Volume 7, Issue 6, December 2019, Page: 164-176
Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration
Gang Zhu, School of Education Science, Harbin University, Harbin, P. R. China
Chunyan He, School of Education Science, Harbin University, Harbin, P. R. China
Received: Oct. 25, 2019;       Accepted: Nov. 18, 2019;       Published: Dec. 30, 2019
DOI: 10.11648/j.ajam.20190706.14      View  473      Downloads  132
A delayed reaction Cdiffusion Rosenzweig-MacArthur model with a constant rate of prey immigration is considered. We derive the characteristic equation through partial differential equation theory, and by analyzing the distribution of the roots of the characteristic equation, the local stability of the positive equilibria is studied, and we get the conditions to determine the stability of the positive equilibria. Furthermore we find that Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values, and we get the conditions under which the Hopf bifurcation occurs and so periodic solutions appear near the positive equilibria. By using the center manifold theory and normal form method, we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, some numerical simulations are carried out to illustrate the analytic results of our study.
Delay, Stability, Bifurcation, Center Manifold, Normal Form
To cite this article
Gang Zhu, Chunyan He, Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration, American Journal of Applied Mathematics. Vol. 7, No. 6, 2019, pp. 164-176. doi: 10.11648/j.ajam.20190706.14
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