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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Dynamics of Modified Leslie-Gower Harvested Predator-Prey Model with Beddington-DeAngelis Functional Response

Discontinuity, Nonlinearity, and Complexity 8(2) (2019) 111--125 | DOI:10.5890/DNC.2019.06.001

R. Sivasamy$^{1}$, M. Sivakumar$^{2}$, K. Sathiyanathan$^{1}$, K. Balachandran$^{2}$

$^{1}$ Department of Mathematics, SRMV College of Arts and Science, Coimbatore - 641020, Tamil Nadu, India

$^{2}$ Department of Mathematics, Bharathiar University, Coimbatore - 641046, Tamil Nadu, India

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This paper considers a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and nonlinear prey harvesting strategy subject to the zero-flux boundary conditions. To understand the dynamics of the considered system, we derive sufficient conditions for permanence analysis, local stability, global stability and Hopf bifurcation of interior equilibrium point. Further we also investigate the existence and non-existence of non-constant positive steady state solutions.


  1. [1]  Lotka, A.J. (1956), Elements of mathematical biology, New York: Dover.
  2. [2]  Leslie, P. H. (1948), Some further notes on the use of matrices in population mathematics, Biometrika, 35(3/4), 213-245.
  3. [3]  Berryman, A.A. (1992), The origin and evolution of predator-prey theory, Ecology, 73(5), 1530-1535.
  4. [4]  Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predator-prey type of interactionbetween two species, Biometrika, 47(3/4), 219-234.
  5. [5]  Holling, C.S. (1965), The functional response of predators to prey density and its role in mimicry and populationregulation, The Memoirs of the Entomological Society of Canada, 97(S45), 5-60.
  6. [6]  Oaten, A. and Murdoch, W.W. (1975), Functional response and stability in predator-prey systems, The AmericanNaturalist, 109(967), 289-298.
  7. [7]  Bie, Q.,Wang, Q. and Yao, Z.A. (2014), Cross-diffusion induced instability and pattern formation for a Holling type-IIpredator-prey model, Applied Mathematics and Computation, 247, 1-12.
  8. [8]  Li, H. and Takeuchi, Y. (2011), Dynamics of the density dependent predator-prey system with Beddington-DeAngelisfunctional response, Journal of Mathematical Analysis and Applications, 374(2), 644-654.
  9. [9]  Aziz-Alaoui,M.A. and Okiye, M.D. (2003), Boundedness and global stability for a predator-preymodel with modifiedLeslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 16(7), 1069-1075.
  10. [10]  Li, Z., Han, M. and Chen, F. (2012), Global stability of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes, International Journal of Biomathematics, 5(06), 1250057.
  11. [11]  Pal, P.J. and Mandal, P.K. (2014), Bifurcation analysis of a modified Leslie-Gower predator-prey model withBeddington-DeAngelis functional response and strong Allee effect, Mathematics and Computers in Simulation, 97,123-146.
  12. [12]  Cao, J. and Yuan, R. (2016), Bifurcation analysis in a modified Lesile-Gower model with Holling type II functionalresponse and delay, Nonlinear Dynamics, 84(3), 1341-1352.
  13. [13]  Beddington, J.R. (1975),Mutual interference between parasites or predators and its effect on searching efficiency, TheJournal of Animal Ecology, 44(1), 331-340.
  14. [14]  DeAngelis, D.L., Goldstein, R.A. and O’neill, R.V. (1975), A model for tropic interaction, Ecology, 56(4), 881-892.
  15. [15]  Yu, S. (2014), Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response,Advances in Difference Equations, 2014(1), 84.
  16. [16]  Indrajaya, D., Suryanto, A., and Alghofari, A.R. (2016), Dynamics of modified Leslie-Gower predator-prey modelwith Beddington-DeAngelis functional response and additive Allee effect, International Journal of Ecology and Development,31(3), 60-71.
  17. [17]  Abid, W., Yafia, R., Alaoui, M.A., Bouhafa, H., and Abichou, A. (2015), Global Dynamics on a circular domain ofa diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type, EvolutionEquations and Control Theory, 4, 115-129.
  18. [18]  Li, Y. and Wang, M. (2015), Dynamics of a diffusive predator-prey model with modified Leslie-Gower term andMichaelis-Menten type prey harvesting, Acta ApplicandaeMathematicae, 140(1), 147-172.
  19. [19]  Yuan, R., Jiang, W., and Wang, Y. (2015), Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-preymodel with time-delay and prey harvesting, Journal of Mathematical Analysis and Applications, 422(2), 1072-1090.
  20. [20]  Yang, R. and Zhang, C. (2017), Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delayand prey harvesting, Nonlinear Dynamics, 87(2), 863-878.
  21. [21]  Yin, H., Xiao, X., Wen, X., and Liu, K. (2014), Pattern analysis of a modified Leslie-Gower predator-prey model withCrowley-Martin functional response and diffusion, Computers & Mathematics with Applications, 67(8), 1607-1621.