A three-component prey-predator system with interval number
Year 2023,
Volume: 3 Issue: 1, 1 - 16, 31.03.2023
Dipankar Ghosh
Prasun Kumar Santra
Ghanshaym Singha Mahapatra
Abstract
This paper presents a three-component model consisting of one prey and two predator species using imprecise biological parameters as interval numbers and applied functional parametric form in the proposed prey-predator system. The positivity and boundedness of the model are checked, and a stability analysis of the five equilibrium points is performed. Numerical simulations are performed to study the effect of the interval number and to illustrate analytical studies.
References
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- Gao, Y., & Yang, F. Persistence and extinction of a modified Leslie–Gower Holling-type II two-predator one-prey model with Lévy jumps. Journal of Biological Dynamics, 16(1), 117-143, (2022).
- Gakkhar, S., Singh, B., & Naji, R.K. Dynamical behavior of two predators competing over a single prey. BioSystems, 90(3), 808-817, (2007).
- Lv, S., & Zhao, M. The dynamic complexity of a three species food chain model. Chaos, Solitons & Fractals, 37(5), 1469-1480, (2008).
- Gholami, M., Ghaziani, R.K., & Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41–47, (2022).
- Mukherjee, D. Effect of fear on two predator-one prey model in deterministic and fluctuating environment. Mathematics in Applied Sciences and Engineering, 2(1), 1-71, (2021).
- Mulugeta, B. T., Yu, L., & Ren, J. Bifurcation Analysis of a One-Prey and Two-Predators Model with Additional Food and Harvesting Subject to Toxicity. International Journal of Bifurcation and Chaos, 31(6), 2150089, (2021).
- Sarwardi, S., Mandal, P.K., & Ray, S. Dynamical behaviour of a two-predator model with prey refuge. Journal of Biological Physics, 39(4), 701-722, (2013).
- Alebraheem, J., & Abu-Hasan, Y. Persistence of predators in a two predators-one prey model with non-periodic solution. Applied Mathematical Sciences, 6(19), 943-956, (2012).
- Savitri, D., Suryanto, A., & Kusumawinahyu, W.M. Dynamical behavior of a modified lesliegower one prey-two predators with competition. Mathematics, 8(5), 699, (2020).
- Kharbanda, H., & Kumar, S. Asymptotic stability of one prey and two predators model with two functional responses. Ricerche Di Matematica, 68(2), 435–452, (2019).
- Pal, D., Santra, P., & Mahapatra, G.S. Dynamical behavior of three species predator-prey system with mutual support between non refuge prey. Ecological Genetics and Genomics, 3(5), 1-6, (2017).
- Vijaya, S., & Rekha, E. Prey–predator three species model using predator harvesting Holling type II functional. Biophysical Reviews and Letters, 11(2), 87-104, (2016).
- Laurie, H., & Venturino, E. A two-predator one-prey model of population dynamics influenced by herd behaviour of the prey. Theoretical Biology Forum, 111(1–2), 27–47, (2019).
- Wang, J., & Wang, M. Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis. Zeitschrift Fur Angewandte Mathematik Und Physik, 69(3), (2018).
- Pal, D., Mahapatra, G.S. Dynamic behavior of a predator–prey system of combined harvesting with interval-valued rate parameters. Nonlinear Dynamics, 83(4), 2113–2123, (2016).
- Pal, D., Mahaptra, G.S., Samanta, G.P. Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model. Mathematical Biosciences, 241(2), 181–187, (2013).
- Santra, P.K. & Mahapatra, G.S. Dynamical study of discrete-time prey-predator model with constant prey refuge under imprecise biological parameters. Journal of Biological Systems, 28(3), 681–699, (2020).
- Ghosh, D., Santra, P.K., & Mahapatra, G.S. Fear effect on a discrete-time prey predator model with imprecise biological parameters. In AIP Conference Proceedings (Vol. 2246), American Institute of Physics Inc. (2020).
- Santra, P., & Mahapatra, G.S. Discrete prey–predator model with square root functional response under imprecise biological parameters.In Springer Proceedings in Mathematics and Statistics, Springer, 320, 211–225, (2020).
- Mondal, B., Rahman, M.S., Sarkar, S., & Ghosh, U. Studies of dynamical behaviours of an imprecise predator-prey model with Holling type II functional response under interval uncertainty. European Physical Journal Plus, 137(1), (2022).
- Mahata, A., Mondal, S.P., Roy, B., & Alam, S. Study of two species prey-predator model in imprecise environment with MSY policy under different harvesting scenario. Environment, Development & Sustainability, 23(10), 14908–14932, (2021).
- Vargas-De-León, C. On the global stability of SIS, SIR and SIRS epidemic models with standard incidence. Chaos, Solitons and Fractals, 44(12), 1106–1110, (2011).
- Beretta, E. & Capasso, V. On the general structure of epidemic systems. Global asymptotic stability, Computers & Mathematics with Applications, 12(6), 677–694, (1986).
- Korobeinikov, A. & Wake, G.C. Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Applied Mathematics Letters, 15(8), 955–960, (2002).
- Goh, B.S. Global stability in two species interactions. Journal of Mathematical Biology, 3(3–4), 313–318, (1976).
- Korobeinikov, A. Lyapunov functions and global properties for SEIR and SEIS epidemic models. Mathematical Medicine and Biology, 21(2), 75–83, (2004).
