Please enter verification code
Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form
Mathematical Modelling and Applications
Volume 5, Issue 3, September 2020, Pages: 156-166
Received: May 1, 2020; Accepted: Jun. 18, 2020; Published: Aug. 4, 2020
Views 327      Downloads 90
Geremew Kenassa Edessa, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Purnachandra Rao Koya, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Article Tools
Follow on us
In this paper, a three species eco system, involving three pairs is considered modeled to examine the stability. Among the three species, one plays dual roles which are a host and an enemy with Monod response. In the first place model assumptions and formulation was carried out for investigations. The biological feasibility of the system is checked. That is positivity and boundedness of the model is verified. It is shown that biologically valid. The dynamical behavior of the proposed model system was analyzed qualitatively. The dynamical analysis includes the determination of all possible equilibrium points and their stability properties. All the equilibrium states are identified and the local asymptotic stability of some of the equilibrium states is examined by considering the set criteria. It is observed that among the states, the state in which the Prey and its Host species are exist is stable and the state where the Predator/Ammensal species is washed out is asymptotically stable. The global stability of the co-existence of the species was investigated by constructing a suitable Lyapunov function. To support our analytical studies, some numerical simulations was performed susing some mathematical software and the results were forwarded in the last section.
Prey, Predator, Ammensal, Commensal, Host, Continuous Time, Stability, Numerical Simulation
To cite this article
Geremew Kenassa Edessa, Purnachandra Rao Koya, Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form, Mathematical Modelling and Applications. Vol. 5, No. 3, 2020, pp. 156-166. doi: 10.11648/
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lotka A. J., “Elements of physical Biology, William & Wilking Baltimore, 1925.
Volterra V., Leconssen La Theorie Mathematique De LaLeitte Pou Lavie, Gautheir Villara, Paris, 1931.
Freedman, H. I., “Deterministic Mathematical Models in population Ecology Marcel-Decker, New York, 1980.
Kapur J. N., “Mathematical modeling, Wiley Eastern, 1985.
Meyer W. J., “Concepts of Mathematical modeling”, Mc. Grawhil, 1985.
Paul Colinvaux, Ecology, John Wiley and Sons Inc., New York, 1986.
Srinivas N. C., “Some Mathematical aspects of modeling in Bio-medical sciences”. PhD thesis, Kakatiya university, 1991.
Cushing J. M., Integro-Differnetial Equations and Delay Models in Population Dynamics, Lecture Notes in Bio-Mathematics, 20, Springer Verlag, 1997.
Lakshmi Narayana. K., “A mathematical study of a preypredator ecological model with a partial cover for the prey and alternative food for the predator”, PhD thesis, JNTU, 2005.
Lakshmi Narayan. K & Pattabhi Ramacharyulu N. Ch, 2007, “A prey predator model with cover for prey and alternate food for the predator and time delay”, International Journal of Scientific Computing Vol. 1, pg. 7-14.
Ravindra Reddy., “A study on mathematical models of Ecological mutualism between two interacting species” PhD thesis, O. U, 2008.
Archana Reddy R. Pattabhi Ramacharyulu N. Ch & Krishna Gandhi. B., “A stability analysis of two competitive interacting species with harvesting of both the species at a constant rate” International Journal of scientific computing (1); pp 57-68, 2007.
Bhaskara Rama Sarma & Pattabhi Ramacharyulu N. Ch., “Stability analysis of two species competitive ecosystem”. International Journal of logic based intelligent systems, Vol. 2 No. 1, 2008.
Seshagiri Rao. N & Pattabhi Ramacharyulu N. Ch., “Stability of a syn-ecosystem consisting of a Prey –Predator and host commensal to the prey-I” (With mortality rate of prey), 2009.
D. Ravi kiran, B. Rami reddy, N. Ch. P. Acharyulu, “A Numerical analysis of stability of a three species Eco system consisting of Prey, Predator and a third species which is host to prey and enemy to Predator” ANU Journal of Physical Sciences, Volume 3, No. 1 & 2, 2011.
D. Ravi kiran, B. Rami reddy, ‘Stability of a three species ecological system consisting of Prey, predator species and a third species which is a host to the prey and enemy to the predator”, Journal of Experimental sciences, Vol. 3, No. 12, 2012.
N. Apreutesei, A. Ducrot, V. Volpert, Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3, pp. 1–27, 2008.
. H. Malchow, S. Petrovskii, E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology, Chapman & Hall/CRC Press, Boca Raton, 2008.
S. G. Ruan, D. M. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, pp. 1445–1472, 2001.
R. MacArthur. Geographical Ecology: Patterns in the Distribution of Species. Harper and Row, New York, 1972.
T. W. Schoener. Alternatives to Lotka-Volterra competition: models of intermediate complexity. Theor. Popul. Biol., 1976, 10: 309-333.
N. Apreutesei, G. Dimitriu and R. Strugariu, An optimal control problem for a two-prey and one-predator model with diffusion, Comput. Math. Appl., 67 (2014), 2127{2143.
J. L. Bronstein, U. Dieckmann and R. Ferri`ere, Coevolutionary dynamics and the conservation of mutualisms, in Evolutionary Conservation Biology (eds. R. Ferri`ere, U. Dieckmann and D. Couvet), Cambridge University Press, (2004), 305{326.
A. E. Douglas, The Symbiotic Habit, Princeton University Press, Princeton, 2010.
Chen, F., Xue, Y., Lin, Q., et al.: Dynamic behaviors of a Lotka Volterra commensal symbiosis model with density dependent birth rate. Adv. Differ. Equ. 2018, 296 (2018).
Cruz Vargas-De-Le´on, Chilpancingo and Guillermo G´omez-Alcaraz. Global stability in some ecological models of commensalism between two species. Biomatem´atica 23 (2013), 138–146.
Liu, Y., Xie, X., Lin, Q.: Permanence, partial survival, extinction, and global attractivity of a non autonomous harvesting Lotka Volterra commensalism model incorporating partial closure for the populations. Adv. Differ. Equ. 2018, 211 (2018).
Lin, Q.: Allee effect increasing the final density of the species subject to the Allee effect in a Lotka Volterra commensal symbiosis model. Adv. Differ. Equ. 2018, 196 (2018).
Georgescu, P., Maxin, D.: Global stability results for models of commensalism. Int. J. Biomath. 10 (3), 1750037 (25 pages) (2017).
Liu, Y., Zhao, L., Huang, X., et al.: Stability and bifurcation analysis of two species amensalism model with Michaelis Menten type harvesting and a cover for the first species. Adv. Differ. Equ. 2018, 295 (2018).
Wu, R. X., Lin, L., Zhou, X. Y.: A commensal symbiosis model with Holling type functional response. J. Math. Comput. Sci. 16, 364–371 (2016).
Lei, C.: Dynamic behaviors of a stage structured commensalism system. Adv. Differ. Equ. 2018, 301 (2018).
Yang, L. Y., Xie, X. D., et al.: Dynamic behaviors of a discrete periodic predator prey mutualist system. Discrete Dyn. Nat. Soc. 2015, Article ID 247269 (2015).
Geremew Kenassa Edessa, Boka Kumsa, Purnachandra Rao Koya. Modeling and Simulation Study of the Population Dynamics of Commensal-Host-Parasite System. American Journal of Applied Mathematics. Vol. 6, No. 3, 2018, pp. 97-108.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186