[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
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

Fear and Density Dependent Mortality Control Chaos-Conclusion Drawn from a Tri-Trophic Food Chain

Journal of Applied Nonlinear Dynamics 14(2) (2025) 399--415 | DOI:10.5890/JAND.2025.06.011

Prodip Roy, Krishna Pada Das, Ani Jain, Parimita Roy, Abhishek Sarkar, Kulbhushon Agnihotri

Department of Mathematics, Mahadevananda Mahavidyalaya, Monirampore, Barrackpore, Kolkata, India

Department of Mathematics, Thapar Institute of engineering and technology, Patiala, Punjab, India

Department of Applied Sciences And Humanities, Shaheed Bhagat Singh State University, Ferozpore, Punjab, India

Department of Mathematics, Punjab University, Chandigarh, India

Download Full Text PDF

 

Abstract

Understanding the environmental conditions associated with predators is essential to predator management. A tri-tropic food chain model is analyzed in this paper. The existence of a solution has been analyzed and shown to be uniformly bounded. The threshold number ($R_0$) is obtained, and the occurrence of bifurcation at $R_0=0$ is shown to be possible using central manifold theory. We used the Partial Rank Correlation Coefficient (PRCC) to do a global sensitivity analysis and identify the most sensitive parameters affecting $R_0$, providing information on potential ways to maintain ecological balance. Global stability of non-trivial equilibrium is established. Criteria for diffusion-driven ecological instability caused by local random movements of species are obtained. Detailed analyses of Turing patterns formation selected by the reaction-diffusion system under zero flux boundary conditions are presented. We found that $b_1$ and self-diffusion coefficients have an appreciable influence on the spatial spread of epidemics. Numerical simulation results confirm the analytical finding and generate patterns that indicate that the population and thus ecological balance can be maintained.

References

  1. [1]  Hastings, A. and Powell, T. (1991), Chaos in three-species food chain, Ecology, 72(3), 896-903.
  2. [2]  Rai, V. and Sreenivasan, R. (1993), Period-doubling bifurcations leading to chaos in a model food chain, Ecological Modelling, 69(1-2), 63-77.
  3. [3]  Ruxton, G.D. (1994), Low levels of immigration between chaotic populations can reduce system extinctions by inducing asynchronous cycles, Series B: Biological Sciences, 256(1346), 189-193.
  4. [4]  Ruxton, G.D. (1996), Chaos in a three-species food chain with a lower bound on the bottom population, Ecology, 77(1), 317-319.
  5. [5]  Eisenberg, J.N. and Maszle, D.R. (1995), The structural stability o f a three-species food chain model, Journal of Theoretical Biology, 176, 501-510.
  6. [6] MaCann, K. and Hastings A. (1997), Re-evaluating the omnivory-stability relationship in food webs, Proceedings of the Royal Society of London. Series B: Biological Sciences, 264(1385), 1249-1254.
  7. [7]  Varriale, M.C. and Gomes, A.A., (1998), A study of a three species food chain, Ecological Modelling, 110(2), 119-133.
  8. [8]  Xiao, Y. and Chen, L. (2001), Modelling and analysis of a predator-prey model with disease in the prey, Mathematical Biosciences, 171, 59-82.
  9. [9] Chattopadhyay, J., Pal, S., and EI Abdllaoui, A. (2003), Classical predator-prey system with infection of prey population-a mathematical model, Mathematical Methods in the Applied Sciences, 26, 1211-1222.
  10. [10]  Lonngren, K. E., Bai, E.W., Ucar, A., Dynamics and synchronization of the Hastings-Powell model of the food chain. Chaos, Solitons and Fractals, 20, 387-393, (2004).
  11. [11]  Maionchi, D.O., Reis, S.F.D., and Aguiar, M.A.M.d, Chaos and pattern formation in a spatial tritrophic food chain. Ecological Modelling, 191(2), 291-303, doi:10.1016/j.ecolmodel.2005.04.028.
  12. [12]  Aziz-Alaoui, M.A. (2002), Study of a Leslie–Gower-type tritrophic population model, Chaos, Solitons $\&$ Fractals, 14(8), 1275-1293. https://doi.org/10.1016/S0960-0779(02)00079-6.
  13. [13]  Schaffer, W.M. and Kot, M. (1986), Chaos in ecological systems the coals that Newcastle forgot, Trends in Ecology and Evolution, 1, 58-63.
  14. [14]  Gilpin, M.E. (1979), Spiral chaos in a predator-prey model, The American Naturalist, 113(2), 306-308.
  15. [15]  Olsen, L.F., Truty G.L., and Schaffer W.M. (1988), Oscillations and chaos in epidemics: a non-linear dynamic study of six childhood diseases in Copenhagen, Denmark, Theoretical Population Biology, 33, 344-370.
  16. [16]  Hassell, M.P. and Varley, G.C. (1969), New inductive population model for insect parasites and its bearing on biological control, Nature(London), 223, 1133-1137.
  17. [17]  May, R.M. (1973), Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, New Jersey, USA.
  18. [18]  May, R.M. and Leonard, W.J. (1975), Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29, 243-253.
  19. [19]  Powell, T. and Richerson, P.J. (1985), Temporal variation, spatial heterogeneity, and competition for resources in plankton systems: a theoretical model, The American Naturalist, 125(3), 431-464.
  20. [20] Turchin, P. (2003), Complex Population Dynamics. A Theoretical Empirical Synthesis, Princeton University Press, Princeton, NJ.
  21. [21]  Anderson, R.M. and May, R.M. (1979), Population biology of infectious diseases: Part I, Nature, 280(5721), 361-367.
  22. [22]  Hallegraeff, G.M. (1993), A review of harmful algae blooms and the apparent global increase, Phycologia, 32, 79-99.
  23. [23]  Dwyer, G., Dushoff, J., and Yee, S.H. (2004), Generalist predators, specialist pathogens, and insect outbreaks, Nature, 430, 341-345.
  24. [24]  Hilker, F.M. and Schmitz, K. (2008), Disease-induced stabilization of predator-prey oscillations, Journal of Theoretical Biology, 255(3), 299-306., doi:10.1016/j.jtbi.2008.08.018.
  25. [25]  Rosenzweig, M.L. and MacArthur, R.H. (1963), Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97(895), 209-223.
  26. [26]  LLotka, A.J. (1925), Elements of Physical Biology, Williams and Wilkins.
  27. [27] Diekmann, O., Heesterbeek, J.A.P., and Metz, J.A.J. (1990), On the definition and the computation of the basic reproduction ratio $R _0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28, 365-382.
  28. [28]  Hadeler, K.P. and Freedman, H.I. (1989), Predator-prey populations with parasitic infection, Journal of Mathematical Biology, 27(6), 609-631.
  29. [29]  Han, L., Ma, Z., and Hethcote, H.W. (2001), Four predator prey models with infectious diseases, Mathematical and Computer Modelling, 34(7-8), p849-858.
  30. [30]  Kooi, B.W., Kuijper, L.D.J., Boer, M.P., and Kooijman, S.A.L.M. (2002), Numerical bifurcation analysis of a tri-trophic food web with omnivory, Mathematical Biosciences, 177, 201-228.
  31. [31]  Bandyopadhyay, M., Chatterjee, S., Chakraborty, S., and Chattopadhyay, J. (2008), Density dependent predator death prevalence chaos in a tri-trophic food chain model, Nonlinear analysis: Modelling and Control, 13(3), 305-324. http://dx.doi.org/10.15388/NA.2008.13.3.14559.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates