Determining the stability condition of a predator-prey interaction with a prescribed delay in the system
Keywords:
Predator-Prey Interaction , Stability, Differential Equation, DelayAbstract
This study attempts to model a real life situation involving delay differential equation, in particular the predator-prey interaction. A delay of 0.01 is prescribed into the system to determine whether the system will be stable or otherwise. The results show that when an insignificant delay is introduced, its stability returns to the an ordinary diffrential, but when the delay is significant the delay is significant, it results into solutions of infinite roots.
Downloads
References
Ashine, A.B. (2018) Stability Analysis of Predator-Prey Model with Stage Structure for the Prey: A Review. International Journal of Sociology and Anthropology Research, 6(2):13-25.
Campbell, S.A., Edwards, R and Driessche P.V. (2004). Delayed Coupling between two neutral network loops. SIAM Journal of Applied Mathematics, 65(1):316. https://doi.org/10.1137/S0036139903434833
Cushing, J.M. (1977). Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Note on Biomathematics 20, Springer-Verlag, Heidelberg.
Gopadsamy, K. (1992). Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and Its Applications. 74, Kluwer Academic Publishers, Dordrecht.
Hussein, S. (2010) Predator-Prey Modelling. Undergraduate Journal of Mathematical Modelling: One+Two: Vol. 3: Issue 1, Article 20. DOI: http://dx.doi.org/10.5038/2326-3652.3.1.32
Jin, G., Qi, H., Li, Z., Han, J. and Li, H. (2017). A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays. Mathematical Problems in Engineering, Vol. 2017, Article ID 9490142. https://doi.org/10.1155/2017/9490142
Kuang, Y. (1993). Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York.
Kundu, S. and Maitra, S. (2016) Stability and delay in a three species predator-prey system. AIP Conference Proceedings 1751, Issue 1. https://doi.org/10.1063/1.4954857
Kundu, S. and Maitra, S. (2018) Permanence and Global Stability Analysis of a delayed three Species Predator-Prey Model with cooperation among preys. Far East Journal of Dynamical Systems, 30(1):11-26. http://dx.doi.org/10.17654/DS030010011
Liu, M. and Fan, M. (2017) Stability in Distribution of a three-species stochastic cascade predator-pre system with time delays. IMA Journal of Applied Mathematics, 82(1):396-423. https://doi.org/10.1093/imamat/hxw057
Liu, Y. and Xin, B. (2011). Numerical Solutions of a Fractional Predator-Prey System. Advances in Difference Equations, Volume 2011, Article ID 190475. doi:10.1155/2011/190475, 1-11
Macdonald N. (1978). Time Lags in Biological Models, Lecture notes in Biomathematics 27, Springer-Verlag, Heidelberg
Mbah, G.C.E. (2002). An Analytical Method of Solution to the Generalized Mathematical Model used for the study of Insulin-Dependent Diabetes Mellitus. Journal of the Nigerian Mathematical Society.
Mohr, M., Barbarossa, M.V. and Kuttler, C. (2014). Predator-Prey Interactions, Age Structures and DelayEquations. Mathematical Modelling of Natural Phenomena, 9(1):92-107.
Nurul, H.G. (2012). Dynamics of populations in fish farm: Analysis of stability and direction of Hopf-bifurcating periodic oscillation. Applied Mathematical Modelling, 36(5):2118-2127. https://doi.org/10.1016/j.apm.2011.08.009
Olgac, N., Elmali, H., Hosek, M. and Renzulli, M. (1997). Active Vibration Control of Distributed Systems using Delayed Resonator with Acceleration Feedback. ASME Journal of Dynamic Systems, Measure Control, 119(3): 380-389. https://doi.org/10.1115/1.2801269.
Pratama, R.A., Ruslau, M.F.V., Nurhayati and Laban, S. (2019). Analysis stability of predator-prey model with Holling type I predation response function and stage structure for predator. IOP Conference Series: Earth and Environmental Science, 343 012161. doi:10.1088/1755-1315/343/1/012161
Suebcharon, T. (2017). Analysis of a Predator-Prey Model with Switching and Stage-Structure for Predator. International Journal of Differential Equations, Volume 2017, Article ID 2653124. https://doi.org/10.1155/2017/2653124
Xia, J., Yu, Z.X. and Zheng, S.W. (2017) Stability and traveling waves in Beddington-DeAngelis type stage-structured predator-prey reaction-diffusion systems with nonlocal delays and harvesting. Advances in Difference Equations, 65:1-24. https://doi.org/10.1186/s13662-017-1093-6
Xu, C. and Li, P. (2012) Dynamical Analysis in a Delayed Predator-Prey Model with two Delays. Discrete Dynamics in Nature and Society, Vol. 2012, Article ID 652947, 1-22. doi:10.1155/2012/652947
Zhao, T. (1995). Global Periodic Solutions for a Differential Delay System Modeling a Microbial Population in the Chemostat. Journal of Mathematical Analysis Applications.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 SAMUEL UGOCHUKWU ENOGWE, Sadik Olaniyi Malik, Bassey Okpo Orie

This work is licensed under a Creative Commons Attribution 4.0 International License.