Discontinuity, Nonlinearity, and Complexity
A Study on Effects of Biotic Resources on a PreyPredator Population
Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 499522  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,
Howrah711103, West Bengal, India
$^3$ Department of Mathematics, Ramsaday College, Amta711401, Howrah, West Bengal, India
$^4$ Department of Mathematics, Nayabasat P.M.Sikshaniketan, Paschim Medinipur721253, 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 preypredator type ecological model in which the prey and predator have different biotic resources for food. Therefore the suggested predatorprey model depends on the ratiodependent 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/EMRII, 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]  Kot, M. (2001), Elements of Mathematical Ecology. ph{Cambridge University Press}.


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


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


[4]  Freedman, H.I. and Waltman, P. (1984), Persistence in models of three interacting predatorprey populations,
ph{Mathematical Biosciences}, 68, 213231.


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


[6]  Umar, M., Sabir, Z., Asif, M., and Raja, Z. (2019)., Intelligent computing for numerical treatment of nonlinear preypredator models, ph{ Applied Soft Computing Journal}, 80, 506524.


[7]  Jana, S. and Kar, T.K. (2013), A mathematical study of a preypredator model in relevance to pest control,
ph{ Nonlinear Dynamics}, 74, 667683.


[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, 235248.


[9]  Jana, S., Guria, S., Das, U., Kar, T.K.,
and Ghorai, A. (2015), Effect of harvesting and infection on predator in a preypredator system, ph{ Nonlinear Dynamics}, 81, 917930.


[10]  Kar, T.K. (2005), Stability analysis of a preypredator model incorporating a prey refuge,
ph{Communications in Nonlinear Science and Numerical Simulation},
10, 681691.


[11]  Wang, Z., Xie, Y., Lu, J., and Li, Y.(2019),
Stability and bifurcation of a delayed generalized fractionalorder
preypredator model with interspecific competition,
ph{Applied Mathematics and Computation}, 347, 360369.


[12]  Hoyle, A. and Bowers, R.G. (2007), When is evolutionary branching in predatorprey systems possible
with an explicit carrying capacity? ph{Math. Biosci.}, 210(1), 116.


[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, 149168.


[14]  Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predatorprey type of
interaction between two species. Biometrika, 47(3/4), 219234.


[15]  Seo, G. and Kot, M. (2008), A comparison of two predatorprey models with Holling type I functional
response, ph{Math. Biosci.}, 212, 161179.


[16]  Saez, E. and Olivares, E.G. (1999), Dynamics of a predatorprey model, ph{SIAM J. Appl. Math.}, 59(5),
18671878.


[17]  Hsu, S.B. and Huang, T.W. (1995), Global stability for a class of predatorprey systems, SIAM J. Appl.
Math., 55(3), 763783.


[18]  Gakkhar, S. and Naji, R.K. (2003), Order and chaos in predator to prey ratiodependent food chain,
Chaos
Solitons Fractals, 18, 229239.


[19]  Polis, G.A. and Holt, R.D. (1992), Intraguild predation the dynamics of complex trophic interactions.
ph{Trends Ecol. Evol.}, 7(5), 151154.


[20]  Holt, R.D. and Polis, G.A. (1997), A theoretical framework for intraguild predation, Am. Nat., 149(4),
745764.


[21]  Safuan, H.M., Sidhu, H.S., Jovanoski, Z., and Towers, I.N. (2013), Impacts of Biotic Resource Enrichment
on a PredatorPrey Population, ph{Bull Math Biol, } 75, 17981812.


[22]  Safuan, H.M., Sidhu, H.S., Jovanoski, Z., and Towers,
I.N. (2014), A twospecies predatorprey model in an environment enriched by a biotic resource. ph{ANZIAM Journal }, 54, 768787.


[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, 20582068.


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


[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, 5271.


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


[27]  Dhooge, A., Govaerts, W., and Kuznetsov, Y. (2003), Matcont: A Matlab package for numerical bifurcation analysis of ODEs,
ph{ACM TOMS}, 29, 141164.


[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), 259276.


[29]  Diehl, S. and Feissel, M.(2001), Intraguild prey suffer from enrichment of their resources: a microcosm
experiment with ciliates, Ecology, 82(11), 29772983.


[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), 701714.
