Discontinuity, Nonlinearity, and Complexity
Study of a PredatorPrey System with MonodHaldane Functional Response and Harvesting
Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 229243  DOI:10.5890/DNC.2020.06.005
N.H. Gazi$^{1}$, M.R. Mandal$^{2}$, S. Sarwardi$^{1}$
$^{1}$ Department of Mathematics and Statistics, Aliah University IIA/27, New Town, Kolkata  700160, West Bengal, India
$^{2}$ Department of Mathematics, Siliguri College, Siliguri, Darjeeling, West Bengal – 734001, India
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Abstract
In the present work we study a predatorprey harvesting model with MonodHaldane functional response. The mathematical features of the model have been analyzed in terms of stability, bifurcations and harvesting. Threshold values for some parameters indicating the feasibility and stability conditions of all equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations are investigated. Optimal harvesting criteria and the net economic revenue are analyzed. Numerical illustrations are performed finally in order to confirm the analytical findings.
References

[1]  Murray, J.D. (2002), Mathematical Biology, 3rd ed, New York, NY: SpringerVerlag. 

[2]  May, R.M. (2001), Stability and Complexity in Model Ecosystems, Princeton University Press. 

[3]  Anderson, R.M. and May, R.M. (1981), The population dynamics of microparasites and their invertebrates hosts, Proc R Soc London, 291, 451463. 

[4]  Kar, T.K. and Misra, S. (2006), Influence of prey reserve in a preypredator fishery, Nonlinear Analysis, 17251735. 

[5]  Xiao, Y. and Chen, L. (2001), Modeling and analysis of a predatorprey model with disease in prey, Math. Biosci., 171, 5982. 

[6]  Clark, C.W. (1973), Profit maximization and the extinction of animal species, J. Pol. Econ., Vol. 81, 950961. 

[7]  Clark, C.W. (1976), Mathematical Bioeconomics: The optimal management of renewable resources, (New York: Wiley), Princeton Univ. Press. 

[8]  Clark, C.W. (1979), Mathematical models in the economics of renewable resources, SIAM Rev., 21, 8199. 

[9]  Clark, C.W. and De Pree, J.D. (1979), A simple linear model for optimal exploitation of renewable resources. J. Appl. Math. Optimization., 5, 181196. 

[10]  Clark, C.W. (1985), Bieconomic modelling and fisheries management, Wiley, New York. 

[11]  Kar, T.K. and Ghosh, B. (2013), Impacts of maximum sustainable yield policy to preypredator systems, Ecological Modelling, 250, 134142. 

[12]  Holling, C.S. (1959), The components of predation as revealed by a study of small predation of the European pine sawfly, Canadian Entomologist., 91, 293320. 

[13]  May, R.M. (1972), Limit cycles in predatorprey communities, Science, 177, 900902. 

[14]  Kuang, Y. and Freedman, H.I. (1988), Uniqueness of limit cycles in Gausetype models of predator prey systems, Math. Biosci., 88, 6784. 

[15]  Gazi, N.H. and Das, K. (2010), Structural stability analysis of an algal bloom mathematical model in tropic interaction, Nonl. Anal.: Real World Applications., 11, 21912206. 

[16]  Das, K. and Gazi, N.H. (2011), Random excitations in modelling of algal blooms in estuarine system. Ecological Modelling, 222, 24952501. 

[17]  Andrews, J.F. (1968), A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioengg., 10, 707723. 

[18]  Boon, B. and Landelout, H. (1992), Kinetics of nitrite oxidation by nitrobacter winogradski. J. Biochem., 85, 440447. 

[19]  Edwards, V.H. (1970), Influence of high substrate concentrations on microbial kinetics, Biotechnol. Bioeng., 12, 679 712. 

[20]  Freedman, H.I. and Wolkowicz, G.S.K. (1986), PredatorPrey systems with group defense: the paradox of enrichment revisited, Bull. Math. Biol., 48, 493508. 

[21]  Ruan, S.G. and Xiao, D.M.(2001), Global analysis in a predatorprey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 14451472. 

[22]  Zhang, L.Y. (2015), Hopf bifurcation analysis in a MonodHaldane predatorprey model with delays and diffusion, Appl. Math. Model., 39, 13691382. 

[23]  Haque, M. and Venturino, E. (2006), The role of transmissible diseases in the HollingTanner predatorprey model. Theor. Popul. Biol., 70, 273288 

[24]  Birkhoff, G. and Rota, G.C. (1982), Ordinary Differential Equations. Ginn Boston. 

[25]  Gard, T.C. and Hallam, T.G. (1979), Persistence in Food web1, LotkaVolterra food chains. Bull. Math. Biol.; 41, 877891. 

[26]  Perko, L. (2001), Differential equations and dynamical systems, Springer, New York. 

[27]  Hale, J.K. (1989), Ordinary Differential Equations. Krieger Publisher Company, Malabar. 

[28]  Arrow, K.J. and Kurz, M. (1970), Public Investment, The Rate of Return and Optimal Fiscal Policy, Johns Hopkins, Baltimore. 

[29]  Pontryagin, L.S., Boltyanskii, V.S., Gamkrelidze, R.V., and Mishchencko, E.F. (1962), The Mathematical theory of optimal processes. Wiley, New York. 

[30]  Chonacky, N. andWinch, D. (2005), MAPLE, MATHEMATICA, and MATLAB: The 3 M?s without the tape, Comput. Sci. Engrg., 7, 816. 

[31]  Sarwardi, S., Haque, M., and Venturino, E. (2010), Global stability and persistence in LGHolling typeII diseased predators ecosystems. J. Biol. Phys., 37, 91106. 

[32]  Sarwardi, S., Mandal, P.K., and Ray, S. (2012), Analysis of a competitive preypredator system with a prey refuge, Biosystems, 110, 133148. 