STUDY OF A PREY-PREDATOR DYNAMICS UNDER THE SIMULTANEOUS EFFECT OF TOXICANT AND DISEASE
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Authors
SUDIPA SINHA
- School of Mathematics and Allied Sciences, Jiwaji University Gwalior (M.P.)- 474011, INDIA..
O.P. MISRA
- School of Mathematics and Allied Sciences, Jiwaji University Gwalior (M.P.)- 474011, INDIA..
JOYDIP DHAR
- Department of Applied Sciences, ABV-Indian Institute of Information Technology and management, Gwalior(M.P.)-474011, INDIA..
Abstract
A mathematical model is proposed to study the simultaneous effects of toxicant and infectious disease on Lotka-Volterra prey-redator system.
It is considered in the model that only the prey population is being affected by
disease and toxicant both, and the susceptible and infected prey populations
are being predated by predator. All the feasible equilibrium of the model are
obtained and the condition for the existence of interior equilibrium point is also
been determined. The criteria for both local stability and instability involving ecotoxicological and epidemiological parameters are derived. The global
stability of the interior equilibrium point is discussed using Lyapunov's direct
method. The results are compared with the case when environmental toxicant
is absent. Moreover, threshold conditions depending upon toxicant, disease
and predation related parameters for the non-linear stability of the model is
determined. Finally, the numerical verifications of analytic results are carried
out.
Share and Cite
ISRP Style
SUDIPA SINHA, O.P. MISRA, JOYDIP DHAR, STUDY OF A PREY-PREDATOR DYNAMICS UNDER THE SIMULTANEOUS EFFECT OF TOXICANT AND DISEASE, Journal of Nonlinear Sciences and Applications, 1 (2008), no. 2, 102-117
AMA Style
SINHA SUDIPA, MISRA O.P., DHAR JOYDIP, STUDY OF A PREY-PREDATOR DYNAMICS UNDER THE SIMULTANEOUS EFFECT OF TOXICANT AND DISEASE. J. Nonlinear Sci. Appl. (2008); 1(2):102-117
Chicago/Turabian Style
SINHA, SUDIPA, MISRA , O.P., DHAR, JOYDIP. "STUDY OF A PREY-PREDATOR DYNAMICS UNDER THE SIMULTANEOUS EFFECT OF TOXICANT AND DISEASE." Journal of Nonlinear Sciences and Applications, 1, no. 2 (2008): 102-117
Keywords
- Prey-Predator Dynamics
- Disease
- Toxicant
- Stability
- Simulation.
MSC
References
-
[1]
R. M. Anderson, R. M. May , Epidemiology and genetics in the Coevolution of parasites and hosts, Proc.R.Soc.Lond., B 219 (1983), 281
-
[2]
R. M. Anderson, R. M. May , The invasion, persistence and spread of infections disease within animal and plant communities, Philos. Trans. R. Soc. Lond., B 314 (1986), 533
-
[3]
R. M. Anderson, R. M. May , The population dynamics of the microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond., B 291 (1981), 451
-
[4]
R. A. Adler, R. C. Brunet , The dynamics of the simultaneous infections with altered susceptibilities, Theoret. Popul. Biol., 40 (1991), 369-410
-
[5]
M. Begon, R. G. Bowers , Beyond host pathogen dynamics, in : B.T. Grenfell, A.P. Dooson (Editors), Ecology of Infectious Diseases in Natural Population, Cambridge University, P. 478 ., Cambridge (1995)
-
[6]
M. Begon, R. G. Bowers , Host-host pathogen models and microbial pest control : the effect of host self-regulation, J. Theoret. Biol., 169 (1995), 275-287
-
[7]
S. Busenberg, P. Van den, P. Driessche , Analysis of a disease transmission model in a population with varying size, J. Math Biol. , 28 (1990), 257-270
-
[8]
Joydip Dhar , A Prey-Predator model with diffusion and a supplementry resource for the prey in a two patch environment, J.math. modl. and Analy. , 9 (2004), 9-24
-
[9]
B. Dubey , Modelling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl., 246 (2000), 58-79
-
[10]
B. Dubey, , Modelling the depletion and conservation of resources : effect of two interacting populations, Ecol. Model. , 101 (1997), 123-136
-
[11]
B. Dubey, J. Hussain, Nonlinear models for the survival of two competing species dependent on resource in industrial environments, Non-lin. Anal. Rl. Worl. Appl. , 4 (2003), 21-44
-
[12]
H. I. Freedman, J. B. Shukla, Models for the effect of toxicant in single species and predator-prey system, J. Math. Biol. , 30 (1991), 15-30
-
[13]
L. Q. Gao, H. W. Hethcote , Disease transmission models with density-dependent demographics , J. Math. Biol. , 30 (1992), 717-731
-
[14]
A. G. Gordon, E. Gorham , Ecological aspects of air pollution from iron sintering plant of Wawa, Can. J.Bot. , 41 (1963), 1063-1078
-
[15]
J. K. Hale, Ordinary differential equations, 2nd Ed., Kriegor, Basel, (1980)
-
[16]
R. D. Holt, J. Pickering, Infectious disease and species co-existence : A model of Lokta- Voltera form, Am. Nat., 126 (1986), 196
-
[17]
K. P. Hadeler, H. I. Freeman , Predator-Prey Population with parasitic infection, J. Math. Biol. , 27 (1989), 609-631
-
[18]
H. W. Hethcote , Three basic epidemiological models, in : S.A. Levin, T.G. Hallom, L.J. Gross (Eds.), Applied Mathematical Ecology, Biomath., 18 (1989), 119-144
-
[19]
S. A. Levin, Some approaches to the modelling of co-evolutionary interactions, in : M. Nitecki (Ed.), Coevolution, University of Chicago, Chicago (1983)
-
[20]
S. A. Levin, Coevolution, in : H.I. Freedman, C. Strobeck (Eds), Population Biology, lect. not. in Biomath., Vol. 52 , Springer-Heidelberg, (1983)
-
[21]
B. Liu, L. Chen, Y. Zhang, The effect of impulsive toxicant input on a population in a polluted environment, Journ. of biol.Sys. , 11(3) (2003), 265-274
-
[22]
D. Mukherjee, Persistence and global stability of a population in a polluted environment with delay, J. Biol. Sys., 10 (2002), 225-232
-
[23]
J. Mena-Lorca, H. W. Hethcote , Dynamic models of infections diseases as regulator of population sizes, J. Math. Biol., 30 (1992), 693-717
-
[24]
M. N. Rao, H. V. N. Rao , Air pollution, Tata McGraw Hill publishing Co. Ltd., New Del, (1989)
-
[25]
J. B. Shukla, B. Dubey , Simultaneous effect of two toxicants on biological species : A mathematical model, J. Biol. System, 4 (1996), 109-130
-
[26]
J. B. Shukla, B. Dubey, Modelling the depletion and conservation of forestry resources : Effect of population and pollution, J. Math. Biol. , 36 (1997), 71-94
-
[27]
S. C. Santra, A. C. Samual, Air quality of Kalyani township (Nadia, West Bengal) and its impact on surrounding vegetation, Ind. J. Env. Healt. , 44(1) (2000), 71-76