STUDY OF A PREY-PREDATOR DYNAMICS UNDER THE SIMULTANEOUS EFFECT OF TOXICANT AND DISEASE

SINHA, SUDIPA¹ ; MISRA , O.P.¹ ; DHAR, JOYDIP²

¹ School of Mathematics and Allied Sciences, Jiwaji University Gwalior (M.P.)- 474011, INDIA.
² Department of Applied Sciences, ABV-Indian Institute of Information Technology and management, Gwalior(M.P.)-474011, INDIA.

Journal of nonlinear sciences and its applications, Tome 1 (2008) no. 2, p. 102-117.

Résumé

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.

Détail

DOI : 10.22436/jnsa.001.02.06

Classification : 92., 92A.
Keywords: Prey-Predator Dynamics, Disease, Toxicant, Stability, Simulation.

Bibliographie

[1] Anderson, R. M.; R. M. May Epidemiology and genetics in the Coevolution of parasites and hosts, Proc.R.Soc.Lond., Volume B 219 (1983), pp. 1-281 | DOI | Zbl

[2] Anderson, R. M.; R. M. May The invasion, persistence and spread of infections disease within animal and plant communities, Philos. Trans. R. Soc. Lond., Volume B 314 (1986), pp. 1-533 | DOI

[3] Anderson, R. M.; R. M. May The population dynamics of the microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond., Volume B 291 (1981), pp. 1-451

[4] Adler, R. A.; R. C. Brunet The dynamics of the simultaneous infections with altered susceptibilities, Theoret. Popul. Biol., Volume 40 (1991), pp. 369-410 | Zbl | DOI

[5] Begon, M.; 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 | Zbl

[6] Begon, M.; R. G. Bowers Host-host pathogen models and microbial pest control : the effect of host self-regulation, J. Theoret. Biol., Volume 169 (1995), pp. 275-287 | DOI

[7] Busenberg, S.; den, P. Van; P. Driessche Analysis of a disease transmission model in a population with varying size, J. Math Biol. , Volume 28 (1990), pp. 257-270 | DOI

[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. , Volume 9 (2004), pp. 9-24

[9] B. Dubey Modelling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl., Volume 246 (2000), pp. 58-79 | Zbl | DOI

[10] Dubey, B. , Modelling the depletion and conservation of resources : effect of two interacting populations, Ecol. Model. , Volume 101 (1997), pp. 123-136 | DOI

[11] Dubey, B.; Hussain, J. Nonlinear models for the survival of two competing species dependent on resource in industrial environments, Non-lin. Anal. Rl. Worl. Appl. , Volume 4 (2003), pp. 21-44 | Zbl | DOI

[12] Freedman, H. I.; Shukla, J. B. Models for the effect of toxicant in single species and predator-prey system, J. Math. Biol. , Volume 30 (1991), pp. 15-30 | DOI

[13] Gao, L. Q.; H. W. Hethcote Disease transmission models with density-dependent demographics , J. Math. Biol. , Volume 30 (1992), pp. 717-731 | Zbl | DOI

[14] Gordon, A. G.; E. Gorham Ecological aspects of air pollution from iron sintering plant of Wawa, Can. J.Bot. , Volume 41 (1963), pp. 1063-1078 | DOI

[15] Hale, J. K. Ordinary differential equations, 2nd Ed., Kriegor, Basel, , 1980

[16] Holt, R. D.; Pickering, J. Infectious disease and species co-existence : A model of Lokta- Voltera form, Am. Nat., Volume 126 (1986), pp. 1-196 | DOI

[17] Hadeler, K. P.; H. I. Freeman Predator-Prey Population with parasitic infection, J. Math. Biol. , Volume 27 (1989), pp. 609-631 | DOI

[18] H. W. Hethcote Three basic epidemiological models, in : S.A. Levin, T.G. Hallom, L.J. Gross (Eds.), Applied Mathematical Ecology, Biomath., Volume 18 (1989), pp. 119-144

[19] Levin, S. A. Some approaches to the modelling of co-evolutionary interactions, in : M. Nitecki (Ed.), Coevolution, University of Chicago, Chicago, 1983

[20] Levin, S. A. Coevolution, in : H.I. Freedman, C. Strobeck (Eds), Population Biology, lect. not. in Biomath., Vol. 52 , Springer-Heidelberg, , 1983

[21] Liu, B.; Chen, L.; Zhang, Y. The effect of impulsive toxicant input on a population in a polluted environment, Journ. of biol.Sys. , Volume 11(3) (2003), pp. 265-274 | Zbl | DOI

[22] Mukherjee, D. Persistence and global stability of a population in a polluted environment with delay, J. Biol. Sys., Volume 10 (2002), pp. 225-232 | DOI | Zbl

[23] Mena-Lorca, J.; H. W. Hethcote Dynamic models of infections diseases as regulator of population sizes, J. Math. Biol., Volume 30 (1992), pp. 693-717 | DOI

[24] Rao, M. N.; H. V. N. Rao Air pollution, Tata McGraw Hill publishing Co. Ltd., New Del, , 1989

[25] Shukla, J. B.; B. Dubey Simultaneous effect of two toxicants on biological species : A mathematical model, J. Biol. System, Volume 4 (1996), pp. 109-130 | DOI

[26] Shukla, J. B.; Dubey, B. Modelling the depletion and conservation of forestry resources : Effect of population and pollution, J. Math. Biol. , Volume 36 (1997), pp. 71-94 | DOI

[27] Santra, S. C.; Samual, A. C. Air quality of Kalyani township (Nadia, West Bengal) and its impact on surrounding vegetation, Ind. J. Env. Healt. , Volume 44(1) (2000), pp. 71-76