The Dynamics and Analysis of Stage-Structured Predator-Prey Model With Prey Refuge and Harvesting Involving Disease in Prey Population

Authors

  • Azhar Abbas Majeed Department of Mathematic, Collage of Science, University of Baghdad, Baghdad

DOI:

https://doi.org/10.26713/cma.v10i3.1031

Keywords:

Eco-epidemiological model, SI epidemic disease, Prey-predator model, Refuge, Stage structure, Harvest, Lyapunove function

Abstract

In this paper, a mathematical model consisting of the prey-predator model with SI infectious disease in prey is proposed and analyzed. The model includes harvesting on the infected prey population, it is assume that the disease is not transmitted from prey to predator. In addition, the disease spread by contact between susceptible individuals and infected individuals, the mature predator only can predate the susceptible and infected prey which are outside refuge according to Lotka-Volterra type of functional response. While, the immature predator depends completely in it's feeding on the mature predator. The existence, uniqueness and boundedness of the solution are discussed. The stability analysis of all possible equilibrium points is studied. Also, Lyapunove function is used to study the global dynamics of the model. Further, the effect of the disease, refuge and harvest on the dynamical of the system is discussed using numerical simulation.

Downloads

Download data is not yet available.

References

K. B. Agnihotri and S. Gakkhar, The dynamics of disease transmission in a prey-predator model system with harvesting of prey, International Journal of Advanced Research in Computer Engineering and Technology 1 (2012), 1 – 17.

W. G. Aiello and H. I. Freedman, A time delay model of single species growth with stage structure, Math. Biosci. (1990), 101 – 139, DOI: 10.1016/0025-5564(90)90019-U.

W. Arild, An analysis of discrete stage-structured prey and prey-predator population models, Discrete Dynamics in Nature and Society 2017 (2017), Article ID 9475854, 11 pages, DOI: 10.1155/2017/9475854.

A. B. Ashine, Dynamics of prey-predator model with Holling-type II and modified Leslie-Gower schemes with prey refuge, Journal of Mathematics and Computer 4(1) (2017), 1 – 12.

A. B. Ashine, Prey-predator model with Holling-type II and modified Leslie-Gower schemes with prey refuge, African Journal of Basic and Applied Sciences 8(5) (2016), 270 – 275.

P. Auger, R. McHich, T. Chowdhury, G. Sallet, M. Tchuente and J. Chattopadhyay, Effects of a disease affecting a predator on the dynamics of a predator-prey system, Journal of Theoretical Biology 258(3) (2009), 344 – 351, DOI: 10.1016/j.jtbi.2008.10.030.

N. Bairagi, S. Chaudhuri and J. Chattopadhyay, Harvesting as a disease control measure in an eco-epidemiological system A theoretical study, Mathematical Biosciences 217 (2009), 134 – 144, DOI: 10.1016/j.mbs.2008.11.002.

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 11 (2010), 3824 – 3833, DOI: 10.1016%2Fj.nonrwa.2010.02.012.

F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting, J. Math. Biol. 12 (1981), 101 – 114, DOI: 10.1007/BF00275206.

M. Falconi, M. Huenchucona and C. Vidal, Stability and global dynamic of a stage-structured predator-prey model with group defense mechanism of the prey, Applied Mathematics and Computation 270 (2015), 47 – 61, DOI: 10.1016/j.amc.2015.07.109.

R. P. Gupta and P. Chandra, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete and Continuous Dynamical Systems Series B 20(2) (2015), 423 – 443, DOI: 10.3934/dcdsb.2015.20.423.

J. K. Hale, Ordinary Differential Equation, Wiley-Interscience, New York (1969).

K. Q. Khalif, A. A. Mated and R. K. Naji, The dynamics of an SIS epidemic disease with contact and external source, Journal of Mathematical Theory and Modeling 5 (2015), 184 – 197.

A. A. Majeed and I. I. Shawka, The stability analysis of eco-epidemiological system with disease, Gen. Math. Notes (2016), 52 – 72.

T. Das, R. N. Mukherjee and K. S. Chaudhuri, Bioeconomic harvesting of a prey-predator Shery, J. Biol. Dyn. 3 (2009), 447 – 462, DOI: 10.1080/17513750802560346.

R. K. Naji and A. N. Mustafa, The dynamics of an eco-epidemiological model with nonlinear incidence rate, Journal of Applied Mathematics, 2012 (2012), Article ID 852631, 24 pages, DOI: 10.1155/2012/852631.

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol. 49 (1987), 253.

M. Roy and R. D. Holt, Effect of predation on host-pathogen dynamics in SIR models, Theoretical Population Biology 73(3) (2008), 319 – 331, DOI: 10.1016/j.tpb.2007.12.008.

H. A. Satar, The effect of disease and harvesting on the dynamics of prey-predator system, Journal of Science Iraqi 57 (2016), 693 – 704.

K. Shashi and K. Vivek, Dynamical behavior of a stage structured eco-epidemiologcal model, Differential Equations and Applications 7(4) (2015), 503 – 515, DOI: 10.7153/dea-07-29.

K. Sujatha and M. Gunasekaran, Dynamics in a harvested prey-predator model with susceptibleinfected-susceptible (SIS) epidemic disease in the prey, Advances in Applied Mathematical Biosciences 1 (2016), 23 – 31.

S. A. Wuhaiba and Y. A. Hasan, A predator-infected prey model with harvesting of infected prey, Science Asia 39S (2013), 37 – 41, DOI: 10.2306/scienceasia1513-1874.2013.39S.037.

Y. N. Xiao and L. S. Chen, Global stability of a predator-prey system with stage structure for the predator, Acta Mathematica Sinica-English Series 20(1) (2004), 63 – 70, DOI: 10.1007/s10114-002-0234-2.

L. Yang and S. Zhang, Global stability of a stage-structured predator-prey model with stochastic perturbation, Discrete Dynamics in Nature and Society 2014 (2014), Article ID 512817, 8 pages, DOI: 10.1155/2014/512817.

Downloads

Published

30-09-2019
CITATION

How to Cite

Majeed, A. A. (2019). The Dynamics and Analysis of Stage-Structured Predator-Prey Model With Prey Refuge and Harvesting Involving Disease in Prey Population. Communications in Mathematics and Applications, 10(3), 337–359. https://doi.org/10.26713/cma.v10i3.1031

Issue

Vol. 10 No. 3 (2019)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.