The modeling investigation in this paper discusses the system level effects of a toxicant on a three species food chain system and the state variables of the models are prey and predator densities, concentration of toxicant in the environment and the concentration of toxicant in the prey population. In the models, we have assumed that the presence of top predator reduces the predatory ability of the intermediate predator. The stability analysis of the models is carried out and the sufficient conditions for the existence and extinction of the populations under the stress of toxicant are obtained. Further, it is also found that the predation rate of the intermediate predator is a bifurcating parameter and Hopf-bifurcation occurs at some critical value of this parameter. Finally, numerical simulation is carried out to support the analytical results.
Raveendra Babu. A , O. P. Misra, Chhatrapal Singh, Preety Kalra
Stability, Bifurcation, Lyapunov function.
- A.A. Gomes, E. Manica and M.C. Varriale, 2008, Applications of chaos control techniques to a three-species food chain, Chaos, Solitons and Fractals 36: 1097-1107.
- Alan Hastings and Thosmas Powell, 1991, Chaos in a Three-species food chain, Ecology 72(3) : 896-903.
- Chengjun Sun and Michel Loreau, 2009, Dynamics of a three-species food chain model with adaptive traits, Chaos, Solitons and Fractals 41: 2812-2819.
- De Luna J.T. and Hallam T.G., 1987, Effect of toxicants on population: a qualitative approach IV. Resource - Consumer - Toxicant models, Ecol Modelling 35: 249-273.
- Debaldev Jana, Rashmi Agrawal, Ranjit Kumar Upadhyay, 2014, Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain, Chaos, Solitons and Fractals 69:50 - 63.
- Fengyan Wang and Guoping Pang, 2008, Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain, Chaos, Solitons and Fractals 36: 1366-1376.
- Gakkhar S and Naji M.A., 2003, Order and Chaos in a predator to prey ratio-dependent food chain, Chaos, Solitons and Fractals 18: 229-239.
- Gakkhar S and Singh B., 2006. Dynamics of a modified Leslie-Gower-type prey-predator models with seasonally varying parameters, Chaos, Solitons & Fractals 27: 1239-1255.
- Gakkhar S, Singh B and Naji R.K., 2007, Dynamical behavior of two predators competing over a single prey, Bio Systems 90:808-817.
- George A.K. van Voorn, Bob W. Kooi and Martin P. Boer., 2010, Ecological consequences of global bifurcations in some food chain models, Mathematical Biosciences 226:120-133.
- H.I. Freedman and J.B. Shukla., 1991, Models for the effect of toxicant in single-species and predator-prey systems, Jornal of Mathematical Biology 30:15-30.
- H. I. Freedman and Paul Waltman, 1977, Mathematical Analysis of Some Three-Species Food-Chain Models, Mathematical Biosciences 33:257-276.
- Huitao Zhao, Yiping Lin and Yunxian Dai, 2011, Bifurcation analysis and control of chaos for a hybrid ratio-dependent three species food chain, Applied Mathematics and Computation 218:1533-1546.
- J.B. Shukla and B. Dubey, 1996, Simultaneous effect of two toxicants on biological species: A Mathematical Model, Journal of Biological Systems 4:109-130.
- J.B. Shukla, A.K. Agrawal, B. Dubey and P.Sinha, 2001, Existence and Survival of Two Competing Species in a Polluted Environment: A Mathematical Model, Journal Biological Systems 9:89-103.
- Kejun Zhuang and Zhaohui Wen, 2011, Dynamics of a Discrete Three Species Food Chain System, Int. J. of Comp. & Math. Sci. 5.
- Liu, W.M., 1994, Criterion of Hopf-bifurcations without using eigenvalues, J. Math. Anal. Appl. 182: 250-256.
- Mada Sanjaya Waryano Sunaryo, Mustafa Mamat, Zabidin Salleh, Ismail Mohd and Noor Maizura Mohamad Noor, 2011, Numerical Simulation Dynamical Model of Three Species Food Chain with Holling Type-II Functional Response, Malaysian Journal of Mathematical Sciences 5: 1-12.
- Mainul Haque, Nijamuddin Ali, Santabrata Chakravarty, 2013, Study of a tri-trophic prey-dependent food chain model of interacting populations, Mathematical Biosciences 246: 55 - 71.
- Manju Agarwal and Sapna Devi, 2011, A resource-dependent competition model: Effects of toxicants emitted from external sources as well as formed by precursors of competing species, Nonlinear Analysis: Real World Applications 12:751-766.
- Nitu Kumari, 2013, Pattern Formation in Spatially Extended Tritrophic Food Chain Model Systems: Generalist versus Specialist Top Predator, Hindawi Publishing Corporation, ISRN Biomathematics,Article ID 198185, 12 pages.
- O.P.Misra and P.Sinha, 2007, Effect of Pollution on the Photosynthate Partitioning during Plant Growth: A three Compartment Model, Research Hunt 2.
