DYNAMICAL BEHAVIOURS OF A DISCRETIZED MODEL WITH MICHAELIS-MENTEN HARVESTING RATE

Year 2022, , 159 - 176, 31.07.2022

Nihal Özdoğan , Mevlüde Yakıt Ongun

https://doi.org/10.33773/jum.1134168

Cited By: 2

Abstract

In this paper, we introduced nonstandard finite difference scheme (NSFD) for solving the continuos model with Michaelis-Menten harvesting rate. We have seen that the proposed scheme preserve local stability and positivity. Stability analysis of each fixed point of the discrete time model has been proven. Also, numerical comparisons were made between the nonstandard finite difference method and the other methods.

Keywords

Nonstandard finite difference scheme, harvesting, equilibrium points.

References

• [1] A.A. Berryman, The origins and evolution of predator-prey theory, Ecology, 73 (5), 1530-1535, (1992).
• [2] L.I. Roege, G. Lahondy, Dynamically consistent discrete Lotka-Volterra competition systems, Journal of Difference Equations and Applications, 19 (2), 191-200, (2015).
• [3] M. Sajjad, Q. Din, M. Safeer, M.A. Khan, K. Ahmad, A dynamically consistent nonstandard finite difference scheme for a predator-prey model, Advances in Difference Equations, 2019:381, (2019).
• [4] A. Lotka, L.I. Dublin. On the True Rate of Natural Increase: As Exemplified by the Population of the United States. Journal of American Statistical Association, 150: 305-339, (1925)
• [5] V. Volterra, V.F.D. Numero, D’individui in Specie Animali Conviventi, Editoria Web design, Multimedia, (1927)
• [6] Y.O. El-Dib, J.H. He, Homotopy perturbation method with three expansions, Journal of Mathematical Chemistry, 1139-1150, (2021).
• [7] C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley Sons, New York, (1966).
• [8] A. Gkana, L. Zachilas, Incorporating prey refuge in a prey-predator model with a Holling type-I funtional response: random dynamics and population outbreaks, Journal of Biological Physics, 39: 587-606, (2013).
• [9] T.K. Kar, K.S. Chaudhuri, On non-selective harvesting of a multipecies fishery, International Journal of Mathematical Education in Science and Technology, 543-556, (2010).
• [10] C.S. Holling, Some characteristics of simple types of Predation and Parasitism, The Canadian Entomologist, Ottowa, Canada, (1959).
• [11] C.S. Holling, The Functional Response of Predators to Prey Density and Its Role in Mimicry and Population Regulation, The Memoirs of the Entomological Society of Canada, 97: 5-60, (1965).
• [12] R. Arditi, L.R. Ginzburg, Coupling in Predator-Prey Dynamics: Ratio-Dependence. Journal of Theoretical Biology, 139 (3): 311-326, (1989).
• [13] M. Haque, Ratio Dependent Predator-Prey Models of Interacting Populations, Bulletin of Mathematical Biology, 71: 430-452, (2009).
• [14] R.F. Luck, Evolution of natural enemies for biological control: a behaviour approach. Trends in Ecology and Evolution, 5 (6): 196-199, (1990).
• [15] D. Xiao, S. Ruan, Global dynamics of a ratio dependent predator-prey system. Journal of Mathematical Biology, 43: 268–290, (2001).
• [16] F. Berezovskaya, G. Karev, R. Arditi, Paremetric analysis of the ratio dependent predator-prey model, Journal of Mathematical Biology, 43: 221-246, (2001).
• [17] C. Jost, O. Arino, R. Arditi, About Deterministic Extinction in Ratio Dependent Predator-Prey Models, Bulletin of Mathematical Biology, 61 (1) :19-32, (1999).
• [18] S.B. Hsu, T.W. Hwang, Y. Kuang, Global Analysis of the Michaelis-Menten Type Ratio Dependent Predator-Prey System, Journal of Mathematical Biology, 42: 489-506, (2001).
• [19] N. Ozdogan, Nonstandard numerical approximations for ratio-dependent ecological models, Doctoral Dissertation, SDU Graduate School of Natural and Applied Sciene, (2018).
• [20] N. Bairagi, S. Chakraborty, S. Pal, Heteroclinic Bifurcation and Multistability in a Ratio dependent Predator-Prey System with Michaelis-Menten Type Harvesting Rate, World Congress on Engineering, London, 3-8, (2012).
• [21] S. Chakraborty, S. Pal, N. Bairagi, Predator-prey interaction with harvesting: mathematical study with biological ramifications, Applied Mathematical Modelling, 36 (9): 4044-4059, (2012).
• [22] R.E. Mickens, Nonstandard finite difference model of differential equations, World Scientific Publishing Co. Pte. Ltd., Singapore, (1994).
• [23] D.T. Dimitrov, H.V. Kojouharov, Nonstandard Finite Difference Methods For Predator-Prey Models With General Functional Response. Mathematics and Computers in Simulation, 78 (1): 1-11, (2008).
• [24] R.E. Mickens, Difference Equations: Theory, Applications and Advanced Topics, 3rd Edition, CRC Press, Atlanta, (2015).
• [25] R.E. Mickens, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications, 11 (7): 645-653, (2005).
• [26] M.Y. Ongun, N. Ozdogan, A nonstandard numerical scheme for a predator-prey model with Allee effect, Journal of Nonlinear Science and Applications, 10: 713-723, (2017).
• [27] D.T. Dimitrov, H.V. Kojouharov, Positive and Elementary Stable Nonstandard Numerical Methods with Applications to Predator-Prey Models, Journal of Computational and Applied Mathematics, 189 (1-2): 98-108, (2009).
• [28] P. Saha, N. Bairagi, M. Biswas, On the dynamic consistency of a discrete predator-prey model, Centre for Mathematical Biology and Ecology, (2019).
• [29] A. Shakri, M.M. Khalsaraei, M. Molayi, Dynamically consistent NSFD methods for predator-prey system, Journal of Applied and Computational Mechanics, 1-10, (2021).
• [30] M. Biswas, N. Bairagi, On the dynamic consistency of a two-species competitive discrete system with toxicity: Local and global analysis, Journal of computational and applied mathematics, 363: 145-155, (2020).
• [31] D.T. Dimitrov, H.V. Kojouharov, Nonstandard finite difference schemes for general two-dimensional autonomous dynamical systems, Applied Mathematics Letters, 18 (7): 769-774, (2005).
• [32] D.T. Dimitrov, H.V. Kojouharov, Nonstandard Numerical Methods for a Class of Predator-Prey Models with Predator Interference, Electronic Journal of Differential Equations, 15: 67-75, (2007).
• [33] J.H. He, F.Y. Ji, H. Mohammad, Difference equation vs differential equation on different scales, Journal of Numerical Methods for Heat and Fluid Flow, 31 (1): 391-401, (2021).
• [34] M. Kocabıyık, N. Ozdogan, M.Y. Ongun, Nonstandard Finite Difference Scheme for a Computer Virus Model, Journal of Innovative Science and Engineering, 4: 96-108, (2020).