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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Functional Responses of Prey-Predator Models in Population Dynamics; A Survey

Journal of Applied Nonlinear Dynamics 13(1) (2024) 83--96 | DOI:10.5890/JAND.2024.03.007

M. Sambath$^1$, K. Balachandran$^2$, M. S. Surendar$^3$

$^1$ Department of Mathematics, Periyar University, Salem 636 011, India

$^2$ Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

$^3$ Department of Mathematics, B V Raju Institute of Technology, Narsapur, Medak, Telangana 502 313, India

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Abstract

This paper presents a survey of research on the study of the impact of different functional responses on prey-predator models in ecology. These functional responses impact the qualitative behaviours of prey-predator models. Further stability and bifurcation analysis of these models are discussed. Graphical representation through numerical simulations are presented.

References

  1. [1]  Lotka, A.J. (1925), Elements of Physical Biology, Williams and Wilkins, Baltimore. [new edition, Elements of Mathemacal Biology, Dover, New York, 1956].
  2. [2]  Volterra, V. (1926), Fluctuations in the abundance of species considered mathematically, Nature, 118, 558-560.
  3. [3]  Kot, M. (2001), Elements of Mathematical Ecology, Cambridge University Press.
  4. [4]  Holling, C.S. (1966), The functional response of invertebrate predators to prey density, Memoirs of the Entomological Society of Canada, 98(S48), 5-86.
  5. [5]  Royama, T. (1971), A comparative study of models for predation and parasitism, Researches on Population Ecology, 13(S1), 1-91.
  6. [6] Solomon, M.E. (1949), The natural control of animal populations, The Journal of Animal Ecology, 1-35.
  7. [7]  Holling, C.S. (1959), The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist, 91(5), 293-320.
  8. [8]  Shen, C. (2009), Permanence and global attractivity of the food chain system with Holling type IV functional response, Applied Mathematics and Computation, 215, 179-185.
  9. [9]  Kazarinov, N.D. and Van den Driessche, P. (1978), A model predator-prey system with functional response, Mathematical Biosciences, 39(1-2), 125-134.
  10. [10]  Kooij, R.E. and Zegeling, A. (1997), Qualitative properties of two-dimensional predator-prey systems, Nonlinear Analysis: Theory, Methods $\&$ Applications, 29(6), 693-715.
  11. [11]  Arie, A., Kottagoda, C., and Shan, C. (2022), A predator-prey system with generalized Holling type IV functional response and Allee effects in prey, Journal of Differential Equations, 309, 704-740.
  12. [12]  Gokila, C., Sambath, M., Balachandran, K., and Ma, Y.K. (2020), Analysis of stochastic predator-prey model with disease in the prey and Holling type II functional response, Advances in Mathematical Physics, 2020, 3632091(17 pages).
  13. [13]  Sambath, M., Gnanavel, S., and Balachandran, K. (2013), Stability and Hopf bifurcation of a diffusive predatorprey model with predator saturation and competition, Applicable Analysis, 92(12), 2439-2456.
  14. [14]  Sambath, M. and Balachandran, K. (2015), Bifurcations in a diffusive predator-prey model with predator saturation and competition response, Mathematical Methods in the Applied Sciences, 38(5), 785-798.
  15. [15]  Sambath, M., Balachandran, K., and Suvinthra, M. (2016), Stability and Hopf bifurcation of a diffusive predator-prey model with hyperbolic mortality, Complexity, 21(S1), 34-43.
  16. [16]  Sarkar, K., Ali, N., and Guin, L.N. (2021), Fear effect in a tri-trophic food chain model with Holling type IV functional response, Journal of Environmental Accounting and Management, 9(3), 235-253.
