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REFERENCES
Abrams, P. A., Ginzburg L. R., 2000: The nature of predation: prey dependent, ratio
dependent or neither, Trends in Ecology and Evolution, 15, 337-341.
Abbas, S., Banerjee, M., and Haungerbuhler, N., 2010: Existence, uniqueness and
stability analysis of allelopathic stimulatory phytoplankton models, J. Math. Anal. Appl.,
367, 249-259.
Agarwal, M., Devi, S., 2011: Harvesting of the vegetation biomass and grazer population
with its effects on predator population: a mathematical model, Int. J. Ecol. Econ. Stat.,
20, 14-31.
Akcakaya, H. R., Arditi R., and Ginzburg L. R., 1995: Ratio-dependent predition: an
abstraction that works, Ecology, 76, 995-1004.
Andrews, J.F., 1968: A mathematical model for the continuous culture of
microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng, 10, 707-723.
Arditi R., Ginzburg L. R. and Akcakaya H. R., 1991: Variation in plankton density
among lakes: a case of ratio dependent models, American Naturalist, 138, 1287- 1296.
Arditi, R., Saiah, H., 1992: Empirical evidence of the role of heterogeneity in ratio
dependent consumption, Ecology, 73, 1544-1551.
Arditi, R., Michalski, J., 1996: Nonlinear food web models and their responses to
increased basal productivity. In G. A. Polis and K. O. Winemiller, eds., Integration of
Patterns and Dynamics, 122–133. Chapman & Hall, London.
Arditi, R., Ginzburg, L.R., 1989: Coupling in predator-prey dynamics: ratio-dependence,
J. Theoretical Biology, 139, 311-326.
227
Bambaradeniya, C.N.B., Edirisinghe, J.P., Silva, D.N., Gunatilleke, C.V.S, Ranawana,
K.B., and Wijekoon, S., 2004: Biodiversity associated with rice agro-ecosystem in Sri
Lanka, Biodiversity Conservation, 13, 1715–1753.
Baek, H., 2010: Dynamic complexities of a three-species Beddington-DeAngelis system
with impulsive control strategy, Acta Applicandae Mathematicae, 110(1), 23-38.
Beck, M.W., 1998: Comparison of the measurement and the effects of habitat structure
on gastropods in rocky intertidal and mangrove habitats, Marine Ecology Progress
Series, 169, 165-178.
Bell, S.S., McCoy, E.D. and Mushinsky, H.R., 1991: Habitat Structure the physical
arrangement of objects in space, London, Chapman and Hall.
Beddington, J. R., 1975: Mutual interference between parasites or predator and its effect
on searching efficiency, J. Animal Ecol., 44, 331-340.
Bhattacharya, D. K., Begum, S., 1996: Bionomic equilibrium of two species system,
Math. Biosci., 135(2), 111-127.
Birkhoff, G., Rota,G.C., 1982: Ordinary Differential Equations (Glnn).
Braza P. A., 2008: A dominant predator, a predator, and a prey, Mathematical
Biosciences and Engineering, 5(1), 61-73.
Cao, J.D., 2000: Periodic oscillation solution of biderictional associative memory
networks with delay, Phys. Rev. E, 59, 1825–1828.
Cantrell, R.S., Consner, C., 2001: On the dynamics of predator–prey Models with the
Beddington– DeAngelis Functional response, J. Math Anal. Appl., 257, 206–22.
Celik, C., 2008: The stability and Hopf bifurcation of a predator–prey system with time
delay, Chaos, Solitons Fract, 37(1), 87–99.
228
Celik, C., 2009: Hopf bifurcation of a ratio-dependent predator–prey system with time
delay, Chaos, Solitons Fract, 42, 1474–1484.
Chaudhuri, K.S., Ray, S.S., 1996: On the combined harvesting of a prey-predator system,
J. Biol. Sys., 4, 373-389.
Chaudhuri, K.S., 1986: A bioeconomic model of harvesting a multispecies fishery, Ecol.
Model., 32, 267–279.
Clark, C.W., 1990: Mathematical Bioeconomic: The Optimal Management of Renewable
Resources, 2nd ed., John Wiley and Sons.
Cooke, K.L., Grossman, Z., 1982: Discrete delay, distributed delay and stability switches,
J. Math Anal Appl., 86(2), 592–627.
Dai, G., Tang, M., 1998: Coexistence region and global dynamics of a harvested
predator-prey system, SIAM J Appl. Math., 58, 193-210.
DeAngelis, D. L., Goldstein, R. A. and O’Neill, R. V., 1975: A model for trophic
interaction, Eocolgy, 56, 881-892.
Dubey B., Upadhyay R. K., 2004: Persistence and extinction of one prey and two
predator system, J. Nonlinear Analysis. Modelling and Control, 9(4), 307-329.
Dubey, B., Chandra, P., and Sinha, P., 2002: A resource dependent fishery model with
optimal harvestig policy, J. Biol. Syst.,10,1-13.
Eckmann, J.P., 1981: Rev.Mod.Phys. , 53, 643.
