Qualitative Study On Several Kinds Of Predator-Prey Systems Under Allee Effect

Allee effect is a unique biological phenomenon existing in the biological world.Allee effect will lead to the decrease of the average individual growth rate of the population,so it will affect the survival or reproduction of the population to a certain extent.In this paper,several kinds of predator-prey systems with Allee effect are established to consider the effects of different forms of Allee effect on the ecosystem in different biological backgrounds.By studying the dynamic properties of the system,including the existence and stability of constant steady-state solutions,local and global stability,the existence and properties of Hopf bifurcation and Bogdanov-Takens bifurcation,it is found that the existence of Allet effect can increase the complexity of the dynamic behavior of the system and change the conditions for the stable survival of the population.The initial size of the population will affect the sustainable survival of the population,and the static stability of the final size of the population may change into dynamic stability,or even become unstable and become extinct.Specific research contents are as follows:Firstly,in the context of pest management,a predator-prey system with time delay and mate selection Allee effect was established based on the Allee effect of pest population induced by sterile pests.The qualitative study of this system shows that the ultimate stable survival of pests is always local,which means that if the number of natural enemies is maintained and the initial scale of pests is reduced to a certain range,the pests will eventually become extinct.Compared with traditional methods(spraying and releasing natural enemies),pest control methods combined with the Allee effect of mate selection can not only reduce the need for pesticides but also reduce the environmental damage caused by spraying.Secondly,a stage structure predator-prey system with mate selection Allee effect,constant casting and general functional response function of predator is established.The functional response function in this system is universal and covers several classical types.This system can be regarded as the extension of some literatures on stage-structured predator-prey systems.The results show that when the feeding amount is small,Allee effect has a great influence on the bait,and with the increase of Allee effect,the number of bait decreases gradually,and finally the bait exists stably at a certain value.With the increase of Allee effect,the number of adult predators gradually decreases,and finally the prey and predator coexist.Strong Allee effect will make the existing periodic solution disappear.Thirdly,a syrphoma-aphid system was established in which the prey(aphids)had a mate selection Allee effect and the predator(syrphomys)had a time delay and a phase structure.Through qualitative analysis,the results showed that the addition of Allee effect again changed the stability conditions of the system and had a certain influence on the final number of aphid population.With the increase of Allee effect,the final quantity of prey changes from stable to periodic oscillation,and finally the prey becomes extinct.Finally,a predator-prey system with both protected and unprotected areas was established,and a strong Allee effect was found in both protected areas and protected areas.Different from the previous mate selection Allee effect,the Allee effect here is not a constituent Allee effect,but a population Allee effect.According to the biological significance of prey and predator and the time scale of speed and speed of some parameters,the system can be divided into fast system and slow system.Through qualitative analysis,the results show that the strong Allee effect in the protected area can also change the conditions for the coexistence of the two species,and the initial size of the population becomes an important condition for the sustainable survival of the population.