A BIFURCATION STUDY: EFFECT OF A TOXICANT ON A BIOLOGICAL SPECIES, EMITTED BY ITSELF IN ITS OWN ENVIRONMENT
DOI:
https://doi.org/10.53555/ephijse.v7i3.189Keywords:
Biological population, mathematical model, toxicant, supercritical hopfbifurcationAbstract
In this paper, a model based on effect of toxicant on a biological species [1, chap. 2] is analyzed for the existence and nature of hopf-bifurcation. The biological population is logistically growing in its own environment and toxicant is being emitted by the population itself. The hopf-bifurcation analysis of model shows that when the emission rate of toxicant by the biological population increases in the environment, the density level of biological population decreases and after crossing a critical value of emission rate, the density of biological population starts oscillating and never settle down to its equilibrium level. The hopf-bifurcation analysis of model increases the validity of model. The dynamic behavior of the model for the emission rate of toxicant is described by providing numerical simulation.
References
. Agrawal, A.K., (1999). Effects of Toxicants on Biological Species: Some non-linear mathematical models and their analyses. Ph. D. Thesis, Department of Mathematics, I.I.T. Kanpur, INDIA.
. Agrawal, A.K., & Shukla, J.B. (2012) Effect of a toxicant on a biological population causing severe symptoms on a subclass. South Pacific Journal of Pure and Applied Mathematics, 1(1):12–27.
. Agrawal, A.K., Sinha, P., Dubey, B. & Shukla, J.B. (2000). Effects of two or more toxicants on a biological species: A non-linear mathematical model and its analysis. In Mathematical Analysis and Applications. A.P. Dwivedi (Ed), Narosa Publishing House, New Delhi, INDIA: 97 – 113.
. Dhooge, A., Govaerts, W., & Kuznetsov, Y.A. (2003). MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software (TOMS), 29(2), 141-164.
. Freedman, H.I. & Shukla J.B. (1991). Models for the effect of toxicant in single species and predator prey systems. J. Math. Biol., 30:15 – 30.
. Hassard, B.D., Kazarinoff, N.D. & Wan, Y. (1981). Theory and Application of Hopf Bifurcation. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 41.
. Kumar, A., Agrawal, A.K., Hasan, A. & Misra, A.K. (2016). Modeling the effect of toxicant on the deformity in a subclass of a biological species. Model. Earth Syst. Environ., 2(1).
. Kumar, A., Khan, A.W. & Agrawal A.K. (2015). Modeling the simultaneous effect of two toxicants causing deformity in a subclass of biological population. IOSR Journal of Mathematics, 11(6):70–82.
. Kuznetsov, Y. (2004). Elements of Applied Bifurcation Theory. NewYork: Springerverlag.
. Liu, W. (1994). Criterion of Hopf Bifurcation without using Eigenvalues. J. Math. Anal. and App., 182:250–256.
. Shukla, J.B. & Agrawal, A.K. (1999). Some mathematical models in ecotoxicology; Effects of toxicants on biological species. Sadhana, 24:25 – 40.
. Shukla, J.B. & Dubey, B. (1996). Simultaneous effects of two toxicants on biological species: A mathematical model. J. Biol. Systems, 4:109 – 130.
Downloads
Published
Issue
Section
License
Copyright (c) 2021 EPH - International Journal of Science And Engineering (ISSN: 2454 - 2016)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
