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RESEARCH PAPERS

On the Bifurcation Pattern and Normal Form in a Modified Predator–Prey Nonlinear System

[+] Author and Article Information
Dibakar Ghosh

High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700032, Indiadrghosẖ_chaos@yahoo.com

A. Roy Chowdhury

High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700032, Indiaasesẖr@yahoo.com

J. Comput. Nonlinear Dynam 2(3), 267-273 (Jan 15, 2007) (7 pages) doi:10.1115/1.2727496 History: Received September 15, 2006; Revised January 15, 2007

Detailed bifurcation pattern and stability structure is studied in a modified predator–prey system, with nonmonotonic response function. It is observed that almost all the parameters of the system have a positive influence as far as bifurcation is concerned. The analysis is done with the help of the package MATCONT. In the second stage of the analysis the detailed structure of the normal form is obtained after the corresponding position of Hopf bifurcation and Bogdanov–Takens bifurcation are identified with the help of a modified approach recently proposed by Kuznetsov (1995, Elements of Bifurcation Theory, Springer, New York, Chap. 8). It is important to note that the positions of Hopf and Bogdanov–Taken bifurcation as obtained from the analytic studies in this approach coincides exactly with those obtained from MATCONT.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

(a) Nullclines in the x1-x2 plane for a=5, λ=2.8, m=12, β=3.2, α=1.5, δ=3, μ=0.25, c=4; (b) regions of stability of the system (1) with the variation of the parameters “α” and “μ”. The black dotted portions are locally stable, “+” region represent unstable and other parameter values are the same as (a)

Grahic Jump Location
Figure 2

Bifurcation curves of equilibrium with the variation of the parameter α and μ of the variables (a) x1, (b) x2, and (c) x1, (d) x2, respectively. Here and in all the following figures dashed lines denote unstable branch and solid line stable branch.

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Figure 3

Bifurcation curves of equilibrium with the variation of the parameter m, δ, λ, and β of the variables x1 and x2, respectively, by (a)–(h)

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Figure 4

(a) Family of limit cycles bifurcating from the Hopf point in Fig. 2 in x1-x2-μ plane, and (b) period of the cycle as a function of α. (c) Parameter region between α and μ started from limit point in Fig. 2.

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Figure 5

(a) Phase space of limit cycles started from the Hopf point with the variation of μ in Fig. 2, and (b) variation of period with respect to μ, starting from Hopf point. (c) A curve of limit cycles with a fold bifurcation of limit cycles.

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