Araştırma Makalesi
PDF Mendeley EndNote BibTex Kaynak Göster

Mathematical Analysis of Discrete Fractional Prey-Predator Model with Fear Effect and Square Root Functional Response

Yıl 2022, Cilt 43, Sayı 1, 72 - 76, 30.03.2022
https://doi.org/10.17776/csj.934046

Öz

This paper investigates the dynamics of a discrete fractional prey-predator system. The prey-predator interaction is modelled using the square root functional response, which appropriately models systems in which the prey exhibits a strong herd structure, implying that the predator generally interacts with the prey along the herd's outer corridor. Some recent field experiments and studies show that predators affect prey by directly killing and inducing fear in prey, reducing prey species' reproduction rate. Considering these facts, we propose a mathematical model to study herd behaviour and fear effect in the prey-predator system. We show algebraically equilibrium points and their stability condition. Condition for Neimark-Sacker bifurcation, Flip bifurcation and Fold bifurcation are given. Phase portraits and bifurcation diagrams are portraits that depict the model's behaviour based on some hypothetical data. Numerical simulations reveal the model's rich dynamics as a result of fear and fractional order.

Kaynakça

  • [1] Mahapatra, G.S., Santra, P., Prey-predator model for Optimal Harvesting with Functional Response Incorporating Prey Refuge, International Journal of Biomathematics, 09 (2016) ID1650014.
  • [2] 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 (2017) 1-6.
  • [3] Al-Basyouni, K.S., Khan, A.Q., Discrete-Time Predator-Prey Model with Bifurcations and Chaos, Mathematical Problems in Engineering, 2020 (2020) ID8845926.
  • [4] Kangalgil, F., Isik, S., Controlling chaos and neimark-sacker bifurcation in a discrete-time predator-prey system, Hacettepe Journal of Mathematics and Statistics, 49(5) (2020) 1761-1776.
  • [5] Liu, W., Jiang, Y., Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with harvesting, International Journal of Biomathematics, 13(1) (2020) ID1950093.
  • [6] Owolabi K.M., Dynamical behaviour of fractional-order predator-prey system of Holling-type, Discrete and Continuous Dynamical Systems - Series S, 13 (3) (2020) 823-834.
  • [7] Nosrati K., Shafiee M., Dynamic analysis of fractional-order singular Holling type-II predator--prey system, Applied Mathematics and Computation, 313 (2017) 159-179.
  • [8] El-Saka H.A.A., Lee S., Jang B., Dynamic analysis of fractional-order predator--prey biological economic system with Holling type II functional response, Nonlinear Dynamics, 96 (1) (2019) 407-416.
  • [9] Bonyah E., Atangana A., Elsadany A.A., A fractional model for predator-prey with omnivore, Chaos, 29 (1) (2019).
  • [10] Shi Y., Ma Q., Ding X., Dynamical behaviors in a discrete fractional-order predator-prey system, Filomat, 32(17) (2018) 5857-5874.
  • [11] Din Q., Complexity and chaos control in a discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 49 (2017) 113-134.
  • [12] Zhao M., Du Y., Stability of a Discrete-Time Predator-Prey System with Allee Effect, Nonlinear Analysis and Differential Equations, 4 (5) (2016) 225 - 233.
  • [13] 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) (2020) 681-699.
  • [14] Santra P. K., Mahapatra G. S., Phaijoo G. R., Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley--Martin Functional Response Incorporating Proportional Prey Refuge, Mathematical Problems in Engineering, (2020) ID 5309814.
  • [15] Baydemir P., Merdan H., Karaoglu E., Sucu G., Complex Dynamics of a Discrete-Time Prey-Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos, International Journal of Bifurcation and Chaos, 30(10) (2020) ID2050149.
  • [16] Santra P.K., Mahapatra G.S., Phaijoo G.R., Bifurcation analysis and chaos control of discrete prey–predator model incorporating novel prey–refuge concept, Computational and Mathematical Methods, (2021) e1185.
  • [17] Rech P.C., On Two Discrete-Time Counterparts of a Continuous-Time Prey-Predator Model, Brazilian Journal of Physics, 50(2) (2020) 119-123.
  • [18] Singh A., Deolia P., Dynamical analysis and chaos control in discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 90 (2020) ID105313.
  • [19] Khan A. Q., Khalique T., Bifurcations and chaos control in a discrete-Time biological model, International Journal of Biomathematics, 13(4) (2020).
  • [20] Rozikov U.A., Shoyimardonov S.K., Leslie's prey-predator model in discrete time, International Journal of Biomathematics, 13(6) (2020) ID2050053.
  • [21] Khan A.Q., Khalique T., Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka-Volterra model, Mathematical Methods in the Applied Sciences, 43(9) (2020) 5887-5904.
  • [22] Chakraborty P., Ghosh U., Sarkar S., Stability and bifurcation analysis of a discrete prey-predator model with square-root functional response and optimal harvesting, Journal of Biological Systems, 28 (1) (2020) 91-110.
  • [23] Santra P.K., Discrete-time prey-predator model with logistic growth for prey incorporating square root functional response, Jambura Journal of Biomathematics, 1 (2) (2020) 41-48.
  • [24] Ma R., Bai Y., Wang F., Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor, Journal of Applied Analysis and Computation, 10 (4) (2020) 1683-1697.
  • [25] Wang X., Zanette L., Zou X., Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73 (5) (2016) 1-26.
  • [26] Elaydi S. N., Discrete chaos: with applications in science and engineering, Chapman & Hall/CRC, 2007