- O.P. Misra and Raveendra Babu.A, 2014, A model for the effect of toxicant on a three species food-chain system with “food-limited" growth of prey population, Global Journal of Mathematical Analysis, 2 (3) :120-145.
- O.P. Misra and V.P. Saxena, 1991, Effects of environmental pollution on the growth and existence of biological populations: Modelling and stability analysis, Indian J. pure appl. Math. 22: 805-815.
- Qihua Huang, Laura Parshotam, HaoWang, Caroline Bampfylde and Mark A. Lewis, 2013, A model for the impact of contaminants on fish population dynamics, Journal of Theoretical Biology 334: 71 - 79.
- R.K. Naji and A.T. Balasim, 2007, On the dynamical behavior of three species food web model, Chaos, Solitons and Fractals 34 :1636-1648.
- Raid Kamel Naji, Ranjit Kumar Upadhyay and Vikas Rai, 2010, Dynamical consequences of predator interference in a tri-trophic model food chain, Nonlinear Analysis: Real World Applications 11: 809-818.
- Robert V. Thomann, Daniel S. Szumski, Dominic M.Ditoto and Donald J O’Connor, 1984, A food chain model of cadmium in western lake Erie, Wat. Res. 8: 841-849.
- Robert V. Thomann and John P. Connolly, 1984, Model of PCB in the Lake Michigan lake trout these food chain, Environ. Sci. Technol. 18:65-71.
- Songjuan Lv and Min Zhao, 2008, The dynamic complexity of a three species food chain model, Chaos, Solitons and Fractals 37:1469-1480.
- Steven J. Hamilton, 2004, Review of selenium toxicity in the aquatic food chain, Science of the Total Environment 326: 1-31.
- Sudipa Sinha, O.P. Misra and J. Dhar, 2010, Modelling a Predator-Prey System with Infected Prey in Polluted Environment, Applied Mathematical Modelling 34:1861-1872.
- Sudipa Sinha, O.P. Misra and Joydip Dhar, 2010, A two species competition model under the simultaneous effect of toxicant and disease, Nonlinear Analysis: Real World Applications 11:1131-1142.
- Swati Khare, O. P. Misra, Chhatrapal Singh and Joydip Dhar, 2011, Role of Delay on Planktonic Ecosystem in the Presence of a Toxic Producing Phytoplankton, Hindawi Publishing Corporation, International Journal of Differential Equations, ID 603183, 16 pages
- T. Das, R.N. Mukherjee and K.S. Chaudhuri, 2009, Harvesting of a prey–predator fishery it the presence of toxicity, Appl. Math. Model. 33: 2282-2292.
- T.G. Hallam and C.E. Clark, 1982, Non-autonomous logistic equations as models of populations in a deteriorating environment, J. Theor. Biol. 93:303-311.
- T.G. Hallam and J.T. De. Luna, 1984, Effects of toxicants on Populations: a Qualitative Approach III. Environmental and Food Chain Pathways, Academic Press Inc. (London) Ltd.
- T.G. Hallam, Clark C.E. and Lassiter R.R., 1983, Effects of toxicants on population: a qualitative approach I. Equilibrium environmental exposure, Ecol Modelling 18 :291-304.
- T.G. Hallam, Clark C.E. and Jordan G.S., 1983, Effects of toxicants on population: a qualitative approach II. First order kinetics, J. Math. Biol. 18:25-37.
- Xitao Wang and Min Zhao., 2011, Chaos in a Hybrid Three-Species Food Chain with Beddington-Deangelis Functional Response, Procedia Environmental Sciences 10:128-134.
- Zhao M and Lv S., 2009a, Chaos in a three-species food chain model with a Beddington-DeAngelis functional response, Chaos, Solitons and Fractals 40: 2305-2316.
- Mini Ghosh, Peeyush Chandra and Prawal Sinha, 2002, A Mathematical Model To Study The Effect Of Toxic Chemicals On A Prey-Predator Type Fishery, Journal of Biological Systems, Vol. 10, No. 2: 97-105.
- Graeme M. Smith, Judith S. Weis, 1997, Predator-prey relationships in mummichogs (Fundulus heteroclitus (L.)): Effects of living in a polluted environment, Journal of Experimental Marine Biology and Ecology, 209: 75-87.
- Jes Jessen Rasmussena, Ulrik Norum, Morten Rygaard Jerris, Peter Wiberg-Larsen, Esben Astrup Kristensen, Nikolai Friberg, 2013, Pesticide impacts on predator–prey interactions across two levels of organisation, Aquatic Toxicology 140– 141: 340– 345.
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Cite This Article
Raveendra Babu. A , O. P. Misra, Chhatrapal Singh, Preety Kalra, "Model for the Dynamical Study of a Three-Species Food-Chain System under Toxicant Stress", International Journal of Scientific Research in Science, Engineering and Technology(IJSRSET), Print ISSN : 2395-1990, Online ISSN : 2394-4099, Volume 1, Issue 2, pp.493-513, March-April-2015.
URL : http://ijsrset.com/IJSRSET1522134.php