Browse > Article
http://dx.doi.org/10.14317/jami.2022.1199

DYNAMICS OF A PREY-PREDATOR INTERACTION WITH HASSELL-VARLEY TYPE FUNCTIONAL RESPONSE AND HARVESTING OF PREY  

BHATTACHARYYA, ANINDITA (Department of Mathematics, Amity University)
MONDAL, ASHOK (Department of Mathematics, Regent Education and Research Foundation)
PAL, A.K. (Department of Mathematics, Seth Anandram Jaipuria College)
SINGH, NIKHITA (Department of Mathematics, Amity University)
Publication Information
Journal of applied mathematics & informatics / v.40, no.5_6, 2022 , pp. 1199-1215 More about this Journal
Abstract
This article aims to study the dynamical behaviours of a two species model in which non-selective harvesting of a prey-predator system by using a reasonable catch-rate function instead of usual catch-per-unit-effort hypothesis is used. A system of two ordinary differential equations(ODE's) has been proposed and analyzed with the predator functional response to prey density is considered as Hassell-Varley type functional responses to study the dynamics of the system. Positivity and boundedness of the system are studied. We have discussed the existence of different equilibrium points and stability of the system at these equilibrium points. We also analysed the system undergoes a Hopf-bifurcation around interior equilibrium point for a various parametric values which has very significant ecological impacts in this work. Computer simulation are carried out to validate our analytical findings. The biological implications of analytical and numerical findings are discussed critically.
Keywords
Hassell-Varley type functional response; stability; harvesting; Hopf-bifurcation;
Citations & Related Records
Times Cited By KSCI : 4  (Citation Analysis)
연도 인용수 순위
1 A. Mondal, A.K. Pal and G.P. Samanta, Stability and Bifurcation Analysis of a Delayed Three Species Food Chain Model with Crowley-Martin Response Function, Appl. and Appl. Maths.(AAM) 13 (2018), 709-749.
2 R. Arditi and L.R. Ginzburg, Coupling in predator - prey dynamics: ratio dependence, J. Theor. Biol. 139 (1989), 311-326.   DOI
3 S. Chakraborty, S. Pal and N. Bairagi, Predator-Prey interaction with harvesting: mathematical study with biological ramifications, Appl. Math. Model. 36 (2012), 4044-4059.   DOI
4 T. Das, R.N. Mukherjee and K.S. Chaudhuri, Bioeconomic Harvesting of a Prey-Predator Fishery, J. of Biol. Dyna. 3 (2009), 447- 462.   DOI
5 M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional, J. of Math. Anal. and Appli. 295 (2004), 15-39.   DOI
6 M.P. Hassell and G.C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature 223 (1969), 1133-1137.   DOI
7 A. Mondal, A.K. Pal and G.P. Samanta, Evolutionary Dynamics of a Single-Species Population Model with Multiple Delays in a Polluted Environment, Discontinuity, Nonlinearity, and Complexity 9 (2020), 433-459.   DOI
8 L. Chen, X. Song and Z. Lu, Mathematical Models and Methods in Ecology, Scient. and Tech. Publisher of Sichuan, Chengdu, 2003.
9 G. Zeng, F. Wang and J.J. Nieto, Complexity of a delayed predator-prey model with impulsive harvest and Holling Type II functional response, Adv. in Compl. Syst. 11 (2008), 77-97.   DOI
10 C.S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawy, Can. Entomol. 91 (1959), 293-320.   DOI
11 S. Pathak, A. Maiti and G.P. Samanta, Rich Dynamics of a food chain model with HassellVarley type functional response, Appl. Math. and Comput. bf 208 (2009), 303-317.
12 Y. Li, S. Huang and T. Zhang, Dynamics of a non-selective harvesting predator - prey model with Hassell-Varley type functional response and impulsive effects, Math. Meth. Appl. Sci. 2015.
13 A. Mondal, A.K. Pal and G.P. Samanta, On the dynamics of evolutionary Leslie-Gower predator-prey eco-epidemiological model with disease in predator, Ecol. Genet. and Geono. 10 (2019), 100034.
14 J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
15 V. Volterra, Variazioni e uttauazionidelnumero d individual in species animals conviventi, Mem. Acad. Lincei 2 (1926), 31-33.
16 Rui Yuan, Weihua Jiang and Yong Wang, Saddle-node-Hopf Bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, J. Math. Anal. Appl. 422 (2015), 1072-1090.   DOI
17 S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. on Appl. Maths. 66 (2006), 1101-1129.   DOI
18 S.B. Hsu, T.W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten type ratiodependent predator-prey system, J. of Math. Biol. 42 (2001), 489-506.   DOI
19 H.F. Huo and W.T. Li, Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model, Math. and Comput. Model. 40 (2004), 261-269.   DOI
20 A.J. Lotka, Elements of Physical Biology, Williams and Wilkins Co. Inc., Baltimore, 1924.
21 Z.N. Ma, Modelling, Mathematical and Study of Species Ecology, Anhui Education Publishing Company, Hefei, China, 1996.
22 A. Mondal, A.K. Pal and G.P. Samanta, Analysis of a Delayed Eco-Epidemiological Pest- Plant Model with Infected Pest, Biophysical Reviews and Letters 14 (2019), 141-170.   DOI
23 A. Mondal, A.K. Pal, R. Kar and A.K. Shaw, Analysis of a complex four species food-web system: A mathematical model, Math. in Eng., Sci. and Aeros. 12 (2021), 1-21.
24 D. Wang, On a Non-Selective Harvesting Prey-Predator Model with Hassell-Varley type functional response, Applied Mathematics and Computation 246 (2014), 678-695.   DOI
25 P. Yang and Y. Wang, Periodic Solutions of a Delayed Eco-Epidemiological Model with Infection-Age Structure and Holling Type II Functional Response, Int. J. of Bif. and Chaos 30 (2020), 2050011.   DOI
26 Z. Xiao, X. Xie and Y. Xue, Stability and bifurcation in a Holling type II predator-prey model with Allee effect and time delay, Adv Differ Equ 288 (2018).
27 R.K Naji and N.A. Mustafa, The dynamics of a eco- epidemiological model with nonlinear incidence rate, J. of Appl. Math. 2012.
28 A.K. Pal, Stability analysis of a delayed predator-prey model with nonlinear harvesting efforts using imprecise biological parameters, Z. Naturforschung A 76 (2021), DOI:10.1515/ZNA-2021-0131.   DOI
29 F. Rao, S. Jiang, Y. Li and H. Liu, Stochastic Analysis of a Hassell-Varley Type Predation Model, Abstract and Applied Analysis, 2013.
30 A. Mondal, A.K. Pal and G.P. Samanta, Rich dynamics of non-toxic phytoplankton, toxic phytoplankton and zooplankton system with multiple gestation delays, Int. J. of Dyna. and Cont. 8 (2020), 112-131.   DOI
31 C. Raymond, A.A. Hugo and M. Kung'aro, Modelling dynamics of prey- predator fishery model with harvesting: a bioeconomic model, J. of Appl. Maths. 2019.
32 X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol. 73 (2016), 11-79.