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International Journal of Mathematical Research

April 2015, Volume 4, 2, pp 64-75

A Genralist Predator Prey Mathematical Model Analysis on A Cusp Point and Bogdanov-Takens Bifurcation Point

Temesgen Tibebu Mekonnen

Temesgen Tibebu Mekonnen 1

  1. Department of Mathematics Debre Berhan University Ethiopia, Ethiopia 1

Pages: 64-75

DOI: 10.18488/journal.24/2015.4.2/24.2.64.75

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Abstract:

This study considers a generalist predator-prey system. We investigate a local bifurcation namely Bogdanov-Takens (co dimension 2) bifurcations and a cusp point on two significant points in the division of parameter space. These points show the place where Bogdanov-Takens point and a cusp point exist. The existence of these bifurcations proved analytically by Normal form derivation. To reach the analysis we first studied the steady state solutions and their dependence on parameters and then investigate a parameter space which is divided into subregions based on the number of equilibrium points. We identified three vital parameters 𝛼 stands for the maximum uptake rate of the generalist predator; 𝜃 stands for half saturation value and 𝜂 such that 𝜇/𝜂 is the conversion efficiency of the generalist predator where 𝜇 is the intrinsic growth rate of the predator.
Contribution/ Originality
This study contributes new findings in the field of a mathematical model. We investigated two bifurcation points namely Bogdanov-Takens bifurcation point and the cusp point on the generalist predator prey mathematical model by observing and analyzing the change of the behavior of the solutions of the couple of the differential equations if small change appears on the parameters of the considered model by using the normal form of derivation method.

Keywords:


Reference:

  1. N. Ali, Nonlinear dynamical systems and chaos. Englewood Cliffs: Prentice Hall, 1983.
  2. Temesgen Tibebu Mekonnen, "Bifurcation analysis on the dynamics of a generalist predator pre system," International Journal of Ecosystem, USA, vol. 2, pp. 38-43, 2012.
  3. F. George Simmons, Differential equations. New Delhi: McGraw-HILL Publishing Company Ltd, 1972.
  4. H. W. Turnbull, Theory of equations. New York: Intercience Publishers Inc, 1952.
  5. P. Lawrence, Differential equations and dynamical systems, 3rd ed. New York: Springer. Available: http://journal.sapub.org/ajb, 2001.
  6. A. Yuri Kuzentsov, Elements of applied bifurcation theory, 2nd ed. New York: Springer, 1998.

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