- McCluskey, C.C. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences and Engineering, 3(4), 603–614, (2006).
Year 2023,
Volume: 3 Issue: 1, 1 - 16, 31.03.2023
Dipankar Ghosh
Prasun Kumar Santra
Ghanshaym Singha Mahapatra
References
- Dubey, B., & Upadhyay, R.K. Persistence and extinction of one-prey and two-predators system. Nonlinear Analysis: Modelling and Control, 9(4), 307-329, (2004).
- Gao, Y., & Yang, F. Persistence and extinction of a modified Leslie–Gower Holling-type II two-predator one-prey model with Lévy jumps. Journal of Biological Dynamics, 16(1), 117-143, (2022).
- Gakkhar, S., Singh, B., & Naji, R.K. Dynamical behavior of two predators competing over a single prey. BioSystems, 90(3), 808-817, (2007).
- Lv, S., & Zhao, M. The dynamic complexity of a three species food chain model. Chaos, Solitons & Fractals, 37(5), 1469-1480, (2008).
- Gholami, M., Ghaziani, R.K., & Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41–47, (2022).
- Mukherjee, D. Effect of fear on two predator-one prey model in deterministic and fluctuating environment. Mathematics in Applied Sciences and Engineering, 2(1), 1-71, (2021).
- Mulugeta, B. T., Yu, L., & Ren, J. Bifurcation Analysis of a One-Prey and Two-Predators Model with Additional Food and Harvesting Subject to Toxicity. International Journal of Bifurcation and Chaos, 31(6), 2150089, (2021).
- Sarwardi, S., Mandal, P.K., & Ray, S. Dynamical behaviour of a two-predator model with prey refuge. Journal of Biological Physics, 39(4), 701-722, (2013).
- Alebraheem, J., & Abu-Hasan, Y. Persistence of predators in a two predators-one prey model with non-periodic solution. Applied Mathematical Sciences, 6(19), 943-956, (2012).
- Savitri, D., Suryanto, A., & Kusumawinahyu, W.M. Dynamical behavior of a modified lesliegower one prey-two predators with competition. Mathematics, 8(5), 699, (2020).
- Kharbanda, H., & Kumar, S. Asymptotic stability of one prey and two predators model with two functional responses. Ricerche Di Matematica, 68(2), 435–452, (2019).
- Pal, D., Santra, P., & Mahapatra, G.S. Dynamical behavior of three species predator-prey system with mutual support between non refuge prey. Ecological Genetics and Genomics, 3(5), 1-6, (2017).
- Vijaya, S., & Rekha, E. Prey–predator three species model using predator harvesting Holling type II functional. Biophysical Reviews and Letters, 11(2), 87-104, (2016).
- Laurie, H., & Venturino, E. A two-predator one-prey model of population dynamics influenced by herd behaviour of the prey. Theoretical Biology Forum, 111(1–2), 27–47, (2019).
- Wang, J., & Wang, M. Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis. Zeitschrift Fur Angewandte Mathematik Und Physik, 69(3), (2018).
- Pal, D., Mahapatra, G.S. Dynamic behavior of a predator–prey system of combined harvesting with interval-valued rate parameters. Nonlinear Dynamics, 83(4), 2113–2123, (2016).
- Pal, D., Mahaptra, G.S., Samanta, G.P. Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model. Mathematical Biosciences, 241(2), 181–187, (2013).
- Santra, P.K. & Mahapatra, G.S. Dynamical study of discrete-time prey-predator model with constant prey refuge under imprecise biological parameters. Journal of Biological Systems, 28(3), 681–699, (2020).
- Ghosh, D., Santra, P.K., & Mahapatra, G.S. Fear effect on a discrete-time prey predator model with imprecise biological parameters. In AIP Conference Proceedings (Vol. 2246), American Institute of Physics Inc. (2020).
- Santra, P., & Mahapatra, G.S. Discrete prey–predator model with square root functional response under imprecise biological parameters.In Springer Proceedings in Mathematics and Statistics, Springer, 320, 211–225, (2020).
- Mondal, B., Rahman, M.S., Sarkar, S., & Ghosh, U. Studies of dynamical behaviours of an imprecise predator-prey model with Holling type II functional response under interval uncertainty. European Physical Journal Plus, 137(1), (2022).
- Mahata, A., Mondal, S.P., Roy, B., & Alam, S. Study of two species prey-predator model in imprecise environment with MSY policy under different harvesting scenario. Environment, Development & Sustainability, 23(10), 14908–14932, (2021).
- Vargas-De-León, C. On the global stability of SIS, SIR and SIRS epidemic models with standard incidence. Chaos, Solitons and Fractals, 44(12), 1106–1110, (2011).
- Beretta, E. & Capasso, V. On the general structure of epidemic systems. Global asymptotic stability, Computers & Mathematics with Applications, 12(6), 677–694, (1986).
- Korobeinikov, A. & Wake, G.C. Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Applied Mathematics Letters, 15(8), 955–960, (2002).
- Goh, B.S. Global stability in two species interactions. Journal of Mathematical Biology, 3(3–4), 313–318, (1976).
- Korobeinikov, A. Lyapunov functions and global properties for SEIR and SEIS epidemic models. Mathematical Medicine and Biology, 21(2), 75–83, (2004).
- McCluskey, C.C. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences and Engineering, 3(4), 603–614, (2006).