Dynamics of Infected Predator-Prey System with Nonlinear Incidence Rate and Prey in Refuge

  • Adin Lazuardy Firdiansyah STAI Muhammadiyah Probolinggo
Keywords: Predator-prey system; Nonlinear incidence rate; Refuge; Stability

Abstract

A predator-prey system with nonlinear incidence rate and refuging in prey is proposed to describe behavior change of certain infected diseases on healthy prey when the number of infected prey is getting large, while predator can predate prey by accessing refuging in prey. Therefore, this paper discusses the dynamics behavior predator-prey model with the spread of infected disease that is denoted by nonlinear incidence rate and adding prey refuge. We find the existence of eight non-negative equilibrium in the model, which their local stability has been determined. Furthermore, we also observe the prey refuge properties in the model. We find that prey refuge can prevent extinction in prey populations. In the end, some numerical solutions are carried out to illustrate our analytic results. For future work, we can investigate the harvesting effect in both populations, which is disease control in the predator-prey model with the spread of infected disease.

Downloads

Download data is not yet available.

References

W. O. Kermack and A. G. McKendrick, “A Contribution to the Mathematical Theory of Epidemics,” Proc. R. Soc. London, vol. 115, no. 772, p. 22, 1927.

R. M. Anderson and R. M. May, “The invasion, persistence and spread of infectious diseases within animal and plant communities,” Philos. Trans. R. Soc. Lond. B. Biol. Sci., vol. 314, no. 1167, pp. 533–570, 1986, doi: 10.1098/rstb.1986.0072.

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.

S. Kant and V. Kumar, “Analysis of an eco-epidemiological model with migrating and refuging prey,” Springer Proc. Math. Stat., vol. 143, pp. 17–36, 2015, doi: 10.1007/978-81-322-2485-3_2.

A. K. Pal and G. P. Samanta, “Stability analysis of an eco-epidemiological model incorporating a prey refuge,” Nonlinear Anal. Model. Control, vol. 15, no. 4, pp. 473–491, 2010, doi: 10.15388/na.15.4.14319.

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.

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.

S. Kant and V. Kumar, “Stability analysis of predator–prey system with migrating prey and disease infection in both species,” Appl. Math. Model., vol. 42, pp. 509–539, 2017, doi: 10.1016/j.apm.2016.10.003.

A. Pusparani, W. M. Kusumawinahyu, and Trisilowati, “Dynamical Analysis of Infected Predator-Prey Model with Saturated Incidence Rate,” IOP Conf. Ser. Mater. Sci. Eng., vol. 546, no. 5, p. 8, 2019, doi: 10.1088/1757-899X/546/5/052055.

V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Math. Biosci., vol. 42, no. 1–2, pp. 43–61, 1978, doi: 10.1016/0025-5564(78)90006-8.

W. min Liu, S. A. Levin, and Y. Iwasa, “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,” J. Math. Biol., vol. 23, no. 2, pp. 187–204, 1986, doi: 10.1007/BF00276956.

S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” J. Differ. Equ., vol. 188, no. 1, pp. 135–163, 2003, doi: 10.1016/S0022-0396(02)00089-X.

R. K. Naji and A. N. Mustafa, “The dynamics of an eco-epidemiological model with nonlinear incidence rate,” J. Appl. Math., vol. 2012, p. 24, 2012, doi: 10.1155/2012/852631.

A. P. Maiti, C. Jana, and D. K. Maiti, “A delayed eco-epidemiological model with nonlinear incidence rate and Crowley–Martin functional response for infected prey and predator,” Nonlinear Dyn., vol. 98, no. 2, pp. 1137–1167, 2019, doi: 10.1007/s11071-019-05253-6.

S. Sharma and G. P. Samanta, “A Leslie-Gower predator-prey model with disease in prey incorporating a prey refuge,” Chaos, Solitons and Fractals, vol. 70, no. 1, pp. 69–84, 2015, doi: 10.1016/j.chaos.2014.11.010.

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.

M. Onana, B. Mewoli, and J. J. Tewa, “Hopf bifurcation analysis in a delayed Leslie–Gower predator–prey model incorporating additional food for predators, refuge and threshold harvesting of preys,” Nonlinear Dyn., vol. 100, no. 3, pp. 3007–3028, 2020, doi: 10.1007/s11071-020-05659-7.

Y. Cai, C. Zhao, W. Wang, and J. Wang, “Dynamics of a Leslie-Gower predator-prey model with additive Allee effect,” Appl. Math. Model., vol. 39, no. 7, pp. 2092–2106, 2015, doi: 10.1016/j.apm.2014.09.038.

CROSSMARK
Published
2020-10-31
DIMENSIONS
How to Cite
FirdiansyahA. L. (2020). Dynamics of Infected Predator-Prey System with Nonlinear Incidence Rate and Prey in Refuge. Jurnal Matematika MANTIK, 6(2), 123-134. https://doi.org/10.15642/mantik.2020.6.2.123-134