Local Analysis of a Competitive Problem with Toxicants

This study aims to explain the dynamics of a competitive problem affected by toxicants. The effect of toxicants on ecological systems is an interesting topic for mathematical modelling. Discretization of the nonlinear problem is inevitable for right approximation of its solutions due to the difficulty of finding analytical solutions. In this work, a continuous time two species competitive problem was transformed into a discrete time problem. Because, it is very important to create a discrete model that will protect the properties of the original continuous model and the dynamics will be independent of step size. Also, in this study, the dynamic behaviour of a competitive system under the influence of toxicants were investigated. Lastly, the stability properties of each fixed point of the corresponding discrete problem have been examined using some theoretical results.

Anahtar Kelimeler:

Stability, Toxicants, Fixed point, Competitive system.

Local Analysis of a Competitive Problem with Toxicants

Keywords:

Stability, Toxicants, Fixed point, Competitive system.,

PDF

___

Freedman, H. I., Shukla J. B. (1991). Models for the effect of toxicant in single species and predator-prey systems. J. Math. Biol. , 15-30.
Chattopadhyay, J. (1994). Effect of toxic substances on a two-species competitive system. Eco. Modell, 287-289.
Samanta, G. P. (2010). A two-species competitive system under the influence of toxic substances. Applied Mathematics and Computation, 291-299.
Biswas, M., Bairagi, N. (2020). On the dynamic consistency of a two-species competitive discrete system with toxicity: Local and Global Analysis. Journal of Computational and Applied Mathematics, 145-155.
Das, T., Mukherje, R.N., Chaudhuri, K.S. (2009). Harvesting of a prey-predator fishery it the presence of toxicity. Appl. Math. Model., 2282-2292. Dimitrov, D.T., Kojouharov, HV. (2008). Nonstandard Finite Difference Methods For Predator-Prey Models With General Functional Response. Mathematics and Computers in Simulation, 78(1), 1-11.
Dimitrov, DT., Kojouharov, HV. (2007). Nonstandard Numerical Methods for a Class of Predator-Prey Models with Predator Interference. Electronic Journal of Differential Equations, 15, 67-75.
Dimitrov, D.T., Kojouharov, H.V. (2009). Positive and Elementary Stable Nonstandard Numerical Methods with Applications to Predator-Prey Models. Journal of Computational and Applied Mathematics, 189 (1-2), 98-108.
Dimitrov, D.T., Kojouharov, H.V. (2005). Nonstandard finite difference schemes for general two-dimensional autonomous dynamical systems. Applied Mathematics Letters, 18 (7), 769-774.
Shokri, A., Khalsaraei M.M., Molayi, M. (2021). Dynamically consistent NSFD methods for predator-prey system. Journal of Applied and Computational Mechanics, 1-10.
Banda, H., Chapwanya, M., Dumani, P. (2022). Pattern formation in the Holling-Tanner predator-prey model with predator-taxis. A nonstandard finite difference approach. Mathematics and Computers in Simulation, 336-353.
Sajjad, M., Din, Q., Safeer, M., Khan, M.A., Ahmad, K. (2019). A dynamically consistent nonstandard finite difference scheme for a predator-prey model. Advances in Difference Equations, 2019:381.
Mickens, R.E. (1994). Nonstandard finite difference model of differential equations. World Scientific Publishing Co. Pte. Ltd., Singapore.
Maynard, J. (1974). Models in Ecology. Cambridge University Press. Cambridge.
Mickens, R.E. (2005). Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations. Journal of Difference Equations and Applications, 11 (7), 645-653.
Anguelov, R., Lubuma, J.M.-S. (2003). Nonstandard finite difference method by nonlocal approximation. Mathematics and Computers in Simulation, 61 (3-6), 465-475.
Xiao, D., Ruan, S. (2001). Global dynamics of a ratio dependent predator-prey system. Journal of Mathematical Biology, 43, 268–290.
Ongun, M.Y., Ozdogan, N. (2017). A nonstandard numerical scheme for a predator-prey model with Allee effect. Journal of Nonlinear Science and Applications, 10, 713-723.