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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:

Dynamic Response of Allee Effect and Refuge on the Interacting Species Model System

Journal of Environmental Accounting and Management 11(2) (2023) 193--224 | DOI:10.5890/JEAM.2023.06.006

Lakshmi Narayan Guin, Deepabali Datta, Santabrata Chakravarty

Department of Mathematics, Visva-Bharati, Santiniketan-731 235, West Bengal, India

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The present concern is to explore the complexity of the dynamics of a Leslie-Gower predator-prey model with the inclusion of the Allee effect and a constant proportion of prey refuge. Due attention is paid on the non-negativity, dissipativity, uniform boundedness and permanence of the dynamical system as well. Theoretical investigation is carried out on the existence of feasible equilibria of the system followed by the deductions imperative for the conditions of stability of the interior equilibrium. The analysis corresponding to the global stability of equilibrium using a suitable Lyapunov function is, however, not ruled out from the present pursuit. Moreover, the analytical conditions for the occurrence of bifurcation phenomena are ascertained both for a saddle-node bifurcation and a Hopf bifurcation. Numerical simulations are implemented finally at the end in order to validate the theoretical outcomes together with the concluding remarks relevant to the biological implications.


The authors are thankful to the anonymous referees and the Editor, Journal of Environmental Accounting and Management (JEAM), L \& H Scientific Publishing, LLC for their careful reading, valuable comments, and helpful suggestions, which have helped us to improve the presentation of this research work significantly. The first author gratefully acknowledges the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.$510$ / $3$ / DRS-III / $2015$ (SAP-I)).


