Establishment And Analysis Of Two Kinds Of Population Models Considering Harvest

Ecosystem stability is a basic condition for human survival and an important source of economic prosperity and material production.Harvest is one of the important behaviors of human being involved in the ecosystem,which can regulate the stability of the ecosystem,but blind harvest will destroy the balance of the ecosystem.Therefore,it is of great significance to study the impact of harvest on the ecosystem.In this paper,the effect of harvest on population behavior is considered,and the corresponding mathematical model is established and its dynamics are analyzed.In the second part of this article,the harvesting behavior is considered to be affected by the fixed number of fishing vessels and harvesting tools in the actual situation,so the harvest function should have a non-zero minimum and maximum value,and the harvest function should increase nonlinearly with the increase of harvesting effort.None of the existing harvest functions can describe this phenomenon.Therefore,this chapter establishes a predator-prey model considering a new harvest function and makes a theoretical analysis of it.Firstly,the existence of the equilibrium point of the system is analyzed.The system always has an extinction equilibrium point,and there may be zero one or two boundary equilibrium points and zero one or two positive equilibrium points under different conditions.Through qualitative analysis,it is obtained that the nature of the equilibrium point may be the focus center or saddle node.On this basis,it is found that saddle node bifurcation,Hopf bifurcation and codimension 2 Bogdanov-Tarken bifurcation occur in the system,and the conditions of bifurcation are given.Finally,the numerical simulation of the system shows that the bifurcation phenomenon is abundant due to the new harvest function,and the change of harvest effort and the intrinsic growth rate of the prey population can change the stability of the system.Therefore,the optimal coexistence of two populations can be achieved by controlling the amount of harvesting effort.The third part of this article considers the phenomenon that predators and prey often share resources in ecosystems and the effects of harvesting on prey populations of predators.A 3D ecological model of predator and prey sharing resources with Holling II functional response function of proportional harvesting is established and analyzed.Firstly,the existence of the equilibrium point of the system is analyzed.The equilibrium point of the prey-free population,the equilibrium point of the predator-free population and the coexistence equilibrium point exist in the system.Through qualitative analysis,the equilibrium point of the prey-free population will appear under different conditions,and the equilibrium point of the predator-free population is a saddle point.Then the global asymptotic stability of coexistence equilibrium is simulated with appropriate parameters.Finally,the numerical simulation of the system shows that harvest can change the stable state of the system,and reduce harvest under certain conditions,so that the population can coexist at a higher level.Therefore,it is theoretically possible to control the harvest to achieve the desired optimal state of the population.