Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

A BRIEF REVIEW OF PREDATOR-PREY MODELS FOR AN ECOLOGICAL SYSTEM WITH A DIFFERENT TYPE OF BEHAVIORS

  • Kuldeep Singh (Department of Mathematics, J.C. Bose University of Science and Technology, YMCA) ;
  • Teekam Singh (Department of Mathematics, Graphic Era (Deemed to be) University) ;
  • Lakshmi Narayan Mishra (Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology) ;
  • Ramu Dubey (Department of Mathematics, J.C. Bose University of Science and Technology, YMCA) ;
  • Laxmi Rathour (Department of Mathematics, National Institute of Technology)
  • Received : 2023.12.12
  • Accepted : 2024.09.02
  • Published : 2024.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

The logistic growth model was developed with a single population in mind. We now analyze the growth of two interdependent populations, moving beyond the one-dimensional model. Interdependence between two species of animals can arise when one (the "prey") acts as a food supply for the other (the "predator"). Predator-prey models are the name given to models of this type. While social scientists are mostly concerned in human communities (where dependency hopefully takes various forms), predator-prey models are interesting for a variety of reasons. Some variations of this model produce limit cycles, an interesting sort of equilibrium that can be found in dynamical systems with two (or more) dimensions. In terms of substance, predator-prey models have a number of beneficial social science applications when the state variables are reinterpreted. This paper provides a quick overview of numerous predator-prey models with various types of behaviours that can be applied to ecological systems, based on a survey of various types of research publications published in the last ten years. The primary source for learning about predator-prey models used in ecological systems is historical research undertaken in various circumstances by various researchers. The review aids in the search for literature that investigates the impact of various parameters on ecological systems. There are also comparisons with traditional models, and the results are double-checked. It can be seen that several older predator-prey models, such as the Beddington-DeAngelis predator-prey model, the stage-structured predator-prey model, and the Lotka-Volterra predator-prey model, are stable and popular among academics. For each of these scenarios, the results are thoroughly checked.

Keywords

References

  1. Baurmann M., Gross T., Feudel U., Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, Journal of Theoretical Biology 245 (2) (2007), 220-229 https://dx.doi.org/10.1016/j.jtbi.2006.09.036
  2. Neuhauser C. M., Mathematical Challenges in Spatial Ecology, Notices of the American Mathematical Society 48 (11) (2001) 1304-1314.
  3. Levin S. A., Grenfell B., Hastings A., Perelson A. S., Mathematical and computational challenges in population biology and ecosystems science, Science 275 (5298) (1997), 334-343. https://dx.doi.org/10.1126/science.275.5298.334
  4. May R. M., Simple mathematical models with very complicated dynamics, Nature 261 (1976), 459-467.
  5. Berryman A. A., The Origins and Evolution of Predator-Prey Theory, Ecology 73 (5) (1992), 1530-1535. https://dx.doi.org/10.2307/1940005
  6. Kuang Y., Beretta E., Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology 36 (1998), 389-406.
  7. Jost C., Comparing predator-prey models qualitatively and quantitatively with ecological time-series data, PhD Thesis, Institut National Agronomique Paris-Grignon (1998).
  8. Quan S. G., Jin Z., Xing L. Q., and Li L., Pattern formation induced by cross-diffusion in a predator-prey system, Chinese Physics B 17 (11) (2008), 3936-3941. https://dx.doi.org/10.1088/1674-1056/17/11/003
  9. Sun G. Q., Zhang G., Jin Z., Li L., Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dynamics 58 (1) (2009), 75-84. https://dx.doi.org/10.1007/s11071-008-9462-z
  10. Sun G.-Q., Jin Z., Li L., and Li B.-L., Self-organized wave pattern in a predator-prey model, Nonlinear Dynamics 60 (2010), 265-275. https://dx.doi.org/10.1007/s11071-009-9594-9
  11. Sengupta A., Kruppa T., and Lowen H., Chemotactic predator-prey dynamics, Physical Review E 83 (3) (2011). https://dx.doi.org/10.1103/PhysRevE.83.031914
  12. Lai Y.-M., Newby J., and Bressloff P. C., Effects of demographic noise on the synchronization of metacommunities by a fluctuating environment, Physical Review Letters 107 (2011).
