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Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model

Published online by Cambridge University Press:  06 June 2012

P.S. Mandal  and
M. Banerjee
P.S. Mandal
Affiliation:
Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur - 208016, INDIA
M. Banerjee*
Affiliation:
Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur - 208016, INDIA
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103–112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopf-bifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.

Keywords

Eco-epidemiologystabilityHopf-bifurcationchaosstochasticityextinction
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 3: Epidemiology , 2012 , pp. 99 - 116
Copyright
© EDP Sciences, 2012

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References

E. Allen. Modeling with Itô Stochastic Differential Equations. Springer, The Netherlands, 2007.
L. J. S. Allen. An Introduction to Stochastic Processes with Applications to Biology. Pearson Eduction Inc., New Jercy, 2003.
Allen, L. J. S., Jones, M. A., Martin, C. F.. A discrete-time model with vaccination for a measles epidemic. Math. Biosci., 105 (1991), 111–131. CrossRefGoogle ScholarPubMed
Arino, O., Abdllaoui, A. El., Mikram, J., Chattopadhyay, J.. Infection on prey population may act as a biological control in ratio-dependent predator-prey model. Nonlinearity, 17 (2004), 1101-1116. CrossRefGoogle Scholar
Allen, E. J., Allen, L. J. S., Arciniega, A., Greenwood, P.. Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl., 26 (2008) 274-297. CrossRefGoogle Scholar
Ball, F. G.. Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci., 156 (1999) 41–67. CrossRefGoogle Scholar
Beltrami, E., Carroll, T. O.. Modelling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol., 32 (1994) 857-863. CrossRefGoogle Scholar
F. Brauer, C. Castillo-Chàvez. Mathematical Models in Population Biolgy and Epidemiology Springer-Verlag, New York, 2001.
Britton, T.. Stochastic epidemic models : A survey. Math. Biosci., 225 (2010) 24–35. CrossRefGoogle ScholarPubMed
Britton, T., Lindenstrand, D.. Epidemic modelling : Aspects where stochasticity matters. Math. Biosci., 222 (2009) 109-116. CrossRefGoogle ScholarPubMed
Chattopadhyay, J., Bairagi, N.. Pelicans at risk in Salton Sea - an eco-epidemiological model. Ecol. Model., 136 (2001) 103–112. CrossRefGoogle Scholar
Chan, M. S., Isham, V. S.. A stochastic model of schistosomiasis immuno-epidemiology. Math. Biosci., 151 (1998) 179–198. CrossRefGoogle ScholarPubMed
Freedman, H. I.. A model of predator-prey dynamics as modified by the action of parasite. Math. Biosci., 99 (1990) 143–155. CrossRefGoogle ScholarPubMed
T. C. Gard. Introduction to Stochastic Differential Equations. Marcel Decker, New York, 1987.
C. W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1983.
Gillespie, D. T.. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phy., 22 (1976) 403–434. CrossRefGoogle Scholar
Gillespie, D. T.. The chemical Langevin equation. J. Chem. Phy., 113 (2000) 297–306. CrossRefGoogle Scholar
N. S. Goel, N. Richter-Dyn. Stochastic Models in Biology. Academic Press, New York, 1974.
Greenhalgh, D., Griffiths, M.. Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model. J. Math. Biol., 59 (2009) 1–36. CrossRefGoogle Scholar
Hadeler, K. P., Freedman, H. I.. Predator-prey population with parasitic infection. J. Math. Biol., 27 (1989) 609–631. CrossRefGoogle ScholarPubMed
Haque, M., Greenhalgh, D.. A predator-prey model with disease in prey species only. M2AS, 30 (2006) 911–929. Google Scholar
Higham, D. J.. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43 (2001) 525–546. CrossRefGoogle Scholar
Kermack, W. O., McKendrick, A. G.. A Contribution to the Mathematical Theory of Epidemics. Proc. Roy. Soc. Lond. A. 115 (1927) 700–721. CrossRefGoogle Scholar
P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1999.
M. Kot. Elements of Mathematical Biology. Cambridge University Press, Cambridge, 2001.
Y. A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, Berlin, 1997.
A. J. Lotka. Elements of physical biology. Williams & Wilkins Co., Baltimore, 1925.
J. Marsden, M. McCracken. The Hopf Bifurcation and its Applications. Springer, New York, 1976.
H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal Patterns in Ecology and Epidemiology : Theory, Models and Simulations. Chapman & Hall, London, 2008.
J. D. Murray. Mathematical Biology. Springer, New York, 1993.
R. J. Serfling. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York, 1980.
Stiefs, D., Venturino, E., Feudel, U.. Evidence of chaos in eco-epidemic models. Math. Biosci. Eng., 6 (2009) 857–873. CrossRefGoogle ScholarPubMed
Upadhyay, R. K., Bairagi, N., Kundu, K., Chattopadhyay, J.. Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Appl. Math. Comput., 196 (2008) 392–401. Google Scholar
Venturino, E.. The influence of diseases on Lotka-Volterra systems. Rocky Mountain Journal of Mathematics. 24 (1994) 381–402. CrossRefGoogle Scholar
E. Venturino. Epidemics in predator-prey models : disease in the prey, In ‘Mathematical Population Dynamics, Analysis of Heterogeneity’. 1, O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds), Wnertz Publisher Ltd, Canada, 381–393, 1995.
V. Volterra. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. 2. Mem. R. Accad. Naz. dei Lincei. Ser. VI, 1926.