  17. [17]  Shang, Z. and Qiao, Y. (2022), Bifurcation analysis of a Leslie-type predator-prey system with simplified Holling type IV functional response and strong Allee effect on prey, Nonlinear Analysis: Real World Applications, 64, 103453(34pages).
  18. [18]  Surendar, M.S., and Sambath, M. (2021), Qualitative analysis for a phytoplankton-zooplankton model with Allee effect and Holling type II response, Discontinuity, Nonlinearity, and Complexity, 10(1), 1-18.
  19. [19]  Guin, L.N. (2020), The effects of unequal diffusion coefficients on spatiotemporal pattern formation in prey harvested, Mathematical Analysis and Applications in Modeling: ICMAAM 2018, 302, 279.
  20. [20]  Guin, L.N. and Baek, H. (2018), Comparative analysis between prey-dependent and ratio-dependent predator-prey systems relating to patterning phenomenon, Mathematics and Computers in Simulation, 146, 100-117.
  21. [21] Guin, L.N. (2018), The effects of unequal diffusion coefficients on spatiotemporal pattern formation in prey harvested reaction-diffusion systems, in International Workshop of Mathematical Analysis and Applications in Modeling, 279-292.
  22. [22]  Mohd, M.H. and Mohd, M.S. (2021), Local dispersal, trophic interactions and handling times mediate contrasting effects in prey-predator dynamics, Chaos, Solitons $\&$ Fractals, 142, 110497.
  23. [23]  Mohd, M.H., Mohamed, N.A.F.K., and Abdullah, N.D.M. (2019), Dynamical systems analysis of prey-predator interactions with group defence, AIP Conference Proceedings, 2184(1), 060029.
  24. [24]  Mohd, M.H. and Hasana, Y.A. (2013), Modelling the spatiotemporal dynamics of diffusive prey-predator interactions: Pattern formation and ecological implications, Science Asia, 39, 31-36.
  25. [25]  Leslie, P.H. (1948), Some further notes on the use of matrices in population mathematics, Biometrika, 35(3/4), 213-245.
  26. [26]  Leslie, P.H. (1958), A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45(1-2), 16-31.
  27. [27]  Zhu, C.R. and Lan, K.Q. (2010), Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete $\&$ Continuous Dynamical Systems - B, 14(1), 289-306.
  28. [28]  Mena-Lorca, J., Gonz´alez-Olivares, E., and Gonzalez-Ya~nez, B. (2006), The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, in Proceedings of the 2006 International Symposium on Mathematical and Computational Biology Biomat, 105-132.
  29. [29]  Aziz-Alaoui, M.A. and Daher Okiye, M. (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.
  30. [30]  Gupta, R.P. and Chandra, P. (2013), Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, Journal of Mathematical Analysis and Applications, 398(1), 278-295.
  31. [31]  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.
  32. [32]  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, 400-408.
  33. [33]  Din, Q. (2017), Complexity and chaos control in a discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 49, 113-134.
  34. [34]  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(1), 27-64.
  35. [35]  Ivlev, V.S. (1961), Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, CT.
  36. [36]  Rosenzweig, M.L. (1971), Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science, 171(3969), 385-387.
  37. [37]  May, R.M. (1972), Limit cycles in predator-prey communities, Science, 177(4052), 900-902.
  38. [38]  Cheng, K.S., Hsu, S.B., and Lin, S.S. (1982), Some results on global stability of a predator-prey system, Journal of Mathematical Biology, 12(1), 115-126.
  39. [39]  Gasull, A., Llibre, J., and Sotomayor, J. (1990), On the phase portrait of a predator prey system, Preprint.
  40. [40]  Wang, W., Zhang, L., Wang, H., and Li, Z. (2010), Pattern formation of a predator-prey system with Ivlev-type functional response, Ecological Modelling, 221(2), 131-140.
  41. [41]  Wang, X. and Wei, J. (2013), Diffusion-driven stability and bifurcation in a predator-prey system with Ivlev-type functional response, Applicable Analysis, 92(4), 752-775.