  13. Shivam, Kumar M., Singh T., Dubey R., and Singh K., Analytical study of food-web system via Turing patterns, AIP Conference Proceedings 2481 (1) (2022).
  14. Shivam, Singh K., Kumar M., Dubey R., and Singh T., Untangling role of cooperative hunting among predators and herd behavior in prey with a dynamical systems approach, Chaos, Solitons & Fractals 162 (2022), 112420.
  15. Yuan S., Xu C., Zhang T., Spatial dynamics in a predator-prey model with herd behavior, Chaos: An Interdisciplinary Journal of Nonlinear Science 23 (3) (2013). https://dx.doi.org/10.1063/1.4812724
  16. Braza P. A., Predator-prey dynamics with square root functional responses, Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp.1837-1843, 2012, DOI:10.1016/j.nonrwa.2011.12.014.
  17. Wang W., Liu Q. X., Jin Z., Spatiotemporal complexity of a ratio-dependent predator-prey system, Physical Review E 75 (2007). https://dx.doi.org/10.1103/PhysRevE.75.051913
  18. Zhang X. C., Sun G. Q., Jin Z., Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response, Physical Review E 85 (2012). https://dx.doi.org/10.1103/PhysRevE.85.021924
  19. Dutt A. K., Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model, AIP Advances 2 (2012). https://dx.doi.org/10.1063/1.4765650
  20. Jana S., Chakraborty M., Chakraborty K., Kar T. K., Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge, Mathematics and Computers in Simulation 85 (2012), 57-77. https://dx.doi.org/10.1016/j.matcom.2012.10.003
  21. Tewa J. J., Djeumen V. Y., Bowong S., Predator-Prey model with Holling response function of type II and SIS infectious disease, Applied Mathematical Modelling 37 (2013), 4825-4841. https://dx.doi.org/10.1016/j.apm.2012.10.003
  22. Chen Y., Zhang F., Dynamics of a delayed predator-prey model with predator migration, Applied Mathematical Modelling 37 (2013), 1400-1412. https://dx.doi.org/10.1016/j.apm.2012.04.012
  23. Gambino G., Lombardo M. C., Sammartino M., Pattern formation driven by cross-diffusion in a 2D domain, Nonlinear Analysis: Real World Applications 14 (2013), 1755-1779. https://dx.doi.org/10.1016/j.nonrwa.2012.11.009
  24. Zuo W., Global stability and Hopf bifurcations of a Beddington-DeAngelis type predator-prey system with diffusion and delays, Applied Mathematics and Computation 223 (2013), 423-435. https://dx.doi.org/10.1016/j.amc.2013.08.029
  25. Xu S., Dynamics of a general prey-predator model with prey-stage structure and diffusive effects, Computers and Mathematics with Applications 68 (2014), 405-423. https://dx.doi.org/10.1016/j.camwa.2014.06.016
  26. Lonnstedt O. M., Ferrari M. C. O., Chivers D. P., Lionfish predators use flared fin displays to initiate cooperative hunting, Biology Letters 10 (2014). https://dx.doi.org/10.1098/rsbl.2014.0281
  27. Aguirre P., Flores J. D., Olivares E. G., Bifurcations and global dynamics in a predator-prey model with a strong Allee effect on the prey, and a ratio-dependent functional response, Nonlinear Analysis: Real World Applications 16 (2014), 235-249. https://dx.doi.org/10.1016/j.nonrwa.2013.10.002
  28. Khajanchi S., Banerjee S., Stability and bifurcation analysis of delay induced tumor immune interaction model, Applied Mathematics and Computation 248 (2014), 652-671. https://dx.doi.org/10.1016/j.amc.2014.10.009
  29. Kuznetsov V. A., Makalkin I. A., Taylor M. A., Perelson A. S., Non-linear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology 56 (2) (1994), 295-321. https://dx.doi.org/10.1007/BF02460644
  30. Lett C., Semeria M., Thiebault A., Tremblay Y., Effects of successive predator attacks on prey aggregations, Theoretical Ecology 7 (3) (2014), 239-252. https://dx.doi.org/10.1007/s12080-014-0213-0
  31. Sharma S., Samanta G. P., A ratio-dependent predator-prey model with Allee effect and disease in prey, Journal of Applied Mathematics and Computing 47 (2015), 345-364. https://dx.doi.org/10.1007/s12190-014-0779-0
  32. Zhang T., Xing Y., Zang H., Han M., Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality, Nonlinear Dynamics 78 (2014), 265-277. https://dx.doi.org/10.1007/s11071-014-1438-6
  33. Nagano S., Maeda Y., Phase transitions in predator-prey systems, Physical Review E 85 (2012). https://dx.doi.org/10.1103/PhysRevE.85.011915
  34. Brentnall S. J., Richards K. J., Murphy E., Brindley J., Plankton patchiness and its effect on larger-scale productivity, Journal of Plankton Research 25 (2) (2003), 121-140.
