Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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DIFFUSIVE AND STOCHASTIC ANALYSIS OF LOKTA-VOLTERRA MODEL WITH BIFURCATION

  • C.V. PAVAN, KUMAR (Department of Mathematics, Apex Math Excel Center) ;
  • G. RANJITH, KUMAR (Department of Mathematics, ANURAG University) ;
  • KALYAN, DAS (Department of Basic and Applied Sciences, National Institute of Food Technology Entrepreneurship and Management, HSIIDC Industrial Estate) ;
  • K. SHIVA, REDDY (Department of Mathematics, ANURAG University) ;
  • MD. HAIDER ALI, BISWAS (Department of Mathematics, Science Engineering and Technology School, Khulna University)
  • Received : 2021.05.01
  • Accepted : 2022.07.19
  • Published : 2023.01.30
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Abstract

The paper presents a critical analysis of selected topics related to the modeling of interacting species in which prey has nonlinear reproduction, which is in competition with predator. The mathematical model's stochastic stability is investigated. The method of designing appropriate Lyapunov functions is used to identify permanence conditions among the parameters of the model and conditions for the structure to no longer be extinct. The system's two-dimensional diffusive stability is regarded and studied. The system experiences the process of saddle-node bifurcation by varying the death rate of predator parameter. Further effects of parameters that undergo inherent oscillations are numerically investigated, revealing that as the intensity of predation parameter b is increased, the device encounters non-periodic and damped oscillations.

Keywords

Acknowledgement

The Department of Basic and Applied Sciences, NIFTEM Knowledge Centre, NIFTEM, India, Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, India and Mathematics Discipline of Khulna University, Khulna, Bangladesh has all provided invaluable assistance.

