The Stability And Bifurcations Analysis Of A Prey-predator Model For A Third Order Runge-Kutta Method

In the ecological dynamic system, the traditional Lotta-Votarra model and the ratiodependent predator-prey model are two kinds of deep research models, which have beenwidely used in ecological capture and resource management. However, for the populationwith long life but non-overlapping generation, or with short life but overlapping generation,differential equations are always applied.The paper investigate the stability and bifurcations of a discrete ecosystem in R8. First,by applying a third order Runge-Kutta method to a ratio-dependent predator-prey system,a discrete ecosystem is proposed. As varying the parameter in the some range, the localstability and the existing conditions and directions of bifurcations around the fixed points ofthe proposed model are discussed by using the bifurcation theory, the normal form of thediscrete ecosystem and the center manifold theorem.This paper is composed of five sections:The first section introduces the developing process, the modeling process, the resultsand points has been innovated.The second section mainly analyze the local stability and the bifurcations conditions ofthe discrete ecosystem under the varying of the model parameter e.The third section mainly investigate the bifurcation directions of the discrete ecosystemas varying the harvesting effort e in some range.The fourth section shows the numerical simulation, which not only verify the rationalityand correctness of our basic results, but also testify the complex dynamic behaviors of thesystem.The last section of this paper provides a summary and an conclusion.