Elettreby, M. F., 2009: Two prey one predator model, Chaos, Solitons and Fractals,
39(5), 2018-2027.
Eubanks, M.D., and Denno, R.F., 2000: Host plants mediate omnivore-herbivore
interactions and influencee prey suppression, Ecology, 81, 936-947.
229
Freedman, H. I., 1980: Deterministic mathematical models in population ecology, M.
Dekker, New York.
Freedman H.I., Waltman P., 1984: Persistence in model of three interacting predator-prey
population, Math. Biosci., 68, 213-231.
Freedman, H.I., 1980: Deterministic mathematical models in population ecology, New
York, Marcel Dekker.
Freedman, H.I., Waltman, P., 1977: Mathematical analysis of some three-species foodchain models, Math. Biosci., 33, 257-276.
Freedman, H.I., So, J.W.-H., 1985: Global stability and persistence of simple food chains,
Math. Biosci., 76, 69-86.
Gao, H., Wei, H., Sun, W. and Zhai, X., 2000: Functions used in biological models and
their influence on simulations, Indian Journal of Marine Science, 29, 230-237.
Gakkhar, S., Naji, R.K., 2002: Chaos in three species ratio-dependent food chain, Chaos,
Solitons & Fractals, 14,771-778.
Gakkhar, S., Naji, R.K., 2003: Chaos in seasonally perturbed ratio-dependent, Chaos,
Solitons and Fractals, 15,107-118.
Gilpin, M.E., 1979: Spiral chaos in a predator-prey model, American Naturalist, 107,
306-308.
Gopalsamy, K., 1992: Stability and Oscillations in Delay Differential Equations of
Population Dynamics, Kluwer Academics, Dordrecht.
González-Olivares, E., Rojas-Palma, A., 2011: Multiple limit cycles in a Gause type
predator–prey model with Holling type III functional response and Allee effect on prey,
Bull. Math. Biol., 73, 1378–1397.
230
Guckenheimer, J., Holmes, P., 1983: Nonlinear oscillations, dynamical systems and
bifurcations of vector fields, Springer,Verlag, New York, USA.
Hale, J., 1977: Theory of functional differential equations, New York: Springer.
Hasting, A., Powell, T., 1991: Chaos in three species food chain, Ecology, 72, 896-903.
Holling, C.S., 1959: Some characteristics of simple types of predation and parasitism,
Can. Entomologist, 91, 385–398.
Hsu, S. B., Hwang, T. W., Kuang Y., 2001: Rich dynamics of a ratio-dependent one
prey- two predators model, J. Math. Biol., 43, 377-396.
Hughes, A.R., Grabowski, J.H., 2006: Habitat context influences predator interference
interactions and the strength of resource partitioning, Oecologia, 149, 256–264.
Kesh, D., Sarkar, A., Kand, and Roy, A. B., 2000: Persistence of two preys – one
predator system with ratio dependent predator influence, Math. Appl. Sci., 23, 347-356.
Kar, T. K., Chaudhury, K. S., 2004: Harvesting in a two-prey one-predator fishery,
ANZIAM J., 45, 443-456.
Kar, T. K., Chakraborty, K., and Pahari, U. K., 2010: A prey-predator model with
alternative prey: Mathematical model and analysis, Canadian Applied Mathematics
Quarterly, 18(2), 137-168.
Kar, T .K. , Misra, S. K, and Mukhopadhyay, B., 2006: A Bioeconomic model of a ratiodependent prey-predator system and optimal harvesting, J. Appl. Math. and Comp., 22 (12), 387- 401.
Kar, T. K. , Chakroborty, K., 2010: Bio economic modeling of a prey-predator system
using differential algebraic equations, Int. J. Eng., Sci. and Tech., 2(1), 13-34.
231
Kuang, Y., 1993: Delay Differential Equations with Applications in Population
Dynamics. Academic Press, New York.
Kuang, Y., 1990: Global stability of Gause-type predator-prey systems, J. Math. Biol.,
28, 463 – 474.
Kuang, Y., 2002: Basic Properties of Mathematical Population Models, J of
Biomathematics, 17, 129-142.
Leung, A., 1995: Optimal harvesting co-efficient control of steady state prey-predator
diffusive Volterra-Lotka systems, Appl. Math. Optim., 31, 219.
Macdonald, N., 1989: Biological Delay Systems: Linear Stability Theory, Cambridge
Univ. Press, Cambridge.
May, R.M., 1974: Stability and Complexity in Model Ecosystem, Princeton Univ. Press,
Princeton.
Meng, X.Y., Huo, H. F., and Xiang, H., 2011: Hopf bifurcation in a three-species system
with delays, J. Appl. Math. Comput., 35, 635–661.
Murray, J.D., 1989: Mathematical Biology, Springer-Verlag, New York.
Murray, J.D., 1993: Mathematical Biology, Second Corrected Ed., Springer, Heidelberg.
