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Stability analysis of a food chain model consisting of two competitive preys and one predator

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Abstract

The present article deals with the intra-specific competition among predator populations of a prey-dependent three-component food chain model system consisting of two competitive preys and one predator. The behaviour of the system near the biologically feasible equilibria is thoroughly analysed. Boundedness and dissipativeness of the system are established. The stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present system experiences Hopf–Andronov bifurcation for suitable choice of parameters. The results of this investigation reveal that the intra-specific competition among predator populations can be beneficial for the survival of predator. The ecological implications of both the analytical and the numerical findings are discussed at length towards the end.

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Acknowledgments

The authors Mr. Nijamuddin Ali and Professor Santabrata Chakravarty gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-II) sponsored by the University Grants Commission (UGC), New Delhi, India. They are thankful to the unanimous reviewers for making fruitful comments and suggestions in order to improve the quality of the work done.

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Authors and Affiliations

  1. Department of Education, Katwa College, Katwa, West Bengal, India

    Nijamuddin Ali

  2. Department of Mathematics, Visva-Bharati, Santiniketan, 731235, West Bengal, India

    Santabrata Chakravarty

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  1. Nijamuddin Ali
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  2. Santabrata Chakravarty
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Correspondence to Santabrata Chakravarty.

Appendix

Appendix

A seventh degree equation has seven roots in the complex domain. For the sake of brevity, its coefficients are not written explicitly in view of their complexity. We now find sufficient conditions for it to have at least a positive root. Since the degree of the equation is odd, by Descartes’ rule of sign, we get a real root and correspondingly a linear factor of the equation. The seven-degree equation can then be factorized as

$$\begin{aligned} \sum _{i=0}^{6}B_ix^i= & {} B_7(x+p)(x^6+A_1x^5+A_2x^4+A_3x^3\nonumber \\&+\,A_4x^2+A_5x+A_6)\nonumber \\= & {} B_7[x^7+(p+A_1)x^6+(pA_1+A_2)x^5\nonumber \\&+\,(pA_2+A_3)x^4+(pA_3+A_4)x^3\nonumber \\&+\,(pA_4+A_5)x^2+(pA_5+A_6)x+pA_6]\nonumber \\ \end{aligned}$$
(6.1)

where p is to be determined. By equating coefficients of like powers of x on the left and the right, we find \(A_1+p=\frac{B_6}{B_7}\), \(A_2+pA_1=\frac{B_5}{B_7}\), \(A_3+pA_2=\frac{B_4}{B_7}\), \(A_4+pA_3=\frac{B_3}{B_7}\), \(A_5+pA_4=\frac{B_2}{B_7}\), \(A_6+pA_5=\frac{B_1}{B_7}\), \(pA_6=\frac{B_0}{B_7}\), from which we have \(p=\frac{B_0}{B_7A_6}\). One root of Eq. (6.1) is thus found as \(x_6=-p\). By imposing conditions \( p< 0\), we obtain \(x_6>0\). This ensures that the feasible coexistence equilibrium is unique.

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Ali, N., Chakravarty, S. Stability analysis of a food chain model consisting of two competitive preys and one predator. Nonlinear Dyn 82, 1303–1316 (2015). https://doi.org/10.1007/s11071-015-2239-2

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