Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting
American Journal of Applied Mathematics
Volume 6, Issue 3, June 2018, Pages: 109-116
Received: May 25, 2018; Accepted: Jun. 26, 2018; Published: Jul. 31, 2018
Views 776      Downloads 81
Solomon Tolcha, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Boka Kumsa, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Purnachandra Rao Koya, Department of Mathematics, Hawassa University, Hawassa, Ethiopia
Article Tools
Follow on us
This paper deals with the study of mutuality interactions between two species population with type II functional response and also with the inclusion of harvesting. Harvesting functions are introduced to express the rate of reductions of the species separately. Mathematical model have been constructed and considered for the analysis and results. In this model, the first population species benefited according to type II functional response and the second species benefited from the first according to type I functional response and also harvested proportional to its density. It is shown that the model has positive and bounded solutions. Stability analysis is carried out. The local and global stability of biologically interested equilibrium point are considered and analyzed. Numerical examples supporting theoretical results such as phase plane and simulation study using DSolver are also included. Assumptions and results are presented and discussed lucidly in the text of the paper.
Mutualism, Functional Response, Harvesting, Phase Plane Analysis, Positivity and Boundedness
To cite this article
Solomon Tolcha, Boka Kumsa, Purnachandra Rao Koya, Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting, American Journal of Applied Mathematics. Vol. 6, No. 3, 2018, pp. 109-116. doi: 10.11648/j.ajam.20180603.12
Copyright © 2018 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.
L. H. Erbe, V. S. H. Rao and H. I. Freedman, “Three-species food chain models with mutual interference and time delays”, Math Biosci 80 (1986), 57–80.
J. L. Bronstein, (1994). “Our current understands of mutualism”. Quarterly Review of Biology. 69 (1): 31–51.
M. Begon, J. L. Harper, and C. R. Townsend. 1996. “Ecology: individuals, populations, and communities”, Third Edition. Blackwell Science Ltd., Cambridge, Massachusetts, USA.
Morin. PJ, (2011). “Community ecology”, John Wiley and Sons, Hoboken, USA.
Levins, R. 1966. “The strategy of model building in population biology”. American Scientist 54:421–431.
MacArthur, R. H., and R. Levins. 1967. “The limiting similarity, convergence, and divergence of coexisting species”. American Naturalist 101:377–385.
P. F. Verhulst, Recherches mathematiques sur la loi d'accroissement de la population [Mathematical researches into the law of population growth increase], Nouveaux Memoires de VAcademic Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1—42.
Freedman, H. I., “Deterministic Mathematical Models in Population Ecology”, Marcel Dekker, New York, USA 1980.
D. H. Wright, “A simple, stable model of mutualism incorporating handling time”, The American Naturalist, 134 (1989), 664-667.
J. Ollerton, 2006. “Biological Barter: Interactions of Specialization Compared across Different Mutualisms”. pp. 411–435 in: Waser, N. M. & Ollerton, J. (Eds) Plant-Pollinator Interactions: From Specialization to Generalization. University of Chicago Press.
C. W. Clark, “Bio economic Modeling and Fisheries Management”, Wiley, New York, 1985.
C. W. Clark, “Mathematical Bio economics: The Optimal Management of Renewable Resources”, Wiley, New York, 1990.
D. R. Jana, Agrawal, R. K. Upadhyay and G. P. Samanta, “Ecological dynamics of age selective harvesting of fish population: Maximum sustainable yield and its control strategy”, Chaos, Solitons and Fractals 93 (2016), 111–122.
D. Jana, and G. P. Samanta, “Role of Multiple Delays in ratio-dependent prey-predator system with prey harvesting under stochastic environment”, Neural, Parallel and scientific Computations 22 (2014), 205–222.
T. R. Das, N. Mukherjee and K. S. Chaudhuri, “Harvesting of a prey–predator fishery in the presence of toxicity”, Appl Math Model 33 (2009), 2282–2292.
M. Kot, “Elements of Mathematical Ecology”, Cambridge University Press, Cambridge, 2001.
R. Ouncharoen, Pinjai S, Dumrongpokaphan T, and Lenbury Y (2012). “Global stability analysis of predator-prey model with harvesting and delay”. Thai Journal of Mathematics, 8(3): 589-605.
Rusliza Ahmad (2017). “Global stability of two-species mutualism model with proportional harvesting”. International Journal of Advanced and Applied Sciences, 4(7) 2017, Pages: 74-79.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186