  42. [42]  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, 97(S45), 5-60.
  43. [43]  Rosenzweig, M.L. and MacArthur, R.H. (1963), Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97(895), 209-223.
  44. [44]  Merchant S.M. and Nagata, W. (2011), Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition, Theoretical Population Biology, 80(4), 289-297.
  45. [45]  Banerjee, M. and Volpert, V. (2017), Spatio-temporal pattern formation in Rosenzweig-MacArthur model: Effect of nonlocal interactions, Ecological Complexity, 30, 2-10.
  46. [46]  Mukherjee, N., Ghorai, S., and Banerjee, M. (2019), Cross-diffusion induced Turing and non-Turing patterns in Rosenzweig-MacArthur model, Letters in Biomathematics, 6(1), 1-22.
  47. [47]  Moustafa, M., Mohd, M.H., Ismail, A.I., and Abdullah, F.A., (2018), Dynamical analysis of a fractional-order Rosenzweig-MacArthur model incorporating a prey refuge, Chaos, Solitons $\&$ Fractals, 109, 1-13.
  48. [48]  Moustafa, M., Mohd, M.H., Ismail, A.I., and Abdullah, F.A., (2019), Stage structure and refuge effects in the dynamical analysis of a fractional order Rosenzweig-MacArthur prey-predator model, Progress in Fractional Differentiation and Applications, 5(1), 49-64.
  49. [49]  Mohd, M.H., Halim, N.F.A.A., and Zaidun, F.N. (2019), Combined effects of prey refuge and death rate of predator on the prey-predator population dynamics, AIP Conference Proceedings, 2184(1), 060028.
  50. [50]  Tanner, J.T. (1975), The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56(4), 855-867.
  51. [51]  May, R.M. (2019), Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton.
  52. [52]  Murray, J.D. (2002), Mathematical Biology: I. An Introduction, Springer, New York.
  53. [53]  Hsu, S.B. and Huang, T.W. (1995), Global stability for a class of predator-prey systems, SIAM Journal on Applied Mathematics, 55(3), 763-783.
  54. [54] Arrowsmith, D.K. and Place, C.M. (1992), Dynamical Systems, Campman and Hall, London.
  55. [55]  Gasull, A., Kooij, R.E., and Torregrosa, J. (1997), Limit cycles in the Holling-Tanner model, Publicacions Matematiques, 149-167.
  56. [56]  Surendar, M.S., Sambath, M., and Balachandran, K. (2020), Bifurcation on diffusive Holling-Tanner predator-prey model with stoichiometric density dependence, Nonlinear Analysis: Modelling and Control, 25(2), 225-244.
  57. [57]  Sambath, M., Balachandran, K., and Jung, I.H. (2019), Dynamics of a modified Holling-Tanner predator-prey model with diffusion, Journal of the Korean Society for Industrial and Applied Mathematics, 23(2), 139-155.
  58. [58]  Pal, S., Banerjee, M., and Ghorai, S. (2019), Spatio-temporal pattern formation in Holling- Tanner type model with nonlocal consumption of resources, International Journal of Bifurcation and Chaos, 29(01), 1930002.
  59. [59]  Banerjee, M. and Banerjee, S. (2012), Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Mathematical Biosciences, 236(1), 64-76.
  60. [60]  Duan, D., Niu, B., and Wei, J. (2021), Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations, Discrete $\&$ Continuous Dynamical Systems-B, 27(7), 3683-3706.
  61. [61]  Ajraldi, V., Puravino, M., and Venturino, E. (2011), Modelling herd behavior in population systems, Nonlinear Analysis: Real World Applications, 12, 2319-2338.
  62. [62] Braza, P.A. (2012), Predator-prey dynamics with square root functional response, Nonlinear Analysis: Real World Applications, 13, 1837-1843.
  63. [63]  Beddington, J.R. (1975), Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, 44(1), 331-340.
  64. [64]  DeAngelis, D.L., Goldstein, R.A., and O'Neill, R.V. (1975), A model for tropic interaction, Ecology, 56(4), 881-892.