Yıl 2022, Cilt 43, Sayı 1, 72 - 76, 30.03.2022
https://doi.org/10.17776/csj.934046

Öz

Kaynakça

  • [1] Mahapatra, G.S., Santra, P., Prey-predator model for Optimal Harvesting with Functional Response Incorporating Prey Refuge, International Journal of Biomathematics, 09 (2016) ID1650014.
  • [2] 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 (2017) 1-6.
  • [3] Al-Basyouni, K.S., Khan, A.Q., Discrete-Time Predator-Prey Model with Bifurcations and Chaos, Mathematical Problems in Engineering, 2020 (2020) ID8845926.
  • [4] Kangalgil, F., Isik, S., Controlling chaos and neimark-sacker bifurcation in a discrete-time predator-prey system, Hacettepe Journal of Mathematics and Statistics, 49(5) (2020) 1761-1776.
  • [5] Liu, W., Jiang, Y., Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with harvesting, International Journal of Biomathematics, 13(1) (2020) ID1950093.
  • [6] Owolabi K.M., Dynamical behaviour of fractional-order predator-prey system of Holling-type, Discrete and Continuous Dynamical Systems - Series S, 13 (3) (2020) 823-834.
  • [7] Nosrati K., Shafiee M., Dynamic analysis of fractional-order singular Holling type-II predator--prey system, Applied Mathematics and Computation, 313 (2017) 159-179.
  • [8] El-Saka H.A.A., Lee S., Jang B., Dynamic analysis of fractional-order predator--prey biological economic system with Holling type II functional response, Nonlinear Dynamics, 96 (1) (2019) 407-416.
  • [9] Bonyah E., Atangana A., Elsadany A.A., A fractional model for predator-prey with omnivore, Chaos, 29 (1) (2019).
  • [10] Shi Y., Ma Q., Ding X., Dynamical behaviors in a discrete fractional-order predator-prey system, Filomat, 32(17) (2018) 5857-5874.
  • [11] Din Q., Complexity and chaos control in a discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 49 (2017) 113-134.
  • [12] Zhao M., Du Y., Stability of a Discrete-Time Predator-Prey System with Allee Effect, Nonlinear Analysis and Differential Equations, 4 (5) (2016) 225 - 233.
  • [13] 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) (2020) 681-699.
  • [14] Santra P. K., Mahapatra G. S., Phaijoo G. R., Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley--Martin Functional Response Incorporating Proportional Prey Refuge, Mathematical Problems in Engineering, (2020) ID 5309814.
  • [15] Baydemir P., Merdan H., Karaoglu E., Sucu G., Complex Dynamics of a Discrete-Time Prey-Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos, International Journal of Bifurcation and Chaos, 30(10) (2020) ID2050149.
  • [16] Santra P.K., Mahapatra G.S., Phaijoo G.R., Bifurcation analysis and chaos control of discrete prey–predator model incorporating novel prey–refuge concept, Computational and Mathematical Methods, (2021) e1185.
  • [17] Rech P.C., On Two Discrete-Time Counterparts of a Continuous-Time Prey-Predator Model, Brazilian Journal of Physics, 50(2) (2020) 119-123.
  • [18] Singh A., Deolia P., Dynamical analysis and chaos control in discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 90 (2020) ID105313.
  • [19] Khan A. Q., Khalique T., Bifurcations and chaos control in a discrete-Time biological model, International Journal of Biomathematics, 13(4) (2020).
  • [20] Rozikov U.A., Shoyimardonov S.K., Leslie's prey-predator model in discrete time, International Journal of Biomathematics, 13(6) (2020) ID2050053.
  • [21] Khan A.Q., Khalique T., Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka-Volterra model, Mathematical Methods in the Applied Sciences, 43(9) (2020) 5887-5904.
  • [22] Chakraborty P., Ghosh U., Sarkar S., Stability and bifurcation analysis of a discrete prey-predator model with square-root functional response and optimal harvesting, Journal of Biological Systems, 28 (1) (2020) 91-110.
  • [23] Santra P.K., Discrete-time prey-predator model with logistic growth for prey incorporating square root functional response, Jambura Journal of Biomathematics, 1 (2) (2020) 41-48.
  • [24] Ma R., Bai Y., Wang F., Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor, Journal of Applied Analysis and Computation, 10 (4) (2020) 1683-1697.
  • [25] Wang X., Zanette L., Zou X., Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73 (5) (2016) 1-26.
  • [26] Elaydi S. N., Discrete chaos: with applications in science and engineering, Chapman & Hall/CRC, 2007

Ayrıntılar

Birincil Dil İngilizce
Konular Ekoloji, Matematik, Çevre Bilimleri
Bölüm Natural Sciences
Yazarlar

Prasun Kumar SANTRA Bu kişi benim (Sorumlu Yazar)
India

Yayımlanma Tarihi 30 Mart 2022
Başvuru Tarihi 6 Mayıs 2021
Kabul Tarihi 24 Şubat 2022
Yayınlandığı Sayı Yıl 2022, Cilt 43, Sayı 1

Kaynak Göster

APA Santra, P. K. (2022). Mathematical Analysis of Discrete Fractional Prey-Predator Model with Fear Effect and Square Root Functional Response . Cumhuriyet Science Journal , 43 (1) , 72-76 . DOI: 10.17776/csj.934046