| In the ecosystem,the dynamic relationship between predator and prey has always existed.Because of its universal existence and importance,the analysis of this relationship has become one of the important research topics in the biological mathematics.At present,depending on the predator’s Hassell-Varley function response predator and prey model is one of the research topics.In particular,much attention has been paid to the dynamics of dynamic systems that combine other functional responses with Hassell-Varley functional responses.In order to more accurately reflect the evolution law of predator and prey system in the real ecosystem,this paper established the predator and prey model coupled with white noise,Levy jump and state pulse and Hassell-Varley functional response,and analyzed the dynamic behavior of the system.Firstly,the living environment based on biological population is disturbed by random factors.For example,all kinds of systems are inevitably disturbed by weather change,predation of natural enemies,disease transmission and other factors,all of which will contribute to the growth rate of the population.Therefore,we introduce random perturbation in the form of white noise into the growth rate,and analyze the dynamic properties of a class of stochastic predation systems with Hassell-Varley functional response.By constructing Lyapunov function,we analyze the existence and uniqueness of the positive solution of the system for any initial value,and then get the conclusion that the positive solution of the system is stochastic bounded.Secondly,in ecological biological systems,due to sudden,discontinuous and highly random disturbances caused by tsunamis,earthquakes and droughts,it is difficult to truly reflect the changes of the population by adding white noise interference.Therefore,we established a mathematical model with Levy jump to describe the stochastic dynamic system.For this model,we discuss the existence and uniqueness of the global positive solution for any initial-value system,and discuss the condition of the permanent existence of the system solution.Combined with the stochastic comparison theorem and martingale theory,we derive the sufficient conditions for the persistence and random extinction of the system in the sense of population mean.Finally,based on the significance of Allee effect in the study of biological invasion and natural ecosystem,we established the predation and predation model of Hassell-Varley functional response with Allee effect,and discussed the existence and stability conditions of system solutions.In order to maintain the continuity of the population in the system,we applied pulse control,and then considered the dynamic behavior of the system,such as the Heteroclinic cycle and the order-1 periodic solution,and obtained the best protection strategy for the system. |
