Hopf Bifurcation For A Predator-prey Biological Economic System With Holling Type Ⅲ Functional Response

In this paper，we studed the stability and Hopf bifurcation of a kind of biological economicsystem which considers a prey-predator system with Holling type Ⅲfunctional response andharvest effort on prey through the use of the differential algebraic system theory and Hopfbifurcation theory. At first，we obtained the necessary conditions for the existence of the positiveequilibrium through calculating and discussing. And then，we discussed the condition of stabilityof the system on the positive equilibrium point，and further we could obtained stability theoremof the system. After that，we discussed the turning up condition of the Hopf bifurcation on thesystem，and then we obtained Hopf bifurcation theorem of the system. The full text is dividedinto five chapters.The first chapter was the introduction. At first we introduced the background of raising ofthe research problem. Then we introduced the research status of this field.The second chapter was prerequisites. We introduced the relative theory knowledge of thestability， the relative theory knowledge of the Hopf bifurcation and the relative theoryknowledge of the differential algebraic system.The third chapter discussed the stability and Hopf bifurcation of a kind of biologicaleconomic system which considers a prey-predator system with Holling type Ⅲfunctionalresponse and harvest effort on prey through the relative theory knowledge of the stability，therelative theory knowledge of the Hopf bifurcation and the relative theory knowledge of thedifferential algebraic system. And then，we made a numerical simulation.The fouth chapter discussed the stability and Hopf bifurcation of a kind of biologicaleconomic system which considers a prey-predator system with Holling type Ⅲfunctionalresponse and harvest effort on prey and delay. We selected the delay for branch parameters，anddiscussed the stability on the positive equilibrium point and existence of the Hopf bifurcationusing of ordinary differential equation qualitative and stability theory and method. At last anumerical example was given.The fifth chapter gave a review to our researched problem，and looked the distance of theresearch direction and the research content of the field.