Analysis And Optimal Control Of Predator-prey Models With Allee Effect,Fear Effect And Harvesting

The predator-prey relationship is a typical pattern of interspecies interactions.The study of the predator-prey model is one of the important contents of population dynamics.The interacting predator-prey population will produce rich dynamic behavior.In nature,ecological effects such as Allee effect,fear effect,and delay are real and will have an impact on population dynamics.In this paper,we establish a predator-prey model with Allee effect,fear effect,and gestation delay.In the first place,the stability of the equilibrium point is analyzed,the transcritical bifurcation is discussed according to the Sotomayor theorem,and the Poincare-Andronov-Hopf theorem is applied to study the existence of Hopf bifurcation.The direction of Hopf bifurcation of the delay differential equation also have discussed.There is one more point,for a reaction-diffusion-predatorprey model with Allee effect and harvesting,the conditions and amplitude equations of Turing instability are calculated.The last but not the least,the harvesting term is considered in the model,the Latin hypercube sampling and partial rank correlation coefficients are used to analyze the sensitivity of the parameters of the predator population.The optimal control problem of the model is analyzed with the harvesting intensity as the control variable.The research in this paper shows that the Allee effect and delay can destroy the stability of the model,as well as the fear effect can promote the stability of the delay model while reducing the number of prey populations.Allee effect and harvesting intensity will change the shape of the pattern,and the increase of the Allee effect makes the spot pattern into a hole pattern as shown in the figure and the possibility of extinction of the prey population is increased.The harvesting of the prey makes the distribution of the prey population more concentrated.In addition,the optimal control harvesting intensity is obtained and in the meantime the number of both groups has increased.These results have been verified by numerical simulations.