Skip Navigation Links
Journal of Environmental Accounting and Management
António Mendes Lopes (editor), Jiazhong Zhang(editor)
António Mendes Lopes (editor)

University of Porto, Portugal


Jiazhong Zhang (editor)

School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China

Fax: +86 29 82668723 Email:

Dynamical Analysis of a Crowley-Martin Predator-Prey Model with Prey Harvesting and Discrete Time-Delay

Journal of Environmental Accounting and Management 11(1) (2023) 1--22 | DOI:10.5890/JEAM.2023.03.001

Anindita Bhattacharyya$^1$, Ashok Mondal$^2$, Sanghita Bose$^1$, A. K. Pal$^3$

Download Full Text PDF



The present study deals with the dynamical response of a predator-prey model in which the prey has been subjected to harvesting. The proposed model considers a Crowley-Martin response function which is subjected to Michaelis-Menten type prey harvesting. It is first shown that the system is bounded and the conditions of existence and stability of the equilibria of the proposed model have been furnished. Next the presence of Hopf bifurcation and limit cycles have been shown to explain the transition of the model from a stable to an unstable one. It is seen that harvesting effort has a remarkable effect on the dynamics of the system and can make the system undergo unstability by reaching beyond a critical value. Considering the discrete time delay as bifurcating parameters, the conditions for existence of limit cycle under which the system admits a Hopf-bifurcation are investigated. The detailed study for direction of Hopf-bifurcation have been derived with the use of both the normal form and the central manifold theory.


The authors are grateful to the anonymous referees and the Associate Editor (Prof. Dimitri Volchenkov) for their careful reading, valuable comments and helpful suggestion, which have helped them to improve the presentation of this work significantly.


