---

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 499--522 | DOI:10.5890/DNC.2021.09.011

Manotosh Mandal$^{1,2}$, Soovoojeet Jana $^3$ , Swapan Kumar Nandi$^4$, T. K. Kar$^2$

$^{1}$ Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk -721636 , West Bengal, India

$^2$ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India

$^3$ Department of Mathematics, Ramsaday College, Amta-711401, Howrah, West Bengal, India

$^4$ Department of Mathematics, Nayabasat P.M.Sikshaniketan, Paschim Medinipur-721253, West Bengal, India

Download Full Text PDF

 

Abstract

The environmental carrying capacities for the prey population and the predator population are restricted by their availability of foods. In this article, we introduce a prey-predator type ecological model in which the prey and predator have different biotic resources for food. Therefore the suggested predator-prey model depends on the ratio-dependent ecological model which can be applied in the study of food chains. The details dynamical behaviour of the proposed model has been carried out. The different bifurcations and numerical analyzes are demonstrated to illustrate the dynamical behavior of our proposed model system.

Acknowledgments

Research of T. K. Kar is supported by the Council of Scientific and Industrial Research(CSIR), India (File No.25(300)/19/EMR-II, dated:16th May, 2019). Moreover, the authors are very much grateful to the anonymous reviewers and the editor Prof. Dimitri Volchenkov for their constructive comments and useful suggestions to improve the quality and presentation of the manuscript significantly.

References

1. [1]	Kot, M. (2001), Elements of Mathematical Ecology. ph{Cambridge University Press}.

2. [2]	Britton, N.F. (2003), Essential Mathematical Biology, ph{Springer}.

3. [3]	Huang, C., Zhang, H., Cao, J., and Hu, H. (2019), Stability and Hopf Bifurcation of a Delayed Prey-Predator Model with Disease in the Predator, ph{International Journal of Bifurcation and Chaos}, 29(07), 1950091.

4. [4]	Freedman, H.I. and Waltman, P. (1984), Persistence in models of three interacting predator-prey populations, ph{Mathematical Biosciences}, 68, 213-231.

5. [5]	Schwarzl, M., Godec, A., Oshanin, G., and Metzler, R. (2016), A single predator charging a herd of prey: effects of self volume and predator-prey decision making, ph{Journal of Physics A: Mathematical and Theoretical}, 49(22), 225601.

6. [6]	Umar, M., Sabir, Z., Asif, M., and Raja, Z. (2019)., Intelligent computing for numerical treatment of nonlinear prey-predator models, ph{ Applied Soft Computing Journal}, 80, 506-524.

7. [7]	Jana, S. and Kar, T.K. (2013), A mathematical study of a prey-predator model in relevance to pest control, ph{ Nonlinear Dynamics}, 74, 667-683.

8. [8]	Jana, S., Ghorai, A., Guria, S., Das, U., and Kar, T.K. (2015), Global dynamics of a predator, weaker prey and stronger prey system., ph{ Nonlinear Dynamics}, 250, 235-248.

9. [9]	Jana, S., Guria, S., Das, U., Kar, T.K., and Ghorai, A. (2015), Effect of harvesting and infection on predator in a prey-predator system, ph{ Nonlinear Dynamics}, 81, 917-930.

10. [10]	Kar, T.K. (2005), Stability analysis of a prey-predator model incorporating a prey refuge, ph{Communications in Nonlinear Science and Numerical Simulation}, 10, 681-691.

11. [11]	Wang, Z., Xie, Y., Lu, J., and Li, Y.(2019), Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, ph{Applied Mathematics and Computation}, 347, 360-369.

12. [12]	Hoyle, A. and Bowers, R.G. (2007), When is evolutionary branching in predator-prey systems possible with an explicit carrying capacity? ph{Math. Biosci.}, 210(1), 1-16.

13. [13]	Collings, J.B. (1997) , The effects of the functional response on the bifurcation behavior of a mite predator- prey interaction model, J. Math. Biol., 36, 149-168.

14. [14]	Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika, 47(3/4), 219-234.

15. [15]	Seo, G. and Kot, M. (2008), A comparison of two predator-prey models with Holling type I functional response, ph{Math. Biosci.}, 212, 161-179.

16. [16]	Saez, E. and Olivares, E.G. (1999), Dynamics of a predator-prey model, ph{SIAM J. Appl. Math.}, 59(5), 1867-1878.

17. [17]	Hsu, S.B. and Huang, T.W. (1995), Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55(3), 763-783.

18. [18]	Gakkhar, S. and Naji, R.K. (2003), Order and chaos in predator to prey ratio-dependent food chain, Chaos Solitons Fractals, 18, 229-239.

19. [19]	Polis, G.A. and Holt, R.D. (1992), Intraguild predation- the dynamics of complex trophic interactions. ph{Trends Ecol. Evol.}, 7(5), 151-154.

20. [20]	Holt, R.D. and Polis, G.A. (1997), A theoretical framework for intraguild predation, Am. Nat., 149(4), 745-764.

21. [21]	Safuan, H.M., Sidhu, H.S., Jovanoski, Z., and Towers, I.N. (2013), Impacts of Biotic Resource Enrichment on a Predator-Prey Population, ph{Bull Math Biol, } 75, 1798-1812.

22. [22]	Safuan, H.M., Sidhu, H.S., Jovanoski, Z., and Towers, I.N. (2014), A two-species predator-prey model in an environment enriched by a biotic resource. ph{ANZIAM Journal }, 54, 768-787.

23. [23]	Kar, T.K. and Mondal, P.K. (2011), Global dynamics and bifurcation in delayed SIR epidemic model, ph{Nonlinear Analysis: Real World Applications}, 12, 2058-2068.

24. [24]	Guckenheimer, G. and Holmes, P. (1983), Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, ph{ Springer Verlag: New York.}

25. [25]	Khajanchia, S., Das, D.K., and Kar, T.K. (2018), Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation, ph{Physica A: Statistical Mechanics and its Applications}, 497, 52-71.

26. [26]	Lynch, S. (2014), Dynamical Systems with Applications using MATLAB$^\circledR$, ph{Springer International Publishing}.

27. [27]	Dhooge, A., Govaerts, W., and Kuznetsov, Y. (2003), Matcont: A Matlab package for numerical bifurcation analysis of ODEs, ph{ACM TOMS}, 29, 141-164.

28. [28]	Mylius, S.D., Klumpers, K., de Roos, A.M., and Persson, L. (2001), Impact of intraguild predation and stage structure on simple communities along a productivity gradient, Am. Nat., 158(3), 259-276.

29. [29]	Diehl, S. and Feissel, M.(2001), Intraguild prey suffer from enrichment of their resources: a microcosm experiment with ciliates, Ecology, 82(11), 2977-2983.

30. [30]	Hin, V., Schellekens, T., Persson, L., and Roos, A.M. (2011), Coexistence of predator and prey in intraguild predation systems with ontogenetic niche shifts, Am. Nat., 178(6), 701-714.