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

P.S. Mandal; M. Banerjee

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 3, page 99-116
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Mandal, P.S., and Banerjee, M.. "Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model." Mathematical Modelling of Natural Phenomena 7.3 (2012): 99-116. <http://eudml.org/doc/222371>.

@article{Mandal2012,
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.},
author = {Mandal, P.S., Banerjee, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Eco-epidemiology; stability; Hopf-bifurcation; chaos; stochasticity; extinction; eco-epidemiology},
language = {eng},
month = {6},
number = {3},
pages = {99-116},
publisher = {EDP Sciences},
title = {Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model},
url = {http://eudml.org/doc/222371},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Mandal, P.S.
AU - Banerjee, M.
TI - Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/6//
PB - EDP Sciences
VL - 7
IS - 3
SP - 99
EP - 116
AB - 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.
LA - eng
KW - Eco-epidemiology; stability; Hopf-bifurcation; chaos; stochasticity; extinction; eco-epidemiology
UR - http://eudml.org/doc/222371
ER -

References

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  1. E. Allen. Modeling with Itô Stochastic Differential Equations. Springer, The Netherlands, 2007.  Zbl1130.60064
  2. L. J. S. Allen. An Introduction to Stochastic Processes with Applications to Biology. Pearson Eduction Inc., New Jercy, 2003.  Zbl1205.60001
  3. L. J. S. Allen, M. A. Jones, C. F. Martin. A discrete-time model with vaccination for a measles epidemic. Math. Biosci., 105 (1991), 111–131.  Zbl0728.92023
  4. O. Arino, A. El. Abdllaoui, J. Mikram, J. Chattopadhyay. Infection on prey population may act as a biological control in ratio-dependent predator-prey model. Nonlinearity, 17 (2004), 1101-1116.  Zbl1062.34050
  5. E. J. Allen, L. J. S. Allen, A. Arciniega, P. Greenwood. Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl., 26 (2008) 274-297.  Zbl1137.60026
  6. F. G. Ball. Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci., 156 (1999) 41–67.  Zbl0979.92033
  7. E. Beltrami, T. O. Carroll. Modelling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol., 32 (1994) 857-863.  Zbl0825.92122
  8. F. Brauer, C. Castillo-Chàvez. Mathematical Models in Population Biolgy and Epidemiology Springer-Verlag, New York, 2001.  Zbl0967.92015
  9. T. Britton. Stochastic epidemic models : A survey. Math. Biosci., 225 (2010) 24–35.  Zbl1188.92031
  10. T. Britton, D. Lindenstrand. Epidemic modelling : Aspects where stochasticity matters. Math. Biosci., 222 (2009) 109-116.  Zbl1178.92046
  11. J. Chattopadhyay, N. Bairagi. Pelicans at risk in Salton Sea - an eco-epidemiological model. Ecol. Model., 136 (2001) 103–112.  
  12. M. S. Chan, V. S. Isham. A stochastic model of schistosomiasis immuno-epidemiology. Math. Biosci., 151 (1998) 179–198.  Zbl0933.92029
  13. H. I. Freedman. A model of predator-prey dynamics as modified by the action of parasite. Math. Biosci., 99 (1990) 143–155.  Zbl0698.92024
  14. T. C. Gard. Introduction to Stochastic Differential Equations. Marcel Decker, New York, 1987.  Zbl0628.60064
  15. C. W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1983.  Zbl0515.60002
  16. D. T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phy., 22 (1976) 403–434.  
  17. D. T. Gillespie. The chemical Langevin equation. J. Chem. Phy., 113 (2000) 297–306.  
  18. N. S. Goel, N. Richter-Dyn. Stochastic Models in Biology. Academic Press, New York, 1974.  
  19. D. Greenhalgh, M. Griffiths. Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model. J. Math. Biol., 59 (2009) 1–36.  Zbl1204.92059
  20. K. P. Hadeler, H. I. Freedman. Predator-prey population with parasitic infection. J. Math. Biol., 27 (1989) 609–631.  Zbl0716.92021
  21. M. Haque, D. Greenhalgh. A predator-prey model with disease in prey species only. M2AS, 30 (2006) 911–929.  Zbl1115.92049
  22. D. J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43 (2001) 525–546.  Zbl0979.65007
  23. W. O. Kermack, A. G. McKendrick. A Contribution to the Mathematical Theory of Epidemics. Proc. Roy. Soc. Lond. A.115 (1927) 700–721.  Zbl53.0517.01
  24. P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1999.  Zbl0752.60043
  25. M. Kot. Elements of Mathematical Biology. Cambridge University Press, Cambridge, 2001.  
  26. Y. A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, Berlin, 1997.  
  27. A. J. Lotka. Elements of physical biology. Williams & Wilkins Co., Baltimore, 1925.  Zbl51.0416.06
  28. J. Marsden, M. McCracken. The Hopf Bifurcation and its Applications. Springer, New York, 1976.  Zbl0346.58007
  29. H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal Patterns in Ecology and Epidemiology : Theory, Models and Simulations. Chapman & Hall, London, 2008.  Zbl1298.92004
  30. J. D. Murray. Mathematical Biology. Springer, New York, 1993.  Zbl0779.92001
  31. R. J. Serfling. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York, 1980.  Zbl0538.62002
  32. D. Stiefs, E. Venturino, U. Feudel. Evidence of chaos in eco-epidemic models. Math. Biosci. Eng., 6 (2009) 857–873.  Zbl1178.92057
  33. R. K. Upadhyay, N. Bairagi, K. Kundu, J. Chattopadhyay. Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Appl. Math. Comput., 196 (2008) 392–401.  Zbl1131.92070
  34. E. Venturino. The influence of diseases on Lotka-Volterra systems. Rocky Mountain Journal of Mathematics.24 (1994) 381–402.  Zbl0799.92017
  35. 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.  
  36. V. Volterra. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. 2. Mem. R. Accad. Naz. dei Lincei. Ser. VI, 1926.  Zbl52.0450.06

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