Study On The Harvesting Predator-prey System With Density Restriction

Population ecology is branch of biological mathematics which develops more completely and systematically, it mainly researched dynamics properties between predator population and prey population, which achieved great success in recent years. Based on the previous research achievements, to reflect the relationship between the most optimal economic gains and the ecological balance better, this paper comprehensive consideration nature differences between species and artificial interference factors, constructing three different predator-prey system models:The first two kinds predator-prey systems harvest modes which were the three species with double density restriction and the two population autonomous predator-prey systems harvest model with single density-dependent were disscussed. Using differential equation qualitative, stability theory research system stability and the optimal harvest strategy, using Routh-Hurwitz discriminant method to prove the the equilibrium point of system is the local asymptotic stability, using Dulac function study two populations of predator-prey systems of limit cycle existence, and at the same time, using the Pontryagin maximum principle gives three species predator-prey systems survive the optimal harvest strategy.The third type was a predator-prey harvest model with double linear density restriction, using differential equation qualitative and stability theory system research global quality and the optimal harvest strategy, by constructing Lyapunov function and using Barbalat lemma, we get system uniform persistence and unique positive periodic solution of the global asymptotic stability.Three types of predator-prey systems harvest model was discussed in this paper. We discuss the system stability and limit ring nonexistence, use the Pontryagin maximum principle to presen system most rewarding strategy, and solve the problem of protecting ecological balance on the basis of the optimal capture problem, which provide theoretical basics to obtain the biggest economic benefits for the human.