  35. Kelley J. L., Fitzpatrick J. L., Merilaita S., Spots and stripes: ecology and colour pattern evolution in butterflyfishes, Proceedings of the Royal Society B: Biological Sciences 280 (2013). https://dx.doi.org/10.1098/rspb.2012.2730
  36. Huang T., Zhang H., Yang H., Wang N., Zhang F., Complex patterns in a space- and time-discrete predator-prey model with Beddington-DeAngelis functional response, Communications in Nonlinear Science and Numerical Simulation 43 (2017), 182-199. https://dx.doi.org/10.1016/j.cnsns.2016.07.004
  37. Ma X., Shao Y., Wang Z., Luo M., Fang X., Ju Z., An impulsive two-stage predator-prey model with stage-structure and square root functional responses, Mathematics and Computers in Simulation 119 (2016), 91-107. https://dx.doi.org/10.1016/j.matcom.2015.08.009
  38. Jiao J. J., Meng X. Z., Chen L. S., Harvesting policy for a delayed stage-structured Holling II predator-prey model with impulsive stocking prey, Chaos, Solitons & Fractals 41 (2009), 103-112. https://dx.doi.org/10.1016/j.chaos.2007.11.015
  39. Mbava W., Mugisha J. Y. T., Gonsalves J. W., Prey, predator and super-predator model with disease in the super-predator, Applied Mathematics and Computation 297 (2016), 92-114. https://dx.doi.org/10.1016/j.amc.2016.10.034
  40. Wang C., Chang L., Liu H., Spatial Patterns of a Predator-Prey System of Leslie Type with Time Delay, PLOS ONE 11 (3) (2016). https://dx.doi.org/10.1371/journal.pone.0150503
  41. Rao F., Kang Y., The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecological Complexity 28 (2016), 123-144. https://dx.doi.org/10.1016/j.ecocom.2016.07.001
  42. Xu C., Yuan S., Global dynamics of a predator-prey model with defence mechanism for prey, Applied Mathematics Letters 62 (2016), 42-48. https://dx.doi.org/10.1016/j.aml.2016.06.013
  43. Banerjee M., Ghorai S., Mukherjee N., Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations, Applied Mathematical Modelling 55 (2017), 383-399. https://dx.doi.org/10.1016/j.apm.2017.11.005
  44. Chen S., Wei J., Zhang J., Dynamics of a Diffusive Predator-Prey Model: The Effect of Conversion Rate, Journal of Dynamics and Differential Equations 30 (2018), 1683-1701.
  45. Chen S., Yu J., Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator, Journal of Differential Equations 260 (11) (2016), 7923-7939. https://dx.doi.org/10.1016/j.jde.2016.02.007
  46. Wang L., Yao P., Feng G., Mathematical analysis of an eco-epidemiological predator-prey model with stage-structure and latency, Journal of Applied Mathematics and Computing 57 (2018), 211-228. https://dx.doi.org/10.1007/s12190-017-1102-7
  47. Huang C., Zhang H., Cao J., Hu H., Stability and Hopf Bifurcation of a Delayed Prey-Predator Model with Disease in the Predator, International Journal of Bifurcation and Chaos 29 (7) (2019). https://dx.doi.org/10.1142/S0218127419500913
  48. Tan W., Yu W., Hayat T., Alsaadi F., Fardoun H. M., Turing Instability and Bifurcation in a Diffusion Predator-Prey Model with Beddington-DeAngelis Functional Response, International Journal of Bifurcation and Chaos 28 (9) (2018). https://dx.doi.org/10.1142/S021812741830029X
  49. Jiang H., Wang L., Yao R., Numerical simulation and qualitative analysis for a predator-prey model with B-D functional response, Mathematics and Computers in Simulation 117 (2015), 39-53. https://dx.doi.org/10.1016/j.matcom.2015.05.006
  50. Zhang L., Fu S., Non-Constant Positive Steady States for a Predator-Prey Cross-Diffusion Model with Beddington-DeAngelis Functional Response, Boundary Value Problems 2011 (404696), 1-19. https://dx.doi.org/10.1155/2011/404696
