Abstract
References
- Allen LJS. An Introduction to Mathematical Biology. Prentice Hall, Upper Saddle River, NJ,2007.
- Cao Q, Wu J, Wang Y. Bifurcation solutions in the di usive minimal sediment. Computers and Mathematics with Applications. 77, 2019, 888–906.
- Kaper TJ, Vo T. Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-di usion equations. Chaos, Interdiscip. J. Nonlinear Sci. 28(9), 2018, 91–103.
- Kot M. Elements of Mathematical Ecology. Cambridge University Press, Cambridge, 2001.
- Li F, Li H. Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey. Math. Comput. Model. 55(3–4), 2012, 672–679.
- Song Y, Xiao W, Qi X. Stability and Hopf bifurcation of a predator-prey model with stage structure and time delay for the prey. Nonlinear Dyn. 83(3), 2016, 1409–1418.
- Sotomayor J. Generic bifurcations of dynamical systems. Dyn. Syst. 1973, 561–582.
- Wu F, Jiao Y. Stability and Hopf bifurcation of a predator-prey model, Boundary Value Problems, 129, 2019, 1–11.
- Xiao D, Ruan S. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61(4), 2001, 1445–1472.
- Xiao Y, Chen L. A ratio-dependent predator-prey model with disease in the prey. Appl. Math. Comput. 131(2–3), 2002, 397–414.
- Zhang L, Liu J, Banerjee M. Hopf and steady state bifurcation analysis in a ratio-dependent predator–prey model. Commun. Nonlinear Sci. Numer. Simulat. 44, 2017, 52–73.
Abstract
This paper is concerned with the Hopf bifurcation and steady state analysis of a predator-prey
model. Firstly, by analyzing the characteristic equation, the local stability of the nonnegative equilibriums
is discussed. Then the Hopf bifurcation around the positive equilibrium is obtained, and the direction
and the stability of the Hopf bifurcation are investigated. Finally, some numerical simulations are given to
support the theoretical results.
Keywords
Predator-prey model Steady state analysis Existence of Hopf bifurcation Supercritical bifurcation
References
- Allen LJS. An Introduction to Mathematical Biology. Prentice Hall, Upper Saddle River, NJ,2007.
- Cao Q, Wu J, Wang Y. Bifurcation solutions in the di usive minimal sediment. Computers and Mathematics with Applications. 77, 2019, 888–906.
- Kaper TJ, Vo T. Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-di usion equations. Chaos, Interdiscip. J. Nonlinear Sci. 28(9), 2018, 91–103.
- Kot M. Elements of Mathematical Ecology. Cambridge University Press, Cambridge, 2001.
- Li F, Li H. Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey. Math. Comput. Model. 55(3–4), 2012, 672–679.
- Song Y, Xiao W, Qi X. Stability and Hopf bifurcation of a predator-prey model with stage structure and time delay for the prey. Nonlinear Dyn. 83(3), 2016, 1409–1418.
- Sotomayor J. Generic bifurcations of dynamical systems. Dyn. Syst. 1973, 561–582.
- Wu F, Jiao Y. Stability and Hopf bifurcation of a predator-prey model, Boundary Value Problems, 129, 2019, 1–11.
- Xiao D, Ruan S. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61(4), 2001, 1445–1472.
- Xiao Y, Chen L. A ratio-dependent predator-prey model with disease in the prey. Appl. Math. Comput. 131(2–3), 2002, 397–414.
- Zhang L, Liu J, Banerjee M. Hopf and steady state bifurcation analysis in a ratio-dependent predator–prey model. Commun. Nonlinear Sci. Numer. Simulat. 44, 2017, 52–73.
Details
| Primary Language | English |
|---|---|
| Journal Section | Volume V Issue III 2020 |
| Authors | |
| Publication Date | December 30, 2020 |
| Published in Issue | Year 2020 Volume: 5 Issue: 3 |