References

  1. A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci. 237 (1952), 37-72.  https://doi.org/10.1098/rstb.1952.0012
  2. J.M. Cushing, tability and Instability in Predator-Prey Models with Growth Rate Response Delays, Rocky Mountain Journal of Mathematics 9 (1979), 43-50.  https://doi.org/10.1216/RMJ-1979-9-1-43
  3. B. Roy, S.K. Roy and M.H.A. Biswas, Effects on prey-predator with different functional responses, Int. J. Biomath. 10 (2017), 1750113-22.  https://doi.org/10.1142/S1793524517501133
  4. S. Akter, M.S. Islam, M.H.A. Biswas and S. Mandal, tability and Instability in Predator-Prey Models with Growth Rate Response Delays, Communication in Mathematical Modeling and Applications 4 (2019), 84-94. 
  5. M.N. Hasan, M.S. Uddin, and M.H.A. Biswas, Interactive Effects of Disease Transmission on Predator-Prey Model, Journal of Applied Nonlinear Dynamics 9 (2020), 401-413.  https://doi.org/10.5890/jand.2020.09.005
  6. A. Aldurayhim, A. Elsonbaty, and A.A. Elsadany, Dynamics of diffusive modified Previte-Hoffman food web model, Mathematical Biosciences and Engineering 17 (2020), 4225-4256.  https://doi.org/10.3934/mbe.2020234
  7. X. Zhang, Y. Huang, and P. Weng, Permanence and stability of a diffusive predator-prey model with disease in the prey, Computers and Mathematics with Applications 68 (2014), 1431-1445.  https://doi.org/10.1016/j.camwa.2014.09.011
  8. X. Feng, Y. Song, J. Liu, and G. Wang, Permanence, stability, and coexistence of a diffusive predator-prey model with modified Leslie-Gower and B-D functional response, Adv. Differ. Equ. 68 (2018), 314. 
  9. K. Manna, V. Volpert, and M. Banerjee, Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species, Mathematics 8 (2020), 1-28.  https://doi.org/10.3390/math8010001
  10. R. Yang,and J. Wei, Stability and bifurcation analysis of a diffusive prey-predator system in Holling type III with a prey refuge, Nonlinear Dyn. 79 (2015), 631-646.  https://doi.org/10.1007/s11071-014-1691-8
  11. L.N. Guin, and S. Acharya, Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting, Nonlinear Dyn. 88 (2017), 1501-1533.  https://doi.org/10.1007/s11071-016-3326-8
  12. W. Wang, Y. Cai, Y. Zhu, and Z. Guo, Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model, Abstract and Applied Analysis 487810 (2013), 1-10. 
  13. A. Hastings, Disturbance, coexistence, history, and competition for space, Theoretical Population Biology 18 (1980), 363-373.  https://doi.org/10.1016/0040-5809(80)90059-3
  14. A.J. Lotka, Elements of Mathematical Biology, Dover Publications, Mineola., 1956. 
  15. A.J. Nicholson, and V.A. Bailey, The balance of animal populations, Proceedings of the Zool. Soc. of London 1 (1935), 551-598.  https://doi.org/10.1111/j.1096-3642.1935.tb01680.x
  16. W.T. Jamieson, and J. Reis, Global behavior for the classical Nicholson-Bailey model, Journal of Mathematical Analysis and Applications 461 (2018), 492-499.  https://doi.org/10.1016/j.jmaa.2017.12.071
  17. M. Edmunds, Defense in Animals: A Survey of Anti-predator Defenses, Longman, Harlow, 1974. 
  18. E. Gonzalez-Olivares, and R. Ramos-Jiliberto, Dynamics consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling 106 (2003), 135-146.  https://doi.org/10.1016/S0304-3800(03)00131-5
  19. L.B. Kats, and L.M. Dill, . The scent of death: Chemosensory assessment of predation risk by prey animals, Ecoscience 5 (1998), 361-394.  https://doi.org/10.1080/11956860.1998.11682468
  20. C.S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can. 45 (2003), 5-60. 
  21. A.K. Sardar, M. Hanif, M. Asaduzzaman, and M.H.A. Biswas, Mathematical Analysis of the Two Species Lotka-Volterra Predator-Prey Inter-specific Game Theoretic Competition Model, Advanced Modeling and Optimization 18 (2016), 231-242. 
  22. V.N. Afanasev, V.B. Kolmanowski, and V.R. Nosov, Mathematical Theory of Global Systems Design, Kluwer Academic, Dordrecht. 1996. 
  23. K. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal. 12 (1981), 541-548.  https://doi.org/10.1137/0512047
  24. C.B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations, Hilgardia 27 (1958), 343-383.  https://doi.org/10.3733/hilg.v27n14p343
  25. C.B. Huffaker, K.P. Shea and S.G. Herman, Experimental studies on predation: complex dispersion and levels of food in an acarine predator-prey interaction, Hilgardia 34 (1963), 305-330.  https://doi.org/10.3733/hilg.v34n09p305
  26. B. Dubey, and A. Patra, Optimal management of a renewable resource utilized by a population with taxation as a control variable, Nonlinear Analysis: Modelling and Control 18 (2013), 37-52.  https://doi.org/10.15388/NA.18.1.14030
  27. J.B. Shukla, and B. Dubey, Modelling the depletion and conservation of forestry resource: Effect of population, J. Math. Biol. 36 (1997), 71-94.  https://doi.org/10.1007/s002850050091
  28. I. Perfecto, and J. Vandermeer, Spatial pattern and ecological process in the coffee agro-forestry system, Ecology 89 (2008), 915-920.  https://doi.org/10.1890/06-2121.1
  29. J. Crank, The Mathematics of Diffusion, Oxford:Clarendon Press, London, 1979. 
  30. L. Zhang, and S. Fu, SNonlinear instability for a Leslie-Gower predator-prey model with cross diffusion, Abstract and Applied Analysis 2013 (2013), Article ID 854862, 13 pages. 
  31. J.F. Zhang, W.T. Li, and Y.X. Wang, Turing patterns of a strongly coupled predator-prey system with diffusion effects, Nonlinear Analysis. Theory, Methods Applications 74 (2011), 847-858.  https://doi.org/10.1016/j.na.2010.09.035
  32. K.N. Chueh, C.C. Conley, and J.A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana University Mathematics Journal 26 (1977), 373-392.  https://doi.org/10.1512/iumj.1977.26.26029
  33. C.V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Analysis 26 (1996), 1889-1903.  https://doi.org/10.1016/0362-546X(95)00058-4
  34. W.F. Morris, M. Mangel, and F.R. Adler, Mechanisms of pollen deposition by insect pollinators, volutionary Ecology 9 (1995), 304-317. 
  35. L.A. Segel, and J.L. Jackson, Theoretical analysis of chemotactic movement in bacteria, Journal of Mechanochemistry and Cell Motility 2 (1973), 25-34. 
  36. J.M. Smith, Mathematical Ideas in Biology, Cambridge University Press, Cambridge, 1968. 
  37. Li. Shangzhi, Guo. Shangjiang, Permanence of a stochastic prey-predator model with a general functional response, Mathematics and Computers in Simulation 187 (2021), 308-336.  https://doi.org/10.1016/j.matcom.2021.02.025
  38. Kar. Tapan Kumar, K.S. Chaudhuri, On Selective Harvesting of Two Competing Fish Species In The Presence Of Environmental Fluctuation, Natural Resource Modeling 17 (2004). 
  39. R. Arditi, and L.R. Ginzburg, Coupling in predator-prey dynamics : ratio-dependence, J. Theort. Biol. 139 (1989), 311-326.  https://doi.org/10.1016/S0022-5193(89)80211-5
  40. B. Dubey, and J. Hussain, Modelling the Interaction of Two Biological Species in a Polluted Environment, Journal of Mathematical Analysis and Applications 246 (2000), 58-79.  https://doi.org/10.1006/jmaa.2000.6741
  41. M. Bandyopadhyay, and C.G. Chakrabarti, Deterministic and stochastic analysis of a non-linear prey-predator system, Journal of Biological Systems 11 (2003), 161-172.  https://doi.org/10.1142/S0218339003000816
  42. M. Bandyopadhyay, R. Bhattacharya, and G.C. Chakrabarti, A nonlinear two species oscillatory system : bifurcation and stability analysis, Int. Jr. Math. Math. Sci. 2003 (2003), 1981-1991.  https://doi.org/10.1155/S0161171203201174
  43. R.M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jercy, 2001. 
  44. V.N. Afanas'ev, V.B. Kolmanowski, V.R. Nosov, Mathematical Theory of Global Systems Design, Kluwer Academic, Dordrecht, 1996.