Stability and Hopf bifurcation of a predator-prey model
Tóm tắt
In this paper, we study a class of predator-prey model with Holling-II functional response. Firstly, by using linearization method, we prove the stability of nonnegative equilibrium points. Secondly, we obtain the existence, direction, and stability of Hopf bifurcation by using Poincare–Andronov Hopf bifurcation theorem. Finally, we demonstrate the validity of our results by numerical simulation.
Tài liệu tham khảo
Caughley, G., Lawton, J.H.: Plant-Herbivore Systems, Theoretical Ecology, pp. 132–166. Sinauer Associates, Sunderland (1989)
Haque, M.: Ratio-dependent predator-prey models of interacting populations. Bull. Math. Biol. 71, 430–452 (2009)
Hassard, B.D., Kazarinoff, N.D.: Theory and Applications of Hopf Bifurcation. CUP Archive, (1981)
Hsu, S.B., Huang, T.W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55(3), 763–783 (1995)
Kaper, T.J., Vo, T.: Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-diffusion equations. Chaos, Interdiscip. J. Nonlinear Sci. 28(9), 091103 (2018)
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), 672–679 (2012)
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), 1409–1418 (2016)
Sotomayor, J.: Generic bifurcations of dynamical systems. Dyn. Syst. 561–582 (1973)
Xiao, D., Ruan, S.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61(4), 1445–1472 (2001)
Xiao, Y., Chen, L.: A ratio-dependent predator-prey model with disease in the prey. Appl. Math. Comput. 131(2–3), 397–414 (2002)