## Journal of Applied Mathematics

### A Study of a Diseased Prey-Predator Model with Refuge in Prey and Harvesting from Predator

#### Abstract

In this paper, a mathematical model of a prey-predator system with infectious disease in the prey population is proposed and studied. It is assumed that there is a constant refuge in prey as a defensive property against predation and harvesting from the predator. The proposed mathematical model is consisting of three first-order nonlinear ordinary differential equations, which describe the interaction among the healthy prey, infected prey, and predator. The existence, uniqueness, and boundedness of the system’ solution are investigated. The system's equilibrium points are calculated with studying their local and global stability. The persistence conditions of the proposed system are established. Finally the obtained analytical results are justified by a numerical simulation.

#### Article information

Source
J. Appl. Math., Volume 2018 (2018), Article ID 2952791, 17 pages.

Dates
Accepted: 14 November 2018
First available in Project Euclid: 10 January 2019

https://projecteuclid.org/euclid.jam/1547089319

Digital Object Identifier
doi:10.1155/2018/2952791

Mathematical Reviews number (MathSciNet)
MR3892179

#### Citation

Abdulghafour, Ahmed Sami; Naji, Raid Kamel. A Study of a Diseased Prey-Predator Model with Refuge in Prey and Harvesting from Predator. J. Appl. Math. 2018 (2018), Article ID 2952791, 17 pages. doi:10.1155/2018/2952791. https://projecteuclid.org/euclid.jam/1547089319

#### References

• M. R. May, Stability and Complexity in Model Ecosystems, Princeton University, Princeton, NJ, USA, 1973.
• W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences, vol. 115, no. 772, pp. 700–721, 1927.
• R. M. Anderson and R. M. May, “The invasion, persistence and spread of infectious diseases within animal and plant communities,” Philosophical Transactions of the Royal Society B: Biological Sciences, vol. 314, no. 1167, pp. 533–570, 1986.
• J. Chattopadhyay and O. Arino, “A predator-prey model with disease in the prey,” Nonlinear Analysis: Theory, Methods & Applications, vol. 36, pp. 747–766, 1999.
• Z. Z. Ma, F. D. Chen, C. Q. Wu, and W. L. Chen, “Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference,” Applied Mathematics and Computation, vol. 219, no. 15, pp. 7945–7953, 2013.
• P. Jyoti Pal and P. K. Mandal, “Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect,” Mathematics and Computers in Simulation, vol. 97, pp. 123–146, 2014.
• R. K. Naji and S. J. Majeed, “The dynamical analysis of a prey-predator model with a refuge-stage structure prey population,” International Journal of Differential Equations, vol. 2016, Article ID 2010464, 10 pages, 2016.
• A. S. Abdulghafour and R. K. Naji, “The impact of refuge and harvesting on the dynamics of prey-predator system,” Science International (Lahore), vol. 30, no. 2, pp. 315–323, 2018.
• D. Greenhalgh, Q. J. Khan, and F. I. Lewis, “Hopf bifurcation in two SIRS density dependent epidemic models,” Mathematical and Computer Modelling, vol. 39, no. 11-12, pp. 1261–1283, 2004.
• T. Zhang, J. Liu, and Z. Teng, “Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 1, pp. 293–306, 2010.
• P. Yongzhen, S. Li, C. Li, and S. Chen, “The effect of constant and pulse vaccination on an SIR epidemic model with infectious period,” Applied Mathematical Modelling, vol. 35, no. 8, pp. 3866–3878, 2011.
• R. K. Naji and B. H. Abdulateef, “The dynamics of SICIR model with nonlinear incidence rate and saturated treatment function,” Science International (Lahore), vol. 29, no. 6, pp. 1223–1236, 2017.
• M. S. Rahman and S. Chakravarty, “A predator-prey model with disease in prey,” Nonlinear Analysis: Modelling and Control, vol. 18, no. 2, pp. 191–209, 2013.
• S. Jana and T. K. Kar, “Modeling and analysis of a prey-predator system with disease in the prey,” Chaos, Solitons & Fractals, vol. 47, pp. 42–53, 2013.
• S. Kant and V. Kumar, “Dynamics of a prey-predator system with infection in prey,” Electronic Journal of Differential Equations, vol. 2017, no. 209, pp. 1–27, 2017.
• M. Haque and E. Venturino, “Increase of the prey may decrease the healthy predator population in presence of a disease in predator,” HERMIS, vol. 7, pp. 38–59, 2006.
• M. Haque, “A predator-prey model with disease in the predator species only,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2224–2236, 2010.
• K. P. Das, “A mathematical study of a predator-prey dynamics with disease in predator,” ISRN Applied Mathematics, vol. 2011, Article ID 807486, 16 pages, 2011.
• M. V. R. Murthy and D. K. Bahlool, “Modeling and Analysis of a Prey-Predator System with Disease in Predator,” IOSR Journal of Mathematics, vol. 12, no. 1, pp. 21–40, 2016.
• Y. Hsieh and C. Hsiao, “Predator-prey model with disease infection in both populations,” Mathematical Medicine and Biology, vol. 25, no. 3, pp. 247–266, 2008.
• K. P. Das, K. Kundu, and J. Chattopadhyay, “A predator-prey mathematical model with both the populations affected by diseases,” Ecological Complexity, vol. 8, no. 1, pp. 68–80, 2011.
• K. P. Das, S. K. Sasmal, and J. Chattopadhyay, “Disease control through harvesting - conclusion drawn from a mathematical study of a predator-prey model with disease in both the population,” International Journal of Biomathematics and Systems Biology, vol. 1, no. 1, pp. 1–29, 2014.
• S. Kant and V. Kumar, “Stability analysis of predator-prey system with migrating prey and disease infection in both species,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 42, pp. 509–539, 2017.
• N. Bairagi, S. Chaudhuri, and J. Chattopadhyay, “Harvesting as a disease control measure in an eco-epidemiological system–-a theoretical study,” Mathematical Biosciences, vol. 217, no. 2, pp. 134–144, 2009.
• R. Bhattacharyya and B. Mukhopadhyay, “On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3824–3833, 2010.
• S. Gakkhar and K. B. Agnihotri, “The dynamics of disease transmission in a prey predator system with harvesting of prey,” International Journal of Advanced Research in Computer Engineering and Technology, vol. 1, pp. 229–239, 2012.
• Q. Yue, “Dynamics of a modified Leslie–Gower predator–prey model with Holling-type II schemes and a prey refuge,” SpringerPlus, vol. 5, no. 1, 2016.
• J. Ghosh, B. Sahoo, and S. Poria, “Prey-predator dynamics with prey refuge providing additional food to predator,” Chaos, Solitons & Fractals, vol. 96, pp. 110–119, 2017.
• S. Gakkhar and R. K. Naji, “Existence of chaos in two-prey, one-predator system,” Chaos, Solitons & Fractals, vol. 17, no. 4, pp. 639–649, 2003.
• C. W. Clark, “Aggregation and fishery dynamics: a theoretical study of schooling and the Purse Seine tuna fisheries,” NOAA Fisheries Bulletin, vol. 77, no. 2, pp. 317–337, 1979. \endinput