The Stability Of Variable Population Models With Square Root Capture Rate

Population ecology is an important part of ecology, and it is also the most extensive and in-depth study of mathematics in ecology. With the development of science and technology, the application of mathematical knowledge in this field is becoming more and more popular because of its wide application prospect in this field. The research on it also arouses the general concern of the researchers and biologists. Based on the previous studies, this paper mainly studies the local stability and bifurcation of the variable population model with square root capture rate, and the basic structure of this paper is as follows.The first chapter is the introduction, which gives a brief introduction to the development and trend of ecological mathematics, and present situation of the research of the biological population model. Moreover, some basic knowledge of the paper is listed.In the second chapter, based on the local stability of the variable population model with square root capture rate, we use the correlation theory of differential equations to analytically study the local stability of the variable population model with square root capture rate according to the eigenvalue of the system’s matrix and the sign of the slope of the X null-cline and the slope of Y null-cline. Then we get some conclusions and testify them.In the third chapter, based on the stability of the equilibrium point, we also analytically study the problem of the variable biological population with the square root of the band. Because of the existence of the bifurcation phenomenon, the structure of the system is not stable, and it can be seen that there is a close connection between the bifurcation phenomenon and the structural instability. Therefore, the necessary and sufficient condition that may diverge is the Jacobian matrix system with zero eigenvalue. Then, the conditions which need to be met by the system are analyzed, and the conditions for the existence of cross critical differences, saddle node bifurcation and Hopf bifurcation are discussed.In the fourth chapter, the main results of this paper are summarized.