Dynamical of Ratio-Dependent Eco-epidemical Model with Prey Refuge

Adin Lazuardy Firdiansyah

Abstract


This paper discusses the dynamic analysis of three species in the eco-epidemiology model by considering the ratio-dependent function and prey refuge. The prey refuge is applied under the fact that infected prey has protection instincts that allow it to reduce predation risk. Here, we get the boundedness and three equilibrium points where are existence under certain conditions. In the model, three equilibrium points are locally asymptotically stable and one of the equilibrium points is globally asymptotically stable. We find that the system undergoes Hopf bifurcation around the interior equilibrium point by choosing  as a bifurcation parameter. We also find a condition for uniform persistence. Finally, several simulations of numerical are performed not only to illustrate the analytical results but also to illustrate the effect of the prey refuge.              


Keywords


eco-epidemiology model; global stable; Hopf bifurcation; local stable; persistence

Full Text:

PDF

References


C. Maji, D. Kesh, and D. Mukherjee, “Bifurcation and global stability in an eco-epidemic model with refuge,” Energy, Ecol. Environ., vol. 4, no. 3, pp. 103–115, 2019, doi: 10.1007/s40974-019-00117-6.

A. K. Pal and G. P. Samanta, “Stability analysis of an eco-epidemiological model incorporating a prey refuge,” Nonlinear Anal. Model. Control, vol. 2014, 2014, doi: 10.1155/2014/978758.

B. Mukhopadhyay and R. Bhattacharyya, “Effects of deterministic and random refuge in a prey-predator model with parasite infection,” Math. Biosci., vol. 239, no. 1, pp. 124–130, 2012, doi: 10.1016/j.mbs.2012.04.007.

S. P. Bera, A. Maiti, and G. P. Samanta, “A prey-predator model with infection in both prey and predator,” Filomat, vol. 29, no. 8, pp. 1753–1767, 2015, doi: 10.2298/FIL1508753B.

C. Huang, H. Zhang, J. Cao, and H. Hu, “Stability and Hopf Bifurcation of a Delayed Prey-Predator Model with Disease in the Predator,” Int. J. Bifurc. Chaos, vol. 29, no. 7, p. 23, 2019, doi: 10.1142/S0218127419500913.

S. A. Wuhaib and Y. A. B. U. Hasan, “Predator-Prey Interactions With Harvesting of Predator With Prey in Refuge,” Commun. Math. Biol. Neurosci., vol. 2013, no. 1, pp. 1–19, 2013.

N. Apreutesei and G. Dimitriu, “On a prey-predator reactiondiffusion system with Holling type III functional response,” J. Comput. Appl. Math., vol. 235, no. 2, pp. 366–379, 2010, doi: 10.1016/j.cam.2010.05.040.

A. K. Misra and B. Dubey, “A ratio-dependent predator-prey model with delay and harvesting,” J. Biol. Syst., vol. 18, no. 2, pp. 437–453, 2010, doi: 10.1142/S021833901000341X.

U. De Lausanne and S. Brook, “Coupling in predator-prey dynamics: ratio-dependence,” J. Theor. Biol., vol. 139, pp. 311–326, 1989.

M. Verma and A. K. Misra, “Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect,” Bull. Math. Biol., vol. 80, no. 3, pp. 626–656, 2018, doi: 10.1007/s11538-018-0394-6.

S. Wang, Z. Ma, and W. Wang, “Dynamical behavior of a generalized eco-epidemiological system with prey refuge,” Adv. Differ. Equations, vol. 2018, no. 1, pp. 1–20, 2018, doi: 10.1186/s13662-018-1704-x.




DOI: https://doi.org/10.18860/ca.v6i4.10827

Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Adin Lazuardy Firdiansyah

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

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.