  1. [1]  Gupta, R.P. and Chandra, P. (2013), Bifurcation analysis of modified Leslie-Gower predator-preymodel withMichaelis-Menten type prey harvesting, Journal of Mathematical Analysis and Application, 398(1), 278–295.
  2. [2]  He, Z.M. and Lai, X. (2011), Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Analysis: Real World Applications, 12(1), 403–417.
  3. [3]  Ma, Z., Chen, F., Wu, C., and Chen, W. (2013), Dynamic behaviors of a Lotka-Volterra predator-preymodel incorporating a prey refuge and predator mutual interference, Applied Mathematics and Computation, 219(15), 7945–7953.
  4. [4]  Pal, P.J. and Mandal, P.K. (2014), Bifurcation analysis of a modified Leslie-Gower predator-preymodel with Beddington-DeAngelis functional response and strong Allee effect, Mathematics and Computers in Simulation, 97, 123–146.
  5. [5]  Agiza, H.N., ELabbasy, E.M., EL-Metwally, H., and Elsadany, A.A. (2009), Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Analysis: Real World Applications, 10(1), 116–129.
  6. [6]  Hu, Z.Y., Teng, Z.D., and Zhang, L. (2011), Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Analysis: Real World Applications, 12(4), 2356–2377.
  7. [7]  Lotka, A.J. (1920), Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Acadamy of Sciences of the United States of America, 6(7), 410–415.
  8. [8]  Volterra, V. (1926), Fluctuations in the abundance of a species consideredmathematically, Nature, 118(2972), 558–560.
  9. [9]  Stone, L. and He, D. (2007), Chaotic oscillations and cycles in multitrophic ecological systems, Journal of Theoretical Biology, 248(2), 38-390.
  10. [10]  Li, Y. and Ye, Y. (2013), Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms, Communications in Nonlinear Science and Numerical Simulation, 18(11), 3190–3201.
  11. [11]  Holling, C.S. (1965), The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97(45), 1–60.
  12. [12]  Sugie, J., Kohno, R., and Miyazaki, R., On a predator-prey system of Holling type, Proceedings of the American Mathematical Society, 125(7), 2041–2050.
  13. [13]  Skalski, G.T. and Gilliam, J.F. (2001), Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82(11), 3083–3092.
  14. [14]  Beddington, J.R. (1975), Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44(1), 331–340.
  15. [15]  Crowley, P.H. and Martin, E.K. (1989), Functional responses and interference within and between year classes of a dragonfly population, Journal of the North Americal Benthological Society, 8(3), 211–221.
  16. [16]  Hassell, M.P. and Varley, G.C. (1969), New inductive population model for insect parasites and its bearing on biological control, Nature, 223(5211), 1133–1137.
  17. [17]  Upadhyay, R.K. and Naji, R.K. (2009), Dynamics of a three species food chain model with Crowley–Martin type functional response, Chaos Solitons Fractals, 42(3), 1337–1346.
  18. [18]  Wang, Y., Zhou, Y., Braue, R.F., and Heffernan, J.M. (2013), Viral dynamics model with ctl immune response incorporating antiretroviral therapy, Journal of mathematical biology, 67(4), 901–934.
  19. [19]  Wood, S.N., Blythe, S.P., Gurney W.S.C., and Nisbet, R.M. (1989), Instability in mortality estimation schemes related to stage-structure population models, Mathematical Medicine and Biology, 6(1), 47–68.
  20. [20]  Xiao, D. and Ruan, S. (2001), Global analysis in a predator–prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 61(4), 1445–1472.
  21. [21]  Mondal, A., Pal, A.K., and Samanta, G.P. (2018), Stability and bifurcation analysis of a delayed three species food chain model with Crowle-Martin response function, Applications and Applied Mathematics: An International Journal (AAM), 13(2), 709-749.
  22. [22]  Maiti, A.P., Dubey, B., and Tushar, J. (2017), A delayed prey–predator model with Crowley–Martintype functional response including prey refuge, Mathematical Methods in the Applied Sciences, 40(16), 5792–5809.
  23. [23]  Tripathi, J.P., Tyagi, S., and Abbas, S. (2016), Global analysis of a delayed density dependent predator–prey model with Crowley–Martin functional response, Communications in Nonlinear Science and Numerical Simulation, 30(1), 45–69.
  24. [24]  Shiffman, D.S. and Hammerschlag, D. (2016), Preferred conservation policies of shark researchers, Conservation Biology, 30, 805-815.
  25. [25]  Mondal, A., Pal, A.K., and Samanta, G.P. (2021), Role of age-selective harvesting in a delayed predator- prey model together with fear and additional food, Journal of Environmental Accounting and Management, 9(4), 343-375.
  26. [26]  Wang, J., Cheng, H., Liu, H., and Wang, Y. (2018), Periodic solution and control optimization of a prey–predator model with two types of harvesting, Advances in Difference Equations, 41(1).
  27. [27]  Krishna, S.V., Srinivasu, P.D.N., and Kaymakcalan, B.(1998), Conservation of an ecosystem through optimal taxation, Bulletin of Mathematical Biology, 60(3), 569–584.
  28. [28]  Yuan, R., Jiang, W., and Wang, Y. (2015), Saddle-node-Hopf bifurcation in a modified Leslie–Gower predator–prey model with time-delay and prey harvesting, Journal of Mathematical Analysis and Applications, 422(2), 1072–1090.
  29. [29]  Clark, C.W. (2006), Mathematical models in the economics of renewable resources, SIAM Review, 21(1), 81–99.
  30. [30]  Pal, A.K. (2021), Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters, Zeitschrift für Naturforschung A, 76(10).
  31. [31]  Das, T., Mukherjee, R.N., and Chaudhari, K.S. (2009), Bioeconomic harvesting of a prey–predator fishery, Journal of Biological Dynamics, 3(5), 447–462.
  32. [32]  Hsu, S.B., Hwang, T.W., and Kuang, Y. (2003), A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181(1), 55–83.
  33. [33]  Chakraborty, K., Jana, S., and Kar, T.K. (2012), Global dynamics and bifurcation in a stage structured prey–predator fishery model with harvesting, Applied Mathematics and Computation, 218(18), 9271–9290.
  34. [34]  Rojas-Palma, A. and Gonz'alez-Olivares, E. (2012), Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response, Applied Mathematical Modelling, 36, 1864–1874.
  35. [35]  Hu, D.P. and Cao, H.J. (2017), Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting, Nonlinear Analysis: Real World Applications, 33, 58–82.
  36. [36]  Gupta, R.P. and Chandra, P.(2003), Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, Journal of Mathematical Analysis and Application, 398(1), 278–295.
  37. [37]  Huang, J., Gong, Y., and Chen, J. (2013), Multipple bifurcation in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, International Journal of Bifurcation and Chaos, 23(10), 1350164.
  38. [38]  Huang, J., Sanhong, L., Ruan, S., and Zhang, X. (2016), Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting, Communications on Pure and Applied Analysis, 15(2), 1053-1067.
  39. [39]  Wang, Z., Wang, X., Li, Y., and Huang, X. (2017), Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay, International Journal of Bifurcation and Chaos, 27(13), 1750209.
  40. [40]  Mondal, A., Pal, A.K. and Samanta, G.P. (2020), Rich dynamics of non-toxic phytoplankton, toxic phytoplankton and zooplankton system with multiple gestation delays, International Journal of Dynamics and Control, 8(1), 112–131.
  41. [41]  Liu, G., Wang, X., Meng, X., and Gao, S. (2017), Extinction and persistence in mean of a novel delay impulsive stochastic infected predator–prey system with jumps, Complexity, 2017(3), 1–15.
  42. [42]  Li, L., Wang, Z., Li, Y., Shen, H., and Lu, J. (2018), Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays, Applied Mathematics and Computation, 330, 152–169.
  43. [43]  Maiti, A., Pal, A.K., and Samanta, G.P. (2008), Effect of time delay on a food-chain model, Applied Mathematics and Computations, 200, 189-203.
  44. [44]  Mondal, A., Pal, A.K., and Samanta, G.P. (2019), Analysis of a delayed eco-epimediological pest-plant model with infected pest, Biophysical Reviews and Letters, 14(3), 141-170.
  45. [45]  Mondal, A., Pal, A.K., and Samanta, G.P. (2020), Evolutionary Dynamics of a Single-Species Population Model with Multiple Delays in a Polluted Environment, Discontinuity, Nonlinearity, and Complexity, 9(3), 433-459.
  46. [46]  Hale J.K. (1977), Theory of functional differential equations, Springer, Heidelberg.
  47. [47]  Ruan, S. and Wei, J. (2015), On the zeros of transcendental functions with application to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Syatems, Series A: Mathematical Analysis, 10, 863-874.
  48. [48]  Hassard, B., Kazarinof, D., and Wan, Y. (1981), Theory and Application of Hopf Bifurcation, Cambridge: Cambridge University Press.