,and easily verifiable.Finally,we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources,and obtain some new interesting dynamical behaviors of the system.At the same time,the influence of the delays and shelters on the dynamics behavior of the system is also considered by solving numerically the predator-prey models.
基金
the Sichuan Science and Technology Program(Grant No.2018JY0480)of China
the Natural Science Foundation Project of CQ CSTC (Grant No. cstc2015jcyjBX0135) of China
the Science Fund for Distinguished Young Scholars(cstc2014jc yjjq40004) of China
the Scientific Research Plan Projects for Higher Schools in Hebei Province(Grant No.Z2017047) of China
the Postdoctoral Science Foundation(Grant No.2016m602663)of China
the National Nature Science Fund (Project No.61503053) of China.
