---

Journal of Environmental Accounting and Management 9(2) (2021) 127--143 | DOI:10.5890/JEAM.2021.06.003

Lakshmi Narayan Guin$^{1}$, Debdeep Roy$^{1}$, Salih Djilali$^{2,3}$

$^{1}$ Department of Mathematics, Visva-Bharati, Santiniketan-731 235, West Bengal, India

$^{2}$ Department of Mathematics, Universite Hassiba Benbouali de Chlef, Chlef 02000, Algeria

$^3$ Laboratoire d'analyse non-lin'{e}aire et math'{e}matiques appliqu'{e}es. Universit'{e} de Tlemcen, Tlemcen, Alg'{e}rie

Download Full Text PDF

 

Abstract

The present article deals with a state of phase transition from stability to chaos emanated from a three-species food chain model with intra-specific competition. The model takes into account two distinct categories of functional response specifically Holling type II and Beddington-DeAngelis type along with the intra-specific competition among predators as well. The existence of equilibria having ecological feasibility together with stability in their proximity is addressed meticulously for the system. The perception of dissipativeness of the system in the realm of ecological principles is not ruled out however from the present investigation. Following the standard stability analysis, both the local and the global response of the co-existence equilibrium are paid due attention in order to understand the dynamics of the system. The fact that the proposed system experiences Hopf-bifurcation and chaos is note-worthy. Finally, the validity of the analytical results is established through numerical simulation based on model parameter values.

Acknowledgments

The present form of the research manuscript owes much to the constructive suggestions of the referees, whose careful revision we are pleased to acknowledge. We sincerely appreciate for the help from Professor Santabrata Chakravarty, Department of Mathematics, Visva-Bharati while carrying out revision work. 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)). S. Djilali supported partially by DGESTR of Algeria No.C00L03UN130120200004.

References

1. [1]	Ali, N. and Chakravarty, S. (2015), Stability analysis of a food chain model consisting of two competitive preys and one predator, Nonlinear Dynamics, 82(3), 1303-1316.

2. [2]	Batabyal, S., Jana, D., Lyu, J. and Parshad, R.D. (2020), Explosive predator and mutualistic preys: A comparative study, Physica A: Statistical Mechanics and its Applications, 541, 123348.

3. [3]	Braza, P.A. (2003), The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing, SIAM Journal on Applied Mathematics, 63(3), 889-904.

4. [4]	Cao, F. and Chen, L. (1998), Asymptotic behavior of nonautonomous diffusive Lotka-Volterra model, Systems Science and Mathematical Sciences, 11(2), 107-111.

5. [5]	Do, Y., Baek, H., Lim, Y. and Lim, D. (2011), A three-species food chain system with two types of functional responses, Abstract and Applied Analysis, {Article ID 934569}, 1-16.

6. [6]	Djilali, S. (2019), Impact of prey herd shape on the predator-prey interaction, Chaos, Solitons \& Fractals, 120, 139-148.

7. [7]	Djilali, S. (2019), Effect of herd shape in a diffusive predator-prey model with time delay, Journal of Applied Analysis and Computation, 9(2), 638-654.

8. [8]	Djilali, S. (2020), Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition, Mathematical Methods in the Applied Sciences, 43(5), 2233-2250.

9. [9]	Djilali, S. (2020), Spatiotemporal patterns induced by cross-diffusion in predator-prey model with prey herd shape effect, International Journal of Biomathematics, 13(04), 2050030.

10. [10]	Hastings, A. and Powell, T. (1991), Chaos in a three-species food chain, Ecology, 72(3), 896-903.

11. [11]	Klebanoff, A. and Hastings, A. (1994), Chaos in three-species food chains, Journal of Mathematical Biology, 32(5), 427-51.

12. [12]	Layek, G.C. (2015), An introduction to Dynamical System and Chaos, Springer.

13. [13]	Lv, S. and Zhao, M. (2008), The dynamic complexity of a three species food chain model, Chaos, Solitons $\&$ Fractals, 37(5), 1469-1480.

14. [14]	Lynch, S. (2013), Dynamical System with Application using MATLAB, Birkh\"{a}user, Springer.

15. [15]	Nath, B., Kumari, N., Kumar, V. and Das, K. P. (2019), Refugia and Allee effect in prey species stabilize chaos in a tri-trophic food chain model, Differential Equations and Dynamical Systems, https://doi.org/10.1007/s12591-019-00457-z, 1-27.

16. [16]	Nath, B., Lyu, J., Parshad, R. D., Das, K. P. and Singh, A. (2020), A study of chaos and its control in a harvested tri-trophic food chain model with alternative food source and diffusion effect, Nonlinear Studies, 27(2), 505-528.

17. [17]	Pal, N., Samanta, S., and Chattopadhyay, J. (2014), Revisited Hastings and Powell model with omnivory and predator switching, Chaos, Solitons $\&$ Fractals, 66, 58-73.

18. [18]	Perko, L. (2001), Differential Equations and Dynamical Systems, Springer.

19. [19]	Ruan, S. and Xiao, D. (2001), Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 61(4), 1445-1472.

20. [20]	Stogartz, S.H. (1994), Nonlinear Dynamics and Chaos: with application to Physics, Biology, Chemistry and Engineering, Taylor and Francis.

21. [21]	ck and Bolnick (2006)] {9} Svanb\"{a}ck, R. and Bolnick, D.I. (2006), Intra-specific competition drives increased resource use diversity within a natural population, Abstract and Applied Analysis, Proceedings of the Royal Society B: Biological Sciences, 274(1611), 839-844.

22. [22]	Zhao, M. and Lv, S. (2009), Chaos in a three-species food chain model with a Beddington-DeAngelis functional response, Chaos, Solitons $\&$ Fractals, 40(5), 2305-2316.