  65. [65]  Cantrell, R.S. and Cosner, C. (2001), On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, Journal of Mathematical Analysis and Applications, 257(1), 206-222.
  66. [66]  Zou, X., Li, Q., and Lv, J. (2021), Stochastic bifurcations, a necessary and sufficient condition for a stochastic Beddington-DeAngelis predator-prey model, Applied Mathematics Letters, 117, 107069.
  67. [67]  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.
  68. [68]  Guin, L.N., Mondal, B., and Chakravarty, S. (2017), Stationary patterns induced by self- and cross-diffusion in a Beddington-DeAngelis predator-prey model, International Journal of Dynamics and Control, 5, 1051-1062.
  69. [69]  Guin, L.N., Murmu, R., Baek, H., and Kim, K.H. (2020), Dynamical analysis of a Beddington-DeAngelis interacting species system with prey harvesting, Mathematical Problems in Engineering, 2020, Art.ID.7596394(22 pages).
  70. [70]  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 American Benthological Society, 8(3), 211-221.
  71. [71]  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.
  72. [72]  Xiao, D. and Ruan, S. (2000), Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 61(4), 1445-1472.
  73. [73]  Zhu, H., Campbell, S.A., and Wolkowicz, G.S. (2002), Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63(2), 636-682.
  74. [74]  Bazykin, A.D. (1998), Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore.
  75. [75]  Surendar, M.S. and Sambath, M. (2021), Modeling and numerical simulations for a prey-predator model with interference among predators, International Journal of Modeling, Simulation, and Scientific Computing, 12(1), 2050065(24 pages).
  76. [76]  Tripathi, J.P., Jana, D., Devi, N.V., Tiwari, V., and Abbas, S. (2020), Intraspecific competition of predator for prey with variable rates in protected areas, Nonlinear Dynamics, 102(1), 511-535.
  77. [77]  Arditi, R. and Ginzburg, L.R. (1989), Coupling in predator-prey dynamics: Ratio-Dependence, Journal of Theoretical Biology, 139(3), 311-326.
  78. [78]  Abrams, P.A. and Ginzburg, L.R. (2000), The nature of predation: prey dependent, ratio dependent or neither?, Trends in Ecology $\&$ Evolution, 15(8), 337-341.
  79. [79]  Berezovskaya, F., Karev, G., and Arditi, R. (2001), Parametric analysis of the ratio-dependent predator-prey model, Journal of Mathematical Biology, 43(3), 221-246.
  80. [80] Freedman, H.I. and Mathsen, R.M. (1993), Persistence in predator-prey systems with ratio-dependent predator influence, Bulletin of Mathematical Biology, 55(4), 817-827.
  81. [81] Hsu, S.B., Hwang, T.W., and Kuang, Y. (2001), Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42(6), 489-506.
  82. [82]  Jost, C., Arino, O., and Arditi, R. (1999), About deterministic extinction in ratio-dependent predator-prey models, Bulletin of Mathematical Biology, 61(1), 19-32.
  83. [83]  Kuang Y. and Beretta, E. (1998), Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36(4), 389-406.
  84. [84]  Xiao, D. and Ruan, S. (2001), Global dynamics of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 43(3), 268-290.
  85. [85]  Guin, L.N., Pal, S., Chakravarty, S., and Djilali, S. (2020), Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting, International Journal of Biomathematics, 14(1), 2050084(29 pages).
  86. [86] 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), 1501-1533.
  87. [87]  Guin, L.N. and Mandal, P.K. (2014), Spatial pattern in a diffusive predator-prey model with sigmoid ratio-dependent functional response, International Journal of Biomathematics, 7(5), 1450047.
  88. [88]  Guin, L.N. and Mandal, P.K. (2014), Spatiotemporal dynamics of reaction-diffusion models of interacting populations, Applied Mathematical Modelling, 38, 4417-4427.
  89. [89]  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.

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