Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration
American Journal of Applied Mathematics
Volume 7, Issue 6, December 2019, Pages: 164-176
Received: Oct. 25, 2019; Accepted: Nov. 18, 2019; Published: Dec. 30, 2019
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Authors
Gang Zhu, School of Education Science, Harbin University, Harbin, P. R. China
Chunyan He, School of Education Science, Harbin University, Harbin, P. R. China
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Abstract
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.
Keywords
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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Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
V. A. A. Jansen, The dynamics of two diffusively coupled predator-prey populations, Theoretical Population Biology, 59 (2001) 119-131.
[2]
Y. Kuang, Delay differential Equation with Applications in Population Dynamics, Academic Press, New York, 1993.
[3]
C. S. Holling, The functional response of predator to prey density and its role in minicry and population regulation, Mem. Ent. Sec. Can., 45 (1965) 1-60.
[4]
X. Liu, L. Chen, Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,Chaos, Solitons & Fractals, 16 (2003) 311-320.
[5]
E. Beretta, Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems, Nonlinear Ana. TMA., 32 (1998) 381-408.
[6]
W. Ko, k. Ryu, Qualitative analysis of a predatorprey model with Holling type II functional response incorporating a prey fefuge, J. Differential Equations, 231(2006) 534-550.
[7]
M. Rosinzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interation, American Naturlist, 97 (1963) 209-223.
[8]
S. B. Hsu, On global stability of a predator-prey system. Math. Biosci., 39 (1978) 1C10.
[9]
S. B. Hsu, P. Waltman, Competing predators, SIAM J. Math. Anal., 35 (1978) 617-625.
[10]
S. B. Hsu, T. W. Hwang and Y. Kuang,Global analysis of the Michaelis-Menten-type ratio-dependent predatorprey system. J. Math. Biol., 42 (1978) 489C506.
[11]
D. Xiao, S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001) 268-290.
[12]
X. Tian, R. Xu, Global dynamics of a predatorprey system with Holling type II functional response, Nonlinear Analysis: Modelling and Control., 16 (2011) 242-253.
[13]
K. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.
[14]
Y. Kuang, H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988) 67-84.
[15]
D. Xiao, Z. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey system,Nonlinearity, 16 (2003) 1-17.
[16]
R. Xu, M. A. J. Chaplain, F.A. Davidson, Periodic solutons for a predator-prey model with Holling type II functional response and time delays Appl. Math. Comput., 161 (2005) 637-654.
[17]
M. Banerjeea, V. Volpertbcd, Spatio-temporal pattern formation in RosenzweigCMacArthur model: Effect of nonlocal interactions, Ecological Complexity, 30 (2017) 2-10.
[18]
M. Moustafa, M. H.Mohd, A. I. Ismail, F, A. Abdullah, Dynamical analysis of a fractional-order RosenzweigCMacArthur model incorporating a prey refuge, Chaos, Solitons, Fractals, 109, (2018) 1-13.
[19]
J. Sugie, Y. Saito, Uniqueness of limit cycles in a Rosenzweig-Macarthur model with prey immigration, SIAM J. Appl. Math., 72 (2012) 299-316.
[20]
G. Zhu, J. J. Wei,Global stability and bifurcation analysis of a delayed predatorCprey system with prey immigration, Elec. J. Qual. Theo. Diff. Eqn., (2016).
[21]
J. Hale, Theory of functional differential equation, Springer-Verlag, New York, 1977.
[22]
S. Ruan, J. Wei, On the zeros of transcentental functions with applications to stability if delay differential equations with two delays. Dyn. Contin. Discrete Impus. Syst. Ser. A Math. Anal., 10 (2003) 863-874.
[23]
S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer, New York, 1990.
[24]
T. Faria, L. Magalh~aes, Normal forms for retarded functional differential equations and applications to Bagdanov-Takens singularity, J. Differential Equations, 122 (1995) 201-224.
[25]
T. Faria, L. Magalh~aes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations, 122 (1995) 181-200.
[26]
J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc., 350 (1998) 4799-838.
[27]
B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981.
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