  51. Zhang H., Xu G., Sun H., Biological control of a predator-prey system through provision of an infected predator, International Journal of Biomathematics 11 (8) (2018). https://dx.doi.org/10.1142/S179352451850105X
  52. Singh T., Banerjee S., Spatial Aspect of Hunting Cooperation In Predators with Holling Type II Functional Response, Journal of Biological Systems 26 (2018), 511-531. https://dx.doi.org/10.1142/S0218339018500237
  53. Clements H. S., Tambling C. J., Kerley G. I. H., Prey morphology and predator sociality drive predator prey preferences, Journal of Mammalogy 97 (2016), 909-927. https://dx.doi.org/10.1093/jmammal/gyw017
  54. Rao F., Chavez C. C., Kang Y., Dynamics of a diffusion reaction prey-predator model with delay in prey: Effects of delay and spatial components, Journal of Mathematical Analysis and Applications 461 (2018), 1177-1214. https://dx.doi.org/10.1016/j.jmaa.2018.01.046
  55. Lin M., Chai Y., Yang X., Wang Y., Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System, Mathematical Problems in Engineering 2019 (3907453), 1-10. https://dx.doi.org/10.1155/2019/3907453
  56. Yi F., Wei J., Shi J., Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations 246 (2009), 1944-1977. https://dx.doi.org/10.1016/j.jde.2008.10.024
  57. Singh T., Banerjee S., Spatiotemporal Model of a Predator-Prey System with Herd Behavior and Quadratic Mortality, International Journal of Bifurcation and Chaos 29 (4) (2019). https://dx.doi.org/10.1142/S0218127419500494
  58. Lu J., Zhang X., Xu R., Global stability and Hopf bifurcation of an eco-epidemiological model with time delay, International Journal of Biomathematics 12 (6) (2019). https://dx.doi.org/10.1142/S1793524519500621
  59. Alidousti J., Stability and bifurcation analysis of a fractional prey-predator scavenger model, Applied Mathematical Modelling 81 (2020), 342-355. https://dx.doi.org/10.1016/j.apm.2019.11.025
  60. Bezabih A. F., Edessa G. K., Koya P. R., Mathematical Eco-Epidemiological Model on Prey-Predator System, Mathematical Modelling and Applications 5 (3) (2020), 183-190. https://dx.doi.org/10.11648/j.mma.20200503.17
  61. Singh T., Dubey R., Spatial patterns dynamics of a diffusive predator-prey system with cooperative behavior in predator, Fractals 29 (4) (2021). https://dx.doi.org/10.1142/S0218348X21500857
  62. Clements H. S., Craig J. T., Kerley G. I. H., Prey morphology and predator sociality drive predator prey preferences, Journal of Mammalogy 97 (3) (2016), 919-927. https://dx.doi.org/10.1093/jmammal/gyw017
  63. Haque M., A Predator-Prey model with disease in the Predator species only, Nonlinear Analysis: Real World Applications 11 (4) (2010), 2224-2236. https://dx.doi.org/10.1016/j.nonrwa.2009.06.012
  64. Haque M., Greenhalgh D., When Predator avoids infected Prey: a model based theoretical studies, Mathematical Medicine and Biology: A Journal of the IMA 27 (1) (2009), 75-94.
  65. Bairagi N., Roy P. K., Chattopadhyay J., Role of infection on the stability of a Predator-Prey system with several response functions - A comparative study, Journal of Theoretical Biology 248 (1) (2007), 10-25. https://dx.doi.org/10.1016/j.jtbi.2007.05.005
  66. Das K. P., Kundu K., Chattopadhyay J., A Predator-Prey mathematical model with both populations affected by diseases, Ecological Complexity 8 (1) (2011), 68-80. https://dx.doi.org/10.1016/j.ecocom.2010.04.001
  67. Ajraldi V., Pittavino M., Venturino E., Modeling Herd behavior in population systems, Nonlinear Analysis: Real World Applications 12 (4) (2011), 2319-2338. https://dx.doi.org/10.1016/j.nonrwa.2011.02.002