Research Article
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A three-component prey-predator system with interval number

Year 2023, Volume: 3 Issue: 1, 1 - 16, 31.03.2023
https://doi.org/10.53391/mmnsa.1273908

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

  • 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).
Year 2023, Volume: 3 Issue: 1, 1 - 16, 31.03.2023
https://doi.org/10.53391/mmnsa.1273908

Abstract

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).
There are 28 citations in total.

Details

Primary Language English
Subjects Bioinformatics and Computational Biology, Applied Mathematics
Journal Section Research Articles
Authors

Dipankar Ghosh This is me 0000-0003-4897-0260

Prasun Kumar Santra This is me 0000-0002-1857-135X

Ghanshaym Singha Mahapatra This is me 0000-0002-5225-0445

Publication Date March 31, 2023
Submission Date January 15, 2023
Published in Issue Year 2023 Volume: 3 Issue: 1

Cite

APA Ghosh, D., Santra, P. K., & Mahapatra, G. S. (2023). A three-component prey-predator system with interval number. Mathematical Modelling and Numerical Simulation With Applications, 3(1), 1-16. https://doi.org/10.53391/mmnsa.1273908


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