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This work is designed to transform the fifth stage – fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcher’s rooted trees. This is because the rooted trees and derived equations for the y derivatives and x,y derivatives are the same for the explicit fourth-stage fourth-order methods, hence, we are motivated to analyze the fifth-stage fourth-order method. This idea is also derivable from general graphs and combinatorics.
A Genralist Predator Prey Mathematical Model Analysis on A Cusp Point and Bogdanov-Takens Bifurcation Point
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Temesgen Tibebu Mekonnen (2015). A Genralist Predator Prey Mathematical Model Analysis on A Cusp Point and Bogdanov-Takens Bifurcation Point. International Journal of Mathematical Research, 4(2): 64-75. DOI: 10.18488/journal.24/2015.4.2/184.108.40.206
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.
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.
Contiguous Function Relations and an Integral Representation for Appell k-Series F_(1,k)
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Shahid Mubeen , Sana Iqbal , Gauhar Rahman (2015). Contiguous Function Relations and an Integral Representation for Appell k-Series F_(1,k). International Journal of Mathematical Research, 4(2): 53-63. DOI: 10.18488/journal.24/2015.4.2/220.127.116.11
The main objective of this paper is to derive contiguous function relations or recurrence relations and obtain an integral representation Appell k-series F_(1,k), where k>0.
This study originates a new formula for Appell’s series in the form of a new symbol k>0 and contributes for deriving contiguous function relations, obtaining an integral representation of the Appell’s series in terms of said symbol k>0.