Myerscough, M.R., Gray, B.E., Hograth, W.L., and Norbury, J., 1992: An analysis of an
ordinary differential equation model for a two-species predator-prey system with
harvesting and stocking, J. Math. Biol., 30, 389-411.
Naji, R.K., Balasim, A.T., 2007: Dynamical behavior of a three species food chain model
with Beddington-DeAngelis functional response, Chaos Solitons and Fractals, 32, 18531866.
232
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., 1964:
The Mathematical Theory of Optimal Processes, Pergamon Press, London.
Qu, Y., Wei, J., 2007: Bifurcation analysis in a time-delay model for prey–predator
growth with stage-structure, Nonlinear Dyn., 49, 285–294.
Rojas-Palma, A., Gonzalez-Olivares, E., 2012: Optimal harvesting in a predator–prey
model with Allee effect and sigmoid functional response, Applied Mathematical
Modelling, 36, 1864–1874.
Rosenzweig, M. L., 1969: Why the prey curve has a hump, Amer.Nat.103, 8187.
Schaefer, M.B., 1957: Some considerations of population dynamics and economics in
relation to the management of marine fisheries, J. Fisheries Res. Board Can., 14, 69–68.
Schaffer, W.M., 1985: Order and chaos in ecological systems, Ecology, 66, 93-106.
Schaffer, W.M., Kot, M., 1986a: Chaos in ecological systems, the coals that Newcastle
forgot, Trends in Ecology and Evolution, 1, 58-63.
Song, Y., Wei, J., 2004: Bifurcation analysis for Chen’s system with delayed feedback
and its application to control of chaos, Chaos, Solitons Fract., 22, 75–91.
Srinivasu, P.D.N., Prasad, B.S.R.V., 2011: Role of quantity of additional food to
predators as a control in predator-prey systems with relevance to pest management and
biological conservation, Bull Math Biol., 73, 2249-2276.
Srinivasu, P.D.N., Prasad, B.S.R.V., and Venkatesula, M. 2007: Biological control
through provision of additional food to predators: a theoretical study, Theor. Popul. Biol.,
72, 111-120.
Strogatz, S. H., 2008: Non-linear dynamics and chaos with applications to Physics,
Biology, Chemistry, and Engineering, WestView Press.
233
Sun, C., Lin, Y., and Han, M., 2006: Stability and Hopf bifurcation for an epidemic
disease model with delay, Chaos, Solitons Fract., 30, 204–216.
Tansley, A.G., 1946: Introduction to Plant Ecology, George Allen & Unwin ltd, Museum
Street, London.
Tripathi, J.P., Abbas, S., and Thakur, M., 2012: Stability analysis of two prey one
predator model, Numerical Analysis and Applied Mathematics ICNAAM 2012 AIP Conf.
Proc. 1479, 905-909; doi:10.1063/1.4756288.
Turchin, P., 2003: Complex Population Dynamics, in: A Theoretical/Empirical Synthesis,
Monographs in Population Biology, 35, Princeton University Press.
Upadhyay, R.K., Rai, V., 1997: Why chaos is rarely observed in natural populations,
Chaos, Solitons and Fractals, 8(12), 1933-1989.
Upadhyay, R.K., Rai, V., 2001: Crisis-limited chaotic dynamics in ecological systems,
Chaos, Solitons and Fractals, 12, 205-218.
Vandermeer, J., 1993: Loose coupling of predator-prey cycles: Entrainment, chaos, and
intermittency in the classic MacArthur consumer-resource equations, American
Naturalist, 141,687-716.
Van Rijin, P.C.J., Sabelis, M.W., 1993: Does alternative food always enhance biological
control? The effect of pollen on the interaction between western flower thrips and its
predators, Bulletin IOBC/WPRS, 16, 123 – 125.
Van Rijin, P.C.J., Tanigoshi, L., 1999: The contribution of extrafloral nectar to survival
and reproduction of the predatory mite Iphiseius degenerans on Ricinus communis,
Experimental and Applied Acarology, 23, 281 – 296.
234
Wang, F., Pang, G., 2008: Chaos and Hopf bifurcation of a hybrid ratio-dependent three
species food chain, Chaos Solitons and Fractals, 36, 1366-1376.
Wang, F., Zhang, S., Chen, L., and Sun, L., 2005: Permanence and Complexity of a
Three Species Food Chain with Impulsive Effect on the Top Predator, Int J Nonlinear Sci
Numer Simul, 6(2), 169-180.
Yang, H., Tian, Y., 2005: Hopf bifurcation in REM algorithm with communication delay,
Chaos, Solitons Fract., 25, 1093–1105.
Yang, Y., 2009: Hopf bifurcation in a two-predator, one-prey system with time delay,
Appl. Math. Comput., 214, 228–235.
Yan, X.P., Zhang, C.H., 2008: Hopf bifurcation in a delayed Lokta-Volterra predatorprey system, Nonlinear Anal., Real World Appl., 9, 114–127.
Zhang, J., Fan, M. and Kuang, Y., 2006: Rabbits Killing birds revisited, Mathematical
Biosciences, 203, 100 – 123.
235