  1. [1]  Leslie, P.H. (1948), Some further notes on the use of matrices in population mathematics, Biometrika, 35(3-4), 213-245.
  2. [2]  Leslie, P.H. (1958), A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45(1-2), 16-31.
  3. [3]  Murdoch, W.W., Briggs, C.J., and Nisbet, R.M. (2003), Consumer-resource dynamics, Princetone University Press, New York.
  4. [4]  Holling, C.S. (1965), The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, Cambridge University Press, 45(S45), 5-60.
  5. [5]  Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47(3-4), 219-234.
  6. [6]  Guin, L.N., Das, E., and Sambath, M. (2020), Pattern formation scenario through Turing instability in interacting reaction-diffusion systems with both refuge and nonlinear harvesting, Journal of Applied Nonlinear Dynamics, 9(1), 1-21.
  7. [7]  Han, R., Guin, L.N. and Dai, B. (2020), Cross-diffusion-driven pattern formation and selection in a modified Leslie-Gower predator-prey model with fear effect, Journal of Biological Systems, 28(01), 27-64.
  8. [8]  May, R. M. (1974), Stability and complexity in model ecosystem, Princeton University Press, Princeton.
  9. [9]  Aziz-Alaouli, M.A. and Okiye, M.D. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Applied mathematics Letters, 16(7), 1069-1075.
  10. [10]  Du, Y., Peng, R., and Wang, M. (2009), Effect of a protection zone in the diffusive Leslie predator-prey model, Journal of Differential Equations, 246(10), 3932-3956.
  11. [11]  Zhu, Y. and Wang, K. (2011), Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes, Journal of Mathematical Analysis and Applications, 384(2), 400-408.
  12. [12]  Nindjin, A.F., Aziz-Alaoui, M.A., and Cadivel, M. (2006), Analysis of a predator-prey model with modified Leslie-Gower Holling-type II schemes with time delay, Nonlinear Analysis: Real World Applications, 7(5), 1104-1118.
  13. [13]  McNair, J.N. (1986), The effects of refuges on predator-prey interactions: a reconsideration, Theoretical Population Biology, 29(1), 38-63.
  14. [14]  Guin, L.N., Murmu, R., Baek, H., and Kim, K.H. (2020), Dynamical analysis of a Beddington-DeAngelis interacting species system with harvesting, Mathematical Problems in Engineering, 2020 Article ID 7596394.
  15. [15]  Krivan, V. (1998), Effects of optimal antipredator behaviour of prey on predator-prey dynamics: the role of refuges, Theoretical Population Biology, 53(2), 131-142.
  16. [16]  Han, R., Guin, L. N. and Dai, B. (2021), Consequences of refuge and diffusion in a spatiotemporal predator-prey model, Nonlinear Analysis: Real World Applications, 60, 103311.
  17. [17]  Guin, L.N. (2014), Existence of spatial patterns in a predator-prey model with self- and cross-diffusion, Applied Mathematics and Computation, 226, 320-335.
  18. [18]  Gonzalez-Olivares, E. and Ramos-Jiliberto, R. (2003), Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166(1), 135-146.
  19. [19]  Hawkins, B.A., Thomas, M.B., and Hochberg, M.E. (1993), Refuge theory and biological control, Science: American Association for the Advancement of Science, 262(5138), 1429-1432.
  20. [20]  Guin, L.N. and Acharya, S. (2017), Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting, Nonlinear Dynamics, 88(2) (2017) 1501-1533.
  21. [21]  Guin, L.N., Chakravarty, S., and Mandal, P.K. (2015), Existence of spatial patterns in reaction-diffusion systems incorporating a prey refuge, Nonlinear Analysis: Modelling and Control, 20(04), 509-527.
  22. [22]  Guin, L.N. and Mandal, P.K. (2014), Spatiotemporal dynamics of reaction-diffusion models of interacting populations, Applied Mathematical Modelling, 38(17-18), 4417-4427.
  23. [23]  Guin, L.N., Mondal, B., and Chakravarty, S. (2016), Existence of spatiotemporal patterns in the reaction-diffusion predator-prey model incorporating prey refuge, International Journal of Biomathematics, 9(06), 1650085.
  24. [24]  Harrison, G.W. (1979), Global stability of predator-prey interactions, Journal of Mathematical Biology, 8(2), 159-171.
  25. [25]  Smith, J.M. (1974), Models in Ecology, Cambridge University Press, Cambridge.
  26. [26]  Gonzalez-Olivars, E., Gonzalez-Yanez, B., and Becerra-Klix, R. (2012), Prey refuge use as a function of predator-prey encounters, International Journal of Biomathematics.
  27. [27]  Courchamp, F., Clutton-Brock, T., and Grenfell, B. (1999), Inverse density dependence and the Allee effect, Trends in Ecology $\&$ Evolution, 14(10), 401-405.
  28. [28]  Stephens, P.A., Sutherland, W.J., and Freckleton, R.P. (1999), What is the Allee effect?, Oikos, 87, 185-190.
  29. [29]  Dennis, B. (1989), Allee effects: Population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3(4), 481-538.
  30. [30]  Wang, G., Liang, X.G., and Wang, F.Z. (1999), The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124(2-3), 183-192.
  31. [31]  Wang, J., Shi, J., and Wei, J. (2011), Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Journal of Differential Equations, 251(4-5), 1276-1304.
  32. [32]  Berec, L., Angulo, E., and Courchamp, F. (2007), Multiple Allee effects and population management, Trends in Ecology $\&$ Evolution, 22(4), 185-191.
  33. [33]  Cai, Y., Zhao, C., Wang, W., and Wang, J. (2015), Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Applied Mathematical Modelling, 39(7), 2092-2106.
  34. [34]  Aguirre, P., Gonzalez-Olivares, E., and Saez, E. (2009), Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10(3), 1401-1416.
  35. [35]  Wang, M.H. and Kot, M. (2001), Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences and Engineering, 171(1), 83-97.
  36. [36]  Marsden, J E. and Hoffman, M.J. (1993), Elementary Classical Analysis, Macmillan, New York.
  37. [37]  Huston, V. and Vickers, G.T. (1983), A criterion for permanent coexistence of species, with an application to a two-prey one-predator system, Mathematical Biosciences, 63(2), 253-269.
  38. [38]  Hale, J.K. (1976), Theory of Functional Differential Equations, Springer-Verlag, New York.
  39. [39]  LaSalle, J.P. (1979), The stability of dynamical systems, Society of Industrial and Applied Mathematics, 21(3), 418-420.
  40. [40]  H. Poincar{e}, L'{E}quilibre d'une masse fluide anim{e}e d'un mouvement de rotation, Acta Mathematica, Springer, 7(1) (1885) 259-380.
  41. [41]  Strogatz, S.H. (1994), Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry and Engineering, Taylor and Francis, CRC Press.
  42. [42]  Perko, L. (2001), Differential Equations and Dynamical Systems, Springer.
  43. [43]  Lynch, S. (2013), Dynamical System with Application using Matlab, Springer.
  44. [44]  Layek, G.C. (2015), An Introduction to Dynamical System and Chaos, Springer.
  45. [45]  Sotomayor, J. (1973), Generic Bifurcations of Dynamical Systems, Academic Press, New York.